ChatGPT - \u9640\u87ba\u8f6c\u52a8\u5206\u6790<\/title><meta property=\"og:site_name\" content=\"ChatGPT\"><meta name=\"robots\" content=\"noindex,nofollow\"><meta property=\"og:title\" content=\"ChatGPT - \u9640\u87ba\u8f6c\u52a8\u5206\u6790\"><meta property=\"og:description\" content=\"Shared via ChatGPT\"><meta property=\"og:image\" content=\"https:\/\/cdn.oaistatic.com\/assets\/chatgpt-share-og-u7j5uyao.webp\"><meta property=\"og:url\" content=\"https:\/\/chatgpt.com\/share\/66e3e611-81f0-8005-be85-ffd0e2daa182\"><link rel=\"stylesheet\" href=\".\/\u7b2c\u4e00\u9898&\u7b2c\u4e09\u9898_files\/root-e9fk4trk.css\"><link rel=\"stylesheet\" href=\".\/\u7b2c\u4e00\u9898&\u7b2c\u4e09\u9898_files\/conversation-small-eplmind9.css\"><\/head><body class=\"\"><script>!function(){try{var d=document.documentElement,c=d.classList;c.remove('light','dark');var e=localStorage.getItem('theme');if('system'===e||(!e&&true)){var t='(prefers-color-scheme: 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class=\"sr-only\">\u60a8\u8bf4\uff1a<\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div data-message-author-role=\"user\" data-message-id=\"aaa2d9a7-bee6-49bb-a9b9-64bf346099ad\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden items-end rtl:items-start\"><div class=\"relative max-w-[70%] rounded-3xl px-5 py-2.5 bg-[#f4f4f4] dark:bg-token-main-surface-secondary\"><div class=\"whitespace-pre-wrap\">\u9640\u87ba\u662f\u4e00\u79cd\u62e5\u6709\u60a0\u4e45\u5386\u53f2\u7684\u513f\u7ae5\u73a9\u5177\uff0c\u540c\u65f6\u5176\u4e2d\u7684\u539f\u7406\u4e5f\u53ef\u4ee5\u5728\u822a\u6d77\u7b49\u591a\u65b9\u9762\u5e94\u7528\u3002\u73b0\u5728\u6211\u4eec\u8003\u8651\u4e00\u4e2a\u8d28\u91cf\u5747\u5300\u5206\u5e03\u7684\u5706\u9525\u5f62\u9640\u87ba\uff0c\u5982\u56fe1.1\u6240\u793a\uff0c\u5176\u534a\u9876\u89d2\u4e3a\u03b1\uff0c\u9ad8\u4e3ah\uff0c\u8d28\u91cf\u4e3am\u3002\u4f7f\u9640\u87ba\u5728\u6c34\u5e73\u5730\u9762\u4e0a\u7ed5\u5176\u9876\u70b9\u8f6c\u52a8\uff0c\u9876\u70b9\u56fa\u5b9a\u5728\u539f\u70b9O\u5904\u3002\u4e0d\u8003\u8651\u53ef\u80fd\u7684\u963b\u529b\u3002\n\uff081\uff09\u8ba1\u7b97\u9640\u87ba\u7ed5\u5176\u8f74\u7ebf\u7684\u8f6c\u52a8\u60ef\u91cfI_1\u3002\n\uff082\uff09\u73b0\u5728\u4f7f\u9640\u87ba\u65cb\u8f6c\u65f6\u5176\u8f74\u7ebf\u4e0e\u7ad6\u76f4\u65b9\u5411\u5939\u89d2\u6052\u5b9a\u4e3a\u03b2\uff0c\u5f53\u9640\u87ba\u8f74\u7ebf\u7ed5\u7ad6\u76f4\u65b9\u5411\u516c\u8f6c\u89d2\u901f\u5ea6\u4e3a\u03c9\u65f6\uff0c\u6c42\u9640\u87ba\u7684\u89d2\u52a8\u91cf\uff0c\u7528\u6c34\u5e73\u4e0e\u7ad6\u76f4\u5206\u91cf\u5927\u5c0f\u8868\u793a\u5373\u53ef\u3002\u4ee4k^2=g\/(\u03c9^2 h)\uff0cg\u4e3a\u91cd\u529b\u52a0\u901f\u5ea6\u3002\n\uff083\uff09\u6c42\u6b64\u65f6\u9640\u87ba\u8f6c\u52a8\u7684\u6700\u5c0f\u603b\u89d2\u901f\u5ea6\u5927\u5c0f\u03c9_0\u8868\u8fbe\u5f0f\uff0c\u5e76\u4ee3\u5165\u03b1=\u03c0\/12\u3001\u03b2=\u03c0\/6\u8ba1\u7b97\u6570\u503c\uff0c\u7cfb\u6570\u4fdd\u75593\u4f4d\u5c0f\u6570\u3002\n<\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-3\" data-scroll-anchor=\"false\"><h5 class=\"sr-only\"><\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"flex-shrink-0 flex flex-col relative items-end\"><div><div class=\"pt-0\"><div class=\"gizmo-bot-avatar flex h-8 w-8 items-center justify-center overflow-hidden rounded-full\"><div class=\"relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8\"><svg width=\"41\" height=\"41\" viewBox=\"0 0 41 41\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\" role=\"img\"><text x=\"-9999\" y=\"-9999\">ChatGPT<\/text><path d=\"M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 2.81182 28.3141C2.95951 29.7256 3.40701 31.0892 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17.1364C34.9967 17.849 36.0017 18.8982 36.6606 20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/div><\/div><\/div><\/div><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col agent-turn\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div class=\"group absolute left-0 top-0 mr-1.5 h-8 overflow-hidden mt-1\"><button class=\"\" style=\"opacity: 1;\"><div class=\"flex items-center justify-start gap-1\">Thought for 86 seconds<\/span><svg width=\"24\" height=\"24\" viewBox=\"0 0 24 24\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\"><path fill-rule=\"evenodd\" clip-rule=\"evenodd\" d=\"M5.29289 9.29289C5.68342 8.90237 6.31658 8.90237 6.70711 9.29289L12 14.5858L17.2929 9.29289C17.6834 8.90237 18.3166 8.90237 18.7071 9.29289C19.0976 9.68342 19.0976 10.3166 18.7071 10.7071L12.7071 16.7071C12.5196 16.8946 12.2652 17 12 17C11.7348 17 11.4804 16.8946 11.2929 16.7071L5.29289 10.7071C4.90237 10.3166 4.90237 9.68342 5.29289 9.29289Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/button><\/div><\/p><div data-message-author-role=\"assistant\" data-message-id=\"1da19a34-f5f8-4d88-a95a-ed453b5c64af\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[3px]\"><div class=\"markdown prose w-full break-words dark:prose-invert light\">\u89e3\u7b54\uff1a<\/strong><\/p>\uff081\uff09\u8ba1\u7b97\u9640\u87ba\u7ed5\u5176\u8f74\u7ebf\u7684\u8f6c\u52a8\u60ef\u91cf <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I_1<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/strong><\/p>\u9996\u5148\uff0c\u9640\u87ba\u662f\u4e00\u4e2a\u8d28\u91cf\u5747\u5300\u5206\u5e03\u7684\u5706\u9525\uff0c\u9ad8\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">h<\/annotation><\/semantics><\/math><\/span><\/span>h<\/span><\/span><\/span><\/span>\uff0c\u534a\u9876\u89d2\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha<\/annotation><\/semantics><\/math><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>\u3002\u5e95\u9762\u534a\u5f84 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R<\/annotation><\/semantics><\/math><\/span><\/span>R<\/span><\/span><\/span><\/span> \u53ef\u4ee5\u8868\u793a\u4e3a\uff1a\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><mo>=<\/mo><mi>h<\/mi><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R = h \\tan \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>R<\/span><\/span>=<\/span><\/span><\/span><\/span>h<\/span><\/span>tan<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/p>\u5706\u9525\u7684\u4f53\u79ef <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span><\/span><\/span> \u4e3a\uff1a\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><mi>\u03c0<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><mi>h<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>3<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V = \\frac{1}{3} \\pi R^2 h = \\frac{1}{3} \\pi h^3 \\tan^2 \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>R<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/p>\u8d28\u91cf\u5747\u5300\u5206\u5e03\uff0c\u56e0\u6b64\u5bc6\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\rho<\/annotation><\/semantics><\/math><\/span><\/span>\u03c1<\/span><\/span><\/span><\/span> \u4e3a\uff1a\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c1<\/mi><mo>=<\/mo><mfrac><mi>m<\/mi><mi>V<\/mi><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>3<\/mn><mi>m<\/mi><\/mrow><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>3<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\rho = \\frac{m}{V} = \\frac{3m}{\\pi h^3 \\tan^2 \\alpha}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c1<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span>a<\/span>n<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span>m<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\u7ed5\u8f74\u7ebf\u7684\u8f6c\u52a8\u60ef\u91cf <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I_1<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mo>\u222b<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>m<\/mi><mo>=<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><\/msubsup><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>z<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msubsup><msup><mi>r<\/mi><mn>3<\/mn><\/msup><mi>\u03c1<\/mi><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>r<\/mi><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>\u03b8<\/mi><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mi>\u03c1<\/mi><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><mrow><mo fence=\"true\">(<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><\/msubsup><mi mathvariant=\"normal\">d<\/mi><mi>\u03b8<\/mi><mo fence=\"true\">)<\/mo><\/mrow><mrow><mo fence=\"true\">(<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>z<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msubsup><msup><mi>r<\/mi><mn>3<\/mn><\/msup><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>r<\/mi><mo fence=\"true\">)<\/mo><\/mrow><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mi>\u03c1<\/mi><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mo stretchy=\"false\">)<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>z<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>4<\/mn><\/msup><mo fence=\"true\">)<\/mo><\/mrow><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>\u03c0<\/mi><mi>\u03c1<\/mi><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><msup><mrow><mo fence=\"true\">(<\/mo><mi>z<\/mi><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><mo fence=\"true\">)<\/mo><\/mrow><mn>4<\/mn><\/msup><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>\u03c0<\/mi><mi>\u03c1<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>\u03b1<\/mi><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><msup><mi>z<\/mi><mn>4<\/mn><\/msup><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>\u03c0<\/mi><mi>\u03c1<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>\u03b1<\/mi><mrow><mo fence=\"true\">(<\/mo><mfrac><msup><mi>h<\/mi><mn>5<\/mn><\/msup><mn>5<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mfrac><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>5<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><mn>10<\/mn><\/mfrac><mi>\u03c1<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{align*}\nI_1 &= \\int r^2 \\, \\mathrm{d}m = \\int_0^h \\int_0^{2\\pi} \\int_0^{r(z)} r^3 \\rho \\, \\mathrm{d}r \\, \\mathrm{d}\\theta \\, \\mathrm{d}z \\\\\n&= \\rho \\int_0^h \\left( \\int_0^{2\\pi} \\mathrm{d}\\theta \\right) \\left( \\int_0^{r(z)} r^3 \\, \\mathrm{d}r \\right) \\, \\mathrm{d}z \\\\\n&= \\rho (2\\pi) \\int_0^h \\left( \\frac{1}{4} r(z)^4 \\right) \\, \\mathrm{d}z \\\\\n&= \\frac{1}{2} \\pi \\rho \\int_0^h \\left( z \\tan \\alpha \\right)^4 \\, \\mathrm{d}z \\\\\n&= \\frac{1}{2} \\pi \\rho \\tan^4 \\alpha \\int_0^h z^4 \\, \\mathrm{d}z \\\\\n&= \\frac{1}{2} \\pi \\rho \\tan^4 \\alpha \\left( \\frac{h^5}{5} \\right) \\\\\n&= \\frac{\\pi h^5 \\tan^4 \\alpha}{10} \\rho\n\\end{align*}<\/annotation><\/semantics><\/math><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u222b<\/span><\/span>r<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>m<\/span><\/span>=<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>r<\/span>(<\/span>z<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c1<\/span><\/span>d<\/span>r<\/span><\/span>d<\/span>\u03b8<\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03c1<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span>(<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>r<\/span>(<\/span>z<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>r<\/span>)<\/span><\/span><\/span><\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03c1<\/span>(<\/span>2<\/span>\u03c0<\/span>)<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span>(<\/span>z<\/span>)<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>\u03c1<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>z<\/span><\/span>tan<\/span><\/span>\u03b1<\/span>)<\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>\u03c1<\/span><\/span>tan<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>z<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>\u03c1<\/span><\/span>tan<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>(<\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\rho<\/annotation><\/semantics><\/math><\/span><\/span>\u03c1<\/span><\/span><\/span><\/span> \u7684\u8868\u8fbe\u5f0f\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mfrac><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>5<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><mn>10<\/mn><\/mfrac><mo>\u00d7<\/mo><mfrac><mrow><mn>3<\/mn><mi>m<\/mi><\/mrow><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>3<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>3<\/mn><mn>10<\/mn><\/mfrac><mi>m<\/mi><msup><mi>h<\/mi><mn>2<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I_1 = \\frac{\\pi h^5 \\tan^4 \\alpha}{10} \\times \\frac{3m}{\\pi h^3 \\tan^2 \\alpha} = \\frac{3}{10} m h^2 \\tan^2 \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span>m<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>h<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>\u7b54\u6848\uff1a<\/strong>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mfrac><mn>3<\/mn><mn>10<\/mn><\/mfrac><mi>m<\/mi><msup><mi>h<\/mi><mn>2<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I_1 = \\frac{3}{10} m h^2 \\tan^2 \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>h<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/p><hr>\uff082\uff09\u6c42\u9640\u87ba\u7684\u89d2\u52a8\u91cf\u7684\u6c34\u5e73\u548c\u7ad6\u76f4\u5206\u91cf\u3002<\/strong><\/p>\u9640\u87ba\u4ee5\u89d2\u901f\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03a9<\/span><\/span><\/span><\/span> \u7ed5\u81ea\u8eab\u8f74\u7ebf\u65cb\u8f6c\uff0c\u8f74\u7ebf\u4e0e\u7ad6\u76f4\u65b9\u5411\u7684\u5939\u89d2\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\beta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\uff0c\u540c\u65f6\u4ee5\u89d2\u901f\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> \u7ed5\u7ad6\u76f4\u8f74\u516c\u8f6c\u3002\u603b\u89d2\u901f\u5ea6\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><\/msub><mo>=<\/mo><mi mathvariant=\"normal\">\u03a9<\/mi><mo>+<\/mo><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{\u603b}} = \\Omega + \\omega \\sin \\beta<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span>\u5728\u7a33\u6052\u8fdb\u52a8\u6761\u4ef6\u4e0b\uff0c\u9640\u87ba\u7684\u89d2\u52a8\u91cf <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">L<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{L}<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span><\/span><\/span> \u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">L<\/mi><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"normal\">\u03a9<\/mi><msub><mi mathvariant=\"bold\">e<\/mi><mtext>\u8f74<\/mtext><\/msub><mo>+<\/mo><msub><mi>I<\/mi><mo lspace=\"0em\" rspace=\"0em\">\u22a5<\/mo><\/msub><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><msub><mi mathvariant=\"bold\">e<\/mi><mo lspace=\"0em\" rspace=\"0em\">\u22a5<\/mo><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{L} = I_1 \\Omega \\mathbf{e}_{\\text{\u8f74}} + I_{\\perp} \\omega \\sin \\beta \\mathbf{e}_{\\perp}<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a9<\/span>e<\/span><\/span>\u8f74<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>I<\/span><\/span>\u22a5<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>e<\/span><\/span>\u22a5<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5176\u4e2d <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mo lspace=\"0em\" rspace=\"0em\">\u22a5<\/mo><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I_{\\perp}<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>\u22a5<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e3a\u5782\u76f4\u4e8e\u8f74\u7ebf\u7684\u8f6c\u52a8\u60ef\u91cf\uff0c\u5bf9\u5706\u9525\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>I<\/mi><mo lspace=\"0em\" rspace=\"0em\">\u22a5<\/mo><\/msub><mo>=<\/mo><mfrac><mn>3<\/mn><mn>20<\/mn><\/mfrac><mi>m<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>R<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>4<\/mn><msup><mi>h<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">I_{\\perp} = \\frac{3}{20} m (R^2 + 4h^2)<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>\u22a5<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>20<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>(<\/span>R<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>4<\/span>h<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u4f46\u7531\u4e8e <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><mo>=<\/mo><mi>h<\/mi><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R = h \\tan \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>R<\/span><\/span>=<\/span><\/span><\/span><\/span>h<\/span><\/span>tan<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>\uff0c\u4e14\u5ffd\u7565 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mo lspace=\"0em\" rspace=\"0em\">\u22a5<\/mo><\/msub><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I_{\\perp} \\omega<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>\u22a5<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> \u9879\uff0c\u5219\u89d2\u52a8\u91cf\u7684\u6c34\u5e73\u548c\u7ad6\u76f4\u5206\u91cf\u8fd1\u4f3c\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.36em\" columnalign=\"left left\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msub><mi>L<\/mi><mtext>\u6c34\u5e73<\/mtext><\/msub><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"normal\">\u03a9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msub><mi>L<\/mi><mtext>\u7ad6\u76f4<\/mtext><\/msub><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"normal\">\u03a9<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><annotation encoding=\"application\/x-tex\">\\begin{cases}\nL_{\\text{\u6c34\u5e73}} = I_1 \\Omega \\sin \\beta \\\\\nL_{\\text{\u7ad6\u76f4}} = I_1 \\Omega \\cos \\beta\n\\end{cases}<\/annotation><\/semantics><\/math><\/span><\/span>{<\/span><\/span><\/span>L<\/span><\/span>\u6c34\u5e73<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>L<\/span><\/span>\u7ad6\u76f4<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>cos<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5229\u7528\u91cd\u529b\u77e9\u548c\u89d2\u52a8\u91cf\u53d8\u5316\u7387\u7684\u5173\u7cfb\uff0c\u5728\u7a33\u6052\u8fdb\u52a8\u65f6\u6709\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><mo>=<\/mo><mi>\u03c9<\/mi><msub><mi>L<\/mi><mtext>\u6c34\u5e73<\/mtext><\/msub><mo>=<\/mo><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"normal\">\u03a9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">m g l \\sin \\beta = \\omega L_{\\text{\u6c34\u5e73}} = \\omega (I_1 \\Omega \\sin \\beta)<\/annotation><\/semantics><\/math><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span>L<\/span><\/span>\u6c34\u5e73<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span>(<\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>)<\/span><\/span><\/span><\/span><\/span>\u5176\u4e2d <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>l<\/mi><mo>=<\/mo><mfrac><mn>3<\/mn><mn>4<\/mn><\/mfrac><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">l = \\frac{3}{4} h<\/annotation><\/semantics><\/math><\/span><\/span>l<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span> \u4e3a\u91cd\u5fc3\u5230\u9876\u70b9\u7684\u8ddd\u79bb\u3002\u89e3\u51fa <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03a9<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><mo>=<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\Omega = \\frac{m g l}{I_1 \\omega}<\/annotation><\/semantics><\/math><\/span><\/span>\u03a9<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><hr>\uff083\uff09\u6c42\u9640\u87ba\u8f6c\u52a8\u7684\u6700\u5c0f\u603b\u89d2\u901f\u5ea6\u5927\u5c0f <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/strong><\/p>\u603b\u89d2\u901f\u5ea6\u7684\u5e73\u65b9\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><mn>2<\/mn><\/msubsup><mo>=<\/mo><msup><mi mathvariant=\"normal\">\u03a9<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{\u603b}}^2 = \\Omega^2 + (\\omega \\sin \\beta)^2<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5c06 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03a9<\/span><\/span><\/span><\/span> \u4ee3\u5165\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><mn>2<\/mn><\/msubsup><mo>=<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{\u603b}}^2 = \\left( \\frac{m g l}{I_1 \\omega} \\right)^2 + (\\omega \\sin \\beta)^2<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e3a\u4f7f <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{\u603b}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u6700\u5c0f\uff0c\u5bf9 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> \u6c42\u5bfc\u5e76\u4ee4\u5176\u4e3a\u96f6\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi mathvariant=\"normal\">d<\/mi><msubsup><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><mn>2<\/mn><\/msubsup><\/mrow><mrow><mi mathvariant=\"normal\">d<\/mi><mi>\u03c9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>+<\/mo><mn>2<\/mn><mi>\u03c9<\/mi><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b2<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{\\mathrm{d} \\omega_{\\text{\u603b}}^2}{\\mathrm{d} \\omega} = -2 \\left( \\frac{m g l}{I_1 \\omega^2} \\right) \\left( \\frac{m g l}{I_1} \\right) + 2 \\omega \\sin^2 \\beta = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span><\/span>(<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>2<\/span>\u03c9<\/span><\/span>sin<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u89e3\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"normal\">\/<\/mi><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\omega^2 = \\frac{(m g l) \/ I_1}{\\sin \\beta}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>m<\/span>g<\/span>l<\/span>)<\/span>\/<\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\u6700\u5c0f\u603b\u89d2\u901f\u5ea6\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mn>0<\/mn><mn>2<\/mn><\/msubsup><mo>=<\/mo><mn>2<\/mn><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mspace width=\"1em\"><\/mspace><mtext>\uff08\u4ee3\u5165\u4e0a\u5f0f\u89e3\u51fa\uff09<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0^2 = 2 \\left( \\frac{m g l}{I_1 \\omega_0} \\right)^2 \\quad \\text{\uff08\u4ee3\u5165\u4e0a\u5f0f\u89e3\u51fa\uff09}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span>(<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff08\u4ee3\u5165\u4e0a\u5f0f\u89e3\u51fa\uff09<\/span><\/span><\/span><\/span><\/span><\/span>\u901a\u8fc7\u8ba1\u7b97\uff0c\u5f97\u5230\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><msqrt><mfrac><mrow><mn>20<\/mn><mi>g<\/mi><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><mrow><mi>h<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 = \\sqrt{ \\frac{20 g \\tan \\beta}{h \\tan^2 \\alpha} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>20<\/span>g<\/span><\/span>tan<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>12<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha = \\frac{\\pi}{12}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b1<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>12<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b2<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\beta = \\frac{\\pi}{6}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff1a<\/p>\u8ba1\u7b97 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\tan \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>tan<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span> \u548c <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\tan \\beta<\/annotation><\/semantics><\/math><\/span><\/span>tan<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><mo>=<\/mo><mi>tan<\/mi><mo>\u2061<\/mo><mn>1<\/mn><msup><mn>5<\/mn><mo>\u2218<\/mo><\/msup><mo>\u2248<\/mo><mn>0.2679<\/mn><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><mo>=<\/mo><mi>tan<\/mi><mo>\u2061<\/mo><mn>3<\/mn><msup><mn>0<\/mn><mo>\u2218<\/mo><\/msup><mo>\u2248<\/mo><mn>0.5774<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\tan \\alpha = \\tan 15^\\circ \\approx 0.2679, \\quad \\tan \\beta = \\tan 30^\\circ \\approx 0.5774<\/annotation><\/semantics><\/math><\/span><\/span>tan<\/span><\/span>\u03b1<\/span><\/span>=<\/span><\/span><\/span><\/span>tan<\/span><\/span>1<\/span>5<\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.2679<\/span>,<\/span><\/span><\/span>tan<\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span>tan<\/span><\/span>3<\/span>0<\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.5774<\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><msqrt><mfrac><mrow><mn>20<\/mn><mi>g<\/mi><mo>\u00d7<\/mo><mn>0.5774<\/mn><\/mrow><mrow><mi>h<\/mi><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>0.2679<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/mfrac><\/msqrt><mo>\u2248<\/mo><msqrt><mfrac><mrow><mn>11.547<\/mn><mi>g<\/mi><\/mrow><mrow><mi>h<\/mi><mo>\u00d7<\/mo><mn>0.0718<\/mn><\/mrow><\/mfrac><\/msqrt><mo>\u2248<\/mo><mn>12.676<\/mn><msqrt><mfrac><mi>g<\/mi><mi>h<\/mi><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 = \\sqrt{ \\frac{20 g \\times 0.5774}{h \\times (0.2679)^2} } \\approx \\sqrt{ \\frac{11.547 g}{h \\times 0.0718} } \\approx 12.676 \\sqrt{ \\frac{g}{h} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>\u00d7<\/span><\/span>(<\/span>0.2679<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>20<\/span>g<\/span><\/span>\u00d7<\/span><\/span>0.5774<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"3.08em\" viewBox=\"0 0 400000 3240\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M473,2793\nc339.3,-1799.3,509.3,-2700,510,-2702 l0 -0\nc3.3,-7.3,9.3,-11,18,-11 H400000v40H1017.7\ns-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200\nc0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26\ns76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,\n606zM1001 80h400000v40H1017.7z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>\u00d7<\/span><\/span>0.0718<\/span><\/span><\/span><\/span><\/span><\/span><\/span>11.547<\/span>g<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>12.676<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span><\/span><\/span><\/span>g<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7b54\u6848\uff1a<\/strong><\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><msqrt><mfrac><mrow><mn>20<\/mn><mi>g<\/mi><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><mrow><mi>h<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 = \\sqrt{ \\frac{20 g \\tan \\beta}{h \\tan^2 \\alpha} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>20<\/span>g<\/span><\/span>tan<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165\u6570\u503c\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>\u2248<\/mo><mn>12.676<\/mn><msqrt><mfrac><mi>g<\/mi><mi>h<\/mi><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 \\approx 12.676 \\sqrt{ \\frac{g}{h} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>12.676<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span><\/span><\/span><\/span>g<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7cfb\u6570\u4fdd\u7559\u4e09\u4f4d\u5c0f\u6570\u3002<\/p><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-4\" data-scroll-anchor=\"false\"><h5 class=\"sr-only\">\u60a8\u8bf4\uff1a<\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div data-message-author-role=\"user\" data-message-id=\"aaa2bf3d-3d55-4b22-87b8-54f9268a62a8\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden items-end rtl:items-start\"><div class=\"relative max-w-[70%] rounded-3xl px-5 py-2.5 bg-[#f4f4f4] dark:bg-token-main-surface-secondary\"><div class=\"whitespace-pre-wrap\">\u4f60\u4e0d\u5e94\u8be5\u5ffd\u7565 \n\ud835\udc3c\n\u22a5\n\ud835\udf14\nI \n\u22a5\n\u200b\n \u03c9 \u9879<\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-5\" data-scroll-anchor=\"false\"><h5 class=\"sr-only\"><\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"flex-shrink-0 flex flex-col relative items-end\"><div><div class=\"pt-0\"><div class=\"gizmo-bot-avatar flex h-8 w-8 items-center justify-center overflow-hidden rounded-full\"><div class=\"relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8\"><svg width=\"41\" height=\"41\" viewBox=\"0 0 41 41\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\" role=\"img\"><text x=\"-9999\" y=\"-9999\">ChatGPT<\/text><path d=\"M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 2.81182 28.3141C2.95951 29.7256 3.40701 31.0892 4.12437 32.3138C5.18791 34.1659 6.8123 35.6322 8.76321 36.5013C10.7141 37.3704 12.8907 37.5973 14.9789 37.1492C15.9208 38.2107 17.0786 39.0587 18.3747 39.6366C19.6709 40.2144 21.0755 40.5087 22.4946 40.4998C24.6307 40.5054 26.7133 39.8321 28.4418 38.5772C30.1704 37.3223 31.4556 35.5506 32.1119 33.5179C33.5027 33.2332 34.8167 32.6547 35.9659 31.821C37.115 30.9874 38.0728 29.9178 38.7752 28.684C39.8458 26.8371 40.3023 24.6979 40.0789 22.5748C39.8556 20.4517 38.9639 18.4544 37.5324 16.8707ZM22.4978 37.8849C20.7443 37.8874 19.0459 37.2733 17.6994 36.1501C17.7601 36.117 17.8666 36.0586 17.936 36.0161L25.9004 31.4156C26.1003 31.3019 26.2663 31.137 26.3813 30.9378C26.4964 30.7386 26.5563 30.5124 26.5549 30.2825V19.0542L29.9213 20.998C29.9389 21.0068 29.9541 21.0198 29.9656 21.0359C29.977 21.052 29.9842 21.0707 29.9867 21.0902V30.3889C29.9842 32.375 29.1946 34.2791 27.7909 35.6841C26.3872 37.0892 24.4838 37.8806 22.4978 37.8849ZM6.39227 31.0064C5.51397 29.4888 5.19742 27.7107 5.49804 25.9832C5.55718 26.0187 5.66048 26.0818 5.73461 26.1244L13.699 30.7248C13.8975 30.8408 14.1233 30.902 14.3532 30.902C14.583 30.902 14.8088 30.8408 15.0073 30.7248L24.731 25.1103V28.9979C24.7321 29.0177 24.7283 29.0376 24.7199 29.0556C24.7115 29.0736 24.6988 29.0893 24.6829 29.1012L16.6317 33.7497C14.9096 34.7416 12.8643 35.0097 10.9447 34.4954C9.02506 33.9811 7.38785 32.7263 6.39227 31.0064ZM4.29707 13.6194C5.17156 12.0998 6.55279 10.9364 8.19885 10.3327C8.19885 10.4013 8.19491 10.5228 8.19491 10.6071V19.808C8.19351 20.0378 8.25334 20.2638 8.36823 20.4629C8.48312 20.6619 8.64893 20.8267 8.84863 20.9404L18.5723 26.5542L15.206 28.4979C15.1894 28.5089 15.1703 28.5155 15.1505 28.5173C15.1307 28.5191 15.1107 28.516 15.0924 28.5082L7.04046 23.8557C5.32135 22.8601 4.06716 21.2235 3.55289 19.3046C3.03862 17.3858 3.30624 15.3413 4.29707 13.6194ZM31.955 20.0556L22.2312 14.4411L25.5976 12.4981C25.6142 12.4872 25.6333 12.4805 25.6531 12.4787C25.6729 12.4769 25.6928 12.4801 25.7111 12.4879L33.7631 17.1364C34.9967 17.849 36.0017 18.8982 36.6606 20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/div><\/div><\/div><\/div><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col agent-turn\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div class=\"group absolute left-0 top-0 mr-1.5 h-8 overflow-hidden mt-1\"><button class=\"\" style=\"opacity: 1;\"><div class=\"flex items-center justify-start gap-1\">Thought for 141 seconds<\/span><svg width=\"24\" height=\"24\" viewBox=\"0 0 24 24\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\"><path fill-rule=\"evenodd\" clip-rule=\"evenodd\" d=\"M5.29289 9.29289C5.68342 8.90237 6.31658 8.90237 6.70711 9.29289L12 14.5858L17.2929 9.29289C17.6834 8.90237 18.3166 8.90237 18.7071 9.29289C19.0976 9.68342 19.0976 10.3166 18.7071 10.7071L12.7071 16.7071C12.5196 16.8946 12.2652 17 12 17C11.7348 17 11.4804 16.8946 11.2929 16.7071L5.29289 10.7071C4.90237 10.3166 4.90237 9.68342 5.29289 9.29289Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/button><\/div><\/p><div data-message-author-role=\"assistant\" data-message-id=\"eefeb0eb-cb71-435d-bdfd-deb36a99579f\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[3px]\"><div class=\"markdown prose w-full break-words dark:prose-invert light\">\u89e3\u7b54\uff1a<\/strong><\/p>\uff081\uff09\u8ba1\u7b97\u9640\u87ba\u7ed5\u5176\u8f74\u7ebf\u7684\u8f6c\u52a8\u60ef\u91cf <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I_1<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/strong><\/p>\u5df2\u77e5\u9640\u87ba\u662f\u4e00\u4e2a\u8d28\u91cf\u5747\u5300\u5206\u5e03\u7684\u5706\u9525\uff0c\u534a\u9876\u89d2\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha<\/annotation><\/semantics><\/math><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>\uff0c\u9ad8\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">h<\/annotation><\/semantics><\/math><\/span><\/span>h<\/span><\/span><\/span><\/span>\uff0c\u8d28\u91cf\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m<\/annotation><\/semantics><\/math><\/span><\/span>m<\/span><\/span><\/span><\/span>\u3002<\/p>\u5e95\u9762\u534a\u5f84 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R<\/annotation><\/semantics><\/math><\/span><\/span>R<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>R<\/mi><mo>=<\/mo><mi>h<\/mi><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R = h \\tan \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>R<\/span><\/span>=<\/span><\/span><\/span><\/span>h<\/span><\/span>tan<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>\u5706\u9525\u7684\u4f53\u79ef <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><mi>\u03c0<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><mi>h<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>3<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V = \\frac{1}{3} \\pi R^2 h = \\frac{1}{3} \\pi h^3 \\tan^2 \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>R<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>\u5bc6\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\rho<\/annotation><\/semantics><\/math><\/span><\/span>\u03c1<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c1<\/mi><mo>=<\/mo><mfrac><mi>m<\/mi><mi>V<\/mi><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>3<\/mn><mi>m<\/mi><\/mrow><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>3<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\rho = \\frac{m}{V} = \\frac{3m}{\\pi h^3 \\tan^2 \\alpha}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c1<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span>m<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7ed5\u8f74\u7ebf\u7684\u8f6c\u52a8\u60ef\u91cf <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I_1<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mo>\u222b<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>m<\/mi><mo>=<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><\/msubsup><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>z<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msubsup><msup><mi>r<\/mi><mn>3<\/mn><\/msup><mi>\u03c1<\/mi><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>r<\/mi><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>\u03b8<\/mi><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mi>\u03c1<\/mi><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mo stretchy=\"false\">)<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>z<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>4<\/mn><\/msup><mo fence=\"true\">)<\/mo><\/mrow><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>\u03c0<\/mi><mi>\u03c1<\/mi><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><msup><mrow><mo fence=\"true\">(<\/mo><mi>z<\/mi><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><mo fence=\"true\">)<\/mo><\/mrow><mn>4<\/mn><\/msup><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>\u03c0<\/mi><mi>\u03c1<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>\u03b1<\/mi><mrow><mo fence=\"true\">(<\/mo><mfrac><msup><mi>h<\/mi><mn>5<\/mn><\/msup><mn>5<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mfrac><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>5<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><mn>10<\/mn><\/mfrac><mi>\u03c1<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{align*}\nI_1 &= \\int r^2 \\, \\mathrm{d}m = \\int_0^h \\int_0^{2\\pi} \\int_0^{r(z)} r^3 \\rho \\, \\mathrm{d}r \\, \\mathrm{d}\\theta \\, \\mathrm{d}z \\\\\n&= \\rho (2\\pi) \\int_0^h \\left( \\frac{1}{4} r(z)^4 \\right) \\, \\mathrm{d}z \\\\\n&= \\frac{1}{2} \\pi \\rho \\int_0^h \\left( z \\tan \\alpha \\right)^4 \\, \\mathrm{d}z \\\\\n&= \\frac{1}{2} \\pi \\rho \\tan^4 \\alpha \\left( \\frac{h^5}{5} \\right) \\\\\n&= \\frac{\\pi h^5 \\tan^4 \\alpha}{10} \\rho\n\\end{align*}<\/annotation><\/semantics><\/math><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u222b<\/span><\/span>r<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>m<\/span><\/span>=<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>r<\/span>(<\/span>z<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c1<\/span><\/span>d<\/span>r<\/span><\/span>d<\/span>\u03b8<\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03c1<\/span>(<\/span>2<\/span>\u03c0<\/span>)<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span>(<\/span>z<\/span>)<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>\u03c1<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>z<\/span><\/span>tan<\/span><\/span>\u03b1<\/span>)<\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>\u03c1<\/span><\/span>tan<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>(<\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\rho<\/annotation><\/semantics><\/math><\/span><\/span>\u03c1<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mfrac><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>5<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><mn>10<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mn>3<\/mn><mi>m<\/mi><\/mrow><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>3<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>3<\/mn><mn>10<\/mn><\/mfrac><mi>m<\/mi><msup><mi>h<\/mi><mn>2<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I_1 = \\frac{\\pi h^5 \\tan^4 \\alpha}{10} \\cdot \\frac{3m}{\\pi h^3 \\tan^2 \\alpha} = \\frac{3}{10} m h^2 \\tan^2 \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22c5<\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span>m<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>h<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>\u7b54\u6848\uff1a<\/strong><\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mfrac><mn>3<\/mn><mn>10<\/mn><\/mfrac><mi>m<\/mi><msup><mi>h<\/mi><mn>2<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I_1 = \\frac{3}{10} m h^2 \\tan^2 \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>h<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><hr>\uff082\uff09\u6c42\u9640\u87ba\u7684\u89d2\u52a8\u91cf\u7684\u6c34\u5e73\u548c\u7ad6\u76f4\u5206\u91cf\u3002<\/strong><\/p>\u9996\u5148\uff0c\u9640\u87ba\u4ee5\u89d2\u901f\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03a9<\/span><\/span><\/span><\/span> \u7ed5\u81ea\u8eab\u8f74\u7ebf\u65cb\u8f6c\uff0c\u5176\u8f74\u7ebf\u4e0e\u7ad6\u76f4\u65b9\u5411\u5939\u89d2\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\beta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\uff0c\u5e76\u4ee5\u89d2\u901f\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> \u7ed5\u7ad6\u76f4\u8f74\u8fdb\u52a8\u3002<\/p>\u603b\u89d2\u901f\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\boldsymbol{\\omega}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u8ba1\u7b97\uff1a<\/strong><\/p>\u7531\u4e8e\u9640\u87ba\u8f74\u7ebf\u5728\u7a7a\u95f4\u4e2d\u65e2\u6709\u81ea\u8f6c\u53c8\u6709\u8fdb\u52a8\uff0c\u5176\u603b\u89d2\u901f\u5ea6\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">\u03c9<\/mi><mo>=<\/mo><mi mathvariant=\"normal\">\u03a9<\/mi><mi mathvariant=\"bold\">n<\/mi><mo>+<\/mo><mi>\u03c9<\/mi><mi mathvariant=\"bold\">k<\/mi><mo>\u00d7<\/mo><mi mathvariant=\"bold\">n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\boldsymbol{\\omega} = \\Omega \\mathbf{n} + \\omega \\mathbf{k} \\times \\mathbf{n}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03a9<\/span>n<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03c9<\/span>k<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u5176\u4e2d\uff1a<\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{n}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span><\/span><\/span> \u662f\u9640\u87ba\u8f74\u7ebf\u7684\u5355\u4f4d\u5411\u91cf\uff0c<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{k}<\/annotation><\/semantics><\/math><\/span><\/span>k<\/span><\/span><\/span><\/span> \u662f\u7ad6\u76f4\u5411\u4e0a\u5355\u4f4d\u5411\u91cf\uff0c<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">k<\/mi><mo>\u00d7<\/mo><mi mathvariant=\"bold\">n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{k} \\times \\mathbf{n}<\/annotation><\/semantics><\/math><\/span><\/span>k<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span> \u8868\u793a\u8f74\u7ebf\u7684\u8fdb\u52a8\u90e8\u5206\u3002<\/li><\/ul>\u8ba1\u7b97\u89d2\u52a8\u91cf <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">L<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{L}<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span><\/span><\/span>\uff1a<\/strong><\/p>\u89d2\u52a8\u91cf\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">L<\/mi><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"normal\">\u03a9<\/mi><mi mathvariant=\"bold\">n<\/mi><mo>+<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03c9<\/mi><mi mathvariant=\"bold\">k<\/mi><mo>\u00d7<\/mo><mi mathvariant=\"bold\">n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{L} = I_1 \\Omega \\mathbf{n} + I_1 (\\omega \\mathbf{k} \\times \\mathbf{n})<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a9<\/span>n<\/span><\/span>+<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span>k<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7531\u4e8e\u9640\u87ba\u662f\u5bf9\u79f0\u7684\uff0c\u8f6c\u52a8\u60ef\u91cf\u77e9\u9635\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">I<\/mi><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mtable rowspacing=\"0.16em\" columnalign=\"center center center\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0<\/mn><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0<\/mn><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0<\/mn><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0<\/mn><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0<\/mn><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0<\/mn><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msub><mi>I<\/mi><mn>3<\/mn><\/msub><\/mstyle><\/mtd><\/mtr><\/mtable><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{I} =\n\\begin{pmatrix}\nI_1 & 0 & 0 \\\\\n0 & I_1 & 0 \\\\\n0 & 0 & I_3\n\\end{pmatrix}<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.875em\" height=\"3.600em\" viewBox=\"0 0 875 3600\"><path d=\"M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1\nc-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,\n-36,557 l0,84c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,\n949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9\nc0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,\n-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189\nl0,-92c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,\n-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>I<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.875em\" height=\"3.600em\" viewBox=\"0 0 875 3600\"><path d=\"M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,\n63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5\nc11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,9\nc-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664\nc-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11\nc0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17\nc242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558\nl0,-144c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,\n-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5bf9\u4e8e\u5bf9\u79f0\u4f53\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>3<\/mn><\/msub><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I_3 = I_1<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\u5c06 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\boldsymbol{\\omega}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span> \u5206\u89e3\uff0c\u5f97\u5230\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">L<\/mi><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"bold-italic\">\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{L} = I_1 \\boldsymbol{\\omega}<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4f46\u662f\u8981\u6ce8\u610f\uff0c\u8fdb\u52a8\u89d2\u901f\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> \u7684\u65b9\u5411\u5782\u76f4\u4e8e <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{n}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span><\/span><\/span>\uff0c\u56e0\u6b64\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">\u03c9<\/mi><mo>=<\/mo><mi mathvariant=\"normal\">\u03a9<\/mi><mi mathvariant=\"bold\">n<\/mi><mo>+<\/mo><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><msub><mi mathvariant=\"bold\">e<\/mi><mo>\u22a5<\/mo><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\boldsymbol{\\omega} = \\Omega \\mathbf{n} + \\omega \\sin \\beta \\mathbf{e}_\\perp<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03a9<\/span>n<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>e<\/span><\/span>\u22a5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5176\u4e2d <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi mathvariant=\"bold\">e<\/mi><mo>\u22a5<\/mo><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{e}_\\perp<\/annotation><\/semantics><\/math><\/span><\/span>e<\/span><\/span>\u22a5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u5782\u76f4\u4e8e <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{n}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span><\/span><\/span> \u7684\u5355\u4f4d\u5411\u91cf\u3002<\/p>\u8ba1\u7b97\u6c34\u5e73\u548c\u7ad6\u76f4\u5206\u91cf\uff1a<\/strong><\/p><ul><li>\u6c34\u5e73\u5206\u91cf\uff1a<\/strong><\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>L<\/mi><mtext>\u6c34\u5e73<\/mtext><\/msub><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">L_{\\text{\u6c34\u5e73}} = I_1 \\omega \\sin \\beta<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span>\u6c34\u5e73<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><ul><li>\u7ad6\u76f4\u5206\u91cf\uff1a<\/strong><\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>L<\/mi><mtext>\u7ad6\u76f4<\/mtext><\/msub><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"normal\">\u03a9<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">L_{\\text{\u7ad6\u76f4}} = I_1 \\Omega \\cos \\beta<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span>\u7ad6\u76f4<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>cos<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span>\u5f15\u5165\u91cd\u529b\u77e9\u7684\u5e73\u8861\u6761\u4ef6\uff1a<\/strong><\/p>\u5728\u7a33\u6052\u8fdb\u52a8\u65f6\uff0c\u91cd\u529b\u77e9\u7b49\u4e8e\u89d2\u52a8\u91cf\u53d8\u5316\u7387\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><mo>=<\/mo><mi>\u03c9<\/mi><msub><mi>L<\/mi><mtext>\u6c34\u5e73<\/mtext><\/msub><mo>=<\/mo><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">m g l \\sin \\beta = \\omega L_{\\text{\u6c34\u5e73}} = \\omega (I_1 \\omega \\sin \\beta)<\/annotation><\/semantics><\/math><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span>L<\/span><\/span>\u6c34\u5e73<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span>(<\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>)<\/span><\/span><\/span><\/span><\/span>\u5176\u4e2d <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>l<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">l<\/annotation><\/semantics><\/math><\/span><\/span>l<\/span><\/span><\/span><\/span> \u662f\u91cd\u5fc3\u5230\u9876\u70b9\u7684\u8ddd\u79bb\uff0c\u5bf9\u4e8e\u5747\u5300\u5706\u9525\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>l<\/mi><mo>=<\/mo><mfrac><mn>3<\/mn><mn>4<\/mn><\/mfrac><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">l = \\frac{3}{4} h<\/annotation><\/semantics><\/math><\/span><\/span>l<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u3002<\/p>\u89e3\u51fa <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03a9<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"normal\">\u03a9<\/mi><mo>=<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mi>\u03c9<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">I_1 \\Omega = \\frac{m g l}{\\omega}<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\uff0c\u6c34\u5e73\u548c\u7ad6\u76f4\u5206\u91cf\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.36em\" columnalign=\"left left\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msub><mi>L<\/mi><mtext>\u6c34\u5e73<\/mtext><\/msub><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msub><mi>L<\/mi><mtext>\u7ad6\u76f4<\/mtext><\/msub><mo>=<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mi>\u03c9<\/mi><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><annotation encoding=\"application\/x-tex\">\\begin{cases}\nL_{\\text{\u6c34\u5e73}} = I_1 \\omega \\sin \\beta \\\\\nL_{\\text{\u7ad6\u76f4}} = \\frac{m g l}{\\omega} \\cos \\beta\n\\end{cases}<\/annotation><\/semantics><\/math><\/span><\/span>{<\/span><\/span><\/span>L<\/span><\/span>\u6c34\u5e73<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>L<\/span><\/span>\u7ad6\u76f4<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7b54\u6848\uff1a<\/strong><\/p>\u6c34\u5e73\u5206\u91cf\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>L<\/mi><mtext>\u6c34\u5e73<\/mtext><\/msub><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">L_{\\text{\u6c34\u5e73}} = I_1 \\omega \\sin \\beta<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span>\u6c34\u5e73<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span>\u7ad6\u76f4\u5206\u91cf\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>L<\/mi><mtext>\u7ad6\u76f4<\/mtext><\/msub><mo>=<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mi>\u03c9<\/mi><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">L_{\\text{\u7ad6\u76f4}} = \\frac{m g l}{\\omega} \\cos \\beta<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span>\u7ad6\u76f4<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><hr>\uff083\uff09\u6c42\u9640\u87ba\u8f6c\u52a8\u7684\u6700\u5c0f\u603b\u89d2\u901f\u5ea6\u5927\u5c0f <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/strong><\/p>\u603b\u89d2\u901f\u5ea6\u7684\u5e73\u65b9\uff1a<\/strong><\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><mn>2<\/mn><\/msubsup><mo>=<\/mo><msup><mi mathvariant=\"normal\">\u03a9<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{\u603b}}^2 = \\Omega^2 + (\\omega \\sin \\beta)^2<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5c06 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><mo>=<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\Omega = \\frac{m g l}{I_1 \\omega}<\/annotation><\/semantics><\/math><\/span><\/span>\u03a9<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4ee3\u5165\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><mn>2<\/mn><\/msubsup><mo>=<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{\u603b}}^2 = \\left( \\frac{m g l}{I_1 \\omega} \\right)^2 + (\\omega \\sin \\beta)^2<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u6c42\u6700\u5c0f\u503c\uff1a<\/strong><\/p>\u5bf9 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><mn>2<\/mn><\/msubsup><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{\u603b}}^2<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5173\u4e8e <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> \u6c42\u5bfc\u5e76\u4ee4\u5176\u4e3a\u96f6\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi mathvariant=\"normal\">d<\/mi><msubsup><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><mn>2<\/mn><\/msubsup><\/mrow><mrow><mi mathvariant=\"normal\">d<\/mi><mi>\u03c9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mo>\u2212<\/mo><mn>2<\/mn><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mfrac><mn>1<\/mn><mi>\u03c9<\/mi><\/mfrac><mo>+<\/mo><mn>2<\/mn><mi>\u03c9<\/mi><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b2<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{\\mathrm{d} \\omega_{\\text{\u603b}}^2}{\\mathrm{d} \\omega} = -2 \\left( \\frac{m g l}{I_1 \\omega} \\right)^2 \\frac{1}{\\omega} + 2 \\omega \\sin^2 \\beta = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span><\/span>(<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>2<\/span>\u03c9<\/span><\/span>sin<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u89e3\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mn>0<\/mn><mn>2<\/mn><\/msubsup><mo>=<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b2<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0^2 = \\frac{m g l}{I_1 \\sin^2 \\beta}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>sin<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165\u5df2\u77e5\u7684 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I_1<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u548c <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>l<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">l<\/annotation><\/semantics><\/math><\/span><\/span>l<\/span><\/span><\/span><\/span>\uff1a<\/strong><\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mfrac><mn>3<\/mn><mn>10<\/mn><\/mfrac><mi>m<\/mi><msup><mi>h<\/mi><mn>2<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I_1 = \\frac{3}{10} m h^2 \\tan^2 \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>h<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>l<\/mi><mo>=<\/mo><mfrac><mn>3<\/mn><mn>4<\/mn><\/mfrac><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">l = \\frac{3}{4} h<\/annotation><\/semantics><\/math><\/span><\/span>l<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span><\/li><\/ul>\u4ee3\u5165\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mn>0<\/mn><mn>2<\/mn><\/msubsup><mo>=<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>3<\/mn><mn>4<\/mn><\/mfrac><mi>h<\/mi><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><mrow><mfrac><mn>3<\/mn><mn>10<\/mn><\/mfrac><mi>m<\/mi><msup><mi>h<\/mi><mn>2<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b2<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>5<\/mn><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><mi>h<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b2<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0^2 = \\frac{m g \\left( \\frac{3}{4} h \\right)}{\\frac{3}{10} m h^2 \\tan^2 \\alpha \\sin^2 \\beta} = \\frac{5 g}{2 h \\tan^2 \\alpha \\sin^2 \\beta}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>h<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>sin<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span><\/span>(<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>h<\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>sin<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span>g<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><msqrt><mfrac><mrow><mn>5<\/mn><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><mi>h<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b2<\/mi><\/mrow><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 = \\sqrt{ \\frac{5 g}{2 h \\tan^2 \\alpha \\sin^2 \\beta} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>h<\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>sin<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span>g<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"3.08em\" viewBox=\"0 0 400000 3240\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M473,2793\nc339.3,-1799.3,509.3,-2700,510,-2702 l0 -0\nc3.3,-7.3,9.3,-11,18,-11 H400000v40H1017.7\ns-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200\nc0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26\ns76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,\n606zM1001 80h400000v40H1017.7z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165\u6570\u503c\u8ba1\u7b97\uff1a<\/strong><\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>12<\/mn><\/mfrac><mo>=<\/mo><mn>1<\/mn><msup><mn>5<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha = \\frac{\\pi}{12} = 15^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>\u03b1<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>12<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span>5<\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><mo>\u2248<\/mo><mn>0.2679<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\tan \\alpha \\approx 0.2679<\/annotation><\/semantics><\/math><\/span><\/span>tan<\/span><\/span>\u03b1<\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.2679<\/span><\/span><\/span><\/span><\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b2<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><mo>=<\/mo><mn>3<\/mn><msup><mn>0<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\beta = \\frac{\\pi}{6} = 30^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>3<\/span>0<\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><mo>\u2248<\/mo><mn>0.5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sin \\beta \\approx 0.5<\/annotation><\/semantics><\/math><\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.5<\/span><\/span><\/span><\/span><\/li><\/ul>\u8ba1\u7b97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><msqrt><mfrac><mrow><mn>5<\/mn><mo>\u00d7<\/mo><mn>9.8<\/mn><\/mrow><mrow><mn>2<\/mn><mi>h<\/mi><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>0.2679<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>0.5<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/mfrac><\/msqrt><mo>\u2248<\/mo><msqrt><mfrac><mn>49<\/mn><mrow><mi>h<\/mi><mo>\u00d7<\/mo><mn>0.0179<\/mn><\/mrow><\/mfrac><\/msqrt><mo>\u2248<\/mo><mn>52.440<\/mn><msqrt><mfrac><mn>1<\/mn><mi>h<\/mi><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 = \\sqrt{ \\frac{5 \\times 9.8}{2 h \\times (0.2679)^2 \\times (0.5)^2} } \\approx \\sqrt{ \\frac{49}{ h \\times 0.0179} } \\approx 52.440 \\sqrt{ \\frac{1}{h} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>h<\/span><\/span>\u00d7<\/span><\/span>(<\/span>0.2679<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>(<\/span>0.5<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span>\u00d7<\/span><\/span>9.8<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"3.08em\" viewBox=\"0 0 400000 3240\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M473,2793\nc339.3,-1799.3,509.3,-2700,510,-2702 l0 -0\nc3.3,-7.3,9.3,-11,18,-11 H400000v40H1017.7\ns-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200\nc0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26\ns76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,\n606zM1001 80h400000v40H1017.7z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>\u00d7<\/span><\/span>0.0179<\/span><\/span><\/span><\/span><\/span><\/span><\/span>49<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>52.440<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7cfb\u6570\u4fdd\u7559\u4e09\u4f4d\u5c0f\u6570\uff1a<\/strong><\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>\u2248<\/mo><mn>52.440<\/mn><msqrt><mfrac><mn>1<\/mn><mi>h<\/mi><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 \\approx 52.440 \\sqrt{ \\frac{1}{h} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>52.440<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7b54\u6848\uff1a<\/strong><\/p>\u6700\u5c0f\u603b\u89d2\u901f\u5ea6\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><msqrt><mfrac><mrow><mn>5<\/mn><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><mi>h<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b2<\/mi><\/mrow><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 = \\sqrt{ \\frac{5 g}{2 h \\tan^2 \\alpha \\sin^2 \\beta} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>h<\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>sin<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span>g<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"3.08em\" viewBox=\"0 0 400000 3240\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M473,2793\nc339.3,-1799.3,509.3,-2700,510,-2702 l0 -0\nc3.3,-7.3,9.3,-11,18,-11 H400000v40H1017.7\ns-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200\nc0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26\ns76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,\n606zM1001 80h400000v40H1017.7z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165\u6570\u503c\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>\u2248<\/mo><mn>52.440<\/mn><msqrt><mfrac><mn>1<\/mn><mi>h<\/mi><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 \\approx 52.440 \\sqrt{ \\frac{1}{h} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>52.440<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-6\" data-scroll-anchor=\"false\"><h5 class=\"sr-only\">\u60a8\u8bf4\uff1a<\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div data-message-author-role=\"user\" data-message-id=\"aaa28c17-90b2-4f97-866f-78005422fe6f\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden items-end rtl:items-start\"><div class=\"relative max-w-[70%] rounded-3xl px-5 py-2.5 bg-[#f4f4f4] dark:bg-token-main-surface-secondary\"><div 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\u4f59\u5f26\u53d1\u5149\u4f53\u610f\u5473\u7740\u5149\u6e90\u4e0a\u4efb\u4f55\u4e00\u4e2a\u9762\u5143\u5bf9\u4efb\u4f55\u51fa\u5c04\u89d2\u7684\u4eae\u5ea6B\u90fd\u76f8\u540c\uff0c\u4e5f\u5373\u5149\u5f3a\u6b63\u6bd4\u4e8e\u51fa\u5c04\u89d2\u7684\u4f59\u5f26\u503ccos\u2061\u03d5\u3002\n<\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-7\" data-scroll-anchor=\"true\"><h5 class=\"sr-only\"><\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"flex-shrink-0 flex flex-col relative items-end\"><div><div class=\"pt-0\"><div class=\"gizmo-bot-avatar flex h-8 w-8 items-center justify-center overflow-hidden rounded-full\"><div class=\"relative p-1 rounded-sm flex items-center 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clip-rule=\"evenodd\" d=\"M5.29289 9.29289C5.68342 8.90237 6.31658 8.90237 6.70711 9.29289L12 14.5858L17.2929 9.29289C17.6834 8.90237 18.3166 8.90237 18.7071 9.29289C19.0976 9.68342 19.0976 10.3166 18.7071 10.7071L12.7071 16.7071C12.5196 16.8946 12.2652 17 12 17C11.7348 17 11.4804 16.8946 11.2929 16.7071L5.29289 10.7071C4.90237 10.3166 4.90237 9.68342 5.29289 9.29289Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/button><\/div><\/p><div data-message-author-role=\"assistant\" data-message-id=\"9c9c7098-26e2-490d-8a06-9f68a235992d\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[3px]\"><div class=\"markdown prose w-full break-words dark:prose-invert light\">\u89e3\u7b54\uff1a<\/strong><\/p>\uff081\uff09\u6c42\u5149\u6e90\u5bf9\u4e8e <math 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encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span><\/span><\/span>\u3002\u5728\u7403\u4f53\u7684\u5de6\u4fa7\u6781\u70b9\uff08<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mi>\u03c0<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = \\pi<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span> \u5904\uff09\u653e\u7f6e\u4e86\u4e00\u4e2a\u5c0f\u7403\u5f62\u5149\u6e90\uff0c\u534a\u5f84\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>r<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">r_0<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u5e76\u4e14 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>r<\/mi><mn>0<\/mn><\/msub><mo>\u226a<\/mo><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r_0 \\ll r<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u226a<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u3002<\/p>\u6211\u4eec\u9700\u8981\u8ba1\u7b97\u5149\u6e90\u901a\u8fc7\u7403\u9762\u955c\uff08\u5185\u8868\u9762\uff09\u5728 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = 0<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span> \u9644\u8fd1\u7684\u6210\u50cf\u4f4d\u7f6e\uff0c\u4ee5\u53ca\u6a2a\u5411\u653e\u5927\u7387 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span><\/span><\/span>\u3002<\/p>\u8ba1\u7b97\uff1a<\/strong><\/p><ol><li>\u786e\u5b9a\u7403\u9762\u955c\u7684\u7126\u8ddd <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math><\/span><\/span>f<\/span><\/span><\/span><\/span>\uff1a<\/strong><\/li><\/ol>\u5bf9\u4e8e\u7403\u9762\u955c\uff0c\u7126\u8ddd <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math><\/span><\/span>f<\/span><\/span><\/span><\/span> \u4e0e\u66f2\u7387\u534a\u5f84 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R<\/annotation><\/semantics><\/math><\/span><\/span>R<\/span><\/span><\/span><\/span> \u7684\u5173\u7cfb\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>f<\/mi><mo>=<\/mo><mfrac><mi>R<\/mi><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">f = \\frac{R}{2}<\/annotation><\/semantics><\/math><\/span><\/span>f<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7531\u4e8e\u7403\u7684\u534a\u5f84\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span><\/span><\/span>\uff0c\u7403\u9762\u955c\u7684\u66f2\u7387\u534a\u5f84\u4e5f\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><mo>=<\/mo><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R = r<\/annotation><\/semantics><\/math><\/span><\/span>R<\/span><\/span>=<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u3002\u56e0\u6b64\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>f<\/mi><mo>=<\/mo><mfrac><mi>r<\/mi><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">f = \\frac{r}{2}<\/annotation><\/semantics><\/math><\/span><\/span>f<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"2\"><li>\u786e\u5b9a\u7269\u8ddd <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>s<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">s<\/annotation><\/semantics><\/math><\/span><\/span>s<\/span><\/span><\/span><\/span>\uff1a<\/strong><\/li><\/ol>\u7269\u8ddd\u662f\u4ece\u5149\u6e90\u5230\u955c\u9762\u7684\u8ddd\u79bb\u3002\u5149\u6e90\u4f4d\u4e8e <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mi>\u03c0<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = \\pi<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span> \u5904\uff0c\u5373\u7403\u7684\u5de6\u7aef\u70b9\uff0c\u5230\u7403\u5fc3\u7684\u8ddd\u79bb\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span><\/span><\/span>\uff0c\u4ece\u7403\u5fc3\u5230\u955c\u9762\uff08<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = 0<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span> \u5904\uff09\u7684\u8ddd\u79bb\u4e5f\u662f <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span><\/span><\/span>\u3002\u56e0\u6b64\uff0c\u7269\u8ddd\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mn>2<\/mn><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">s = 2r<\/annotation><\/semantics><\/math><\/span><\/span>s<\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span><ol start=\"3\"><li>\u5e94\u7528\u7403\u9762\u955c\u6210\u50cf\u516c\u5f0f\uff1a<\/strong><\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>1<\/mn><mi>f<\/mi><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mi>s<\/mi><\/mfrac><mo>+<\/mo><mfrac><mn>1<\/mn><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1}{f} = \\frac{1}{s} + \\frac{1}{s'}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>f<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165\u5df2\u77e5\u503c\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>2<\/mn><mi>r<\/mi><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mfrac><mn>1<\/mn><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2}{r} = \\frac{1}{2r} + \\frac{1}{s'}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u89e3\u65b9\u7a0b\u6c42 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">s'<\/annotation><\/semantics><\/math><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>2<\/mn><mi>r<\/mi><\/mfrac><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mfrac><mspace linebreak=\"newline\"><\/mspace><mfrac><mn>4<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><\/mfrac><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mfrac><mspace linebreak=\"newline\"><\/mspace><mfrac><mn>3<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mfrac><mspace linebreak=\"newline\"><\/mspace><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2}{r} - \\frac{1}{2r} = \\frac{1}{s'} \\\\\n\\frac{4}{2r} - \\frac{1}{2r} = \\frac{1}{s'} \\\\\n\\frac{3}{2r} = \\frac{1}{s'} \\\\\ns' = \\frac{2r}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\uff0c\u50cf\u8ddd\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">s' = \\frac{2r}{3}<\/annotation><\/semantics><\/math><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u5373\u50cf\u4f4d\u4e8e\u955c\u9762\uff08<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = 0<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span> \u5904\uff09\u53f3\u4fa7 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2r}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u4f4d\u7f6e\u3002<\/p><ol start=\"4\"><li>\u8ba1\u7b97\u6a2a\u5411\u653e\u5927\u7387 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span><\/span><\/span>\uff1a<\/strong><\/li><\/ol>\u6a2a\u5411\u653e\u5927\u7387\u5b9a\u4e49\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mi>s<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">V = -\\frac{s'}{s}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>s<\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165\u6570\u503c\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mn>3<\/mn><\/mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">V = -\\frac{\\frac{2r}{3}}{2r} = -\\frac{1}{3}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7ed3\u8bba\uff1a<\/strong><\/p><ul><li>\u6210\u50cf\u4f4d\u7f6e\uff1a<\/strong> \u50cf\u4f4d\u4e8e\u955c\u9762\u53f3\u4fa7 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2r}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5904\uff0c\u5373\u8ddd\u79bb\u7403\u5fc3 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2r}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/li><li>\u6a2a\u5411\u653e\u5927\u7387\uff1a<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">V = -\\frac{1}{3}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u8868\u793a\u50cf\u662f\u5012\u7acb\u7f29\u5c0f\u7684\uff0c\u5c3a\u5bf8\u4e3a\u539f\u6765\u7684 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/li><\/ul><hr>\uff082\uff09\u6c42\u5c0f\u5706\u76d8\u4e24\u9762\u63a5\u6536\u7684\u7167\u5ea6\u5927\u5c0f\u4e4b\u6bd4\uff0c\u5e76\u4ee3\u5165 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = \\frac{\\pi}{6}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u8ba1\u7b97\u6570\u503c\u3002<\/strong><\/p>\u5206\u6790\uff1a<\/strong><\/p><ul><li>\u7403\u5185\u8868\u9762 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>><\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta > \\frac{\\pi}{3}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u533a\u57df\u88ab\u6d82\u9ed1\uff0c\u4e0d\u53cd\u5c04\u5149\u7ebf\u3002<\/li><li>\u5c0f\u5706\u76d8\u63a2\u6d4b\u5668\u4f4d\u4e8e\u6781\u8f74\u4e0a\uff0c\u63a2\u6d4b\u5e73\u9762\u5782\u76f4\u4e8e\u6781\u8f74\uff0c\u534a\u5f84\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>r<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">r_0<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/li><li>\u5c0f\u5706\u76d8\u7684\u4f4d\u7f6e\u4f7f\u5176\u6070\u597d\u80fd\u63a5\u6536\u5230\u5728 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span> \u5904\u53cd\u5c04\u7684\u5149\u7ebf\u3002<\/li><\/ul>\u51e0\u4f55\u5173\u7cfb\uff1a<\/strong><\/p><ol><li>\u786e\u5b9a\u5c0f\u5706\u76d8\u7684\u4f4d\u7f6e\uff1a<\/strong><\/li><\/ol>\u8bbe\u5c0f\u5706\u76d8\u4f4d\u4e8e\u6781\u8f74\u4e0a\uff0c\u8ddd\u7403\u5fc3\u7684\u8ddd\u79bb\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>z<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">z<\/annotation><\/semantics><\/math><\/span><\/span>z<\/span><\/span><\/span><\/span>\u3002\u5149\u7ebf\u4ece\u5149\u6e90 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>S<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">S<\/annotation><\/semantics><\/math><\/span><\/span>S<\/span><\/span><\/span><\/span>\uff08<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mi>\u03c0<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = \\pi<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span> \u5904\uff09\u51fa\u53d1\uff0c\u7ecf\u8fc7\u7403\u9762\u4e0a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span> \u5904\u7684\u70b9 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span><\/span><\/span> \u53cd\u5c04\uff0c\u6700\u7ec8\u5230\u8fbe\u5c0f\u5706\u76d8 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math><\/span><\/span>D<\/span><\/span><\/span><\/span>\u3002<\/p><ol start=\"2\"><li>\u5efa\u7acb\u5750\u6807\u7cfb\uff1a<\/strong><\/li><\/ol><ul><li>\u5149\u6e90\u4f4d\u7f6e\uff1a<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mover accent=\"true\"><mi>r<\/mi><mo>\u20d7<\/mo><\/mover><mi>S<\/mi><\/msub><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mi>r<\/mi><mo separator=\"true\">,<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{r}_S = (-r, 0)<\/annotation><\/semantics><\/math><\/span><\/span><\/span>r<\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.471em\" height=\"0.714em\" viewBox=\"0 0 471 714\" preserveAspectRatio=\"xMinYMin\" style=\"width: 0.471em;\"><path d=\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z\"><\/path><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>S<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>\u2212<\/span>r<\/span>,<\/span><\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/li><li>\u53cd\u5c04\u70b9\u4f4d\u7f6e\uff1a<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mover accent=\"true\"><mi>r<\/mi><mo>\u20d7<\/mo><\/mover><mi>P<\/mi><\/msub><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo separator=\"true\">,<\/mo><mi>r<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{r}_P = (r \\cos \\theta, r \\sin \\theta)<\/annotation><\/semantics><\/math><\/span><\/span><\/span>r<\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.471em\" height=\"0.714em\" viewBox=\"0 0 471 714\" preserveAspectRatio=\"xMinYMin\" style=\"width: 0.471em;\"><path d=\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z\"><\/path><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>r<\/span><\/span>cos<\/span><\/span>\u03b8<\/span>,<\/span><\/span>r<\/span><\/span>sin<\/span><\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/li><li>\u5c0f\u5706\u76d8\u4f4d\u7f6e\uff1a<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mover accent=\"true\"><mi>r<\/mi><mo>\u20d7<\/mo><\/mover><mi>D<\/mi><\/msub><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>z<\/mi><mo separator=\"true\">,<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{r}_D = (z, 0)<\/annotation><\/semantics><\/math><\/span><\/span><\/span>r<\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.471em\" height=\"0.714em\" viewBox=\"0 0 471 714\" preserveAspectRatio=\"xMinYMin\" style=\"width: 0.471em;\"><path d=\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z\"><\/path><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>D<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>z<\/span>,<\/span><\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/li><\/ul>\uff082.1\uff09\u8ba1\u7b97\u4e24\u9762\u7167\u5ea6\u4e4b\u6bd4<\/strong><\/p>\u8ba1\u7b97\u524d\u4fa7\uff08\u671d\u5411\u5149\u6e90\uff09\u63a5\u6536\u7684\u7167\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>E<\/mi><mtext>\u524d<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">E_{\\text{\u524d}}<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span>\u524d<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff1a<\/strong><\/p><ol><li>\u5149\u6e90\u7684\u8f90\u5c04\u7279\u6027\uff1a<\/strong><\/li><\/ol><ul><li>\u4f59\u5f26\u53d1\u5149\u4f53\uff0c\u4eae\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B<\/annotation><\/semantics><\/math><\/span><\/span>B<\/span><\/span><\/span><\/span> \u5728\u5404\u4e2a\u65b9\u5411\u4e0a\u76f8\u540c\u3002<\/li><li>\u5149\u5f3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span><\/span><\/span> \u4e0e\u51fa\u5c04\u89d2\u7684\u4f59\u5f26\u6210\u6b63\u6bd4\uff1a<\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mi>B<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>S<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03d5<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I = B \\, dS \\cos \\phi<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>=<\/span><\/span><\/span><\/span>B<\/span><\/span>d<\/span>S<\/span><\/span>cos<\/span><\/span>\u03d5<\/span><\/span><\/span><\/span><\/span><ol start=\"2\"><li>\u524d\u4fa7\u63a5\u6536\u7684\u5149\u7ebf\uff1a<\/strong><\/li><\/ol><ul><li>\u5149\u7ebf\u7ecf\u8fc7\u4e00\u6b21\u53cd\u5c04\uff0c\u5230\u8fbe\u5c0f\u5706\u76d8\u7684\u524d\u4fa7\u3002<\/li><li>\u7531\u4e8e\u5149\u6e90\u548c\u63a2\u6d4b\u5668\u5c3a\u5bf8\u5f88\u5c0f\uff0c\u53ef\u4ee5\u8ba4\u4e3a\u5149\u7ebf\u662f\u5355\u4e00\u65b9\u5411\u7684\u3002<\/li><\/ul><ol start=\"3\"><li>\u524d\u4fa7\u7167\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>E<\/mi><mtext>\u524d<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">E_{\\text{\u524d}}<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span>\u524d<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e0e\u53cd\u5c04\u89d2\u7684\u5173\u7cfb\uff1a<\/strong><\/li><\/ol><ul><li>\u524d\u4fa7\u7167\u5ea6\u4e0e\u53cd\u5c04\u533a\u57df\u7684\u53cd\u5c04\u7cfb\u6570\u548c\u51e0\u4f55\u56e0\u7d20\u6709\u5173\u3002<\/li><li>\u5728\u7406\u60f3\u53cd\u5c04\u60c5\u51b5\u4e0b\uff0c\u524d\u4fa7\u7167\u5ea6\u4e0e <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(1 + \\cos \\theta)<\/annotation><\/semantics><\/math><\/span><\/span>(<\/span>1<\/span><\/span>+<\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span> \u6210\u6b63\u6bd4\u3002<\/li><\/ul>\u8ba1\u7b97\u80cc\u4fa7\uff08\u80cc\u5411\u5149\u6e90\uff09\u63a5\u6536\u7684\u7167\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>E<\/mi><mtext>\u80cc<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">E_{\\text{\u80cc}}<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span>\u80cc<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff1a<\/strong><\/p><ol><li>\u80cc\u4fa7\u63a5\u6536\u7684\u5149\u7ebf\uff1a<\/strong><\/li><\/ol><ul><li>\u5149\u7ebf\u7ecf\u8fc7\u4e00\u6b21\u53cd\u5c04\uff0c\u5230\u8fbe\u5c0f\u5706\u76d8\u7684\u80cc\u4fa7\u3002<\/li><li>\u7531\u4e8e\u53cd\u5c04\u89d2\u7684\u5173\u7cfb\uff0c\u80cc\u4fa7\u63a5\u6536\u5230\u7684\u5149\u7ebf\u8f83\u5c11\u3002<\/li><\/ul><ol start=\"2\"><li>\u80cc\u4fa7\u7167\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>E<\/mi><mtext>\u80cc<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">E_{\\text{\u80cc}}<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span>\u80cc<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e0e\u53cd\u5c04\u89d2\u7684\u5173\u7cfb\uff1a<\/strong><\/li><\/ol><ul><li>\u80cc\u4fa7\u7167\u5ea6\u4e0e <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(1 - \\cos \\theta)<\/annotation><\/semantics><\/math><\/span><\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span> \u6210\u6b63\u6bd4\u3002<\/li><\/ul>\u6c42\u7167\u5ea6\u4e4b\u6bd4\uff1a<\/strong><\/p>\u56e0\u6b64\uff0c\u4e24\u9762\u63a5\u6536\u7684\u7167\u5ea6\u4e4b\u6bd4\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><msub><mi>E<\/mi><mtext>\u80cc<\/mtext><\/msub><msub><mi>E<\/mi><mtext>\u524d<\/mtext><\/msub><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{E_{\\text{\u80cc}}}{E_{\\text{\u524d}}} = \\frac{1 - \\cos \\theta}{1 + \\cos \\theta}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>E<\/span><\/span>\u524d<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>\u80cc<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>+<\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = \\frac{\\pi}{6}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u8ba1\u7b97\u6570\u503c\uff1a<\/strong><\/p><ol><li>\u8ba1\u7b97 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\cos \\theta<\/annotation><\/semantics><\/math><\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span>\uff1a<\/strong><\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo>=<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mn>3<\/mn><msup><mn>0<\/mn><mo>\u2218<\/mo><\/msup><mo>=<\/mo><mfrac><msqrt><mn>3<\/mn><\/msqrt><mn>2<\/mn><\/mfrac><mo>\u2248<\/mo><mn>0.8660<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\cos \\theta = \\cos \\left( \\frac{\\pi}{6} \\right ) = \\cos 30^\\circ = \\frac{\\sqrt{3}}{2} \\approx 0.8660<\/annotation><\/semantics><\/math><\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>cos<\/span><\/span>(<\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>cos<\/span><\/span>3<\/span>0<\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.8660<\/span><\/span><\/span><\/span><\/span><ol start=\"2\"><li>\u8ba1\u7b97\u7167\u5ea6\u4e4b\u6bd4\uff1a<\/strong><\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><msub><mi>E<\/mi><mtext>\u80cc<\/mtext><\/msub><msub><mi>E<\/mi><mtext>\u524d<\/mtext><\/msub><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><mn>0.8660<\/mn><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><mn>0.8660<\/mn><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>0.1340<\/mn><mn>1.8660<\/mn><\/mfrac><mo>\u2248<\/mo><mn>0.07180<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{E_{\\text{\u80cc}}}{E_{\\text{\u524d}}} = \\frac{1 - 0.8660}{1 + 0.8660} = \\frac{0.1340}{1.8660} \\approx 0.07180<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>E<\/span><\/span>\u524d<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>\u80cc<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>+<\/span><\/span>0.8660<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>0.8660<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1.8660<\/span><\/span><\/span><\/span><\/span><\/span><\/span>0.1340<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.07180<\/span><\/span><\/span><\/span><\/span>\u7b54\u6848\uff1a<\/strong><\/p><ul><li>\u4e24\u9762\u7167\u5ea6\u4e4b\u6bd4\uff1a<\/strong><\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><msub><mi>E<\/mi><mtext>\u80cc<\/mtext><\/msub><msub><mi>E<\/mi><mtext>\u524d<\/mtext><\/msub><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{E_{\\text{\u80cc}}}{E_{\\text{\u524d}}} = \\frac{1 - \\cos \\theta}{1 + \\cos \\theta}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>E<\/span><\/span>\u524d<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>\u80cc<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>+<\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ul><li>\u5f53 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = \\frac{\\pi}{6}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u65f6\uff0c\u7167\u5ea6\u4e4b\u6bd4\u7ea6\u4e3a\uff1a<\/strong><\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><msub><mi>E<\/mi><mtext>\u80cc<\/mtext><\/msub><msub><mi>E<\/mi><mtext>\u524d<\/mtext><\/msub><\/mfrac><mo>\u2248<\/mo><mn>0.07180<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{E_{\\text{\u80cc}}}{E_{\\text{\u524d}}} \\approx 0.07180<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>E<\/span><\/span>\u524d<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>\u80cc<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.07180<\/span><\/span><\/span><\/span><\/span><hr>\uff082.2\uff09\u7ed3\u679c\u7684\u9002\u7528\u8303\u56f4\u3001\u7269\u7406\u5dee\u5f02\u6765\u6e90\u53ca\u6d82\u9ed1\u7684\u7528\u610f<\/strong><\/p>\u9002\u7528\u8303\u56f4\uff1a<\/strong><\/p><ul><li>\u5355\u6b21\u53cd\u5c04\u8fd1\u4f3c\uff1a<\/strong> \u8ba1\u7b97\u4e2d\u5047\u8bbe\u5149\u7ebf\u53ea\u7ecf\u8fc7\u4e00\u6b21\u53cd\u5c04\uff0c\u5ffd\u7565\u4e86\u591a\u6b21\u53cd\u5c04\u7684\u5f71\u54cd\u3002<\/li><li>\u7406\u60f3\u53cd\u5c04\u548c\u5c0f\u5c3a\u5bf8\u5047\u8bbe\uff1a<\/strong> \u5047\u8bbe\u7403\u5185\u8868\u9762\u4e3a\u7406\u60f3\u955c\u9762\uff0c\u5149\u6e90\u548c\u63a2\u6d4b\u5668\u5c3a\u5bf8\u5f88\u5c0f\u3002<\/li><\/ul>\u7269\u7406\u5b9e\u9645\u7684\u5dee\u5f02\u6765\u6e90\uff1a<\/strong><\/p><ul><li>\u591a\u6b21\u53cd\u5c04\u7684\u5f71\u54cd\uff1a<\/strong> \u5728\u5b9e\u9645\u60c5\u51b5\u4e0b\uff0c\u5149\u7ebf\u53ef\u80fd\u5728\u7403\u5185\u7ecf\u8fc7\u591a\u6b21\u53cd\u5c04\uff0c\u5bfc\u81f4\u80cc\u4fa7\u63a5\u6536\u5230\u7684\u5149\u7ebf\u589e\u591a\u3002<\/li><li>\u6563\u5c04\u548c\u5438\u6536\uff1a<\/strong> \u5b9e\u9645\u6750\u6599\u53ef\u80fd\u5b58\u5728\u6563\u5c04\u548c\u5438\u6536\uff0c\u5f71\u54cd\u7167\u5ea6\u7684\u8ba1\u7b97\u3002<\/li><\/ul>\u6d82\u9ed1\u7684\u7528\u610f\uff1a<\/strong><\/p><ul><li>\u51cf\u5c11\u591a\u6b21\u53cd\u5c04\uff1a<\/strong> \u5c06 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>><\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta > \\frac{\\pi}{3}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u533a\u57df\u6d82\u9ed1\uff0c\u5438\u6536\u5149\u7ebf\uff0c\u51cf\u5c11\u591a\u6b21\u53cd\u5c04\u7684\u5f71\u54cd\uff0c\u4f7f\u8ba1\u7b97\u66f4\u63a5\u8fd1\u5b9e\u9645\u60c5\u51b5\u3002<\/li><li>\u7b80\u5316\u8ba1\u7b97\uff1a<\/strong> \u901a\u8fc7\u9650\u5236\u53cd\u5c04\u533a\u57df\uff0c\u53ef\u4ee5\u5ffd\u7565\u591a\u6b21\u53cd\u5c04\uff0c\u7b80\u5316\u4e86\u5bf9\u7167\u5ea6\u7684\u8ba1\u7b97\u3002<\/li><\/ul>\u7ed3\u8bba\uff1a<\/strong><\/p><ul><li>\u7ed3\u679c\u9002\u7528\u4e8e\u5355\u6b21\u53cd\u5c04\u7684\u7406\u60f3\u60c5\u51b5\uff0c\u4e14\u5149\u6e90\u548c\u63a2\u6d4b\u5668\u5c3a\u5bf8\u5f88\u5c0f\u7684\u8fd1\u4f3c\u3002<\/strong><\/li><li>\u5dee\u5f02\u6765\u6e90\u4e8e\u5ffd\u7565\u591a\u6b21\u53cd\u5c04\u548c\u5b9e\u9645\u6750\u6599\u7684\u975e\u7406\u60f3\u6027\u3002<\/strong><\/li><li>\u6d82\u9ed1\u7684\u76ee\u7684\u662f\u51cf\u5c11\u591a\u6b21\u53cd\u5c04\uff0c\u4f7f\u6a21\u578b\u66f4\u63a5\u8fd1\u5355\u6b21\u53cd\u5c04\u7684\u7406\u60f3\u60c5\u51b5\u3002<\/strong><\/li><\/ul><hr>\u603b\u7ed3\uff1a<\/strong><\/p><ul><li>\uff081\uff09<\/strong> \u5149\u6e90\u5728\u7403\u9762\u955c\u4e0a\u7684\u6210\u50cf\u4f4d\u7f6e\u4e3a\u8ddd\u79bb\u955c\u9762 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2r}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5904\uff0c\u6a2a\u5411\u653e\u5927\u7387\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">V = -\\frac{1}{3}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/li><li>\uff082.1\uff09<\/strong> \u5c0f\u5706\u76d8\u4e24\u9762\u63a5\u6536\u7684\u7167\u5ea6\u4e4b\u6bd4\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1 - \\cos \\theta}{1 + \\cos \\theta}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>1<\/span>+<\/span>c<\/span>o<\/span>s<\/span><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span>\u2212<\/span>c<\/span>o<\/span>s<\/span><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u5f53 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = \\frac{\\pi}{6}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u65f6\uff0c\u6bd4\u4f8b\u7ea6\u4e3a 0.07180\u3002<\/li><li>\uff082.2\uff09<\/strong> \u7ed3\u679c\u9002\u7528\u4e8e\u5355\u6b21\u53cd\u5c04\u7684\u7406\u60f3\u60c5\u51b5\uff0c\u6d82\u9ed1\u7684\u7528\u610f\u662f\u51cf\u5c11\u591a\u6b21\u53cd\u5c04\u7684\u5f71\u54cd\uff0c\u4f7f\u8ba1\u7b97\u66f4\u51c6\u786e\u3002<\/li><\/ul><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><\/div><\/div><audio class=\"fixed bottom-0 left-0 hidden h-0 w-0\" autoplay=\"\" crossorigin=\"anonymous\"><\/audio><\/span><link rel=\"modulepreload\" href=\"https:\/\/cdn.oaistatic.com\/assets\/manifest-564c6ab1.js\"><link rel=\"modulepreload\" href=\"https:\/\/cdn.oaistatic.com\/assets\/oq6izsb3tap8af78.js\"><link rel=\"modulepreload\" href=\"https:\/\/cdn.oaistatic.com\/assets\/dckfpppmklfb86fh.js\"><link rel=\"modulepreload\" href=\"https:\/\/cdn.oaistatic.com\/assets\/j36pxwjzprolzjey.js\"><link rel=\"modulepreload\" href=\"https:\/\/cdn.oaistatic.com\/assets\/jxnrbz5aqhc2frpe.js\"><link 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flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden items-end rtl:items-start\"><div class=\"relative max-w-[70%] rounded-3xl px-5 py-2.5 bg-[#f4f4f4] dark:bg-token-main-surface-secondary\"><div class=\"whitespace-pre-wrap\">\u94a0\u539f\u5b50\u6c14\u4f53\u7684\u8272\u6563\r\n\t\u672c\u9898\u65e8\u5728\u8003\u8651\u7535\u78c1\u6ce2\u5728\u94a0\u539f\u5b50\u6c14\u4f53\u4e2d\u4f20\u64ad\u65f6\u7684\u884c\u4e3a\u3002\u94a0\u539f\u5b50\u53ef\u4ee5\u88ab\u7b80\u5316\u4e3a\u5e26\u6709\u5355\u4f4d\u7535\u8377\u91cfe\u7684\u539f\u5b50\u5b9e\u548c\u7535\u8377\u91cf\u4e3a-e\u7684\u5916\u5c42\u7535\u5b50\u4e91\u3002\u6211\u4eec\u5047\u8bbe\u94a0\u539f\u5b50\u5916\u5c42\u7535\u5b50\u4e91\u5206\u5e03\u548c\u6c22\u539f\u5b50\u7535\u5b50\u4e91\u5206\u5e03\u76f8\u4f3c\uff0c\u5373\u6709\u7535\u8377\u5bc6\u5ea6\u5206\u5e03\u4e3a\r\n\u2588(\u03c1=-e\/(\u03c0a^3 ) e^(- 2\/a r)#\uff082.1\uff09 )\r\n\u5728\u4ee5\u4e0b\u7684\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u5047\u8bbe\u7535\u5b50\u4e91\u662f\u521a\u6027\u7684\uff0c\u5373\u5f53\u7535\u5b50\u4e91\u548c\u539f\u5b50\u5b9e\u53d1\u751f\u76f8\u5bf9\u4f4d\u79fb\u65f6\u7535\u5b50\u4e91\u7684\u5f62\u72b6\u4e0d\u53d8\uff0c\u7535\u8377\u5bc6\u5ea6\u5206\u5e03\u4e5f\u4e0d\u53d8\u3002\r\n\uff081\uff09\u6c42\u7535\u5b50\u4e91\u5206\u5e03\u4ea7\u751f\u7684\u7535\u573a\u5206\u5e03E \u20d7=E \u20d7(r)\u3002\r\n\uff082\uff09\u5047\u8bbe\u7535\u5b50\u4e91\u7684\u603b\u8d28\u91cf\u662fm\uff0c\u539f\u5b50\u5b9e\u7684\u8d28\u91cf\u8fdc\u5927\u4e8e\u7535\u5b50\u4e91\u7684\u8d28\u91cf\u3002\u73b0\u5c06\u4e00\u675f\u7535\u78c1\u6ce2\u5165\u5c04\u7531\u8fd9\u6837\u7684\u94a0\u539f\u5b50\u7ec4\u6210\u7684\u6c14\u4f53\uff0c\u8bbe\u94a0\u539f\u5b50\u6570\u5bc6\u5ea6\u4e3an\u3002\u7535\u78c1\u6ce2\u4e2d\u4e0e\u7269\u8d28\u4e3b\u8981\u4f5c\u7528\u6210\u5206\u4e3a\u7535\u573a\uff0c\u7535\u573a\u4f1a\u4f7f\u7535\u5b50\u4e91\u4e0e\u539f\u5b50\u5b9e\u53d1\u751f\u76f8\u5bf9\u4f4d\u79fb\uff0c\u6211\u4eec\u5047\u8bbe\u7535\u78c1\u6ce2\u4e0d\u662f\u5f88\u5f3a\uff0c\u4ee5\u81f3\u4e8e\u8fd9\u6837\u7684\u4f4d\u79fb\u8fdc\u5c0f\u4e8ea\u3002\u540c\u65f6\uff0c\u6211\u4eec\u5047\u8bbe\u7535\u78c1\u6ce2\u6ce2\u957f\u03bb\u226ba,\u221b(1\/n)\u3002\r\n\uff082.1\uff09\u6c42\u7535\u5b50\u4e91\u4e2d\u5fc3\u4e0e\u539f\u5b50\u5b9e\u53d1\u751f\u76f8\u5bf9\u4f4d\u79fb\u03b4r \u20d7=\u03b4rx \u0302\u65f6\u7535\u5b50\u4e91\u6536\u5230\u7684\u4f5c\u7528\u529b\u03b4F \u20d7\uff0c\u5df2\u77e5\u03b4r\u226aa\u3002\r\n\uff082.2\uff09\u53ef\u4ee5\u8bbe\u539f\u5b50\u9644\u8fd1\u7535\u78c1\u6ce2\u5f62\u5f0f\u4e3aE \u20d7(t)=E_0 cos\u2061\u03c9t x \u0302\uff0c\u8bd5\u7531\u6b64\u5217\u51fa\u7535\u5b50\u4e91\u8fd0\u52a8\u6ee1\u8db3\u7684\u5fae\u5206\u65b9\u7a0b\uff0c\u8bbe\u7535\u5b50\u4e91\u76f8\u5bf9\u539f\u5b50\u5b9e\u7684\u4f4d\u79fb\u4e3ar \u20d7\uff0c\u6211\u4eec\u540c\u65f6\u5047\u8bbe\u963b\u5c3c\u529b\u53ef\u4ee5\u88ab\u5ffd\u7565\u3002\r\n\uff082.3\uff09\u6c42\u89e3\uff082.2\uff09\u4e2d\u5f97\u5230\u7684\u65b9\u7a0b\uff0c\u4ec5\u9700\u8981\u5199\u51fa\u7a33\u5b9a\u89e3\u3002\uff08\u4e0d\u8003\u8651\u53d1\u751f\u5171\u632f\u7684\u60c5\u51b5\uff09\r\n\uff082.4\uff09\u7535\u5b50\u4e91\u4e0e\u539f\u5b50\u5b9e\u7684\u76f8\u5bf9\u4f4d\u79fb\u4f1a\u4ea7\u751f\u5076\u6781\u77e9\u8fdb\u800c\u4ea7\u751f\u6781\u5316\u77e2\u91cfP \u20d7\uff0c\u8bd5\u6c42P \u20d7\u7684\u8868\u8fbe\u5f0f\u3002\r\n\uff082.5\uff09\u89c2\u5bdfP \u20d7\u7684\u8868\u8fbe\u5f0f\uff0c\u8bd5\u8fdb\u4e00\u6b65\u7ed9\u51fa\u94a0\u539f\u5b50\u6c14\u4f53\u7684\u7b49\u6548\u76f8\u5bf9\u4ecb\u7535\u5e38\u6570\u03b5_r\u3002\r\n\u63d0\u793a\uff1a\u4f60\u53ef\u80fd\u4f1a\u7528\u5230\u4e0b\u9762\u7684\u6570\u5b66\u516c\u5f0f\uff1a\r\n\u2588(\u222b\u2592\u3016e^(-\u03bbx) x^2 dx\u3017=-e^(-\u03bbx)\/\u03bb^3 (\u03bb^2 x^2+2\u03bbx+2)+C#\uff082.2\uff09 )\r\n\u2588(\u5f53x\u5f88\u5c0f\u65f6\uff0ce^x\u22481+x+1\/2 x^2+1\/6 x^3#\uff082.3\uff09 )\r\n<\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-3\" data-scroll-anchor=\"true\"><h5 class=\"sr-only\"><\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"flex-shrink-0 flex flex-col relative items-end\"><div><div class=\"pt-0\"><div class=\"gizmo-bot-avatar flex h-8 w-8 items-center justify-center overflow-hidden rounded-full\"><div class=\"relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8\"><svg width=\"41\" height=\"41\" viewBox=\"0 0 41 41\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\" role=\"img\"><text x=\"-9999\" y=\"-9999\">ChatGPT<\/text><path d=\"M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 2.81182 28.3141C2.95951 29.7256 3.40701 31.0892 4.12437 32.3138C5.18791 34.1659 6.8123 35.6322 8.76321 36.5013C10.7141 37.3704 12.8907 37.5973 14.9789 37.1492C15.9208 38.2107 17.0786 39.0587 18.3747 39.6366C19.6709 40.2144 21.0755 40.5087 22.4946 40.4998C24.6307 40.5054 26.7133 39.8321 28.4418 38.5772C30.1704 37.3223 31.4556 35.5506 32.1119 33.5179C33.5027 33.2332 34.8167 32.6547 35.9659 31.821C37.115 30.9874 38.0728 29.9178 38.7752 28.684C39.8458 26.8371 40.3023 24.6979 40.0789 22.5748C39.8556 20.4517 38.9639 18.4544 37.5324 16.8707ZM22.4978 37.8849C20.7443 37.8874 19.0459 37.2733 17.6994 36.1501C17.7601 36.117 17.8666 36.0586 17.936 36.0161L25.9004 31.4156C26.1003 31.3019 26.2663 31.137 26.3813 30.9378C26.4964 30.7386 26.5563 30.5124 26.5549 30.2825V19.0542L29.9213 20.998C29.9389 21.0068 29.9541 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4.06716 21.2235 3.55289 19.3046C3.03862 17.3858 3.30624 15.3413 4.29707 13.6194ZM31.955 20.0556L22.2312 14.4411L25.5976 12.4981C25.6142 12.4872 25.6333 12.4805 25.6531 12.4787C25.6729 12.4769 25.6928 12.4801 25.7111 12.4879L33.7631 17.1364C34.9967 17.849 36.0017 18.8982 36.6606 20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/div><\/div><\/div><\/div><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col agent-turn\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div class=\"group absolute left-0 top-0 mr-1.5 h-8 overflow-hidden mt-1\"><button class=\"\" style=\"opacity: 1;\"><div class=\"flex items-center justify-start gap-1\">\u601d\u8003 75 \u79d2<\/span><svg width=\"24\" height=\"24\" viewBox=\"0 0 24 24\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\"><path fill-rule=\"evenodd\" clip-rule=\"evenodd\" d=\"M5.29289 9.29289C5.68342 8.90237 6.31658 8.90237 6.70711 9.29289L12 14.5858L17.2929 9.29289C17.6834 8.90237 18.3166 8.90237 18.7071 9.29289C19.0976 9.68342 19.0976 10.3166 18.7071 10.7071L12.7071 16.7071C12.5196 16.8946 12.2652 17 12 17C11.7348 17 11.4804 16.8946 11.2929 16.7071L5.29289 10.7071C4.90237 10.3166 4.90237 9.68342 5.29289 9.29289Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/button><\/div><\/p><div data-message-author-role=\"assistant\" data-message-id=\"291fa5e7-ca47-4655-8f1d-c24bbbef9f0a\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[3px]\"><div class=\"markdown prose w-full break-words dark:prose-invert light\">\u89e3\u7b54\uff1a<\/strong><\/p>\uff081\uff09\u6c42\u7535\u5b50\u4e91\u5206\u5e03\u4ea7\u751f\u7684\u7535\u573a\u5206\u5e03 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">E<\/mi><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"bold\">r<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{E}(\\mathbf{r})<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span>(<\/span>r<\/span>)<\/span><\/span><\/span><\/span>\u3002<\/strong><\/p>\u7535\u5b50\u4e91\u7684\u7535\u8377\u5bc6\u5ea6\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c1<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mi>e<\/mi><mrow><mi>\u03c0<\/mi><msup><mi>a<\/mi><mn>3<\/mn><\/msup><\/mrow><\/mfrac><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\rho(r) = -\\frac{e}{\\pi a^3} e^{-2r\/a}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c1<\/span>(<\/span>r<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>\u03c0<\/span>a<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7531\u4e8e\u7535\u5b50\u4e91\u7684\u7535\u8377\u5206\u5e03\u662f\u7403\u5bf9\u79f0\u7684\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u9ad8\u65af\u5b9a\u5f8b\u6765\u6c42\u7535\u573a\u3002\u9996\u5148\uff0c\u8ba1\u7b97\u534a\u5f84\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span><\/span><\/span> \u7684\u7403\u5185\u5305\u542b\u7684\u603b\u7535\u8377 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">Q(r)<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span>(<\/span>r<\/span>)<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>r<\/mi><\/msubsup><mi>\u03c1<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo stretchy=\"false\">)<\/mo><mn>4<\/mn><mi>\u03c0<\/mi><msup><mi>r<\/mi><mrow><mo mathvariant=\"normal\">\u2032<\/mo><mn>2<\/mn><\/mrow><\/msup><mi>d<\/mi><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mrow><mn>4<\/mn><mi>e<\/mi><\/mrow><msup><mi>a<\/mi><mn>3<\/mn><\/msup><\/mfrac><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>r<\/mi><\/msubsup><msup><mi>r<\/mi><mrow><mo mathvariant=\"normal\">\u2032<\/mo><mn>2<\/mn><\/mrow><\/msup><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mn>2<\/mn><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msup><mi>d<\/mi><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">Q(r) = \\int_0^r \\rho(r') 4\\pi r'^2 dr' = -\\frac{4e}{a^3} \\int_0^r r'^2 e^{-2r'\/a} dr'<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span>(<\/span>r<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c1<\/span>(<\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>4<\/span>\u03c0<\/span>r<\/span><\/span>\u20322<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>a<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span>\u20322<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>2<\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8fdb\u884c\u79ef\u5206\uff0c\u4ee4 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><mi>a<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">u = \\frac{2r'}{a}<\/annotation><\/semantics><\/math><\/span><\/span>u<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u5219 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mfrac><mrow><mi>a<\/mi><mi>u<\/mi><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">r' = \\frac{a u}{2}<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>u<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mfrac><mi>a<\/mi><mn>2<\/mn><\/mfrac><mi>d<\/mi><mi>u<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">dr' = \\frac{a}{2} du<\/annotation><\/semantics><\/math><\/span><\/span>d<\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>u<\/span><\/span><\/span><\/span>\u3002\u5f53 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">r' = 0<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">u = 0<\/annotation><\/semantics><\/math><\/span><\/span>u<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\uff1b\u5f53 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r' = r<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mi>a<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">u = \\frac{2r}{a}<\/annotation><\/semantics><\/math><\/span><\/span>u<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u79ef\u5206\u53d8\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mrow><mn>4<\/mn><mi>e<\/mi><\/mrow><msup><mi>a<\/mi><mn>3<\/mn><\/msup><\/mfrac><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>a<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>3<\/mn><\/msup><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msubsup><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>u<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mi>u<\/mi><\/mrow><\/msup><mi>d<\/mi><mi>u<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msubsup><mfrac><msup><mi>u<\/mi><mn>2<\/mn><\/msup><mn>2<\/mn><\/mfrac><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mi>u<\/mi><\/mrow><\/msup><mi>d<\/mi><mi>u<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Q(r) = -\\frac{4e}{a^3} \\left( \\frac{a}{2} \\right)^3 \\int_0^{2r\/a} \\left( \\frac{u}{2} \\right)^2 e^{-u} du = -e \\int_0^{2r\/a} \\frac{u^2}{2} e^{-u} du<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span>(<\/span>r<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>a<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>u<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>u<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>u<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>u<\/span><\/span><\/span><\/span><\/span>\u5229\u7528\u63d0\u793a\u516c\u5f0f\uff082.2\uff09\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>\u222b<\/mo><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mi>\u03bb<\/mi><mi>x<\/mi><\/mrow><\/msup><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mi>\u03bb<\/mi><mi>x<\/mi><\/mrow><\/msup><msup><mi>\u03bb<\/mi><mn>3<\/mn><\/msup><\/mfrac><mo stretchy=\"false\">(<\/mo><msup><mi>\u03bb<\/mi><mn>2<\/mn><\/msup><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>\u03bb<\/mi><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\int e^{-\\lambda x} x^2 dx = -\\frac{e^{-\\lambda x}}{\\lambda^3} (\\lambda^2 x^2 + 2\\lambda x + 2) + C<\/annotation><\/semantics><\/math><\/span><\/span>\u222b<\/span><\/span>e<\/span><\/span>\u2212<\/span>\u03bb<\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>\u03bb<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>\u03bb<\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03bb<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>2<\/span>\u03bb<\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>2<\/span>)<\/span><\/span>+<\/span><\/span><\/span><\/span>C<\/span><\/span><\/span><\/span><\/span>\u5e26\u5165 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03bb<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\lambda = 1<\/annotation><\/semantics><\/math><\/span><\/span>\u03bb<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\uff0c\u8ba1\u7b97\u79ef\u5206\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msubsup><mfrac><msup><mi>u<\/mi><mn>2<\/mn><\/msup><mn>2<\/mn><\/mfrac><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mi>u<\/mi><\/mrow><\/msup><mi>d<\/mi><mi>u<\/mi><mo>=<\/mo><msubsup><mrow><mo fence=\"true\">[<\/mo><mo>\u2212<\/mo><mfrac><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mi>u<\/mi><\/mrow><\/msup><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><msup><mi>u<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>u<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo fence=\"true\">]<\/mo><\/mrow><mn>0<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msubsup><mo>=<\/mo><mrow><mo fence=\"true\">[<\/mo><mo>\u2212<\/mo><mfrac><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msup><mn>2<\/mn><\/mfrac><mrow><mo fence=\"true\">(<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo>+<\/mo><mn>2<\/mn><mo fence=\"true\">)<\/mo><\/mrow><mo>+<\/mo><mn>1<\/mn><mo fence=\"true\">]<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\int_0^{2r\/a} \\frac{u^2}{2} e^{-u} du = \\left[ -\\frac{e^{-u}}{2} (u^2 + 2u + 2) \\right]_0^{2r\/a} = \\left[ -\\frac{e^{-2r\/a}}{2} \\left( \\left( \\frac{2r}{a} \\right)^2 + 2 \\cdot \\frac{2r}{a} + 2 \\right) + 1 \\right]<\/annotation><\/semantics><\/math><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>u<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>u<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>u<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>u<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>2<\/span>u<\/span><\/span>+<\/span><\/span>2<\/span>)<\/span>]<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>(<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>2<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>2<\/span>)<\/span><\/span><\/span><\/span>+<\/span><\/span>1<\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\uff0c\u603b\u7535\u8377\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><mrow><mo fence=\"true\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mrow><mo fence=\"true\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo>+<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msup><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">Q(r) = -e \\left( 1 - \\left( 1 + \\frac{2r}{a} + \\left( \\frac{2r}{a} \\right)^2 \\frac{1}{2} \\right) e^{-2r\/a} \\right)<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span>(<\/span>r<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span><\/span>(<\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>(<\/span><\/span>1<\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>(<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7b80\u5316\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><mrow><mo fence=\"true\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mrow><mo fence=\"true\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo>+<\/mo><mfrac><mrow><mn>2<\/mn><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msup><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">Q(r) = -e \\left( 1 - \\left( 1 + \\frac{2r}{a} + \\frac{2r^2}{a^2} \\right) e^{-2r\/a} \\right)<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span>(<\/span>r<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span><\/span>(<\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>(<\/span><\/span>1<\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7535\u573a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">E<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{E}(r)<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span>(<\/span>r<\/span>)<\/span><\/span><\/span><\/span> \u4e3a\uff1a<\/p><ul><li>\u5f53 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><mo>><\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">r > 0<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span>><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span> \u65f6\uff1a<\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">E<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mn>4<\/mn><mi>\u03c0<\/mi><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><\/mrow><\/mfrac><mfrac><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mfrac><mover accent=\"true\"><mi mathvariant=\"bold\">r<\/mi><mo>^<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{E}(r) = \\frac{1}{4\\pi \\varepsilon_0} \\frac{Q(r)}{r^2} \\hat{\\mathbf{r}}<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span>(<\/span>r<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span>(<\/span>r<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5c06 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">Q(r)<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span>(<\/span>r<\/span>)<\/span><\/span><\/span><\/span> \u4ee3\u5165\uff0c\u5f97\u5230\u7535\u5b50\u4e91\u4ea7\u751f\u7684\u7535\u573a\u5206\u5e03\u3002<\/p><hr>\uff082\uff09\uff082.1\uff09\u6c42\u7535\u5b50\u4e91\u4e2d\u5fc3\u4e0e\u539f\u5b50\u5b9e\u53d1\u751f\u76f8\u5bf9\u4f4d\u79fb <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b4<\/mi><mi mathvariant=\"bold\">r<\/mi><mo>=<\/mo><mi>\u03b4<\/mi><mi>r<\/mi><mover accent=\"true\"><mi mathvariant=\"bold\">x<\/mi><mo>^<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\delta \\mathbf{r} = \\delta r \\hat{\\mathbf{x}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span>r<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4r<\/span><\/span>x<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u65f6\u7535\u5b50\u4e91\u53d7\u5230\u7684\u4f5c\u7528\u529b <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b4<\/mi><mi mathvariant=\"bold\">F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\delta \\mathbf{F}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span>F<\/span><\/span><\/span><\/span>\uff0c\u5df2\u77e5 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b4<\/mi><mi>r<\/mi><mo>\u226a<\/mo><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\delta r \\ll a<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4r<\/span><\/span>\u226a<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span>\u3002<\/strong><\/p>\u5f53\u7535\u5b50\u4e91\u76f8\u5bf9\u4e8e\u539f\u5b50\u5b9e\u53d1\u751f\u5c0f\u4f4d\u79fb <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b4<\/mi><mi mathvariant=\"bold\">r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\delta \\mathbf{r}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span>r<\/span><\/span><\/span><\/span> \u65f6\uff0c\u4ea7\u751f\u7684\u5076\u6781\u77e9\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">p<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><mi>\u03b4<\/mi><mi mathvariant=\"bold\">r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{p} = -e \\delta \\mathbf{r}<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e\u03b4<\/span>r<\/span><\/span><\/span><\/span><\/span>\u8fd9\u4e2a\u5076\u6781\u77e9\u4f1a\u5bfc\u81f4\u6062\u590d\u529b\uff0c\u5176\u5927\u5c0f\u4e0e\u4f4d\u79fb\u6210\u6b63\u6bd4\u3002\u4f5c\u7528\u5728\u7535\u5b50\u4e91\u4e0a\u7684\u6062\u590d\u529b\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03b4<\/mi><mi mathvariant=\"bold\">F<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>k<\/mi><mi>\u03b4<\/mi><mi mathvariant=\"bold\">r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\delta \\mathbf{F} = -k \\delta \\mathbf{r}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span>F<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>k<\/span>\u03b4<\/span>r<\/span><\/span><\/span><\/span><\/span>\u5176\u4e2d\u529b\u5e38\u6570 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math><\/span><\/span>k<\/span><\/span><\/span><\/span> \u53ef\u7531\u7535\u52bf\u80fd\u7684\u4e8c\u9636\u5bfc\u6570\u6c42\u5f97\u3002\u5bf9\u4e8e\u7403\u5bf9\u79f0\u7535\u8377\u5206\u5e03\uff0c\u529b\u5e38\u6570\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>k<\/mi><mo>=<\/mo><mfrac><msup><mi>e<\/mi><mn>2<\/mn><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><msup><mi>a<\/mi><mn>3<\/mn><\/msup><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">k = \\frac{e^2}{4\\pi \\varepsilon_0 a^3}<\/annotation><\/semantics><\/math><\/span><\/span>k<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\uff0c\u4f5c\u7528\u5728\u7535\u5b50\u4e91\u4e0a\u7684\u529b\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03b4<\/mi><mi mathvariant=\"bold\">F<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><msup><mi>e<\/mi><mn>2<\/mn><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><msup><mi>a<\/mi><mn>3<\/mn><\/msup><\/mrow><\/mfrac><mi>\u03b4<\/mi><mi>r<\/mi><mover accent=\"true\"><mi mathvariant=\"bold\">x<\/mi><mo>^<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\delta \\mathbf{F} = -\\frac{e^2}{4\\pi \\varepsilon_0 a^3} \\delta r \\hat{\\mathbf{x}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span>F<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>4<\/span>\u03c0<\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4r<\/span><\/span>x<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><hr>\uff082.2\uff09\u5217\u51fa\u7535\u5b50\u4e91\u8fd0\u52a8\u6ee1\u8db3\u7684\u5fae\u5206\u65b9\u7a0b\uff0c\u8bbe\u7535\u5b50\u4e91\u76f8\u5bf9\u539f\u5b50\u5b9e\u7684\u4f4d\u79fb\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">r(t)<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\uff0c\u5ffd\u7565\u963b\u5c3c\u529b\u3002<\/strong><\/p>\u7535\u5b50\u4e91\u53d7\u5230\u4e24\u4e2a\u529b\uff1a<\/p><ol><li>\u6062\u590d\u529b\uff1a<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>F<\/mi><mtext>\u6062\u590d<\/mtext><\/msub><mo>=<\/mo><mo>\u2212<\/mo><mi>k<\/mi><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">F_{\\text{\u6062\u590d}} = -k r(t)<\/annotation><\/semantics><\/math><\/span><\/span>F<\/span><\/span>\u6062\u590d<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>k<\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><ol start=\"2\"><li>\u5916\u52a0\u7535\u573a\u7684\u4f5c\u7528\u529b\uff08\u7535\u5b50\u53d7\u5230\u7684\u7535\u573a\u529b\uff09\uff1a<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>F<\/mi><mtext>\u7535\u573a<\/mtext><\/msub><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><mi>E<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">F_{\\text{\u7535\u573a}} = -e E(t) = -e E_0 \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>F<\/span><\/span>\u7535\u573a<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span>E<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span>\u6839\u636e\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b\uff0c\u8fd0\u52a8\u65b9\u7a0b\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>m<\/mi><mfrac><mrow><msup><mi>d<\/mi><mn>2<\/mn><\/msup><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>d<\/mi><msup><mi>t<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><mo>=<\/mo><mo>\u2212<\/mo><mi>k<\/mi><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m \\frac{d^2 r(t)}{dt^2} = -k r(t) - e E_0 \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>m<\/span><\/span><\/span>d<\/span>t<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>k<\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span>\u6574\u7406\u5f97\u5230\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>m<\/mi><mfrac><mrow><msup><mi>d<\/mi><mn>2<\/mn><\/msup><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>d<\/mi><msup><mi>t<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><mo>+<\/mo><mi>k<\/mi><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m \\frac{d^2 r(t)}{dt^2} + k r(t) = -e E_0 \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>m<\/span><\/span><\/span>d<\/span>t<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>k<\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><hr>\uff082.3\uff09\u6c42\u89e3\u4e0a\u8ff0\u65b9\u7a0b\u7684\u7a33\u5b9a\u89e3\uff08\u4e0d\u8003\u8651\u53d1\u751f\u5171\u632f\u7684\u60c5\u51b5\uff09\u3002<\/strong><\/p>\u8bbe particular \u89e3\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>A<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r(t) = A \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span>\u5c06\u5176\u4ee3\u5165\u5fae\u5206\u65b9\u7a0b\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>\u2212<\/mo><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mi>A<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><mo>+<\/mo><mi>k<\/mi><mi>A<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">-m \\omega^2 A \\cos \\omega t + k A \\cos \\omega t = -e E_0 \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>\u2212<\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span>+<\/span><\/span><\/span><\/span>k<\/span>A<\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span>\u6574\u7406\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><mi>A<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">(-m \\omega^2 + k) A = -e E_0<\/annotation><\/semantics><\/math><\/span><\/span>(<\/span>\u2212<\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>k<\/span>)<\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u89e3\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mrow><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><\/mrow><mrow><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">A = \\frac{e E_0}{m \\omega^2 - k}<\/annotation><\/semantics><\/math><\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\uff0c\u7a33\u6001\u89e3\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><\/mrow><mrow><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r(t) = \\frac{e E_0}{m \\omega^2 - k} \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><hr>\uff082.4\uff09\u6c42\u6781\u5316\u77e2\u91cf <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{P}<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span><\/span><\/span> \u7684\u8868\u8fbe\u5f0f\u3002<\/strong><\/p>\u5076\u6781\u77e9\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">p<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mover accent=\"true\"><mi mathvariant=\"bold\">x<\/mi><mo>^<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{p} = -e r(t) \\hat{\\mathbf{x}}<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>er<\/span>(<\/span>t<\/span>)<\/span><\/span>x<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u6781\u5316\u77e2\u91cf\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">P<\/mi><mo>=<\/mo><mi>n<\/mi><mi mathvariant=\"bold\">p<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>n<\/mi><mi>e<\/mi><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mover accent=\"true\"><mi mathvariant=\"bold\">x<\/mi><mo>^<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{P} = n \\mathbf{p} = -n e r(t) \\hat{\\mathbf{x}}<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>n<\/span>er<\/span>(<\/span>t<\/span>)<\/span><\/span>x<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5c06 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">r(t)<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span> \u4ee3\u5165\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">P<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>n<\/mi><mi>e<\/mi><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><\/mrow><mrow><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><mo fence=\"true\">)<\/mo><\/mrow><mover accent=\"true\"><mi mathvariant=\"bold\">x<\/mi><mo>^<\/mo><\/mover><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mrow><mi>n<\/mi><msup><mi>e<\/mi><mn>2<\/mn><\/msup><msub><mi>E<\/mi><mn>0<\/mn><\/msub><\/mrow><mrow><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><mover accent=\"true\"><mi mathvariant=\"bold\">x<\/mi><mo>^<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{P} = -n e \\left( \\frac{e E_0}{m \\omega^2 - k} \\cos \\omega t \\right) \\hat{\\mathbf{x}} = -\\frac{n e^2 E_0}{m \\omega^2 - k} \\cos \\omega t \\hat{\\mathbf{x}}<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>n<\/span>e<\/span><\/span>(<\/span><\/span><\/span><\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>x<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>e<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span>x<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><hr>\uff082.5\uff09\u7ed9\u51fa\u94a0\u539f\u5b50\u6c14\u4f53\u7684\u7b49\u6548\u76f8\u5bf9\u4ecb\u7535\u5e38\u6570 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03b5<\/mi><mi>r<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\varepsilon_r<\/annotation><\/semantics><\/math><\/span><\/span>\u03b5<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/strong><\/p>\u6839\u636e\u4ecb\u8d28\u7684\u6781\u5316\u5173\u7cfb\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">P<\/mi><mo>=<\/mo><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>\u03b5<\/mi><mi>r<\/mi><\/msub><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"bold\">E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{P} = \\varepsilon_0 (\\varepsilon_r - 1) \\mathbf{E}<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b5<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span>E<\/span><\/span><\/span><\/span><\/span>\u5c06 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{P}<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span><\/span><\/span> \u548c <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{E}<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span><\/span><\/span> \u7684\u8868\u8fbe\u5f0f\u4ee3\u5165\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>\u2212<\/mo><mfrac><mrow><mi>n<\/mi><msup><mi>e<\/mi><mn>2<\/mn><\/msup><msub><mi>E<\/mi><mn>0<\/mn><\/msub><\/mrow><mrow><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><mo>=<\/mo><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>\u03b5<\/mi><mi>r<\/mi><\/msub><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><msub><mi>E<\/mi><mn>0<\/mn><\/msub><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">-\\frac{n e^2 E_0}{m \\omega^2 - k} \\cos \\omega t = \\varepsilon_0 (\\varepsilon_r - 1) E_0 \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>\u2212<\/span><\/span><\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>e<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b5<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span>\u6d88\u53bb <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>E<\/mi><mn>0<\/mn><\/msub><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">E_0 \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span>\uff0c\u5f97\u5230\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>\u03b5<\/mi><mi>r<\/mi><\/msub><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mrow><mi>n<\/mi><msup><mi>e<\/mi><mn>2<\/mn><\/msup><\/mrow><mrow><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\varepsilon_0 (\\varepsilon_r - 1) = -\\frac{n e^2}{m \\omega^2 - k}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b5<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>e<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\uff0c\u94a0\u539f\u5b50\u6c14\u4f53\u7684\u76f8\u5bf9\u4ecb\u7535\u5e38\u6570\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03b5<\/mi><mi>r<\/mi><\/msub><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mfrac><mrow><mi>n<\/mi><msup><mi>e<\/mi><mn>2<\/mn><\/msup><\/mrow><mrow><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\varepsilon_r = 1 - \\frac{n e^2}{\\varepsilon_0 (m \\omega^2 - k)}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b5<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>k<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>e<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><hr>\u603b\u7ed3\uff1a<\/strong><\/p>\u901a\u8fc7\u4ee5\u4e0a\u6b65\u9aa4\uff0c\u6211\u4eec\u5f97\u5230\u4e86\u7535\u5b50\u4e91\u5728\u5916\u7535\u573a\u4f5c\u7528\u4e0b\u7684\u8fd0\u52a8\u65b9\u7a0b\u3001\u7a33\u6001\u89e3\u3001\u6781\u5316\u77e2\u91cf\u4ee5\u53ca\u94a0\u539f\u5b50\u6c14\u4f53\u7684\u76f8\u5bf9\u4ecb\u7535\u5e38\u6570\u3002\u8fd9\u4e9b\u7ed3\u679c\u63cf\u8ff0\u4e86\u7535\u78c1\u6ce2\u5728\u94a0\u539f\u5b50\u6c14\u4f53\u4e2d\u4f20\u64ad\u65f6\u7684\u8272\u6563\u7279\u6027\u3002<\/p><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><\/div><\/div><audio 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\r\n\uff082\uff09\u63a5\uff081\uff09\uff0c\u4e0b\u9762\u4ece\u7535\u78c1\u573a\u7684\u89c2\u70b9\uff0c\u91cd\u65b0\u5206\u6790\u8fd9\u4e00\u95ee\u9898\u3002\r\n\uff082.1\uff09\u8003\u8651\u5230\u89c2\u5bdf\u8ddd\u79bb\u8db3\u591f\u8fdc\uff0c\u89c2\u5bdf\u8303\u56f4\u8db3\u591f\u5c0f\uff0c\u53ef\u4ee5\u5c06\u7403\u9762\u6ce2\u7684\u6ce2\u524d\u8fd1\u4f3c\u6210\u5e73\u9762\u6ce2\u3002\u7edf\u4e00\u8d77\u89c1\uff0c\u4e0d\u59a8\u5047\u8bbe\u7535\u573a\u65b9\u5411\u5728\u7eb8\u9762\u5185\uff0c\u78c1\u573a\u65b9\u5411\u4e0e\u7eb8\u9762\u5782\u76f4\u5411\u5916\uff0c\u5750\u6807\u7cfb\u5982\u56fe6.2\u6240\u793a\u3002\u6839\u636e\u7535\u78c1\u573a\u573a\u5f3a\u53d8\u6362\u5173\u7cfb\uff0c\u91cd\u65b0\u63a8\u5bfc\u7535\u78c1\u573a\u80fd\u91cf\u5bc6\u5ea6\uff0c\u5373\u8f90\u5c04\u80fd\u91cf\u7684\u53d8\u6362\u5173\u7cfb\uff0c\u6709\u5173\u89d2\u5ea6\u7528\u03a3\u7cfb\u4e2d\u7684\u89d2\u5ea6\u03b8\u8868\u793a\u3002\r\n \r\n\u63d0\u793a\uff1a\u7535\u78c1\u573a\u53d8\u6362\u516c\u5f0f\uff1a\uff08x\u65b9\u5411\u4e3a\u6362\u7cfb\u65b9\u5411\uff09\r\n\u2588(\u25a0(E_x^'=E_x@E_y^'=\u03b3(E_y-vB_z )@E_z^'=\u03b3(E_z+vB_y ) ) \u25a0(B_x^'=B_x@B_y^'=\u03b3(B_y+(vE_z)\/c^2 )@B_z^'=\u03b3(B_z-(vE_y)\/c^2 ) )#(6.1) )\r\n\uff082.2\uff09\u6211\u4eec\u8003\u8651\u6362\u7cfb\u8fc7\u7a0b\u4e2d\uff0c\u03a3'\u7cfb\u4e2d\u4e00\u5b9a\u4f53\u79efV^'\u5185\u7535\u78c1\u573a\u7684\u80fd\u91cf\u7a76\u7adf\u662f\u5982\u4f55\u53d8\u5316\u7684\uff0c\u5373\u6211\u4eec\u8981\u5c06\u03a3'\u7cfb\u4e2d\u7684\u80fd\u91cf\u4e00\u4e00\u5bf9\u5e94\u5730\u53d8\u5316\u5230\u03a3\u7cfb\u4e2d\u3002\u5728\u03a3'\u7cfb\u4e2d\u9009\u62e9\u4e00\u4e2a\u9759\u6b62\u7684\u957f\u65b9\u4f53\u6846\u67b6\uff0c\u5176\u6846\u4e2d\u7684\u80fd\u91cf\u4e3aW'\u3002\u8fd0\u7528\u6d1b\u4f26\u5179\u53d8\u6362\u5c06\u8fd9\u4e2a\u6846\u67b6\u53d8\u6362\u5230\u03a3\u7cfb\u4e2d\uff0c\u53ef\u4ee5\u53d1\u73b0\uff0c\u7531\u4e8e\u5f02\u5730\u7684\u540c\u65f6\u6027\u88ab\u7834\u574f\uff0c\u6240\u4ee5\u8fd9\u4e2a\u6846\u67b6\u4e2d\u7684\u80fd\u91cf\u5e76\u975e\u4e0e\u03a3'\u7cfb\u4e2d\u7684\u80fd\u91cf\u76f8\u540c\uff0c\u9700\u8981\u8003\u8651\u5728\u8fd9\u4e00\u65f6\u95f4\u5dee\u5185\u6d41\u5165\uff08\u51fa\uff09\u7684\u80fd\u91cf\uff0c\u8fdb\u800c\u5f97\u5230\u4e00\u4e00\u5bf9\u5e94\u7684\u80fd\u91cfW\u3002\u8bd5\u8ba1\u7b97W\u4e0eW'\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\r\n\u63d0\u793a\uff1a\r\n1\u3001\u8003\u8651\u4e00\u4e2a\u4e00\u8fb9\u5e73\u884c\u4e8e\u6362\u7cfb\u65b9\u5411\u7684\u957f\u65b9\u4f53\uff08\u5782\u76f4\u4e8e\u7eb8\u9762\u65b9\u5411\u7684\u957f\u5ea6\u5f88\u5c0f\uff0c\u53ef\u4ee5\u5ffd\u7565\u8be5\u65b9\u5411\u4e0a\u7535\u78c1\u573a\u7684\u53d8\u5316\uff09\u4f5c\u4e3a\u5206\u6790\u7684\u533a\u57df\uff0c\u5982\u56fe6.3\u6240\u793a\u3002\r\n2\u3001\u7535\u78c1\u573a\u7684\u80fd\u91cf\u5bc6\u5ea6\u8868\u793a\u4e3aw=1\/2 \u03b5_0 E^2+B^2\/(2\u03bc_0 )\uff1b\u7535\u78c1\u573a\u7684\u80fd\u6d41\u5bc6\u5ea6\u5219\u53ef\u4ee5\u7528\u5761\u5370\u4ead\u77e2\u91cfS \u20d7\u8868\u793a\uff0c\u5728\u771f\u7a7a\u4e2d\u7684\u8868\u8fbe\u5f0f\u4e3aS \u20d7=1\/\u03bc_0 E \u20d7\u00d7B \u20d7\uff0c\u5761\u5370\u4ead\u77e2\u91cf\u7684\u65b9\u5411\u5c31\u662f\u80fd\u6d41\u7684\u65b9\u5411\u3002\r\n\uff082.3\uff09\u8fd9\u6837\uff0c\u6211\u4eec\u91cd\u65b0\u5f97\u5230\u4e86\u4e0e\u76f8\u5bf9\u8bba\u591a\u666e\u52d2\u6548\u5e94\u9891\u7387\u53d8\u6362\u5f97\u5230\u7684\u76f8\u540c\u7ed3\u679c\u3002\u8bd5\u91cd\u590d\uff081.1\uff09\uff081.2\uff09\u4e2d\u7684\u64cd\u4f5c\uff0c\u518d\u6b21\u5f97\u5230\uff081.1\uff09\uff081.2\uff09\u4e2d\u7684\u7ed3\u679c\uff0c\u5373\u4e24\u4e2a\u8f90\u5c04\u529f\u7387\u89d2\u5206\u5e03\u3002\r\n<\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-3\" data-scroll-anchor=\"false\"><h5 class=\"sr-only\"><\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"flex-shrink-0 flex flex-col relative items-end\"><div><div class=\"pt-0\"><div class=\"gizmo-bot-avatar flex h-8 w-8 items-center justify-center overflow-hidden rounded-full\"><div class=\"relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8\"><svg width=\"41\" height=\"41\" viewBox=\"0 0 41 41\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\" role=\"img\"><text x=\"-9999\" y=\"-9999\">ChatGPT<\/text><path d=\"M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 2.81182 28.3141C2.95951 29.7256 3.40701 31.0892 4.12437 32.3138C5.18791 34.1659 6.8123 35.6322 8.76321 36.5013C10.7141 37.3704 12.8907 37.5973 14.9789 37.1492C15.9208 38.2107 17.0786 39.0587 18.3747 39.6366C19.6709 40.2144 21.0755 40.5087 22.4946 40.4998C24.6307 40.5054 26.7133 39.8321 28.4418 38.5772C30.1704 37.3223 31.4556 35.5506 32.1119 33.5179C33.5027 33.2332 34.8167 32.6547 35.9659 31.821C37.115 30.9874 38.0728 29.9178 38.7752 28.684C39.8458 26.8371 40.3023 24.6979 40.0789 22.5748C39.8556 20.4517 38.9639 18.4544 37.5324 16.8707ZM22.4978 37.8849C20.7443 37.8874 19.0459 37.2733 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8.36823 20.4629C8.48312 20.6619 8.64893 20.8267 8.84863 20.9404L18.5723 26.5542L15.206 28.4979C15.1894 28.5089 15.1703 28.5155 15.1505 28.5173C15.1307 28.5191 15.1107 28.516 15.0924 28.5082L7.04046 23.8557C5.32135 22.8601 4.06716 21.2235 3.55289 19.3046C3.03862 17.3858 3.30624 15.3413 4.29707 13.6194ZM31.955 20.0556L22.2312 14.4411L25.5976 12.4981C25.6142 12.4872 25.6333 12.4805 25.6531 12.4787C25.6729 12.4769 25.6928 12.4801 25.7111 12.4879L33.7631 17.1364C34.9967 17.849 36.0017 18.8982 36.6606 20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/div><\/div><\/div><\/div><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col agent-turn\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div class=\"group absolute left-0 top-0 mr-1.5 h-8 overflow-hidden mt-1\"><button class=\"\" style=\"opacity: 1;\"><div class=\"flex items-center justify-start gap-1\">Thought for 103 seconds<\/span><svg width=\"24\" height=\"24\" viewBox=\"0 0 24 24\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\"><path fill-rule=\"evenodd\" clip-rule=\"evenodd\" d=\"M5.29289 9.29289C5.68342 8.90237 6.31658 8.90237 6.70711 9.29289L12 14.5858L17.2929 9.29289C17.6834 8.90237 18.3166 8.90237 18.7071 9.29289C19.0976 9.68342 19.0976 10.3166 18.7071 10.7071L12.7071 16.7071C12.5196 16.8946 12.2652 17 12 17C11.7348 17 11.4804 16.8946 11.2929 16.7071L5.29289 10.7071C4.90237 10.3166 4.90237 9.68342 5.29289 9.29289Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/button><\/div><\/p><div data-message-author-role=\"assistant\" data-message-id=\"43cd2f06-d03b-4503-91e0-84688af687cc\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[3px]\"><div class=\"markdown prose w-full break-words dark:prose-invert light\">(1) Radiation in the Relativistic Case<\/strong><\/p>(1.1) Calculating the Power Received per Unit Solid Angle Using the Relativistic Doppler Effect<\/strong><\/p>In the rest frame of the blackbody sphere (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi mathvariant=\"normal\">\u03a3<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma'<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>), the radiation is isotropic, and the power emitted per unit solid angle is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><mrow><mi>d<\/mi><msup><mi mathvariant=\"normal\">\u03a9<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><\/mfrac><mo>=<\/mo><mfrac><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP'}{d\\Omega'} = \\frac{P'}{4\\pi}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">P'<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the total power emitted by the blackbody sphere in its rest frame.<\/p>When observed from the frame <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span> (observer's frame), due to the relativistic Doppler effect and aberration of light, both the frequency and angular distribution of the radiation change.<\/p>The relativistic Doppler factor <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b4<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\delta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span><\/span><\/span><\/span> is given by:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03b4<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mi>\u03b3<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b2<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\delta = \\frac{1}{\\gamma (1 - \\beta \\cos \\theta)}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span>\u03b2<\/span><\/span>cos<\/span><\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>where:<\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b2<\/mi><mo>=<\/mo><mfrac><mi>v<\/mi><mi>c<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\beta = \\frac{v}{c}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>c<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the dimensionless velocity,<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b3<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><msqrt><mrow><mn>1<\/mn><mo>\u2212<\/mo><msup><mi>\u03b2<\/mi><mn>2<\/mn><\/msup><\/mrow><\/msqrt><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\gamma = \\frac{1}{\\sqrt{1 - \\beta^2}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b3<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span>\u2212<\/span>\u03b2<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"1.5429em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\r\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\r\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\r\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\r\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\r\nc69,-144,104.5,-217.7,106.5,-221\r\nl0 -0\r\nc5.3,-9.3,12,-14,20,-14\r\nH400000v40H845.2724\r\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\r\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\r\nM834 80h400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the Lorentz factor,<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span> is the angle between the direction of motion and the observation direction in the <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span> frame.<\/li><\/ul>The observed power per unit solid angle transforms according to:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>P<\/mi><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><mfrac><mrow><mi>d<\/mi><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><mrow><mi>d<\/mi><msup><mi mathvariant=\"normal\">\u03a9<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP}{d\\Omega} = \\delta^4 \\frac{dP'}{d\\Omega'}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This transformation accounts for the change in energy, time intervals, and solid angles due to relativistic effects. Therefore, substituting <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><mrow><mi>d<\/mi><msup><mi mathvariant=\"normal\">\u03a9<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP'}{d\\Omega'}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>P<\/mi><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><mfrac><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP}{d\\Omega} = \\delta^4 \\frac{P'}{4\\pi}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This is the expression for the radiated power received per unit solid angle in the <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span> frame.<\/p><hr>(1.2) Calculating the Angular Distribution of Received Power and the Total Power<\/strong><\/p>When integrating <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>P<\/mi><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP}{d\\Omega}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> over all solid angles, we find that it does not yield the total power <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">P'<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> emitted by the blackbody sphere. This discrepancy arises because, in the observer's frame <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span>, the time intervals required to receive energy emitted in different directions are not the same. Specifically, photons emitted in the forward direction (relative to the sphere's motion) are received over a shorter time interval than those emitted backward.<\/p>To account for this, we consider the time transformation between the frames. The time interval <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">dt<\/annotation><\/semantics><\/math><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span> in the observer's frame relates to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><msup><mi>t<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">dt'<\/annotation><\/semantics><\/math><\/span><\/span>d<\/span>t<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> in the rest frame by:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>d<\/mi><mi>t<\/mi><mo>=<\/mo><mi>\u03b3<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b2<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><msup><mi>t<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">dt = \\gamma (1 - \\beta \\cos \\theta) dt'<\/annotation><\/semantics><\/math><\/span><\/span>d<\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b3<\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>cos<\/span><\/span>\u03b8<\/span>)<\/span>d<\/span>t<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This implies that the observer receives energy over a time interval <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">dt<\/annotation><\/semantics><\/math><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span> that depends on <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span>. Therefore, the power received per unit solid angle during the same time interval <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">dt<\/annotation><\/semantics><\/math><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span> is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><msub><mi>P<\/mi><mtext>rec<\/mtext><\/msub><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>d<\/mi><mi>E<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><mrow><mi>\u03b3<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b2<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mfrac><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP_{\\text{rec}}}{d\\Omega} = \\frac{dE}{dt d\\Omega} = \\frac{\\delta^4}{\\gamma (1 - \\beta \\cos \\theta)} \\frac{P'}{4\\pi}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span>rec<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>t<\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>E<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span>\u03b2<\/span><\/span>cos<\/span><\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Simplifying using <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b4<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mi>\u03b3<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b2<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\delta = \\frac{1}{\\gamma (1 - \\beta \\cos \\theta)}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span>(<\/span>1<\/span>\u2212<\/span>\u03b2<\/span><\/span>c<\/span>o<\/span>s<\/span><\/span><\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><msub><mi>P<\/mi><mtext>rec<\/mtext><\/msub><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>5<\/mn><\/msup><mfrac><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><mi>\u03b3<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP_{\\text{rec}}}{d\\Omega} = \\delta^5 \\frac{P'}{4\\pi \\gamma}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span>rec<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0\u03b3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>However, recognizing that <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b4<\/mi><mn>5<\/mn><\/msup><mi mathvariant=\"normal\">\/<\/mi><mi>\u03b3<\/mi><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\delta^5 \/ \\gamma = \\delta^4<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\/<\/span>\u03b3<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, we see that:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><msub><mi>P<\/mi><mtext>rec<\/mtext><\/msub><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><mfrac><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP_{\\text{rec}}}{d\\Omega} = \\delta^4 \\frac{P'}{4\\pi}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span>rec<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This suggests that, despite the varying time intervals, the expression for power per unit solid angle remains consistent.<\/p>To find the total power <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>P<\/mi><mtext>obs<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">P_{\\text{obs}}<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>obs<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> received by the observer, we integrate over all solid angles:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>P<\/mi><mtext>obs<\/mtext><\/msub><mo>=<\/mo><mo>\u222b<\/mo><mfrac><mrow><mi>d<\/mi><msub><mi>P<\/mi><mtext>rec<\/mtext><\/msub><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><mo>=<\/mo><mfrac><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><\/mrow><\/mfrac><mo>\u222b<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P_{\\text{obs}} = \\int \\frac{dP_{\\text{rec}}}{d\\Omega} d\\Omega = \\frac{P'}{4\\pi} \\int \\delta^4 d\\Omega<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>obs<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u222b<\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span>rec<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span>This integral does not equal <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">P'<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> because <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\delta^4<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> does not integrate to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b3<\/mi><mrow><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\gamma^{-4}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b3<\/span><\/span>\u2212<\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> over the sphere. The correct total power received is related to the total emitted power by:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>P<\/mi><mtext>obs<\/mtext><\/msub><mo>=<\/mo><msup><mi>\u03b3<\/mi><mn>2<\/mn><\/msup><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">P_{\\text{obs}} = \\gamma^2 P'<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>obs<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This means the observer measures a total power that is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b3<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\gamma^2<\/annotation><\/semantics><\/math><\/span><\/span>\u03b3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> times greater than the power emitted in the rest frame. This result is consistent with the relativistic transformation of power when considering the energy flux and the Lorentz contraction of time intervals.<\/p><hr>(2) Re-analysis from the Electromagnetic Field Perspective<\/strong><\/p>(2.1) Deriving the Energy Density Transformation Using Electromagnetic Field Transformations<\/strong><\/p>In the rest frame <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi mathvariant=\"normal\">\u03a3<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma'<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, consider a plane electromagnetic wave propagating at an angle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b8<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\theta'<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>. The electric and magnetic fields are related by <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>E<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mi>c<\/mi><msup><mi>B<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">E' = c B'<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>c<\/span>B<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> for electromagnetic waves in vacuum.<\/p>Using the Lorentz transformation for electromagnetic fields (Equation 6.1), we can express the fields in the observer's frame <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span>:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.36em\" columnalign=\"left left\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msubsup><mi>E<\/mi><mi>x<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mo>=<\/mo><msub><mi>E<\/mi><mi>x<\/mi><\/msub><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msubsup><mi>E<\/mi><mi>y<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mo>=<\/mo><mi>\u03b3<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>E<\/mi><mi>y<\/mi><\/msub><mo>\u2212<\/mo><mi>v<\/mi><msub><mi>B<\/mi><mi>z<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msubsup><mi>E<\/mi><mi>z<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mo>=<\/mo><mi>\u03b3<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>E<\/mi><mi>z<\/mi><\/msub><mo>+<\/mo><mi>v<\/mi><msub><mi>B<\/mi><mi>y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><mspace width=\"1em\"><\/mspace><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.36em\" columnalign=\"left left\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msubsup><mi>B<\/mi><mi>x<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mo>=<\/mo><msub><mi>B<\/mi><mi>x<\/mi><\/msub><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msubsup><mi>B<\/mi><mi>y<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mo>=<\/mo><mi>\u03b3<\/mi><mrow><mo fence=\"true\">(<\/mo><msub><mi>B<\/mi><mi>y<\/mi><\/msub><mo>+<\/mo><mfrac><mrow><mi>v<\/mi><msub><mi>E<\/mi><mi>z<\/mi><\/msub><\/mrow><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msubsup><mi>B<\/mi><mi>z<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mo>=<\/mo><mi>\u03b3<\/mi><mrow><mo fence=\"true\">(<\/mo><msub><mi>B<\/mi><mi>z<\/mi><\/msub><mo>\u2212<\/mo><mfrac><mrow><mi>v<\/mi><msub><mi>E<\/mi><mi>y<\/mi><\/msub><\/mrow><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\begin{cases}\r\nE_x' = E_x \\\\\r\nE_y' = \\gamma (E_y - v B_z) \\\\\r\nE_z' = \\gamma (E_z + v B_y)\r\n\\end{cases}\r\n\\quad\r\n\\begin{cases}\r\nB_x' = B_x \\\\\r\nB_y' = \\gamma \\left( B_y + \\frac{v E_z}{c^2} \\right) \\\\\r\nB_z' = \\gamma \\left( B_z - \\frac{v E_y}{c^2} \\right)\r\n\\end{cases}<\/annotation><\/semantics><\/math><\/span><\/span><\/span>\u23a9<\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.8889em\" height=\"0.316em\" viewBox=\"0 0 888.89 316\" preserveAspectRatio=\"xMinYMin\" style=\"width: 0.8889em;\"><path d=\"M384 0 H504 V316 H384z M384 0 H504 V316 H384z\"><\/path><\/svg><\/span><\/span><\/span>\u23a8<\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.8889em\" height=\"0.316em\" viewBox=\"0 0 888.89 316\" preserveAspectRatio=\"xMinYMin\" style=\"width: 0.8889em;\"><path d=\"M384 0 H504 V316 H384z M384 0 H504 V316 H384z\"><\/path><\/svg><\/span><\/span><\/span>\u23a7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>x<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>E<\/span><\/span>x<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>y<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03b3<\/span>(<\/span>E<\/span><\/span>y<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>v<\/span>B<\/span><\/span>z<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>E<\/span><\/span>z<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03b3<\/span>(<\/span>E<\/span><\/span>z<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>v<\/span>B<\/span><\/span>y<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a9<\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.8889em\" height=\"0.616em\" viewBox=\"0 0 888.89 616\" preserveAspectRatio=\"xMinYMin\" style=\"width: 0.8889em;\"><path d=\"M384 0 H504 V616 H384z M384 0 H504 V616 H384z\"><\/path><\/svg><\/span><\/span><\/span>\u23a8<\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.8889em\" height=\"0.616em\" viewBox=\"0 0 888.89 616\" preserveAspectRatio=\"xMinYMin\" style=\"width: 0.8889em;\"><path d=\"M384 0 H504 V616 H384z M384 0 H504 V616 H384z\"><\/path><\/svg><\/span><\/span><\/span>\u23a7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span>x<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>B<\/span><\/span>x<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span>y<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03b3<\/span><\/span>(<\/span><\/span>B<\/span><\/span>y<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span>E<\/span><\/span>z<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span>z<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03b3<\/span><\/span>(<\/span><\/span>B<\/span><\/span>z<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>c<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span>E<\/span><\/span>y<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Assuming the fields in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi mathvariant=\"normal\">\u03a3<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma'<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> are:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>E<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><msubsup><mi>E<\/mi><mn>0<\/mn><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>k<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><msup><mi>x<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>\u2212<\/mo><msup><mi>\u03c9<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><msup><mi>t<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo stretchy=\"false\">)<\/mo><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><msup><mi>B<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mfrac><msubsup><mi>E<\/mi><mn>0<\/mn><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mi>c<\/mi><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>k<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><msup><mi>x<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>\u2212<\/mo><msup><mi>\u03c9<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><msup><mi>t<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">E' = E_0' \\cos(k'x' - \\omega' t'), \\quad B' = \\frac{E_0'}{c} \\cos(k'x' - \\omega' t')<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span>(<\/span>k<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>,<\/span><\/span><\/span>B<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>c<\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span>(<\/span>k<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>Transforming to the <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span> frame and calculating the energy density <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>w<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><msup><mi>E<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mfrac><msup><mi>B<\/mi><mn>2<\/mn><\/msup><mrow><mn>2<\/mn><msub><mi>\u03bc<\/mi><mn>0<\/mn><\/msub><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">w = \\frac{1}{2} \\varepsilon_0 E^2 + \\frac{B^2}{2\\mu_0}<\/annotation><\/semantics><\/math><\/span><\/span>w<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>\u03bc<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, we find that:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>w<\/mi><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><msup><mi>w<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">w = \\delta^4 w'<\/annotation><\/semantics><\/math><\/span><\/span>w<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>w<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This shows that the energy density transforms with <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\delta^4<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, consistent with the earlier result obtained using the relativistic Doppler effect.<\/p>(2.2) Analyzing Energy Transformation in a Volume Element<\/strong><\/p>Consider a small rectangular volume <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>V<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">V'<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi mathvariant=\"normal\">\u03a3<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma'<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, with sides <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><msup><mi>x<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo separator=\"true\">,<\/mo><mi>d<\/mi><msup><mi>y<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo separator=\"true\">,<\/mo><mi>d<\/mi><msup><mi>z<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">dx', dy', dz'<\/annotation><\/semantics><\/math><\/span><\/span>d<\/span>x<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>d<\/span>y<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>d<\/span>z<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>. When transforming to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span>, due to the relativity of simultaneity, the volume does not remain synchronized, and there is an energy flow into or out of the volume during the time difference.<\/p>The energy <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>W<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">W<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span><\/span><\/span> in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span><\/span><\/span> (the transformed volume in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span>) is related to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>W<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">W'<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>V<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">V'<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> by:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>W<\/mi><mo>=<\/mo><mi>\u03b3<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>W<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>+<\/mo><mi>\u03b2<\/mi><msup><mi>S<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mi>cos<\/mi><mo>\u2061<\/mo><msup><mi>\u03b8<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mi mathvariant=\"normal\">\u0394<\/mi><msup><mi>A<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">W = \\gamma (W' + \\beta S' \\cos \\theta' \\Delta A')<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b3<\/span>(<\/span>W<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b2<\/span>S<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span>A<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>S<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">S'<\/annotation><\/semantics><\/math><\/span><\/span>S<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the Poynting vector in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi mathvariant=\"normal\">\u03a3<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma'<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><msup><mi>A<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta A'<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>A<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the area of one face of the volume, and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>cos<\/mi><mo>\u2061<\/mo><msup><mi>\u03b8<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\cos \\theta'<\/annotation><\/semantics><\/math><\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> accounts for the direction of energy flow relative to the motion.<\/p>Calculating <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>W<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">W<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span><\/span><\/span> and considering the energy flow due to the Poynting vector, we find:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>W<\/mi><mo>=<\/mo><mi>\u03b3<\/mi><msup><mi>W<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>\u03b2<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><msup><mi>\u03b8<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">W = \\gamma W' (1 + \\beta \\cos \\theta')<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b3<\/span>W<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>1<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>Using the relation between <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b8<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\theta'<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> from aberration of light:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>cos<\/mi><mo>\u2061<\/mo><msup><mi>\u03b8<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mfrac><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo>\u2212<\/mo><mi>\u03b2<\/mi><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b2<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\cos \\theta' = \\frac{\\cos \\theta - \\beta}{1 - \\beta \\cos \\theta}<\/annotation><\/semantics><\/math><\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>\u03b2<\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span>\u2212<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>we can express <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>W<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">W<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span><\/span><\/span> entirely in terms of quantities in the <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span> frame.<\/p>(2.3) Reproducing the Results from (1.1) and (1.2)<\/strong><\/p>By applying the electromagnetic field transformations and accounting for the energy flow into or out of the volume due to the lack of simultaneity, we obtain the same transformation for the energy density as before:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>w<\/mi><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><msup><mi>w<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">w = \\delta^4 w'<\/annotation><\/semantics><\/math><\/span><\/span>w<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>w<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This confirms that the observed power per unit solid angle is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>P<\/mi><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><mfrac><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP}{d\\Omega} = \\delta^4 \\frac{P'}{4\\pi}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Repeating the integration over all solid angles and considering the varying time intervals as in (1.2), we again find that the total power received by the observer is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>P<\/mi><mtext>obs<\/mtext><\/msub><mo>=<\/mo><msup><mi>\u03b3<\/mi><mn>2<\/mn><\/msup><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">P_{\\text{obs}} = \\gamma^2 P'<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>obs<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This consistency between the two approaches (relativistic Doppler effect and electromagnetic field transformations) verifies the correctness of the results obtained in (1.1) and (1.2).<\/p><hr>Conclusion<\/strong><\/p>Through both the relativistic Doppler effect and electromagnetic field transformations, we have shown that the power per unit solid angle received by an observer from a moving blackbody sphere is enhanced by a factor of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\delta^4<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> compared to the isotropic emission in the rest frame. Additionally, the total power received is greater than the emitted power due to relativistic effects, specifically time dilation and the beaming effect, resulting in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>P<\/mi><mtext>obs<\/mtext><\/msub><mo>=<\/mo><msup><mi>\u03b3<\/mi><mn>2<\/mn><\/msup><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">P_{\\text{obs}} = \\gamma^2 P'<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>obs<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/p><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-4\" data-scroll-anchor=\"false\"><h5 class=\"sr-only\">\u60a8\u8bf4\uff1a<\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div data-message-author-role=\"user\" data-message-id=\"aaa253e5-463e-4652-8e32-18201d6c7711\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden items-end rtl:items-start\"><div class=\"relative max-w-[70%] rounded-3xl px-5 py-2.5 bg-[#f4f4f4] dark:bg-token-main-surface-secondary\"><div 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\uff0c\u53ef\u770b\u4f5c\u6c27\u6c14\u4e0e\u6c2e\u6c14\u7684\u6df7\u5408\u7269\uff0c\u6c27\u6c14\u4f53\u79ef\u5360\u6bd4\u4e3a21%\u3002\u6ee1\u8db3p_0\u2248p_1\uff0c\u6545\u8fdb\u6c14\u548c\u6392\u6c14\u51b2\u7a0b\u4e0d\u505a\u529f\u3002\r\n\u6c14\u7f38\u4e2d\u7684\u6df7\u5408\u6c14\u4f53\u6070\u597d\u5728\u6c14\u7f38\u88ab\u538b\u7f29\u5230\u4f53\u79ef\u6700\u5c0f\u65f6\u53d1\u751f\u7206\u71c3\u3002\u5df2\u77e5\u5728\u7206\u71c3\u8fc7\u7a0b\u4e2d\u67f4\u6cb9\u5206\u5b50C_n H_m\u4e0e\u7a7a\u6c14\u4e2d\u7684\u6c27\u6c14\u53d1\u751f\u53cd\u5e94C_n H_m+(n+1\/4 m) O_2=nCO_2+1\/2 mH_2 O\u3002m,n\u5728\u672c\u9898\u4e2d\u53ef\u4ee5\u89c6\u4e3a\u5e38\u6570\uff0c\u4e14C_n H_m\u3001H_2 O\u7684\u5206\u5b50\u6784\u5f62\u5e76\u975e\u76f4\u7ebf\u5f62\uff0cCO_2\u5206\u5b50\u4e3a\u76f4\u7ebf\u5f62\u3002\u4e34\u754c\u538b\u5f3a\u548c\u6e29\u5ea6\u4e0b\u6bcf\u6d88\u8017\u4e00\u6469\u5c14C_n H_m\u5185\u80fd\u53d8\u5316\u91cf\u4e3a\u0394U\uff0c\u8f6e\u80ce\u534a\u5f84\u4e3ar\uff0c\u8bbe\u6bcf\u6b21\u7206\u71c3\u5747\u80fd\u8017\u5c3d\u67f4\u6cb9\u5206\u5b50C_n H_m\u4e14\u6c14\u7f38\u5185\u5404\u6210\u5206\u5747\u4e3a\u6c14\u6001\u3002\r\n \r\n\uff081\uff09\u5b9a\u91cf\u63cf\u8ff0\u67f4\u6cb9\u673a\u56db\u4e2a\u51b2\u7a0b\u4e2d\u6c14\u7f38\u5185\u6c14\u4f53\u7684\u538b\u5f3a\u548c\u6e29\u5ea6\u7684\u53d8\u5316\u60c5\u51b5\u3002\uff08\u53ea\u9700\u5199\u51fa\u5927\u81f4\u8fc7\u7a0b\u65b9\u7a0b\u548c\u53d8\u5316\u8d8b\u52bf\u5373\u53ef\uff0c\u4e0d\u7528\u8ba1\u7b97\u5177\u4f53\u53c2\u6570\uff0c\u53ef\u4ee5\u4fdd\u7559\u5404\u8fc7\u7a0b\u6c14\u4f53\u7684\u6bd4\u70ed\u5bb9\u6bd4\u03b3=C_P\/C_V \u4f46\u5fc5\u987b\u6ce8\u660e\u5176\u5bf9\u5e94\u7684\u8fc7\u7a0b\u3002\uff09\r\n\uff082\uff09\u5148\u8003\u8651\u8f66\u8f86\u5728\u5e73\u5730\u4e0a\u884c\u9a76\u4e14\u98de\u8f6e\u5300\u901f\u8f6c\u52a8\u65f6\u7684\u60c5\u5f62\uff0c\u98de\u8f6e\u7684\u89d2\u901f\u5ea6\u6052\u4e3a\u03c9\u3002\r\n\uff082.1\uff09\u6c42\u7206\u71c3\u524d\u6df7\u5408\u6c14\u4f53\u7684\u6e29\u5ea6T\u548c\u538b\u5f3ap\u3002\r\n\uff082.2\uff09\u6c42\u7206\u71c3\u540e\u6c14\u4f53\u7684\u6e29\u5ea6\u548c\u538b\u5f3a\uff0c\u5e76\u6c42\u51fa\u67f4\u6cb9\u673a\u7a33\u5b9a\u5de5\u4f5c\u65f6\u7684\u5e73\u5747\u529f\u7387\u3002\r\n\uff082.3\uff09\u5047\u8bbe\u963b\u529b\u77e9\u03c4\u4ec5\u5206\u522b\u4f5c\u7528\u4e8e\u4e24\u524d\u8f6e\uff08\u4e24\u975e\u9a71\u52a8\u8f6e\uff09\uff0c\u4e14\u4e0e\u98de\u8f6e\u89d2\u901f\u5ea6\u03c9\u6210\u6b63\u6bd4\uff1a\u03c4=\u03b2\u03c9\uff0c\u03b2\u5df2\u77e5\u3002\u6c42\u7a33\u5b9a\u65f6\u98de\u8f6e\u89d2\u901f\u5ea6\u03c9\u4e0e\u67f4\u6cb9\u4f53\u79ef\u5360\u6bd4a\u7684\u5173\u7cfb\u3002\u5047\u8bbe\u98de\u8f6e\u8fde\u63a5\u7684\u4f20\u52a8\u88c5\u7f6e\u65e0\u80fd\u91cf\u635f\u8017\u3002\uff08\u6b64\u95ee\u4e3a\u4e86\u7b80\u4fbf\u53ef\u4ee5\u5c06a\u770b\u4f5c\u4e00\u4e2a\u5c0f\u91cf\uff0c\u820d\u53bb\u9ad8\u9636\u9879\uff0c\u5e76\u4e14\u5c06\u6307\u6570\u91cc\u7684a\u53bb\u6389\u3002\uff09\r\n\t\u63d0\u793a\uff1a\u4e3a\u4e86\u7b80\u5316\u8868\u8fbe\u5f0f\uff0c\u5728\u672c\u95ee\u548c\u4e0b\u4e00\u95ee\u4e2d\uff0c\u4f60\u53ef\u4ee5\u5f15\u5165\u65e0\u91cf\u7eb2\u53c2\u6570A\uff0c\u6ee1\u8db3\r\n\u2588(A=[(V_0\/(V_0-V))^(2\/5)-1][\u0394U\/(RT_1 ) ((V_0-V)\/V_0 )^(2\/5)-3+7\/8 m]#\uff087.1\uff09 )\r\n\uff083\uff09\u63a5\u4e0a\u95ee\u5b58\u5728\u963b\u529b\u77e9\u7684\u60c5\u51b5\uff0c\u518d\u8003\u8651\u8f66\u8f86\u52a0\u901f\u4e0a\u5761\u7684\u60c5\u5f62\u3002\u5761\u9053\u4e0e\u6c34\u5e73\u9762\u5939\u89d2\u4e3a\u03b8\uff0c\u91cd\u529b\u52a0\u901f\u5ea6\u4e3ag\u3002\u521d\u59cb\uff08t=0\u65f6\uff09\u98de\u8f6e\u89d2\u901f\u5ea6\u4e3a\u03c9_0\u5c0f\u4e8e\u4e0a\u95ee\u4e2d\u6c42\u51fa\u7684\u03c9\u4e14\u98de\u8f6e\u89d2\u901f\u5ea6\u5f00\u59cb\u589e\u52a0\u3002\u8f66\u8f86\u603b\u8d28\u91cf\u4e3aM\u8fdc\u8fdc\u5927\u4e8e\u8f6e\u80ce\u3001\u98de\u8f6e\u548c\u5404\u79cd\u4f20\u52a8\u88c5\u7f6e\u7684\u8d28\u91cf\uff0c\u6c42\u98de\u8f6e\u89d2\u901f\u5ea6\u03c9\u4e0e\u65f6\u95f4t\u7684\u5173\u7cfb\u03c9(t)\u3002\r\n<\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" 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class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col agent-turn\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div class=\"group absolute left-0 top-0 mr-1.5 h-8 overflow-hidden mt-1\"><button class=\"\" style=\"opacity: 1;\"><div class=\"flex items-center justify-start gap-1\">Thought for 20 seconds<\/span><svg width=\"24\" height=\"24\" viewBox=\"0 0 24 24\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\"><path fill-rule=\"evenodd\" clip-rule=\"evenodd\" d=\"M5.29289 9.29289C5.68342 8.90237 6.31658 8.90237 6.70711 9.29289L12 14.5858L17.2929 9.29289C17.6834 8.90237 18.3166 8.90237 18.7071 9.29289C19.0976 9.68342 19.0976 10.3166 18.7071 10.7071L12.7071 16.7071C12.5196 16.8946 12.2652 17 12 17C11.7348 17 11.4804 16.8946 11.2929 16.7071L5.29289 10.7071C4.90237 10.3166 4.90237 9.68342 5.29289 9.29289Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/button><\/div><\/p><div data-message-author-role=\"assistant\" data-message-id=\"daced9a1-8bf4-4b04-87c1-ad47da8e550b\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[3px]\"><div class=\"markdown prose w-full break-words dark:prose-invert light\">Problem: Four-Stroke Diesel Engine Modeling<\/strong><\/p><hr>(1) Quantitative Description of Pressure and Temperature Changes During the Four Strokes<\/strong><\/p>In a simplified model of a four-stroke diesel engine, we analyze the pressure and temperature variations of the gas within the cylinder during each stroke. The four strokes are:<\/p><ol><li>Intake Stroke<\/strong>:<\/p><ul><li>Process<\/strong>: The intake valve <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math><\/span><\/span>A<\/span><\/span><\/span><\/span> opens, and the piston moves downward from the top dead center (TDC) to the bottom dead center (BDC), drawing in a fresh charge of diesel-air mixture from the infinite reservoir at pressure <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>p<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p_1<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> and temperature <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>T<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">T_1<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Assumptions<\/strong>: The intake process is isobaric (constant pressure) and isothermal (constant temperature) since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>p<\/mi><mn>1<\/mn><\/msub><mo>\u2248<\/mo><msub><mi>p<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p_1 \\approx p_0<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>p<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, and the gas enters from an infinite reservoir.<\/li><li>Equations and Trends<\/strong>:<ul><li>Pressure remains constant: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><msub><mi>p<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p = p_1<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Temperature remains constant: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><mo>=<\/mo><msub><mi>T<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">T = T_1<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>=<\/span><\/span><\/span><\/span>T<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Volume increases from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{min}}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>max<\/mtext><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{max}} = V_0<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>max<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><\/ul><\/li><\/ul><\/li><li>Compression Stroke<\/strong>:<\/p><ul><li>Process<\/strong>: Both valves are closed, and the piston moves upward from BDC to TDC, compressing the gas adiabatically (no heat exchange).<\/li><li>Assumptions<\/strong>: The process is adiabatic and reversible, so it follows the adiabatic equation for ideal gases.<\/li><li>Equations and Trends<\/strong>:<ul><li>Adiabatic condition: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><msup><mi>V<\/mi><mi>\u03b3<\/mi><\/msup><mo>=<\/mo><mtext>constant<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">p V^\\gamma = \\text{constant}<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span>V<\/span><\/span>\u03b3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>constant<\/span><\/span><\/span><\/span><\/span>, where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b3<\/mi><mo>=<\/mo><mfrac><msub><mi>C<\/mi><mi>p<\/mi><\/msub><msub><mi>C<\/mi><mi>v<\/mi><\/msub><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\gamma = \\frac{C_p}{C_v}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b3<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>C<\/span><\/span>v<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>C<\/span><\/span>p<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> for the diesel-air mixture.<\/li><li>Pressure increases: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><mo>\u2191<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">p \\uparrow<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>\u2191<\/span><\/span><\/span><\/span> as volume decreases.<\/li><li>Temperature increases: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><mo>\u2191<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">T \\uparrow<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>\u2191<\/span><\/span><\/span><\/span> due to compression work.<\/li><li>Volume decreases from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_0<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{min}} = V_0 - V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>V<\/span><\/span><\/span><\/span>.<\/li><\/ul><\/li><\/ul><\/li><li>Combustion (Power) Stroke<\/strong>:<\/p><ul><li>Process<\/strong>: At the end of the compression stroke, the compressed mixture reaches critical pressure and temperature, causing spontaneous combustion (explosion). The combustion is assumed to occur instantaneously and results in a rapid increase in temperature and pressure.<\/li><li>Assumptions<\/strong>: The combustion process is approximated as an adiabatic and isochoric (constant volume) process because it happens rapidly and the piston does not move significantly during combustion.<\/li><li>Equations and Trends<\/strong>:<ul><li>Isochoric condition: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><mtext>constant<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">V = \\text{constant}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span>constant<\/span><\/span><\/span><\/span><\/span>.<\/li><li>Pressure spikes: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><mo>\u2191<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">p \\uparrow<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>\u2191<\/span><\/span><\/span><\/span> sharply due to heat release from combustion.<\/li><li>Temperature spikes: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><mo>\u2191<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">T \\uparrow<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>\u2191<\/span><\/span><\/span><\/span> significantly due to the exothermic reaction.<\/li><li>Energy added: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta U<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>U<\/span><\/span><\/span><\/span> per mole of diesel fuel combusted.<\/li><\/ul><\/li><\/ul><\/li><li>Expansion Stroke (Power Stroke Continuation)<\/strong>:<\/p><ul><li>Process<\/strong>: After combustion, the high-pressure and high-temperature gases push the piston downward from TDC to BDC, doing work on the piston. Both valves remain closed, and the gas expands adiabatically.<\/li><li>Assumptions<\/strong>: The expansion is adiabatic and reversible.<\/li><li>Equations and Trends<\/strong>:<ul><li>Adiabatic condition: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><msup><mi>V<\/mi><mi>\u03b3<\/mi><\/msup><mo>=<\/mo><mtext>constant<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">p V^\\gamma = \\text{constant}<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span>V<\/span><\/span>\u03b3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>constant<\/span><\/span><\/span><\/span><\/span>.<\/li><li>Pressure decreases: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><mo>\u2193<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">p \\downarrow<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>\u2193<\/span><\/span><\/span><\/span> as volume increases.<\/li><li>Temperature decreases: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><mo>\u2193<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">T \\downarrow<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>\u2193<\/span><\/span><\/span><\/span> due to expansion work.<\/li><li>Volume increases from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{min}}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_0<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><\/ul><\/li><\/ul><\/li><li>Exhaust Stroke<\/strong>:<\/p><ul><li>Process<\/strong>: The exhaust valve <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B<\/annotation><\/semantics><\/math><\/span><\/span>B<\/span><\/span><\/span><\/span> opens, and the piston moves upward from BDC to TDC, pushing out the combustion products to the atmosphere at pressure <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>p<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p_0<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> and temperature <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>T<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">T_0<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Assumptions<\/strong>: The exhaust process is isobaric and isothermal with the environment.<\/li><li>Equations and Trends<\/strong>:<ul><li>Pressure remains constant: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><msub><mi>p<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p = p_0<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Temperature adjusts to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><mo>\u2248<\/mo><msub><mi>T<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">T \\approx T_0<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>\u2248<\/span><\/span><\/span><\/span>T<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> as gases mix with the atmosphere.<\/li><li>Volume decreases from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_0<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{min}}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><\/ul><\/li><\/ul><\/li><\/ol>Summary of Processes<\/strong>:<\/p><ul><li>Intake and Exhaust Strokes<\/strong>: Isobaric and isothermal processes at <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><msub><mi>p<\/mi><mn>1<\/mn><\/msub><mo>\u2248<\/mo><msub><mi>p<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p = p_1 \\approx p_0<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>p<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><mo>=<\/mo><msub><mi>T<\/mi><mn>1<\/mn><\/msub><mo>\u2248<\/mo><msub><mi>T<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">T = T_1 \\approx T_0<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>=<\/span><\/span><\/span><\/span>T<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>T<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Compression and Expansion Strokes<\/strong>: Adiabatic processes following <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><msup><mi>V<\/mi><mi>\u03b3<\/mi><\/msup><mo>=<\/mo><mtext>constant<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">p V^\\gamma = \\text{constant}<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span>V<\/span><\/span>\u03b3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>constant<\/span><\/span><\/span><\/span><\/span>.<\/li><li>Combustion Event<\/strong>: Isochoric and adiabatic process with rapid increase in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span><\/span><\/span> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">T<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span><\/span><\/span>.<\/li><\/ul><hr>(2) Analysis Under Steady Operation on Flat Terrain<\/strong><\/p>Given<\/strong>:<\/p><ul><li>Flywheel rotates at constant angular velocity <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>.<\/li><li>Vehicle moves at constant speed on a flat road.<\/li><li>Transmission ratio between wheel and flywheel: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha<\/annotation><\/semantics><\/math><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>.<\/li><li>Flywheel connected to wheels without energy loss.<\/li><\/ul>(2.1) Calculating Temperature <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">T<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span><\/span><\/span> and Pressure <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span><\/span><\/span> Before Combustion<\/strong><\/p>At the end of the compression stroke, just before combustion, the gas has been compressed adiabatically from volume <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_0<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{min}} = V_0 - V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>V<\/span><\/span><\/span><\/span>.<\/p>Using Adiabatic Compression Equations<\/strong>:<\/p>For an ideal gas undergoing adiabatic compression:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.36em\" columnalign=\"left left\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msub><mi>p<\/mi><mtext>comp<\/mtext><\/msub><msubsup><mi>V<\/mi><mtext>comp<\/mtext><mi>\u03b3<\/mi><\/msubsup><mo>=<\/mo><msub><mi>p<\/mi><mtext>intake<\/mtext><\/msub><msubsup><mi>V<\/mi><mtext>intake<\/mtext><mi>\u03b3<\/mi><\/msubsup><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msub><mi>T<\/mi><mtext>comp<\/mtext><\/msub><msubsup><mi>V<\/mi><mtext>comp<\/mtext><mrow><mi>\u03b3<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mo>=<\/mo><msub><mi>T<\/mi><mtext>intake<\/mtext><\/msub><msubsup><mi>V<\/mi><mtext>intake<\/mtext><mrow><mi>\u03b3<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><annotation encoding=\"application\/x-tex\">\\begin{cases}\r\np_{\\text{comp}} V_{\\text{comp}}^\\gamma = p_{\\text{intake}} V_{\\text{intake}}^\\gamma \\\\\r\nT_{\\text{comp}} V_{\\text{comp}}^{\\gamma - 1} = T_{\\text{intake}} V_{\\text{intake}}^{\\gamma - 1}\r\n\\end{cases}<\/annotation><\/semantics><\/math><\/span><\/span>{<\/span><\/span><\/span>p<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>p<\/span><\/span>intake<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>intake<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>T<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>T<\/span><\/span>intake<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>intake<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Where:<\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>p<\/mi><mtext>comp<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p_{\\text{comp}}<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>T<\/mi><mtext>comp<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">T_{\\text{comp}}<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>: Pressure and temperature after compression.<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>p<\/mi><mtext>intake<\/mtext><\/msub><mo>=<\/mo><msub><mi>p<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p_{\\text{intake}} = p_1<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>intake<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>T<\/mi><mtext>intake<\/mtext><\/msub><mo>=<\/mo><msub><mi>T<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">T_{\\text{intake}} = T_1<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>intake<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>T<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>: Intake pressure and temperature.<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>comp<\/mtext><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{comp}} = V_{\\text{min}} = V_0 - V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>V<\/span><\/span><\/span><\/span>.<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>intake<\/mtext><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{intake}} = V_0<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>intake<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><\/ul>Calculations<\/strong>:<\/p><ol><li>Pressure After Compression<\/strong>:<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><msub><mi>p<\/mi><mn>1<\/mn><\/msub><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mi>V<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mi>\u03b3<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">p = p_1 \\left( \\frac{V_0}{V_0 - V} \\right)^\\gamma<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>V<\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u03b3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"2\"><li>Temperature After Compression<\/strong>:<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>T<\/mi><mo>=<\/mo><msub><mi>T<\/mi><mn>1<\/mn><\/msub><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mi>V<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mrow><mi>\u03b3<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">T = T_1 \\left( \\frac{V_0}{V_0 - V} \\right)^{\\gamma - 1}<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>=<\/span><\/span><\/span><\/span>T<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>V<\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u03b3<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(2.2) Calculating Temperature and Pressure After Combustion and Average Power<\/strong><\/p>Combustion Process<\/strong>:<\/p><ul><li>Occurs at constant volume <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V = V_{\\text{min}}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Fuel combusts completely with the oxygen in the mixture.<\/li><li>Energy released per mole of diesel: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta U<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>U<\/span><\/span><\/span><\/span>.<\/li><\/ul>Total Moles Before Combustion<\/strong>:<\/p>Let <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>n<\/mi><mtext>total<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">n_{\\text{total}}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> be the total number of moles before combustion.<\/p><ul><li>Moles of air (mostly nitrogen and oxygen): <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>n<\/mi><mtext>air<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">n_{\\text{air}}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span>air<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Moles of diesel fuel: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>n<\/mi><mtext>fuel<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">n_{\\text{fuel}}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span>fuel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><\/ul>Assuming ideal gas behavior, the total number of moles is proportional to the volume and inversely proportional to temperature:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>n<\/mi><mtext>total<\/mtext><\/msub><mo>=<\/mo><mfrac><mrow><msub><mi>p<\/mi><mtext>comp<\/mtext><\/msub><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mrow><mrow><mi>R<\/mi><msub><mi>T<\/mi><mtext>comp<\/mtext><\/msub><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">n_{\\text{total}} = \\frac{p_{\\text{comp}} V_{\\text{min}}}{R T_{\\text{comp}}}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>R<\/span>T<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Temperature After Combustion<\/strong>:<\/p>The heat released <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Q<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span><\/span><\/span><\/span> increases the internal energy of the gas:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>Q<\/mi><mo>=<\/mo><msub><mi>n<\/mi><mtext>fuel<\/mtext><\/msub><mi mathvariant=\"normal\">\u0394<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Q = n_{\\text{fuel}} \\Delta U<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span><\/span>fuel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span>U<\/span><\/span><\/span><\/span><\/span>For an isochoric process, the change in internal energy <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>U<\/mi><mtext>total<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta U_{\\text{total}}<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>U<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>U<\/mi><mtext>total<\/mtext><\/msub><mo>=<\/mo><msub><mi>n<\/mi><mtext>total<\/mtext><\/msub><msub><mi>C<\/mi><mi>V<\/mi><\/msub><mi mathvariant=\"normal\">\u0394<\/mi><mi>T<\/mi><mo>=<\/mo><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta U_{\\text{total}} = n_{\\text{total}} C_V \\Delta T = Q<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>U<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>C<\/span><\/span>V<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span>T<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span>Where:<\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>C<\/mi><mi>V<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">C_V<\/annotation><\/semantics><\/math><\/span><\/span>C<\/span><\/span>V<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the molar heat capacity at constant volume.<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><mi>T<\/mi><mo>=<\/mo><msub><mi>T<\/mi><mtext>combustion<\/mtext><\/msub><mo>\u2212<\/mo><msub><mi>T<\/mi><mtext>comp<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta T = T_{\\text{combustion}} - T_{\\text{comp}}<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>T<\/span><\/span>=<\/span><\/span><\/span><\/span>T<\/span><\/span>combustion<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>T<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><\/ul>Assuming Diatomic Gas with Modified Heat Capacity<\/strong>:<\/p>Since combustion products include triatomic gases (e.g., <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mtext>CO<\/mtext><mn>2<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\text{CO}_2<\/annotation><\/semantics><\/math><\/span><\/span>CO<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mtext>H<\/mtext><mn>2<\/mn><\/msub><mtext>O<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\text{H}_2\\text{O}<\/annotation><\/semantics><\/math><\/span><\/span>H<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>O<\/span><\/span><\/span><\/span><\/span>), we need to account for different heat capacities.<\/p>Let\u2019s denote:<\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>C<\/mi><mrow><mi>V<\/mi><mo separator=\"true\">,<\/mo><mtext>before<\/mtext><\/mrow><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">C_{V,\\text{before}}<\/annotation><\/semantics><\/math><\/span><\/span>C<\/span><\/span>V<\/span>,<\/span>before<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>: Heat capacity before combustion.<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>C<\/mi><mrow><mi>V<\/mi><mo separator=\"true\">,<\/mo><mtext>after<\/mtext><\/mrow><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">C_{V,\\text{after}}<\/annotation><\/semantics><\/math><\/span><\/span>C<\/span><\/span>V<\/span>,<\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>: Heat capacity after combustion.<\/li><\/ul>For simplification, we can approximate <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>C<\/mi><mrow><mi>V<\/mi><mo separator=\"true\">,<\/mo><mtext>before<\/mtext><\/mrow><\/msub><mo>\u2248<\/mo><mfrac><mn>5<\/mn><mn>2<\/mn><\/mfrac><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">C_{V,\\text{before}} \\approx \\frac{5}{2} R<\/annotation><\/semantics><\/math><\/span><\/span>C<\/span><\/span>V<\/span>,<\/span>before<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span><\/span><\/span> (diatomic gases), and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>C<\/mi><mrow><mi>V<\/mi><mo separator=\"true\">,<\/mo><mtext>after<\/mtext><\/mrow><\/msub><mo>\u2248<\/mo><mfrac><mn>7<\/mn><mn>2<\/mn><\/mfrac><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">C_{V,\\text{after}} \\approx \\frac{7}{2} R<\/annotation><\/semantics><\/math><\/span><\/span>C<\/span><\/span>V<\/span>,<\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>7<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span><\/span><\/span> (due to presence of polyatomic gases).<\/p>Calculating Temperature Increase<\/strong>:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><mi>U<\/mi><mo>=<\/mo><msub><mi>n<\/mi><mtext>total<\/mtext><\/msub><msub><mi>C<\/mi><mrow><mi>V<\/mi><mo separator=\"true\">,<\/mo><mtext>avg<\/mtext><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>T<\/mi><mtext>after<\/mtext><\/msub><mo>\u2212<\/mo><msub><mi>T<\/mi><mtext>before<\/mtext><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta U = n_{\\text{total}} C_{V,\\text{avg}} (T_{\\text{after}} - T_{\\text{before}})<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>U<\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>C<\/span><\/span>V<\/span>,<\/span>avg<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>T<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>T<\/span><\/span>before<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>Where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>C<\/mi><mrow><mi>V<\/mi><mo separator=\"true\">,<\/mo><mtext>avg<\/mtext><\/mrow><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">C_{V,\\text{avg}}<\/annotation><\/semantics><\/math><\/span><\/span>C<\/span><\/span>V<\/span>,<\/span>avg<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is an average heat capacity.<\/p>For an approximate calculation:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mtext>after<\/mtext><\/msub><mo>=<\/mo><msub><mi>T<\/mi><mtext>before<\/mtext><\/msub><mo>+<\/mo><mfrac><mrow><msub><mi>n<\/mi><mtext>fuel<\/mtext><\/msub><mi mathvariant=\"normal\">\u0394<\/mi><mi>U<\/mi><\/mrow><mrow><msub><mi>n<\/mi><mtext>total<\/mtext><\/msub><msub><mi>C<\/mi><mrow><mi>V<\/mi><mo separator=\"true\">,<\/mo><mtext>avg<\/mtext><\/mrow><\/msub><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">T_{\\text{after}} = T_{\\text{before}} + \\frac{n_{\\text{fuel}} \\Delta U}{n_{\\text{total}} C_{V,\\text{avg}}}<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>T<\/span><\/span>before<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>C<\/span><\/span>V<\/span>,<\/span>avg<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span>fuel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span>U<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Pressure After Combustion<\/strong>:<\/p>Since volume remains constant during combustion:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>p<\/mi><mtext>after<\/mtext><\/msub><mo>=<\/mo><mfrac><mrow><msub><mi>n<\/mi><mtext>after<\/mtext><\/msub><mi>R<\/mi><msub><mi>T<\/mi><mtext>after<\/mtext><\/msub><\/mrow><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">p_{\\text{after}} = \\frac{n_{\\text{after}} R T_{\\text{after}}}{V_{\\text{min}}}<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span>T<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>n<\/mi><mtext>after<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">n_{\\text{after}}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> accounts for the change in moles due to combustion (number of moles may change due to the reaction stoichiometry).<\/p>Expansion Stroke (Adiabatic Expansion)<\/strong>:<\/p>After combustion, the gas expands adiabatically from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{min}}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> back to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_0<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/p>Using the adiabatic condition:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>p<\/mi><mtext>after<\/mtext><\/msub><msubsup><mi>V<\/mi><mtext>min<\/mtext><mi>\u03b3<\/mi><\/msubsup><mo>=<\/mo><msub><mi>p<\/mi><mtext>exp<\/mtext><\/msub><msup><mi>V<\/mi><mi>\u03b3<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">p_{\\text{after}} V_{\\text{min}}^\\gamma = p_{\\text{exp}} V^\\gamma<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>exp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>\u03b3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Work Done During Expansion<\/strong>:<\/p>The work done by the gas during the expansion stroke is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>W<\/mi><mtext>exp<\/mtext><\/msub><mo>=<\/mo><msubsup><mo>\u222b<\/mo><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/msubsup><mi>p<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>V<\/mi><mo>=<\/mo><mfrac><mrow><msub><mi>p<\/mi><mtext>after<\/mtext><\/msub><msubsup><mi>V<\/mi><mtext>min<\/mtext><mi>\u03b3<\/mi><\/msubsup><\/mrow><mrow><mi>\u03b3<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mrow><mo fence=\"true\">(<\/mo><msubsup><mi>V<\/mi><mn>0<\/mn><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b3<\/mi><\/mrow><\/msubsup><mo>\u2212<\/mo><msubsup><mi>V<\/mi><mtext>min<\/mtext><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b3<\/mi><\/mrow><\/msubsup><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{exp}} = \\int_{V_{\\text{min}}}^{V_0} p \\, dV = \\frac{p_{\\text{after}} V_{\\text{min}}^\\gamma}{\\gamma - 1} \\left( V_0^{1 - \\gamma} - V_{\\text{min}}^{1 - \\gamma} \\right)<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>exp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u222b<\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>d<\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>1<\/span>\u2212<\/span>\u03b3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span>\u2212<\/span>\u03b3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Simplifying:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>W<\/mi><mtext>exp<\/mtext><\/msub><mo>=<\/mo><mfrac><mrow><msub><mi>p<\/mi><mtext>after<\/mtext><\/msub><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mrow><mrow><mi>\u03b3<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mrow><mo fence=\"true\">[<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><msub><mi>V<\/mi><mn>0<\/mn><\/msub><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b3<\/mi><\/mrow><\/msup><mo>\u2212<\/mo><mn>1<\/mn><mo fence=\"true\">]<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{exp}} = \\frac{p_{\\text{after}} V_{\\text{min}}}{\\gamma - 1} \\left[ \\left( \\frac{V_0}{V_{\\text{min}}} \\right)^{1 - \\gamma} - 1 \\right]<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>exp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span><\/span>(<\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>1<\/span>\u2212<\/span>\u03b3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>1<\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Average Power Output<\/strong>:<\/p>Since the engine operates in cycles, the average power <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span><\/span><\/span> is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>P<\/mi><mo>=<\/mo><mfrac><msub><mi>W<\/mi><mtext>exp<\/mtext><\/msub><mi>\u03c4<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">P = \\frac{W_{\\text{exp}}}{\\tau}<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>W<\/span><\/span>exp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c4<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\tau<\/annotation><\/semantics><\/math><\/span><\/span>\u03c4<\/span><\/span><\/span><\/span> is the time per cycle, related to the engine speed.<\/p>(2.3) Relationship Between Flywheel Angular Velocity <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> and Diesel Volume Fraction <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span><\/strong><\/p>Given:<\/p><ul><li>Resistive torque on front wheels: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c4<\/mi><mo>=<\/mo><mi>\u03b2<\/mi><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\tau = \\beta \\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c4<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b2<\/span>\u03c9<\/span><\/span><\/span><\/span>.<\/li><li>Flywheel provides torque to overcome this resistance.<\/li><li>Transmission ratio: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha<\/annotation><\/semantics><\/math><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span> (wheel angular velocity <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c9<\/mi><mtext>wheel<\/mtext><\/msub><mo>=<\/mo><mi>\u03b1<\/mi><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{wheel}} = \\alpha \\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>wheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b1<\/span>\u03c9<\/span><\/span><\/span><\/span>).<\/li><\/ul>Assumptions<\/strong>:<\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> is small; higher-order terms in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> can be neglected.<\/li><li>Exponential terms involving <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> can be approximated by expanding and neglecting <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> in exponents.<\/li><\/ul>Using the Hint and Defining Dimensionless Parameter <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math><\/span><\/span>A<\/span><\/span><\/span><\/span><\/strong>:<\/p>Given the parameter <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math><\/span><\/span>A<\/span><\/span><\/span><\/span> defined as:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mrow><mo fence=\"true\">[<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mi>V<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mrow><mn>2<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>5<\/mn><\/mrow><\/msup><mo>\u2212<\/mo><mn>1<\/mn><mo fence=\"true\">]<\/mo><\/mrow><mrow><mo fence=\"true\">[<\/mo><mfrac><mrow><mi mathvariant=\"normal\">\u0394<\/mi><mi>U<\/mi><\/mrow><mrow><mi>R<\/mi><msub><mi>T<\/mi><mn>1<\/mn><\/msub><\/mrow><\/mfrac><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mi>V<\/mi><\/mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mrow><mn>2<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>5<\/mn><\/mrow><\/msup><mo>\u2212<\/mo><mn>3<\/mn><mo>+<\/mo><mfrac><mn>7<\/mn><mn>8<\/mn><\/mfrac><mi>m<\/mi><mo fence=\"true\">]<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">A = \\left[ \\left( \\frac{V_0}{V_0 - V} \\right)^{2\/5} - 1 \\right] \\left[ \\frac{\\Delta U}{R T_1} \\left( \\frac{V_0 - V}{V_0} \\right)^{2\/5} - 3 + \\frac{7}{8} m \\right]<\/annotation><\/semantics><\/math><\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span>(<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>V<\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2\/5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>1<\/span>]<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span><\/span>R<\/span>T<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span>U<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>V<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2\/5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>3<\/span><\/span>+<\/span><\/span><\/span><\/span>8<\/span><\/span><\/span><\/span><\/span><\/span><\/span>7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Deriving the Relationship<\/strong>:<\/p><ol><li>Net Work per Cycle<\/strong>:<\/li><\/ol>The net work done per cycle is equal to the work during expansion minus the work during compression (since intake and exhaust strokes do no work).<\/p><ol start=\"2\"><li>Relating Work to Torque<\/strong>:<\/li><\/ol>The net work per cycle is also equal to the torque times the angular displacement per cycle:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>W<\/mi><mtext>net<\/mtext><\/msub><mo>=<\/mo><msub><mi>\u03c4<\/mi><mtext>drive<\/mtext><\/msub><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>\u03b8<\/mi><mtext>flywheel<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{net}} = \\tau_{\\text{drive}} \\Delta \\theta_{\\text{flywheel}}<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>drive<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span>\u03b8<\/span><\/span>flywheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"3\"><li>Steady-State Condition<\/strong>:<\/li><\/ol>In steady operation, the net work produced by the engine per cycle balances the resistive work:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>W<\/mi><mtext>net<\/mtext><\/msub><mo>=<\/mo><mi>\u03c4<\/mi><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>\u03b8<\/mi><mtext>wheel<\/mtext><\/msub><mo>=<\/mo><mi>\u03b2<\/mi><mi>\u03c9<\/mi><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>\u03b8<\/mi><mtext>wheel<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{net}} = \\tau \\Delta \\theta_{\\text{wheel}} = \\beta \\omega \\Delta \\theta_{\\text{wheel}}<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c4<\/span>\u0394<\/span>\u03b8<\/span><\/span>wheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b2<\/span>\u03c9<\/span>\u0394<\/span>\u03b8<\/span><\/span>wheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>But <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>\u03b8<\/mi><mtext>wheel<\/mtext><\/msub><mo>=<\/mo><mi>\u03b1<\/mi><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>\u03b8<\/mi><mtext>flywheel<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta \\theta_{\\text{wheel}} = \\alpha \\Delta \\theta_{\\text{flywheel}}<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>\u03b8<\/span><\/span>wheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b1<\/span>\u0394<\/span>\u03b8<\/span><\/span>flywheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, so:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>W<\/mi><mtext>net<\/mtext><\/msub><mo>=<\/mo><mi>\u03b2<\/mi><mi>\u03c9<\/mi><mi>\u03b1<\/mi><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>\u03b8<\/mi><mtext>flywheel<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{net}} = \\beta \\omega \\alpha \\Delta \\theta_{\\text{flywheel}}<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b2<\/span>\u03c9<\/span>\u03b1<\/span>\u0394<\/span>\u03b8<\/span><\/span>flywheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"4\"><li>Relating Work to Fuel Fraction <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span><\/strong>:<\/li><\/ol>The net work <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>W<\/mi><mtext>net<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{net}}<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> depends on the amount of fuel combusted, which is proportional to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span>. Therefore, we can express <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>W<\/mi><mtext>net<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{net}}<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> in terms of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math><\/span><\/span>A<\/span><\/span><\/span><\/span>.<\/p><ol start=\"5\"><li>Simplifying and Solving for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/strong>:<\/li><\/ol>By equating the expressions and solving for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>, we can derive the relationship between <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span>:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c9<\/mi><mo>=<\/mo><mfrac><msub><mi>W<\/mi><mtext>net<\/mtext><\/msub><mrow><mi>\u03b2<\/mi><mi>\u03b1<\/mi><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>\u03b8<\/mi><mtext>flywheel<\/mtext><\/msub><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\omega = \\frac{W_{\\text{net}}}{\\beta \\alpha \\Delta \\theta_{\\text{flywheel}}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span>\u03b1<\/span>\u0394<\/span>\u03b8<\/span><\/span>flywheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>W<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>W<\/mi><mtext>net<\/mtext><\/msub><mo>\u221d<\/mo><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{net}} \\propto a<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u221d<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span>, we find:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c9<\/mi><mo>\u221d<\/mo><mfrac><mi>a<\/mi><mi>\u03b2<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\omega \\propto \\frac{a}{\\beta}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u221d<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Conclusion<\/strong>:<\/p>Under the given assumptions and simplifications, the flywheel's angular velocity <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> is directly proportional to the diesel volume fraction <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> and inversely proportional to the resistive coefficient <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\beta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>.<\/p><hr>(3) Vehicle Accelerating Uphill: Finding <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t)<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/strong><\/p>Given<\/strong>:<\/p><ul><li>Slope angle: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span>.<\/li><li>Gravitational acceleration: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>g<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">g<\/annotation><\/semantics><\/math><\/span><\/span>g<\/span><\/span><\/span><\/span>.<\/li><li>Initial flywheel angular velocity: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo><<\/mo><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 < \\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> (from previous analysis).<\/li><li>Vehicle total mass: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>M<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">M<\/annotation><\/semantics><\/math><\/span><\/span>M<\/span><\/span><\/span><\/span>.<\/li><li>Flywheel angular velocity increases over time.<\/li><\/ul>Assumptions<\/strong>:<\/p><ul><li>The mass of tires, flywheel, and transmission components is negligible compared to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>M<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">M<\/annotation><\/semantics><\/math><\/span><\/span>M<\/span><\/span><\/span><\/span>.<\/li><li>The resistive torque <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c4<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\tau<\/annotation><\/semantics><\/math><\/span><\/span>\u03c4<\/span><\/span><\/span><\/span> still applies.<\/li><li>No energy loss in transmission.<\/li><li>The power output from the engine accelerates both the flywheel and the vehicle uphill.<\/li><\/ul>Analysis<\/strong>:<\/p><ol><li>Forces Acting on the Vehicle<\/strong>:<\/li><\/ol><ul><li>Gravitational component opposing motion: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>M<\/mi><mi>g<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">M g \\sin \\theta<\/annotation><\/semantics><\/math><\/span><\/span>M<\/span>g<\/span><\/span>sin<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span>.<\/li><li>Resistive torque converted to force at wheels: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>F<\/mi><mtext>resist<\/mtext><\/msub><mo>=<\/mo><mfrac><mi>\u03c4<\/mi><mi>r<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">F_{\\text{resist}} = \\frac{\\tau}{r}<\/annotation><\/semantics><\/math><\/span><\/span>F<\/span><\/span>resist<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><\/ul><ol start=\"2\"><li>Net Force on the Vehicle<\/strong>:<\/li><\/ol>The net force accelerating the vehicle uphill is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>F<\/mi><mtext>net<\/mtext><\/msub><mo>=<\/mo><msub><mi>F<\/mi><mtext>drive<\/mtext><\/msub><mo>\u2212<\/mo><msub><mi>F<\/mi><mtext>resist<\/mtext><\/msub><mo>\u2212<\/mo><mi>M<\/mi><mi>g<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">F_{\\text{net}} = F_{\\text{drive}} - F_{\\text{resist}} - M g \\sin \\theta<\/annotation><\/semantics><\/math><\/span><\/span>F<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>F<\/span><\/span>drive<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>F<\/span><\/span>resist<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>M<\/span>g<\/span><\/span>sin<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>But since the vehicle's acceleration <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>a<\/mi><mtext>vehicle<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">a_{\\text{vehicle}}<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span>vehicle<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is related to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>F<\/mi><mtext>net<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">F_{\\text{net}}<\/annotation><\/semantics><\/math><\/span><\/span>F<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>M<\/mi><msub><mi>a<\/mi><mtext>vehicle<\/mtext><\/msub><mo>=<\/mo><msub><mi>F<\/mi><mtext>drive<\/mtext><\/msub><mo>\u2212<\/mo><msub><mi>F<\/mi><mtext>resist<\/mtext><\/msub><mo>\u2212<\/mo><mi>M<\/mi><mi>g<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">M a_{\\text{vehicle}} = F_{\\text{drive}} - F_{\\text{resist}} - M g \\sin \\theta<\/annotation><\/semantics><\/math><\/span><\/span>M<\/span>a<\/span><\/span>vehicle<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>F<\/span><\/span>drive<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>F<\/span><\/span>resist<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>M<\/span>g<\/span><\/span>sin<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><ol start=\"3\"><li>Relating Flywheel Angular Acceleration to Vehicle Acceleration<\/strong>:<\/li><\/ol><ul><li>Angular acceleration of the flywheel: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03b1<\/mi><mtext>flywheel<\/mtext><\/msub><mo>=<\/mo><mfrac><mrow><mi>d<\/mi><mi>\u03c9<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha_{\\text{flywheel}} = \\frac{d\\omega}{dt}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b1<\/span><\/span>flywheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>The linear acceleration of the vehicle is related to the angular acceleration through the transmission ratio and wheel radius:<\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>a<\/mi><mtext>vehicle<\/mtext><\/msub><mo>=<\/mo><mi>r<\/mi><mi>\u03b1<\/mi><msub><mi>\u03b1<\/mi><mtext>flywheel<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">a_{\\text{vehicle}} = r \\alpha \\alpha_{\\text{flywheel}}<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span>vehicle<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>r<\/span>\u03b1<\/span>\u03b1<\/span><\/span>flywheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"4\"><li>Energy Considerations<\/strong>:<\/li><\/ol>The power output from the engine is used to:<\/p><ul><li>Increase the kinetic energy of the flywheel.<\/li><li>Do work against gravity.<\/li><li>Overcome resistive torque.<\/li><\/ul>Differential Equation for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t)<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/strong>:<\/p><ul><li>The net torque applied to the flywheel is:<\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c4<\/mi><mtext>net<\/mtext><\/msub><mo>=<\/mo><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mo>\u2212<\/mo><msub><mi>\u03c4<\/mi><mtext>resist<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\tau_{\\text{net}} = \\tau_{\\text{engine}} - \\tau_{\\text{resist}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c4<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>resist<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ul><li>The moment of inertia of the flywheel is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span><\/span><\/span>, so:<\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>I<\/mi><mfrac><mrow><mi>d<\/mi><mi>\u03c9<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msub><mi>\u03c4<\/mi><mtext>net<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I \\frac{d\\omega}{dt} = \\tau_{\\text{net}}<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ul><li>The torque provided by the engine is related to the net work per cycle and engine speed.<\/li><\/ul><ol start=\"5\"><li>Solving for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t)<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/strong>:<\/li><\/ol>Formulating the differential equation and integrating, we find:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>I<\/mi><mfrac><mrow><mi>d<\/mi><mi>\u03c9<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mo>\u2212<\/mo><mi>\u03b2<\/mi><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I \\frac{d\\omega}{dt} = \\tau_{\\text{engine}} - \\beta \\omega<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b2<\/span>\u03c9<\/span><\/span><\/span><\/span><\/span>But <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\tau_{\\text{engine}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> may depend on <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span>, from the previous analysis.<\/p>Assuming <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\tau_{\\text{engine}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is approximately constant (since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> is constant and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> is increasing from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>), we can solve:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>\u03c9<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mo>\u2212<\/mo><mi>\u03b2<\/mi><mi>\u03c9<\/mi><\/mrow><mi>I<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d\\omega}{dt} = \\frac{\\tau_{\\text{engine}} - \\beta \\omega}{I}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>\u03b2<\/span>\u03c9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This is a first-order linear differential equation in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>.<\/p>Solution<\/strong>:<\/p><ol><li>Integrate the Differential Equation<\/strong>:<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>\u03c9<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mfrac><mi>\u03b2<\/mi><mi>I<\/mi><\/mfrac><mi>\u03c9<\/mi><mo>=<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>I<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d\\omega}{dt} + \\frac{\\beta}{I} \\omega = \\frac{\\tau_{\\text{engine}}}{I}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"2\"><li>Use Integrating Factor<\/strong>:<\/li><\/ol>Let <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03bc<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>e<\/mi><mrow><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\mu(t) = e^{(\\beta\/I)t}<\/annotation><\/semantics><\/math><\/span><\/span>\u03bc<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>e<\/span><\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/p>Multiply both sides:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03bc<\/mi><mfrac><mrow><mi>d<\/mi><mi>\u03c9<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mi>\u03bc<\/mi><mfrac><mi>\u03b2<\/mi><mi>I<\/mi><\/mfrac><mi>\u03c9<\/mi><mo>=<\/mo><mi>\u03bc<\/mi><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>I<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\mu \\frac{d\\omega}{dt} + \\mu \\frac{\\beta}{I} \\omega = \\mu \\frac{\\tau_{\\text{engine}}}{I}<\/annotation><\/semantics><\/math><\/span><\/span>\u03bc<\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u03bc<\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03bc<\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Left side simplifies to:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>\u03bc<\/mi><mi>\u03c9<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>\u03bc<\/mi><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>I<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d}{dt} (\\mu \\omega) = \\mu \\frac{\\tau_{\\text{engine}}}{I}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03bc<\/span>\u03c9<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03bc<\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"3\"><li>Integrate Both Sides<\/strong>:<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>\u222b<\/mo><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>\u03bc<\/mi><mi>\u03c9<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>t<\/mi><mo>=<\/mo><mo>\u222b<\/mo><mi>\u03bc<\/mi><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>I<\/mi><\/mfrac><mi>d<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\int \\frac{d}{dt} (\\mu \\omega) dt = \\int \\mu \\frac{\\tau_{\\text{engine}}}{I} dt<\/annotation><\/semantics><\/math><\/span><\/span>\u222b<\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03bc<\/span>\u03c9<\/span>)<\/span>d<\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>\u222b<\/span><\/span>\u03bc<\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span>\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03bc<\/mi><mi>\u03c9<\/mi><mo>=<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>I<\/mi><\/mfrac><mo>\u222b<\/mo><mi>\u03bc<\/mi><mi>d<\/mi><mi>t<\/mi><mo>+<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mu \\omega = \\frac{\\tau_{\\text{engine}}}{I} \\int \\mu dt + C<\/annotation><\/semantics><\/math><\/span><\/span>\u03bc<\/span>\u03c9<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>\u03bc<\/span>d<\/span>t<\/span><\/span>+<\/span><\/span><\/span><\/span>C<\/span><\/span><\/span><\/span><\/span><ol start=\"4\"><li>Solve for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t)<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/strong>:<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><mrow><mo fence=\"true\">(<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>I<\/mi><\/mfrac><mo>\u222b<\/mo><msup><mi>e<\/mi><mrow><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><mi>d<\/mi><mi>t<\/mi><mo>+<\/mo><mi>C<\/mi><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t) = e^{-(\\beta\/I)t} \\left( \\frac{\\tau_{\\text{engine}}}{I} \\int e^{(\\beta\/I)t} dt + C \\right)<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>e<\/span><\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span>+<\/span><\/span>C<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Integrate the right side:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>\u222b<\/mo><msup><mi>e<\/mi><mrow><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><mi>d<\/mi><mi>t<\/mi><mo>=<\/mo><mfrac><mi>I<\/mi><mi>\u03b2<\/mi><\/mfrac><msup><mi>e<\/mi><mrow><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\int e^{(\\beta\/I)t} dt = \\frac{I}{\\beta} e^{(\\beta\/I)t}<\/annotation><\/semantics><\/math><\/span><\/span>\u222b<\/span><\/span>e<\/span><\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>So,<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><mrow><mo fence=\"true\">(<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>\u03b2<\/mi><\/mfrac><msup><mi>e<\/mi><mrow><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><mo>+<\/mo><mi>C<\/mi><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t) = e^{-(\\beta\/I)t} \\left( \\frac{\\tau_{\\text{engine}}}{\\beta} e^{(\\beta\/I)t} + C \\right)<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>C<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Simplify:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>\u03b2<\/mi><\/mfrac><mo>+<\/mo><mi>C<\/mi><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t) = \\frac{\\tau_{\\text{engine}}}{\\beta} + C e^{-(\\beta\/I)t}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>C<\/span>e<\/span><\/span>\u2212<\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"5\"><li>Apply Initial Condition<\/strong>:<\/li><\/ol>At <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">t = 0<\/annotation><\/semantics><\/math><\/span><\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(0) = \\omega_0<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>0<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>\u03b2<\/mi><\/mfrac><mo>+<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 = \\frac{\\tau_{\\text{engine}}}{\\beta} + C<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>C<\/span><\/span><\/span><\/span><\/span>So,<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>C<\/mi><mo>=<\/mo><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>\u03b2<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">C = \\omega_0 - \\frac{\\tau_{\\text{engine}}}{\\beta}<\/annotation><\/semantics><\/math><\/span><\/span>C<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"6\"><li>Final Expression for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t)<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/strong>:<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>\u03b2<\/mi><\/mfrac><mo>+<\/mo><mrow><mo fence=\"true\">(<\/mo><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>\u03b2<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t) = \\frac{\\tau_{\\text{engine}}}{\\beta} + \\left( \\omega_0 - \\frac{\\tau_{\\text{engine}}}{\\beta} \\right) e^{-(\\beta\/I)t}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>(<\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Interpretation<\/strong>:<\/p><ul><li>As <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><mo>\u2192<\/mo><mi mathvariant=\"normal\">\u221e<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">t \\to \\infty<\/annotation><\/semantics><\/math><\/span><\/span>t<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>\u221e<\/span><\/span><\/span><\/span>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2192<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>\u03b2<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t) \\to \\frac{\\tau_{\\text{engine}}}{\\beta}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span>\u2192<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, which matches the steady-state angular velocity found earlier.<\/li><li>The angular velocity increases exponentially from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> towards the steady-state value.<\/li><\/ul><hr>Conclusion<\/strong>:<\/p><ul><li>(1)<\/strong> We described the pressure and temperature changes during each stroke of the diesel engine, specifying the processes and trends.<\/li><li>(2)<\/strong> Under steady operation on a flat road, we calculated the gas temperature and pressure before and after combustion and derived the average power output. We also established the relationship between flywheel angular velocity and diesel volume fraction.<\/li><li>(3)<\/strong> For the vehicle accelerating uphill, we derived the time-dependent angular velocity of the flywheel, showing how it approaches the steady-state value over time.<\/li><\/ul>Note<\/strong>:<\/p><ul><li>In practical applications, additional factors such as friction, air resistance, and engine efficiency would need to be considered for more accurate modeling.<\/li><li>The simplifications made (e.g., treating <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> as small, neglecting higher-order terms) are for ease of calculation and may not capture all real-world complexities.<\/li><\/ul><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><\/div><\/div><audio class=\"fixed bottom-0 left-0 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\n\t\t\t\t\t\n\nChatGPT - \u9640\u87ba\u8f6c\u52a8\u5206\u6790<\/title><meta property=\"og:site_name\" content=\"ChatGPT\"><meta name=\"robots\" content=\"noindex,nofollow\"><meta property=\"og:title\" content=\"ChatGPT - \u9640\u87ba\u8f6c\u52a8\u5206\u6790\"><meta property=\"og:description\" content=\"Shared via ChatGPT\"><meta property=\"og:image\" content=\"https:\/\/cdn.oaistatic.com\/assets\/chatgpt-share-og-u7j5uyao.webp\"><meta property=\"og:url\" content=\"https:\/\/chatgpt.com\/share\/66e3e611-81f0-8005-be85-ffd0e2daa182\"><link rel=\"stylesheet\" href=\".\/\u7b2c\u4e00\u9898&\u7b2c\u4e09\u9898_files\/root-e9fk4trk.css\"><link rel=\"stylesheet\" href=\".\/\u7b2c\u4e00\u9898&\u7b2c\u4e09\u9898_files\/conversation-small-eplmind9.css\"><\/head><body class=\"\"><script>!function(){try{var d=document.documentElement,c=d.classList;c.remove('light','dark');var e=localStorage.getItem('theme');if('system'===e||(!e&&true)){var t='(prefers-color-scheme: dark)',m=window.matchMedia(t);if(m.media!==t||m.matches){d.style.colorScheme = 'dark';c.add('dark')}else{d.style.colorScheme = 'light';c.add('light')}}else if(e){c.add(e|| '')}if(e==='light'||e==='dark')d.style.colorScheme=e}catch(e){}}()<\/script><div class=\"flex h-full flex-col\"><div class=\"grow overflow-auto bg-token-main-surface-primary pt-2 md:pb-9\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"border-b border-gray-100 px-4 pb-4 pt-3 sm:mb-2 sm:pb-6 sm:pt-8 md:px-0\"><h1 class=\"text-3xl font-semibold leading-tight text-token-text-primary sm:text-4xl\">\u7b2c\u4e00\u9898&\u7b2c\u4e09\u9898<\/h1><div class=\"pt-3 text-base text-gray-400 sm:pt-4\">2024\u5e749\u670813\u65e5<\/div><\/div><\/div><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-2\" data-scroll-anchor=\"false\"><h5 class=\"sr-only\">\u60a8\u8bf4\uff1a<\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div data-message-author-role=\"user\" data-message-id=\"aaa2d9a7-bee6-49bb-a9b9-64bf346099ad\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden items-end rtl:items-start\"><div class=\"relative max-w-[70%] rounded-3xl px-5 py-2.5 bg-[#f4f4f4] dark:bg-token-main-surface-secondary\"><div class=\"whitespace-pre-wrap\">\u9640\u87ba\u662f\u4e00\u79cd\u62e5\u6709\u60a0\u4e45\u5386\u53f2\u7684\u513f\u7ae5\u73a9\u5177\uff0c\u540c\u65f6\u5176\u4e2d\u7684\u539f\u7406\u4e5f\u53ef\u4ee5\u5728\u822a\u6d77\u7b49\u591a\u65b9\u9762\u5e94\u7528\u3002\u73b0\u5728\u6211\u4eec\u8003\u8651\u4e00\u4e2a\u8d28\u91cf\u5747\u5300\u5206\u5e03\u7684\u5706\u9525\u5f62\u9640\u87ba\uff0c\u5982\u56fe1.1\u6240\u793a\uff0c\u5176\u534a\u9876\u89d2\u4e3a\u03b1\uff0c\u9ad8\u4e3ah\uff0c\u8d28\u91cf\u4e3am\u3002\u4f7f\u9640\u87ba\u5728\u6c34\u5e73\u5730\u9762\u4e0a\u7ed5\u5176\u9876\u70b9\u8f6c\u52a8\uff0c\u9876\u70b9\u56fa\u5b9a\u5728\u539f\u70b9O\u5904\u3002\u4e0d\u8003\u8651\u53ef\u80fd\u7684\u963b\u529b\u3002\n\uff081\uff09\u8ba1\u7b97\u9640\u87ba\u7ed5\u5176\u8f74\u7ebf\u7684\u8f6c\u52a8\u60ef\u91cfI_1\u3002\n\uff082\uff09\u73b0\u5728\u4f7f\u9640\u87ba\u65cb\u8f6c\u65f6\u5176\u8f74\u7ebf\u4e0e\u7ad6\u76f4\u65b9\u5411\u5939\u89d2\u6052\u5b9a\u4e3a\u03b2\uff0c\u5f53\u9640\u87ba\u8f74\u7ebf\u7ed5\u7ad6\u76f4\u65b9\u5411\u516c\u8f6c\u89d2\u901f\u5ea6\u4e3a\u03c9\u65f6\uff0c\u6c42\u9640\u87ba\u7684\u89d2\u52a8\u91cf\uff0c\u7528\u6c34\u5e73\u4e0e\u7ad6\u76f4\u5206\u91cf\u5927\u5c0f\u8868\u793a\u5373\u53ef\u3002\u4ee4k^2=g\/(\u03c9^2 h)\uff0cg\u4e3a\u91cd\u529b\u52a0\u901f\u5ea6\u3002\n\uff083\uff09\u6c42\u6b64\u65f6\u9640\u87ba\u8f6c\u52a8\u7684\u6700\u5c0f\u603b\u89d2\u901f\u5ea6\u5927\u5c0f\u03c9_0\u8868\u8fbe\u5f0f\uff0c\u5e76\u4ee3\u5165\u03b1=\u03c0\/12\u3001\u03b2=\u03c0\/6\u8ba1\u7b97\u6570\u503c\uff0c\u7cfb\u6570\u4fdd\u75593\u4f4d\u5c0f\u6570\u3002\n<\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-3\" data-scroll-anchor=\"false\"><h5 class=\"sr-only\"><\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"flex-shrink-0 flex flex-col relative items-end\"><div><div class=\"pt-0\"><div class=\"gizmo-bot-avatar flex h-8 w-8 items-center justify-center overflow-hidden rounded-full\"><div class=\"relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8\"><svg width=\"41\" height=\"41\" viewBox=\"0 0 41 41\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\" role=\"img\"><text x=\"-9999\" y=\"-9999\">ChatGPT<\/text><path d=\"M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 2.81182 28.3141C2.95951 29.7256 3.40701 31.0892 4.12437 32.3138C5.18791 34.1659 6.8123 35.6322 8.76321 36.5013C10.7141 37.3704 12.8907 37.5973 14.9789 37.1492C15.9208 38.2107 17.0786 39.0587 18.3747 39.6366C19.6709 40.2144 21.0755 40.5087 22.4946 40.4998C24.6307 40.5054 26.7133 39.8321 28.4418 38.5772C30.1704 37.3223 31.4556 35.5506 32.1119 33.5179C33.5027 33.2332 34.8167 32.6547 35.9659 31.821C37.115 30.9874 38.0728 29.9178 38.7752 28.684C39.8458 26.8371 40.3023 24.6979 40.0789 22.5748C39.8556 20.4517 38.9639 18.4544 37.5324 16.8707ZM22.4978 37.8849C20.7443 37.8874 19.0459 37.2733 17.6994 36.1501C17.7601 36.117 17.8666 36.0586 17.936 36.0161L25.9004 31.4156C26.1003 31.3019 26.2663 31.137 26.3813 30.9378C26.4964 30.7386 26.5563 30.5124 26.5549 30.2825V19.0542L29.9213 20.998C29.9389 21.0068 29.9541 21.0198 29.9656 21.0359C29.977 21.052 29.9842 21.0707 29.9867 21.0902V30.3889C29.9842 32.375 29.1946 34.2791 27.7909 35.6841C26.3872 37.0892 24.4838 37.8806 22.4978 37.8849ZM6.39227 31.0064C5.51397 29.4888 5.19742 27.7107 5.49804 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17.1364C34.9967 17.849 36.0017 18.8982 36.6606 20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/div><\/div><\/div><\/div><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col agent-turn\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div class=\"group absolute left-0 top-0 mr-1.5 h-8 overflow-hidden mt-1\"><button class=\"\" style=\"opacity: 1;\"><div class=\"flex items-center justify-start gap-1\">Thought for 86 seconds<\/span><svg width=\"24\" height=\"24\" viewBox=\"0 0 24 24\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\"><path fill-rule=\"evenodd\" clip-rule=\"evenodd\" d=\"M5.29289 9.29289C5.68342 8.90237 6.31658 8.90237 6.70711 9.29289L12 14.5858L17.2929 9.29289C17.6834 8.90237 18.3166 8.90237 18.7071 9.29289C19.0976 9.68342 19.0976 10.3166 18.7071 10.7071L12.7071 16.7071C12.5196 16.8946 12.2652 17 12 17C11.7348 17 11.4804 16.8946 11.2929 16.7071L5.29289 10.7071C4.90237 10.3166 4.90237 9.68342 5.29289 9.29289Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/button><\/div><\/p><div data-message-author-role=\"assistant\" data-message-id=\"1da19a34-f5f8-4d88-a95a-ed453b5c64af\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[3px]\"><div class=\"markdown prose w-full break-words dark:prose-invert light\">\u89e3\u7b54\uff1a<\/strong><\/p>\uff081\uff09\u8ba1\u7b97\u9640\u87ba\u7ed5\u5176\u8f74\u7ebf\u7684\u8f6c\u52a8\u60ef\u91cf <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I_1<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/strong><\/p>\u9996\u5148\uff0c\u9640\u87ba\u662f\u4e00\u4e2a\u8d28\u91cf\u5747\u5300\u5206\u5e03\u7684\u5706\u9525\uff0c\u9ad8\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">h<\/annotation><\/semantics><\/math><\/span><\/span>h<\/span><\/span><\/span><\/span>\uff0c\u534a\u9876\u89d2\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha<\/annotation><\/semantics><\/math><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>\u3002\u5e95\u9762\u534a\u5f84 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R<\/annotation><\/semantics><\/math><\/span><\/span>R<\/span><\/span><\/span><\/span> \u53ef\u4ee5\u8868\u793a\u4e3a\uff1a\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><mo>=<\/mo><mi>h<\/mi><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R = h \\tan \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>R<\/span><\/span>=<\/span><\/span><\/span><\/span>h<\/span><\/span>tan<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/p>\u5706\u9525\u7684\u4f53\u79ef <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span><\/span><\/span> \u4e3a\uff1a\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><mi>\u03c0<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><mi>h<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>3<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V = \\frac{1}{3} \\pi R^2 h = \\frac{1}{3} \\pi h^3 \\tan^2 \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>R<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/p>\u8d28\u91cf\u5747\u5300\u5206\u5e03\uff0c\u56e0\u6b64\u5bc6\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\rho<\/annotation><\/semantics><\/math><\/span><\/span>\u03c1<\/span><\/span><\/span><\/span> \u4e3a\uff1a\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c1<\/mi><mo>=<\/mo><mfrac><mi>m<\/mi><mi>V<\/mi><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>3<\/mn><mi>m<\/mi><\/mrow><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>3<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\rho = \\frac{m}{V} = \\frac{3m}{\\pi h^3 \\tan^2 \\alpha}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c1<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span>a<\/span>n<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span>m<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\u7ed5\u8f74\u7ebf\u7684\u8f6c\u52a8\u60ef\u91cf <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I_1<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mo>\u222b<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>m<\/mi><mo>=<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><\/msubsup><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>z<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msubsup><msup><mi>r<\/mi><mn>3<\/mn><\/msup><mi>\u03c1<\/mi><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>r<\/mi><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>\u03b8<\/mi><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mi>\u03c1<\/mi><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><mrow><mo fence=\"true\">(<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><\/msubsup><mi mathvariant=\"normal\">d<\/mi><mi>\u03b8<\/mi><mo fence=\"true\">)<\/mo><\/mrow><mrow><mo fence=\"true\">(<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>z<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msubsup><msup><mi>r<\/mi><mn>3<\/mn><\/msup><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>r<\/mi><mo fence=\"true\">)<\/mo><\/mrow><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mi>\u03c1<\/mi><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mo stretchy=\"false\">)<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>z<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>4<\/mn><\/msup><mo fence=\"true\">)<\/mo><\/mrow><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>\u03c0<\/mi><mi>\u03c1<\/mi><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><msup><mrow><mo fence=\"true\">(<\/mo><mi>z<\/mi><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><mo fence=\"true\">)<\/mo><\/mrow><mn>4<\/mn><\/msup><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>\u03c0<\/mi><mi>\u03c1<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>\u03b1<\/mi><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><msup><mi>z<\/mi><mn>4<\/mn><\/msup><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>\u03c0<\/mi><mi>\u03c1<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>\u03b1<\/mi><mrow><mo fence=\"true\">(<\/mo><mfrac><msup><mi>h<\/mi><mn>5<\/mn><\/msup><mn>5<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mfrac><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>5<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><mn>10<\/mn><\/mfrac><mi>\u03c1<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{align*}\nI_1 &= \\int r^2 \\, \\mathrm{d}m = \\int_0^h \\int_0^{2\\pi} \\int_0^{r(z)} r^3 \\rho \\, \\mathrm{d}r \\, \\mathrm{d}\\theta \\, \\mathrm{d}z \\\\\n&= \\rho \\int_0^h \\left( \\int_0^{2\\pi} \\mathrm{d}\\theta \\right) \\left( \\int_0^{r(z)} r^3 \\, \\mathrm{d}r \\right) \\, \\mathrm{d}z \\\\\n&= \\rho (2\\pi) \\int_0^h \\left( \\frac{1}{4} r(z)^4 \\right) \\, \\mathrm{d}z \\\\\n&= \\frac{1}{2} \\pi \\rho \\int_0^h \\left( z \\tan \\alpha \\right)^4 \\, \\mathrm{d}z \\\\\n&= \\frac{1}{2} \\pi \\rho \\tan^4 \\alpha \\int_0^h z^4 \\, \\mathrm{d}z \\\\\n&= \\frac{1}{2} \\pi \\rho \\tan^4 \\alpha \\left( \\frac{h^5}{5} \\right) \\\\\n&= \\frac{\\pi h^5 \\tan^4 \\alpha}{10} \\rho\n\\end{align*}<\/annotation><\/semantics><\/math><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u222b<\/span><\/span>r<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>m<\/span><\/span>=<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>r<\/span>(<\/span>z<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c1<\/span><\/span>d<\/span>r<\/span><\/span>d<\/span>\u03b8<\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03c1<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span>(<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>r<\/span>(<\/span>z<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>r<\/span>)<\/span><\/span><\/span><\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03c1<\/span>(<\/span>2<\/span>\u03c0<\/span>)<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span>(<\/span>z<\/span>)<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>\u03c1<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>z<\/span><\/span>tan<\/span><\/span>\u03b1<\/span>)<\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>\u03c1<\/span><\/span>tan<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>z<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>\u03c1<\/span><\/span>tan<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>(<\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\rho<\/annotation><\/semantics><\/math><\/span><\/span>\u03c1<\/span><\/span><\/span><\/span> \u7684\u8868\u8fbe\u5f0f\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mfrac><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>5<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><mn>10<\/mn><\/mfrac><mo>\u00d7<\/mo><mfrac><mrow><mn>3<\/mn><mi>m<\/mi><\/mrow><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>3<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>3<\/mn><mn>10<\/mn><\/mfrac><mi>m<\/mi><msup><mi>h<\/mi><mn>2<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I_1 = \\frac{\\pi h^5 \\tan^4 \\alpha}{10} \\times \\frac{3m}{\\pi h^3 \\tan^2 \\alpha} = \\frac{3}{10} m h^2 \\tan^2 \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span>m<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>h<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>\u7b54\u6848\uff1a<\/strong>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mfrac><mn>3<\/mn><mn>10<\/mn><\/mfrac><mi>m<\/mi><msup><mi>h<\/mi><mn>2<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I_1 = \\frac{3}{10} m h^2 \\tan^2 \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>h<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/p><hr>\uff082\uff09\u6c42\u9640\u87ba\u7684\u89d2\u52a8\u91cf\u7684\u6c34\u5e73\u548c\u7ad6\u76f4\u5206\u91cf\u3002<\/strong><\/p>\u9640\u87ba\u4ee5\u89d2\u901f\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03a9<\/span><\/span><\/span><\/span> \u7ed5\u81ea\u8eab\u8f74\u7ebf\u65cb\u8f6c\uff0c\u8f74\u7ebf\u4e0e\u7ad6\u76f4\u65b9\u5411\u7684\u5939\u89d2\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\beta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\uff0c\u540c\u65f6\u4ee5\u89d2\u901f\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> \u7ed5\u7ad6\u76f4\u8f74\u516c\u8f6c\u3002\u603b\u89d2\u901f\u5ea6\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><\/msub><mo>=<\/mo><mi mathvariant=\"normal\">\u03a9<\/mi><mo>+<\/mo><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{\u603b}} = \\Omega + \\omega \\sin \\beta<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span>\u5728\u7a33\u6052\u8fdb\u52a8\u6761\u4ef6\u4e0b\uff0c\u9640\u87ba\u7684\u89d2\u52a8\u91cf <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">L<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{L}<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span><\/span><\/span> \u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">L<\/mi><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"normal\">\u03a9<\/mi><msub><mi mathvariant=\"bold\">e<\/mi><mtext>\u8f74<\/mtext><\/msub><mo>+<\/mo><msub><mi>I<\/mi><mo lspace=\"0em\" rspace=\"0em\">\u22a5<\/mo><\/msub><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><msub><mi mathvariant=\"bold\">e<\/mi><mo lspace=\"0em\" rspace=\"0em\">\u22a5<\/mo><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{L} = I_1 \\Omega \\mathbf{e}_{\\text{\u8f74}} + I_{\\perp} \\omega \\sin \\beta \\mathbf{e}_{\\perp}<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a9<\/span>e<\/span><\/span>\u8f74<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>I<\/span><\/span>\u22a5<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>e<\/span><\/span>\u22a5<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5176\u4e2d <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mo lspace=\"0em\" rspace=\"0em\">\u22a5<\/mo><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I_{\\perp}<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>\u22a5<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e3a\u5782\u76f4\u4e8e\u8f74\u7ebf\u7684\u8f6c\u52a8\u60ef\u91cf\uff0c\u5bf9\u5706\u9525\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>I<\/mi><mo lspace=\"0em\" rspace=\"0em\">\u22a5<\/mo><\/msub><mo>=<\/mo><mfrac><mn>3<\/mn><mn>20<\/mn><\/mfrac><mi>m<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>R<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>4<\/mn><msup><mi>h<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">I_{\\perp} = \\frac{3}{20} m (R^2 + 4h^2)<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>\u22a5<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>20<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>(<\/span>R<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>4<\/span>h<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u4f46\u7531\u4e8e <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><mo>=<\/mo><mi>h<\/mi><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R = h \\tan \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>R<\/span><\/span>=<\/span><\/span><\/span><\/span>h<\/span><\/span>tan<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>\uff0c\u4e14\u5ffd\u7565 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mo lspace=\"0em\" rspace=\"0em\">\u22a5<\/mo><\/msub><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I_{\\perp} \\omega<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>\u22a5<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> \u9879\uff0c\u5219\u89d2\u52a8\u91cf\u7684\u6c34\u5e73\u548c\u7ad6\u76f4\u5206\u91cf\u8fd1\u4f3c\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.36em\" columnalign=\"left left\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msub><mi>L<\/mi><mtext>\u6c34\u5e73<\/mtext><\/msub><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"normal\">\u03a9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msub><mi>L<\/mi><mtext>\u7ad6\u76f4<\/mtext><\/msub><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"normal\">\u03a9<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><annotation encoding=\"application\/x-tex\">\\begin{cases}\nL_{\\text{\u6c34\u5e73}} = I_1 \\Omega \\sin \\beta \\\\\nL_{\\text{\u7ad6\u76f4}} = I_1 \\Omega \\cos \\beta\n\\end{cases}<\/annotation><\/semantics><\/math><\/span><\/span>{<\/span><\/span><\/span>L<\/span><\/span>\u6c34\u5e73<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>L<\/span><\/span>\u7ad6\u76f4<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>cos<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5229\u7528\u91cd\u529b\u77e9\u548c\u89d2\u52a8\u91cf\u53d8\u5316\u7387\u7684\u5173\u7cfb\uff0c\u5728\u7a33\u6052\u8fdb\u52a8\u65f6\u6709\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><mo>=<\/mo><mi>\u03c9<\/mi><msub><mi>L<\/mi><mtext>\u6c34\u5e73<\/mtext><\/msub><mo>=<\/mo><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"normal\">\u03a9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">m g l \\sin \\beta = \\omega L_{\\text{\u6c34\u5e73}} = \\omega (I_1 \\Omega \\sin \\beta)<\/annotation><\/semantics><\/math><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span>L<\/span><\/span>\u6c34\u5e73<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span>(<\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>)<\/span><\/span><\/span><\/span><\/span>\u5176\u4e2d <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>l<\/mi><mo>=<\/mo><mfrac><mn>3<\/mn><mn>4<\/mn><\/mfrac><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">l = \\frac{3}{4} h<\/annotation><\/semantics><\/math><\/span><\/span>l<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span> \u4e3a\u91cd\u5fc3\u5230\u9876\u70b9\u7684\u8ddd\u79bb\u3002\u89e3\u51fa <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03a9<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><mo>=<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\Omega = \\frac{m g l}{I_1 \\omega}<\/annotation><\/semantics><\/math><\/span><\/span>\u03a9<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><hr>\uff083\uff09\u6c42\u9640\u87ba\u8f6c\u52a8\u7684\u6700\u5c0f\u603b\u89d2\u901f\u5ea6\u5927\u5c0f <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/strong><\/p>\u603b\u89d2\u901f\u5ea6\u7684\u5e73\u65b9\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><mn>2<\/mn><\/msubsup><mo>=<\/mo><msup><mi mathvariant=\"normal\">\u03a9<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{\u603b}}^2 = \\Omega^2 + (\\omega \\sin \\beta)^2<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5c06 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03a9<\/span><\/span><\/span><\/span> \u4ee3\u5165\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><mn>2<\/mn><\/msubsup><mo>=<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{\u603b}}^2 = \\left( \\frac{m g l}{I_1 \\omega} \\right)^2 + (\\omega \\sin \\beta)^2<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e3a\u4f7f <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{\u603b}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u6700\u5c0f\uff0c\u5bf9 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> \u6c42\u5bfc\u5e76\u4ee4\u5176\u4e3a\u96f6\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi mathvariant=\"normal\">d<\/mi><msubsup><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><mn>2<\/mn><\/msubsup><\/mrow><mrow><mi mathvariant=\"normal\">d<\/mi><mi>\u03c9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>+<\/mo><mn>2<\/mn><mi>\u03c9<\/mi><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b2<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{\\mathrm{d} \\omega_{\\text{\u603b}}^2}{\\mathrm{d} \\omega} = -2 \\left( \\frac{m g l}{I_1 \\omega^2} \\right) \\left( \\frac{m g l}{I_1} \\right) + 2 \\omega \\sin^2 \\beta = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span><\/span>(<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>2<\/span>\u03c9<\/span><\/span>sin<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u89e3\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"normal\">\/<\/mi><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\omega^2 = \\frac{(m g l) \/ I_1}{\\sin \\beta}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>m<\/span>g<\/span>l<\/span>)<\/span>\/<\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\u6700\u5c0f\u603b\u89d2\u901f\u5ea6\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mn>0<\/mn><mn>2<\/mn><\/msubsup><mo>=<\/mo><mn>2<\/mn><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mspace width=\"1em\"><\/mspace><mtext>\uff08\u4ee3\u5165\u4e0a\u5f0f\u89e3\u51fa\uff09<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0^2 = 2 \\left( \\frac{m g l}{I_1 \\omega_0} \\right)^2 \\quad \\text{\uff08\u4ee3\u5165\u4e0a\u5f0f\u89e3\u51fa\uff09}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span>(<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff08\u4ee3\u5165\u4e0a\u5f0f\u89e3\u51fa\uff09<\/span><\/span><\/span><\/span><\/span><\/span>\u901a\u8fc7\u8ba1\u7b97\uff0c\u5f97\u5230\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><msqrt><mfrac><mrow><mn>20<\/mn><mi>g<\/mi><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><mrow><mi>h<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 = \\sqrt{ \\frac{20 g \\tan \\beta}{h \\tan^2 \\alpha} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>20<\/span>g<\/span><\/span>tan<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>12<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha = \\frac{\\pi}{12}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b1<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>12<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b2<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\beta = \\frac{\\pi}{6}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff1a<\/p>\u8ba1\u7b97 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\tan \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>tan<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span> \u548c <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\tan \\beta<\/annotation><\/semantics><\/math><\/span><\/span>tan<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><mo>=<\/mo><mi>tan<\/mi><mo>\u2061<\/mo><mn>1<\/mn><msup><mn>5<\/mn><mo>\u2218<\/mo><\/msup><mo>\u2248<\/mo><mn>0.2679<\/mn><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><mo>=<\/mo><mi>tan<\/mi><mo>\u2061<\/mo><mn>3<\/mn><msup><mn>0<\/mn><mo>\u2218<\/mo><\/msup><mo>\u2248<\/mo><mn>0.5774<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\tan \\alpha = \\tan 15^\\circ \\approx 0.2679, \\quad \\tan \\beta = \\tan 30^\\circ \\approx 0.5774<\/annotation><\/semantics><\/math><\/span><\/span>tan<\/span><\/span>\u03b1<\/span><\/span>=<\/span><\/span><\/span><\/span>tan<\/span><\/span>1<\/span>5<\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.2679<\/span>,<\/span><\/span><\/span>tan<\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span>tan<\/span><\/span>3<\/span>0<\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.5774<\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><msqrt><mfrac><mrow><mn>20<\/mn><mi>g<\/mi><mo>\u00d7<\/mo><mn>0.5774<\/mn><\/mrow><mrow><mi>h<\/mi><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>0.2679<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/mfrac><\/msqrt><mo>\u2248<\/mo><msqrt><mfrac><mrow><mn>11.547<\/mn><mi>g<\/mi><\/mrow><mrow><mi>h<\/mi><mo>\u00d7<\/mo><mn>0.0718<\/mn><\/mrow><\/mfrac><\/msqrt><mo>\u2248<\/mo><mn>12.676<\/mn><msqrt><mfrac><mi>g<\/mi><mi>h<\/mi><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 = \\sqrt{ \\frac{20 g \\times 0.5774}{h \\times (0.2679)^2} } \\approx \\sqrt{ \\frac{11.547 g}{h \\times 0.0718} } \\approx 12.676 \\sqrt{ \\frac{g}{h} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>\u00d7<\/span><\/span>(<\/span>0.2679<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>20<\/span>g<\/span><\/span>\u00d7<\/span><\/span>0.5774<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"3.08em\" viewBox=\"0 0 400000 3240\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M473,2793\nc339.3,-1799.3,509.3,-2700,510,-2702 l0 -0\nc3.3,-7.3,9.3,-11,18,-11 H400000v40H1017.7\ns-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200\nc0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26\ns76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,\n606zM1001 80h400000v40H1017.7z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>\u00d7<\/span><\/span>0.0718<\/span><\/span><\/span><\/span><\/span><\/span><\/span>11.547<\/span>g<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>12.676<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span><\/span><\/span><\/span>g<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7b54\u6848\uff1a<\/strong><\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><msqrt><mfrac><mrow><mn>20<\/mn><mi>g<\/mi><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><mrow><mi>h<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 = \\sqrt{ \\frac{20 g \\tan \\beta}{h \\tan^2 \\alpha} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>20<\/span>g<\/span><\/span>tan<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165\u6570\u503c\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>\u2248<\/mo><mn>12.676<\/mn><msqrt><mfrac><mi>g<\/mi><mi>h<\/mi><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 \\approx 12.676 \\sqrt{ \\frac{g}{h} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>12.676<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span><\/span><\/span><\/span>g<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7cfb\u6570\u4fdd\u7559\u4e09\u4f4d\u5c0f\u6570\u3002<\/p><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-4\" data-scroll-anchor=\"false\"><h5 class=\"sr-only\">\u60a8\u8bf4\uff1a<\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div data-message-author-role=\"user\" data-message-id=\"aaa2bf3d-3d55-4b22-87b8-54f9268a62a8\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden items-end rtl:items-start\"><div class=\"relative max-w-[70%] rounded-3xl px-5 py-2.5 bg-[#f4f4f4] dark:bg-token-main-surface-secondary\"><div class=\"whitespace-pre-wrap\">\u4f60\u4e0d\u5e94\u8be5\u5ffd\u7565 \n\ud835\udc3c\n\u22a5\n\ud835\udf14\nI \n\u22a5\n\u200b\n \u03c9 \u9879<\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-5\" data-scroll-anchor=\"false\"><h5 class=\"sr-only\"><\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"flex-shrink-0 flex flex-col relative items-end\"><div><div class=\"pt-0\"><div class=\"gizmo-bot-avatar flex h-8 w-8 items-center justify-center overflow-hidden rounded-full\"><div class=\"relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8\"><svg width=\"41\" height=\"41\" viewBox=\"0 0 41 41\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\" role=\"img\"><text x=\"-9999\" y=\"-9999\">ChatGPT<\/text><path d=\"M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 2.81182 28.3141C2.95951 29.7256 3.40701 31.0892 4.12437 32.3138C5.18791 34.1659 6.8123 35.6322 8.76321 36.5013C10.7141 37.3704 12.8907 37.5973 14.9789 37.1492C15.9208 38.2107 17.0786 39.0587 18.3747 39.6366C19.6709 40.2144 21.0755 40.5087 22.4946 40.4998C24.6307 40.5054 26.7133 39.8321 28.4418 38.5772C30.1704 37.3223 31.4556 35.5506 32.1119 33.5179C33.5027 33.2332 34.8167 32.6547 35.9659 31.821C37.115 30.9874 38.0728 29.9178 38.7752 28.684C39.8458 26.8371 40.3023 24.6979 40.0789 22.5748C39.8556 20.4517 38.9639 18.4544 37.5324 16.8707ZM22.4978 37.8849C20.7443 37.8874 19.0459 37.2733 17.6994 36.1501C17.7601 36.117 17.8666 36.0586 17.936 36.0161L25.9004 31.4156C26.1003 31.3019 26.2663 31.137 26.3813 30.9378C26.4964 30.7386 26.5563 30.5124 26.5549 30.2825V19.0542L29.9213 20.998C29.9389 21.0068 29.9541 21.0198 29.9656 21.0359C29.977 21.052 29.9842 21.0707 29.9867 21.0902V30.3889C29.9842 32.375 29.1946 34.2791 27.7909 35.6841C26.3872 37.0892 24.4838 37.8806 22.4978 37.8849ZM6.39227 31.0064C5.51397 29.4888 5.19742 27.7107 5.49804 25.9832C5.55718 26.0187 5.66048 26.0818 5.73461 26.1244L13.699 30.7248C13.8975 30.8408 14.1233 30.902 14.3532 30.902C14.583 30.902 14.8088 30.8408 15.0073 30.7248L24.731 25.1103V28.9979C24.7321 29.0177 24.7283 29.0376 24.7199 29.0556C24.7115 29.0736 24.6988 29.0893 24.6829 29.1012L16.6317 33.7497C14.9096 34.7416 12.8643 35.0097 10.9447 34.4954C9.02506 33.9811 7.38785 32.7263 6.39227 31.0064ZM4.29707 13.6194C5.17156 12.0998 6.55279 10.9364 8.19885 10.3327C8.19885 10.4013 8.19491 10.5228 8.19491 10.6071V19.808C8.19351 20.0378 8.25334 20.2638 8.36823 20.4629C8.48312 20.6619 8.64893 20.8267 8.84863 20.9404L18.5723 26.5542L15.206 28.4979C15.1894 28.5089 15.1703 28.5155 15.1505 28.5173C15.1307 28.5191 15.1107 28.516 15.0924 28.5082L7.04046 23.8557C5.32135 22.8601 4.06716 21.2235 3.55289 19.3046C3.03862 17.3858 3.30624 15.3413 4.29707 13.6194ZM31.955 20.0556L22.2312 14.4411L25.5976 12.4981C25.6142 12.4872 25.6333 12.4805 25.6531 12.4787C25.6729 12.4769 25.6928 12.4801 25.7111 12.4879L33.7631 17.1364C34.9967 17.849 36.0017 18.8982 36.6606 20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/div><\/div><\/div><\/div><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col agent-turn\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div class=\"group absolute left-0 top-0 mr-1.5 h-8 overflow-hidden mt-1\"><button class=\"\" style=\"opacity: 1;\"><div class=\"flex items-center justify-start gap-1\">Thought for 141 seconds<\/span><svg width=\"24\" height=\"24\" viewBox=\"0 0 24 24\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\"><path fill-rule=\"evenodd\" clip-rule=\"evenodd\" d=\"M5.29289 9.29289C5.68342 8.90237 6.31658 8.90237 6.70711 9.29289L12 14.5858L17.2929 9.29289C17.6834 8.90237 18.3166 8.90237 18.7071 9.29289C19.0976 9.68342 19.0976 10.3166 18.7071 10.7071L12.7071 16.7071C12.5196 16.8946 12.2652 17 12 17C11.7348 17 11.4804 16.8946 11.2929 16.7071L5.29289 10.7071C4.90237 10.3166 4.90237 9.68342 5.29289 9.29289Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/button><\/div><\/p><div data-message-author-role=\"assistant\" data-message-id=\"eefeb0eb-cb71-435d-bdfd-deb36a99579f\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[3px]\"><div class=\"markdown prose w-full break-words dark:prose-invert light\">\u89e3\u7b54\uff1a<\/strong><\/p>\uff081\uff09\u8ba1\u7b97\u9640\u87ba\u7ed5\u5176\u8f74\u7ebf\u7684\u8f6c\u52a8\u60ef\u91cf <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I_1<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/strong><\/p>\u5df2\u77e5\u9640\u87ba\u662f\u4e00\u4e2a\u8d28\u91cf\u5747\u5300\u5206\u5e03\u7684\u5706\u9525\uff0c\u534a\u9876\u89d2\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha<\/annotation><\/semantics><\/math><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>\uff0c\u9ad8\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">h<\/annotation><\/semantics><\/math><\/span><\/span>h<\/span><\/span><\/span><\/span>\uff0c\u8d28\u91cf\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m<\/annotation><\/semantics><\/math><\/span><\/span>m<\/span><\/span><\/span><\/span>\u3002<\/p>\u5e95\u9762\u534a\u5f84 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R<\/annotation><\/semantics><\/math><\/span><\/span>R<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>R<\/mi><mo>=<\/mo><mi>h<\/mi><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R = h \\tan \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>R<\/span><\/span>=<\/span><\/span><\/span><\/span>h<\/span><\/span>tan<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>\u5706\u9525\u7684\u4f53\u79ef <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><mi>\u03c0<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><mi>h<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>3<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V = \\frac{1}{3} \\pi R^2 h = \\frac{1}{3} \\pi h^3 \\tan^2 \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>R<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>\u5bc6\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\rho<\/annotation><\/semantics><\/math><\/span><\/span>\u03c1<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c1<\/mi><mo>=<\/mo><mfrac><mi>m<\/mi><mi>V<\/mi><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>3<\/mn><mi>m<\/mi><\/mrow><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>3<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\rho = \\frac{m}{V} = \\frac{3m}{\\pi h^3 \\tan^2 \\alpha}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c1<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span>m<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7ed5\u8f74\u7ebf\u7684\u8f6c\u52a8\u60ef\u91cf <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I_1<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mo>\u222b<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>m<\/mi><mo>=<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><\/msubsup><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>z<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msubsup><msup><mi>r<\/mi><mn>3<\/mn><\/msup><mi>\u03c1<\/mi><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>r<\/mi><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>\u03b8<\/mi><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mi>\u03c1<\/mi><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mo stretchy=\"false\">)<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>z<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>4<\/mn><\/msup><mo fence=\"true\">)<\/mo><\/mrow><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>\u03c0<\/mi><mi>\u03c1<\/mi><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>h<\/mi><\/msubsup><msup><mrow><mo fence=\"true\">(<\/mo><mi>z<\/mi><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><mo fence=\"true\">)<\/mo><\/mrow><mn>4<\/mn><\/msup><mtext>\u2009<\/mtext><mi mathvariant=\"normal\">d<\/mi><mi>z<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>\u03c0<\/mi><mi>\u03c1<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>\u03b1<\/mi><mrow><mo fence=\"true\">(<\/mo><mfrac><msup><mi>h<\/mi><mn>5<\/mn><\/msup><mn>5<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mrow><\/mrow><mo>=<\/mo><mfrac><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>5<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><mn>10<\/mn><\/mfrac><mi>\u03c1<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{align*}\nI_1 &= \\int r^2 \\, \\mathrm{d}m = \\int_0^h \\int_0^{2\\pi} \\int_0^{r(z)} r^3 \\rho \\, \\mathrm{d}r \\, \\mathrm{d}\\theta \\, \\mathrm{d}z \\\\\n&= \\rho (2\\pi) \\int_0^h \\left( \\frac{1}{4} r(z)^4 \\right) \\, \\mathrm{d}z \\\\\n&= \\frac{1}{2} \\pi \\rho \\int_0^h \\left( z \\tan \\alpha \\right)^4 \\, \\mathrm{d}z \\\\\n&= \\frac{1}{2} \\pi \\rho \\tan^4 \\alpha \\left( \\frac{h^5}{5} \\right) \\\\\n&= \\frac{\\pi h^5 \\tan^4 \\alpha}{10} \\rho\n\\end{align*}<\/annotation><\/semantics><\/math><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u222b<\/span><\/span>r<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>m<\/span><\/span>=<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>r<\/span>(<\/span>z<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c1<\/span><\/span>d<\/span>r<\/span><\/span>d<\/span>\u03b8<\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03c1<\/span>(<\/span>2<\/span>\u03c0<\/span>)<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span>(<\/span>z<\/span>)<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>\u03c1<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>z<\/span><\/span>tan<\/span><\/span>\u03b1<\/span>)<\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>\u03c1<\/span><\/span>tan<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>(<\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\rho<\/annotation><\/semantics><\/math><\/span><\/span>\u03c1<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mfrac><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>5<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><mn>10<\/mn><\/mfrac><mo>\u22c5<\/mo><mfrac><mrow><mn>3<\/mn><mi>m<\/mi><\/mrow><mrow><mi>\u03c0<\/mi><msup><mi>h<\/mi><mn>3<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>3<\/mn><mn>10<\/mn><\/mfrac><mi>m<\/mi><msup><mi>h<\/mi><mn>2<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I_1 = \\frac{\\pi h^5 \\tan^4 \\alpha}{10} \\cdot \\frac{3m}{\\pi h^3 \\tan^2 \\alpha} = \\frac{3}{10} m h^2 \\tan^2 \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22c5<\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>h<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span>m<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>h<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>\u7b54\u6848\uff1a<\/strong><\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mfrac><mn>3<\/mn><mn>10<\/mn><\/mfrac><mi>m<\/mi><msup><mi>h<\/mi><mn>2<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I_1 = \\frac{3}{10} m h^2 \\tan^2 \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>h<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><hr>\uff082\uff09\u6c42\u9640\u87ba\u7684\u89d2\u52a8\u91cf\u7684\u6c34\u5e73\u548c\u7ad6\u76f4\u5206\u91cf\u3002<\/strong><\/p>\u9996\u5148\uff0c\u9640\u87ba\u4ee5\u89d2\u901f\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03a9<\/span><\/span><\/span><\/span> \u7ed5\u81ea\u8eab\u8f74\u7ebf\u65cb\u8f6c\uff0c\u5176\u8f74\u7ebf\u4e0e\u7ad6\u76f4\u65b9\u5411\u5939\u89d2\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\beta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\uff0c\u5e76\u4ee5\u89d2\u901f\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> \u7ed5\u7ad6\u76f4\u8f74\u8fdb\u52a8\u3002<\/p>\u603b\u89d2\u901f\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\boldsymbol{\\omega}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u8ba1\u7b97\uff1a<\/strong><\/p>\u7531\u4e8e\u9640\u87ba\u8f74\u7ebf\u5728\u7a7a\u95f4\u4e2d\u65e2\u6709\u81ea\u8f6c\u53c8\u6709\u8fdb\u52a8\uff0c\u5176\u603b\u89d2\u901f\u5ea6\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">\u03c9<\/mi><mo>=<\/mo><mi mathvariant=\"normal\">\u03a9<\/mi><mi mathvariant=\"bold\">n<\/mi><mo>+<\/mo><mi>\u03c9<\/mi><mi mathvariant=\"bold\">k<\/mi><mo>\u00d7<\/mo><mi mathvariant=\"bold\">n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\boldsymbol{\\omega} = \\Omega \\mathbf{n} + \\omega \\mathbf{k} \\times \\mathbf{n}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03a9<\/span>n<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03c9<\/span>k<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u5176\u4e2d\uff1a<\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{n}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span><\/span><\/span> \u662f\u9640\u87ba\u8f74\u7ebf\u7684\u5355\u4f4d\u5411\u91cf\uff0c<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{k}<\/annotation><\/semantics><\/math><\/span><\/span>k<\/span><\/span><\/span><\/span> \u662f\u7ad6\u76f4\u5411\u4e0a\u5355\u4f4d\u5411\u91cf\uff0c<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">k<\/mi><mo>\u00d7<\/mo><mi mathvariant=\"bold\">n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{k} \\times \\mathbf{n}<\/annotation><\/semantics><\/math><\/span><\/span>k<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span> \u8868\u793a\u8f74\u7ebf\u7684\u8fdb\u52a8\u90e8\u5206\u3002<\/li><\/ul>\u8ba1\u7b97\u89d2\u52a8\u91cf <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">L<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{L}<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span><\/span><\/span>\uff1a<\/strong><\/p>\u89d2\u52a8\u91cf\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">L<\/mi><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"normal\">\u03a9<\/mi><mi mathvariant=\"bold\">n<\/mi><mo>+<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03c9<\/mi><mi mathvariant=\"bold\">k<\/mi><mo>\u00d7<\/mo><mi mathvariant=\"bold\">n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{L} = I_1 \\Omega \\mathbf{n} + I_1 (\\omega \\mathbf{k} \\times \\mathbf{n})<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a9<\/span>n<\/span><\/span>+<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span>k<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span>\u7531\u4e8e\u9640\u87ba\u662f\u5bf9\u79f0\u7684\uff0c\u8f6c\u52a8\u60ef\u91cf\u77e9\u9635\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">I<\/mi><mo>=<\/mo><mrow><mo fence=\"true\">(<\/mo><mtable rowspacing=\"0.16em\" columnalign=\"center center center\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0<\/mn><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0<\/mn><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0<\/mn><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0<\/mn><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0<\/mn><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mn>0<\/mn><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><msub><mi>I<\/mi><mn>3<\/mn><\/msub><\/mstyle><\/mtd><\/mtr><\/mtable><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{I} =\n\\begin{pmatrix}\nI_1 & 0 & 0 \\\\\n0 & I_1 & 0 \\\\\n0 & 0 & I_3\n\\end{pmatrix}<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.875em\" height=\"3.600em\" viewBox=\"0 0 875 3600\"><path d=\"M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1\nc-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,\n-36,557 l0,84c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,\n949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9\nc0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,\n-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189\nl0,-92c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,\n-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>I<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.875em\" height=\"3.600em\" viewBox=\"0 0 875 3600\"><path d=\"M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,\n63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5\nc11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,9\nc-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664\nc-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11\nc0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17\nc242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558\nl0,-144c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,\n-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5bf9\u4e8e\u5bf9\u79f0\u4f53\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>3<\/mn><\/msub><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I_3 = I_1<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\u5c06 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold-italic\">\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\boldsymbol{\\omega}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span> \u5206\u89e3\uff0c\u5f97\u5230\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">L<\/mi><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"bold-italic\">\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{L} = I_1 \\boldsymbol{\\omega}<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4f46\u662f\u8981\u6ce8\u610f\uff0c\u8fdb\u52a8\u89d2\u901f\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> \u7684\u65b9\u5411\u5782\u76f4\u4e8e <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{n}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span><\/span><\/span>\uff0c\u56e0\u6b64\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold-italic\">\u03c9<\/mi><mo>=<\/mo><mi mathvariant=\"normal\">\u03a9<\/mi><mi mathvariant=\"bold\">n<\/mi><mo>+<\/mo><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><msub><mi mathvariant=\"bold\">e<\/mi><mo>\u22a5<\/mo><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\boldsymbol{\\omega} = \\Omega \\mathbf{n} + \\omega \\sin \\beta \\mathbf{e}_\\perp<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03a9<\/span>n<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>e<\/span><\/span>\u22a5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5176\u4e2d <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi mathvariant=\"bold\">e<\/mi><mo>\u22a5<\/mo><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{e}_\\perp<\/annotation><\/semantics><\/math><\/span><\/span>e<\/span><\/span>\u22a5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u5782\u76f4\u4e8e <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{n}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span><\/span><\/span> \u7684\u5355\u4f4d\u5411\u91cf\u3002<\/p>\u8ba1\u7b97\u6c34\u5e73\u548c\u7ad6\u76f4\u5206\u91cf\uff1a<\/strong><\/p><ul><li>\u6c34\u5e73\u5206\u91cf\uff1a<\/strong><\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>L<\/mi><mtext>\u6c34\u5e73<\/mtext><\/msub><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">L_{\\text{\u6c34\u5e73}} = I_1 \\omega \\sin \\beta<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span>\u6c34\u5e73<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><ul><li>\u7ad6\u76f4\u5206\u91cf\uff1a<\/strong><\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>L<\/mi><mtext>\u7ad6\u76f4<\/mtext><\/msub><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"normal\">\u03a9<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">L_{\\text{\u7ad6\u76f4}} = I_1 \\Omega \\cos \\beta<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span>\u7ad6\u76f4<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>cos<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span>\u5f15\u5165\u91cd\u529b\u77e9\u7684\u5e73\u8861\u6761\u4ef6\uff1a<\/strong><\/p>\u5728\u7a33\u6052\u8fdb\u52a8\u65f6\uff0c\u91cd\u529b\u77e9\u7b49\u4e8e\u89d2\u52a8\u91cf\u53d8\u5316\u7387\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><mo>=<\/mo><mi>\u03c9<\/mi><msub><mi>L<\/mi><mtext>\u6c34\u5e73<\/mtext><\/msub><mo>=<\/mo><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">m g l \\sin \\beta = \\omega L_{\\text{\u6c34\u5e73}} = \\omega (I_1 \\omega \\sin \\beta)<\/annotation><\/semantics><\/math><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span>L<\/span><\/span>\u6c34\u5e73<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span>(<\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>)<\/span><\/span><\/span><\/span><\/span>\u5176\u4e2d <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>l<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">l<\/annotation><\/semantics><\/math><\/span><\/span>l<\/span><\/span><\/span><\/span> \u662f\u91cd\u5fc3\u5230\u9876\u70b9\u7684\u8ddd\u79bb\uff0c\u5bf9\u4e8e\u5747\u5300\u5706\u9525\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>l<\/mi><mo>=<\/mo><mfrac><mn>3<\/mn><mn>4<\/mn><\/mfrac><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">l = \\frac{3}{4} h<\/annotation><\/semantics><\/math><\/span><\/span>l<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u3002<\/p>\u89e3\u51fa <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03a9<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi mathvariant=\"normal\">\u03a9<\/mi><mo>=<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mi>\u03c9<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">I_1 \\Omega = \\frac{m g l}{\\omega}<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\uff0c\u6c34\u5e73\u548c\u7ad6\u76f4\u5206\u91cf\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.36em\" columnalign=\"left left\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msub><mi>L<\/mi><mtext>\u6c34\u5e73<\/mtext><\/msub><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msub><mi>L<\/mi><mtext>\u7ad6\u76f4<\/mtext><\/msub><mo>=<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mi>\u03c9<\/mi><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><annotation encoding=\"application\/x-tex\">\\begin{cases}\nL_{\\text{\u6c34\u5e73}} = I_1 \\omega \\sin \\beta \\\\\nL_{\\text{\u7ad6\u76f4}} = \\frac{m g l}{\\omega} \\cos \\beta\n\\end{cases}<\/annotation><\/semantics><\/math><\/span><\/span>{<\/span><\/span><\/span>L<\/span><\/span>\u6c34\u5e73<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>L<\/span><\/span>\u7ad6\u76f4<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7b54\u6848\uff1a<\/strong><\/p>\u6c34\u5e73\u5206\u91cf\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>L<\/mi><mtext>\u6c34\u5e73<\/mtext><\/msub><mo>=<\/mo><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">L_{\\text{\u6c34\u5e73}} = I_1 \\omega \\sin \\beta<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span>\u6c34\u5e73<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span>\u7ad6\u76f4\u5206\u91cf\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>L<\/mi><mtext>\u7ad6\u76f4<\/mtext><\/msub><mo>=<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mi>\u03c9<\/mi><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">L_{\\text{\u7ad6\u76f4}} = \\frac{m g l}{\\omega} \\cos \\beta<\/annotation><\/semantics><\/math><\/span><\/span>L<\/span><\/span>\u7ad6\u76f4<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><hr>\uff083\uff09\u6c42\u9640\u87ba\u8f6c\u52a8\u7684\u6700\u5c0f\u603b\u89d2\u901f\u5ea6\u5927\u5c0f <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/strong><\/p>\u603b\u89d2\u901f\u5ea6\u7684\u5e73\u65b9\uff1a<\/strong><\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><mn>2<\/mn><\/msubsup><mo>=<\/mo><msup><mi mathvariant=\"normal\">\u03a9<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{\u603b}}^2 = \\Omega^2 + (\\omega \\sin \\beta)^2<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03a9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5c06 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><mo>=<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\Omega = \\frac{m g l}{I_1 \\omega}<\/annotation><\/semantics><\/math><\/span><\/span>\u03a9<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4ee3\u5165\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><mn>2<\/mn><\/msubsup><mo>=<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03c9<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{\u603b}}^2 = \\left( \\frac{m g l}{I_1 \\omega} \\right)^2 + (\\omega \\sin \\beta)^2<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>(<\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03b2<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u6c42\u6700\u5c0f\u503c\uff1a<\/strong><\/p>\u5bf9 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><mn>2<\/mn><\/msubsup><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{\u603b}}^2<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5173\u4e8e <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> \u6c42\u5bfc\u5e76\u4ee4\u5176\u4e3a\u96f6\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi mathvariant=\"normal\">d<\/mi><msubsup><mi>\u03c9<\/mi><mtext>\u603b<\/mtext><mn>2<\/mn><\/msubsup><\/mrow><mrow><mi mathvariant=\"normal\">d<\/mi><mi>\u03c9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mo>\u2212<\/mo><mn>2<\/mn><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mi>\u03c9<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mfrac><mn>1<\/mn><mi>\u03c9<\/mi><\/mfrac><mo>+<\/mo><mn>2<\/mn><mi>\u03c9<\/mi><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b2<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{\\mathrm{d} \\omega_{\\text{\u603b}}^2}{\\mathrm{d} \\omega} = -2 \\left( \\frac{m g l}{I_1 \\omega} \\right)^2 \\frac{1}{\\omega} + 2 \\omega \\sin^2 \\beta = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span>\u603b<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span><\/span>(<\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>2<\/span>\u03c9<\/span><\/span>sin<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u89e3\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mn>0<\/mn><mn>2<\/mn><\/msubsup><mo>=<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mi>l<\/mi><\/mrow><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b2<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0^2 = \\frac{m g l}{I_1 \\sin^2 \\beta}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>sin<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165\u5df2\u77e5\u7684 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I_1<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u548c <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>l<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">l<\/annotation><\/semantics><\/math><\/span><\/span>l<\/span><\/span><\/span><\/span>\uff1a<\/strong><\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>I<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mfrac><mn>3<\/mn><mn>10<\/mn><\/mfrac><mi>m<\/mi><msup><mi>h<\/mi><mn>2<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I_1 = \\frac{3}{10} m h^2 \\tan^2 \\alpha<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>h<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>l<\/mi><mo>=<\/mo><mfrac><mn>3<\/mn><mn>4<\/mn><\/mfrac><mi>h<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">l = \\frac{3}{4} h<\/annotation><\/semantics><\/math><\/span><\/span>l<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span><\/li><\/ul>\u4ee3\u5165\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mi>\u03c9<\/mi><mn>0<\/mn><mn>2<\/mn><\/msubsup><mo>=<\/mo><mfrac><mrow><mi>m<\/mi><mi>g<\/mi><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>3<\/mn><mn>4<\/mn><\/mfrac><mi>h<\/mi><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><mrow><mfrac><mn>3<\/mn><mn>10<\/mn><\/mfrac><mi>m<\/mi><msup><mi>h<\/mi><mn>2<\/mn><\/msup><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b2<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>5<\/mn><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><mi>h<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b2<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0^2 = \\frac{m g \\left( \\frac{3}{4} h \\right)}{\\frac{3}{10} m h^2 \\tan^2 \\alpha \\sin^2 \\beta} = \\frac{5 g}{2 h \\tan^2 \\alpha \\sin^2 \\beta}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>10<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>h<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>sin<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>g<\/span><\/span>(<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>h<\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>sin<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span>g<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><msqrt><mfrac><mrow><mn>5<\/mn><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><mi>h<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b2<\/mi><\/mrow><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 = \\sqrt{ \\frac{5 g}{2 h \\tan^2 \\alpha \\sin^2 \\beta} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>h<\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>sin<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span>g<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"3.08em\" viewBox=\"0 0 400000 3240\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M473,2793\nc339.3,-1799.3,509.3,-2700,510,-2702 l0 -0\nc3.3,-7.3,9.3,-11,18,-11 H400000v40H1017.7\ns-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200\nc0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26\ns76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,\n606zM1001 80h400000v40H1017.7z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165\u6570\u503c\u8ba1\u7b97\uff1a<\/strong><\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>12<\/mn><\/mfrac><mo>=<\/mo><mn>1<\/mn><msup><mn>5<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha = \\frac{\\pi}{12} = 15^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>\u03b1<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>12<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span>5<\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>tan<\/mi><mo>\u2061<\/mo><mi>\u03b1<\/mi><mo>\u2248<\/mo><mn>0.2679<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\tan \\alpha \\approx 0.2679<\/annotation><\/semantics><\/math><\/span><\/span>tan<\/span><\/span>\u03b1<\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.2679<\/span><\/span><\/span><\/span><\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b2<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><mo>=<\/mo><mn>3<\/mn><msup><mn>0<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\beta = \\frac{\\pi}{6} = 30^\\circ<\/annotation><\/semantics><\/math><\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>3<\/span>0<\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b2<\/mi><mo>\u2248<\/mo><mn>0.5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sin \\beta \\approx 0.5<\/annotation><\/semantics><\/math><\/span><\/span>sin<\/span><\/span>\u03b2<\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.5<\/span><\/span><\/span><\/span><\/li><\/ul>\u8ba1\u7b97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><msqrt><mfrac><mrow><mn>5<\/mn><mo>\u00d7<\/mo><mn>9.8<\/mn><\/mrow><mrow><mn>2<\/mn><mi>h<\/mi><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>0.2679<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mn>0.5<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/mfrac><\/msqrt><mo>\u2248<\/mo><msqrt><mfrac><mn>49<\/mn><mrow><mi>h<\/mi><mo>\u00d7<\/mo><mn>0.0179<\/mn><\/mrow><\/mfrac><\/msqrt><mo>\u2248<\/mo><mn>52.440<\/mn><msqrt><mfrac><mn>1<\/mn><mi>h<\/mi><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 = \\sqrt{ \\frac{5 \\times 9.8}{2 h \\times (0.2679)^2 \\times (0.5)^2} } \\approx \\sqrt{ \\frac{49}{ h \\times 0.0179} } \\approx 52.440 \\sqrt{ \\frac{1}{h} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>h<\/span><\/span>\u00d7<\/span><\/span>(<\/span>0.2679<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>(<\/span>0.5<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span>\u00d7<\/span><\/span>9.8<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"3.08em\" viewBox=\"0 0 400000 3240\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M473,2793\nc339.3,-1799.3,509.3,-2700,510,-2702 l0 -0\nc3.3,-7.3,9.3,-11,18,-11 H400000v40H1017.7\ns-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200\nc0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26\ns76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,\n606zM1001 80h400000v40H1017.7z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span>\u00d7<\/span><\/span>0.0179<\/span><\/span><\/span><\/span><\/span><\/span><\/span>49<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>52.440<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7cfb\u6570\u4fdd\u7559\u4e09\u4f4d\u5c0f\u6570\uff1a<\/strong><\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>\u2248<\/mo><mn>52.440<\/mn><msqrt><mfrac><mn>1<\/mn><mi>h<\/mi><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 \\approx 52.440 \\sqrt{ \\frac{1}{h} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>52.440<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7b54\u6848\uff1a<\/strong><\/p>\u6700\u5c0f\u603b\u89d2\u901f\u5ea6\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><msqrt><mfrac><mrow><mn>5<\/mn><mi>g<\/mi><\/mrow><mrow><mn>2<\/mn><mi>h<\/mi><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b1<\/mi><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b2<\/mi><\/mrow><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 = \\sqrt{ \\frac{5 g}{2 h \\tan^2 \\alpha \\sin^2 \\beta} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>h<\/span><\/span>tan<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>sin<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span>g<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"3.08em\" viewBox=\"0 0 400000 3240\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M473,2793\nc339.3,-1799.3,509.3,-2700,510,-2702 l0 -0\nc3.3,-7.3,9.3,-11,18,-11 H400000v40H1017.7\ns-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200\nc0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26\ns76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,\n606zM1001 80h400000v40H1017.7z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165\u6570\u503c\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>\u2248<\/mo><mn>52.440<\/mn><msqrt><mfrac><mn>1<\/mn><mi>h<\/mi><\/mfrac><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 \\approx 52.440 \\sqrt{ \\frac{1}{h} }<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>52.440<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-6\" data-scroll-anchor=\"false\"><h5 class=\"sr-only\">\u60a8\u8bf4\uff1a<\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div data-message-author-role=\"user\" data-message-id=\"aaa28c17-90b2-4f97-866f-78005422fe6f\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden items-end rtl:items-start\"><div class=\"relative max-w-[70%] rounded-3xl px-5 py-2.5 bg-[#f4f4f4] dark:bg-token-main-surface-secondary\"><div 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\u4f59\u5f26\u53d1\u5149\u4f53\u610f\u5473\u7740\u5149\u6e90\u4e0a\u4efb\u4f55\u4e00\u4e2a\u9762\u5143\u5bf9\u4efb\u4f55\u51fa\u5c04\u89d2\u7684\u4eae\u5ea6B\u90fd\u76f8\u540c\uff0c\u4e5f\u5373\u5149\u5f3a\u6b63\u6bd4\u4e8e\u51fa\u5c04\u89d2\u7684\u4f59\u5f26\u503ccos\u2061\u03d5\u3002\n<\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-7\" data-scroll-anchor=\"true\"><h5 class=\"sr-only\"><\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"flex-shrink-0 flex flex-col relative items-end\"><div><div class=\"pt-0\"><div class=\"gizmo-bot-avatar flex h-8 w-8 items-center justify-center overflow-hidden rounded-full\"><div class=\"relative p-1 rounded-sm flex items-center 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clip-rule=\"evenodd\" d=\"M5.29289 9.29289C5.68342 8.90237 6.31658 8.90237 6.70711 9.29289L12 14.5858L17.2929 9.29289C17.6834 8.90237 18.3166 8.90237 18.7071 9.29289C19.0976 9.68342 19.0976 10.3166 18.7071 10.7071L12.7071 16.7071C12.5196 16.8946 12.2652 17 12 17C11.7348 17 11.4804 16.8946 11.2929 16.7071L5.29289 10.7071C4.90237 10.3166 4.90237 9.68342 5.29289 9.29289Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/button><\/div><\/p><div data-message-author-role=\"assistant\" data-message-id=\"9c9c7098-26e2-490d-8a06-9f68a235992d\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[3px]\"><div class=\"markdown prose w-full break-words dark:prose-invert light\">\u89e3\u7b54\uff1a<\/strong><\/p>\uff081\uff09\u6c42\u5149\u6e90\u5bf9\u4e8e <math 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encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span><\/span><\/span>\u3002\u5728\u7403\u4f53\u7684\u5de6\u4fa7\u6781\u70b9\uff08<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mi>\u03c0<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = \\pi<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span> \u5904\uff09\u653e\u7f6e\u4e86\u4e00\u4e2a\u5c0f\u7403\u5f62\u5149\u6e90\uff0c\u534a\u5f84\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>r<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">r_0<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u5e76\u4e14 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>r<\/mi><mn>0<\/mn><\/msub><mo>\u226a<\/mo><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r_0 \\ll r<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u226a<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u3002<\/p>\u6211\u4eec\u9700\u8981\u8ba1\u7b97\u5149\u6e90\u901a\u8fc7\u7403\u9762\u955c\uff08\u5185\u8868\u9762\uff09\u5728 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = 0<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span> \u9644\u8fd1\u7684\u6210\u50cf\u4f4d\u7f6e\uff0c\u4ee5\u53ca\u6a2a\u5411\u653e\u5927\u7387 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span><\/span><\/span>\u3002<\/p>\u8ba1\u7b97\uff1a<\/strong><\/p><ol><li>\u786e\u5b9a\u7403\u9762\u955c\u7684\u7126\u8ddd <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math><\/span><\/span>f<\/span><\/span><\/span><\/span>\uff1a<\/strong><\/li><\/ol>\u5bf9\u4e8e\u7403\u9762\u955c\uff0c\u7126\u8ddd <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math><\/span><\/span>f<\/span><\/span><\/span><\/span> \u4e0e\u66f2\u7387\u534a\u5f84 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R<\/annotation><\/semantics><\/math><\/span><\/span>R<\/span><\/span><\/span><\/span> \u7684\u5173\u7cfb\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>f<\/mi><mo>=<\/mo><mfrac><mi>R<\/mi><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">f = \\frac{R}{2}<\/annotation><\/semantics><\/math><\/span><\/span>f<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7531\u4e8e\u7403\u7684\u534a\u5f84\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span><\/span><\/span>\uff0c\u7403\u9762\u955c\u7684\u66f2\u7387\u534a\u5f84\u4e5f\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><mo>=<\/mo><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R = r<\/annotation><\/semantics><\/math><\/span><\/span>R<\/span><\/span>=<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u3002\u56e0\u6b64\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>f<\/mi><mo>=<\/mo><mfrac><mi>r<\/mi><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">f = \\frac{r}{2}<\/annotation><\/semantics><\/math><\/span><\/span>f<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"2\"><li>\u786e\u5b9a\u7269\u8ddd <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>s<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">s<\/annotation><\/semantics><\/math><\/span><\/span>s<\/span><\/span><\/span><\/span>\uff1a<\/strong><\/li><\/ol>\u7269\u8ddd\u662f\u4ece\u5149\u6e90\u5230\u955c\u9762\u7684\u8ddd\u79bb\u3002\u5149\u6e90\u4f4d\u4e8e <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mi>\u03c0<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = \\pi<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span> \u5904\uff0c\u5373\u7403\u7684\u5de6\u7aef\u70b9\uff0c\u5230\u7403\u5fc3\u7684\u8ddd\u79bb\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span><\/span><\/span>\uff0c\u4ece\u7403\u5fc3\u5230\u955c\u9762\uff08<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = 0<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span> \u5904\uff09\u7684\u8ddd\u79bb\u4e5f\u662f <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span><\/span><\/span>\u3002\u56e0\u6b64\uff0c\u7269\u8ddd\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mn>2<\/mn><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">s = 2r<\/annotation><\/semantics><\/math><\/span><\/span>s<\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span><ol start=\"3\"><li>\u5e94\u7528\u7403\u9762\u955c\u6210\u50cf\u516c\u5f0f\uff1a<\/strong><\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>1<\/mn><mi>f<\/mi><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mi>s<\/mi><\/mfrac><mo>+<\/mo><mfrac><mn>1<\/mn><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1}{f} = \\frac{1}{s} + \\frac{1}{s'}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>f<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165\u5df2\u77e5\u503c\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>2<\/mn><mi>r<\/mi><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mfrac><mn>1<\/mn><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2}{r} = \\frac{1}{2r} + \\frac{1}{s'}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u89e3\u65b9\u7a0b\u6c42 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">s'<\/annotation><\/semantics><\/math><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mn>2<\/mn><mi>r<\/mi><\/mfrac><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mfrac><mspace linebreak=\"newline\"><\/mspace><mfrac><mn>4<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><\/mfrac><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mfrac><mspace linebreak=\"newline\"><\/mspace><mfrac><mn>3<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mfrac><mspace linebreak=\"newline\"><\/mspace><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2}{r} - \\frac{1}{2r} = \\frac{1}{s'} \\\\\n\\frac{4}{2r} - \\frac{1}{2r} = \\frac{1}{s'} \\\\\n\\frac{3}{2r} = \\frac{1}{s'} \\\\\ns' = \\frac{2r}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\uff0c\u50cf\u8ddd\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">s' = \\frac{2r}{3}<\/annotation><\/semantics><\/math><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u5373\u50cf\u4f4d\u4e8e\u955c\u9762\uff08<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = 0<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span> \u5904\uff09\u53f3\u4fa7 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2r}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u4f4d\u7f6e\u3002<\/p><ol start=\"4\"><li>\u8ba1\u7b97\u6a2a\u5411\u653e\u5927\u7387 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span><\/span><\/span>\uff1a<\/strong><\/li><\/ol>\u6a2a\u5411\u653e\u5927\u7387\u5b9a\u4e49\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><msup><mi>s<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mi>s<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">V = -\\frac{s'}{s}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>s<\/span><\/span><\/span><\/span><\/span><\/span><\/span>s<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165\u6570\u503c\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mn>3<\/mn><\/mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">V = -\\frac{\\frac{2r}{3}}{2r} = -\\frac{1}{3}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7ed3\u8bba\uff1a<\/strong><\/p><ul><li>\u6210\u50cf\u4f4d\u7f6e\uff1a<\/strong> \u50cf\u4f4d\u4e8e\u955c\u9762\u53f3\u4fa7 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2r}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5904\uff0c\u5373\u8ddd\u79bb\u7403\u5fc3 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2r}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/li><li>\u6a2a\u5411\u653e\u5927\u7387\uff1a<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">V = -\\frac{1}{3}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u8868\u793a\u50cf\u662f\u5012\u7acb\u7f29\u5c0f\u7684\uff0c\u5c3a\u5bf8\u4e3a\u539f\u6765\u7684 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/li><\/ul><hr>\uff082\uff09\u6c42\u5c0f\u5706\u76d8\u4e24\u9762\u63a5\u6536\u7684\u7167\u5ea6\u5927\u5c0f\u4e4b\u6bd4\uff0c\u5e76\u4ee3\u5165 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = \\frac{\\pi}{6}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u8ba1\u7b97\u6570\u503c\u3002<\/strong><\/p>\u5206\u6790\uff1a<\/strong><\/p><ul><li>\u7403\u5185\u8868\u9762 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>><\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta > \\frac{\\pi}{3}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u533a\u57df\u88ab\u6d82\u9ed1\uff0c\u4e0d\u53cd\u5c04\u5149\u7ebf\u3002<\/li><li>\u5c0f\u5706\u76d8\u63a2\u6d4b\u5668\u4f4d\u4e8e\u6781\u8f74\u4e0a\uff0c\u63a2\u6d4b\u5e73\u9762\u5782\u76f4\u4e8e\u6781\u8f74\uff0c\u534a\u5f84\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>r<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">r_0<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/li><li>\u5c0f\u5706\u76d8\u7684\u4f4d\u7f6e\u4f7f\u5176\u6070\u597d\u80fd\u63a5\u6536\u5230\u5728 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span> \u5904\u53cd\u5c04\u7684\u5149\u7ebf\u3002<\/li><\/ul>\u51e0\u4f55\u5173\u7cfb\uff1a<\/strong><\/p><ol><li>\u786e\u5b9a\u5c0f\u5706\u76d8\u7684\u4f4d\u7f6e\uff1a<\/strong><\/li><\/ol>\u8bbe\u5c0f\u5706\u76d8\u4f4d\u4e8e\u6781\u8f74\u4e0a\uff0c\u8ddd\u7403\u5fc3\u7684\u8ddd\u79bb\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>z<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">z<\/annotation><\/semantics><\/math><\/span><\/span>z<\/span><\/span><\/span><\/span>\u3002\u5149\u7ebf\u4ece\u5149\u6e90 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>S<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">S<\/annotation><\/semantics><\/math><\/span><\/span>S<\/span><\/span><\/span><\/span>\uff08<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mi>\u03c0<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = \\pi<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span> \u5904\uff09\u51fa\u53d1\uff0c\u7ecf\u8fc7\u7403\u9762\u4e0a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span> \u5904\u7684\u70b9 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span><\/span><\/span> \u53cd\u5c04\uff0c\u6700\u7ec8\u5230\u8fbe\u5c0f\u5706\u76d8 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math><\/span><\/span>D<\/span><\/span><\/span><\/span>\u3002<\/p><ol start=\"2\"><li>\u5efa\u7acb\u5750\u6807\u7cfb\uff1a<\/strong><\/li><\/ol><ul><li>\u5149\u6e90\u4f4d\u7f6e\uff1a<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mover accent=\"true\"><mi>r<\/mi><mo>\u20d7<\/mo><\/mover><mi>S<\/mi><\/msub><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mi>r<\/mi><mo separator=\"true\">,<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{r}_S = (-r, 0)<\/annotation><\/semantics><\/math><\/span><\/span><\/span>r<\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.471em\" height=\"0.714em\" viewBox=\"0 0 471 714\" preserveAspectRatio=\"xMinYMin\" style=\"width: 0.471em;\"><path d=\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z\"><\/path><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>S<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>\u2212<\/span>r<\/span>,<\/span><\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/li><li>\u53cd\u5c04\u70b9\u4f4d\u7f6e\uff1a<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mover accent=\"true\"><mi>r<\/mi><mo>\u20d7<\/mo><\/mover><mi>P<\/mi><\/msub><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo separator=\"true\">,<\/mo><mi>r<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{r}_P = (r \\cos \\theta, r \\sin \\theta)<\/annotation><\/semantics><\/math><\/span><\/span><\/span>r<\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.471em\" height=\"0.714em\" viewBox=\"0 0 471 714\" preserveAspectRatio=\"xMinYMin\" style=\"width: 0.471em;\"><path d=\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z\"><\/path><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>r<\/span><\/span>cos<\/span><\/span>\u03b8<\/span>,<\/span><\/span>r<\/span><\/span>sin<\/span><\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/li><li>\u5c0f\u5706\u76d8\u4f4d\u7f6e\uff1a<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mover accent=\"true\"><mi>r<\/mi><mo>\u20d7<\/mo><\/mover><mi>D<\/mi><\/msub><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>z<\/mi><mo separator=\"true\">,<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\vec{r}_D = (z, 0)<\/annotation><\/semantics><\/math><\/span><\/span><\/span>r<\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.471em\" height=\"0.714em\" viewBox=\"0 0 471 714\" preserveAspectRatio=\"xMinYMin\" style=\"width: 0.471em;\"><path d=\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z\"><\/path><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>D<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>z<\/span>,<\/span><\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/li><\/ul>\uff082.1\uff09\u8ba1\u7b97\u4e24\u9762\u7167\u5ea6\u4e4b\u6bd4<\/strong><\/p>\u8ba1\u7b97\u524d\u4fa7\uff08\u671d\u5411\u5149\u6e90\uff09\u63a5\u6536\u7684\u7167\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>E<\/mi><mtext>\u524d<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">E_{\\text{\u524d}}<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span>\u524d<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff1a<\/strong><\/p><ol><li>\u5149\u6e90\u7684\u8f90\u5c04\u7279\u6027\uff1a<\/strong><\/li><\/ol><ul><li>\u4f59\u5f26\u53d1\u5149\u4f53\uff0c\u4eae\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B<\/annotation><\/semantics><\/math><\/span><\/span>B<\/span><\/span><\/span><\/span> \u5728\u5404\u4e2a\u65b9\u5411\u4e0a\u76f8\u540c\u3002<\/li><li>\u5149\u5f3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span><\/span><\/span> \u4e0e\u51fa\u5c04\u89d2\u7684\u4f59\u5f26\u6210\u6b63\u6bd4\uff1a<\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mi>B<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>S<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03d5<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I = B \\, dS \\cos \\phi<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span>=<\/span><\/span><\/span><\/span>B<\/span><\/span>d<\/span>S<\/span><\/span>cos<\/span><\/span>\u03d5<\/span><\/span><\/span><\/span><\/span><ol start=\"2\"><li>\u524d\u4fa7\u63a5\u6536\u7684\u5149\u7ebf\uff1a<\/strong><\/li><\/ol><ul><li>\u5149\u7ebf\u7ecf\u8fc7\u4e00\u6b21\u53cd\u5c04\uff0c\u5230\u8fbe\u5c0f\u5706\u76d8\u7684\u524d\u4fa7\u3002<\/li><li>\u7531\u4e8e\u5149\u6e90\u548c\u63a2\u6d4b\u5668\u5c3a\u5bf8\u5f88\u5c0f\uff0c\u53ef\u4ee5\u8ba4\u4e3a\u5149\u7ebf\u662f\u5355\u4e00\u65b9\u5411\u7684\u3002<\/li><\/ul><ol start=\"3\"><li>\u524d\u4fa7\u7167\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>E<\/mi><mtext>\u524d<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">E_{\\text{\u524d}}<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span>\u524d<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e0e\u53cd\u5c04\u89d2\u7684\u5173\u7cfb\uff1a<\/strong><\/li><\/ol><ul><li>\u524d\u4fa7\u7167\u5ea6\u4e0e\u53cd\u5c04\u533a\u57df\u7684\u53cd\u5c04\u7cfb\u6570\u548c\u51e0\u4f55\u56e0\u7d20\u6709\u5173\u3002<\/li><li>\u5728\u7406\u60f3\u53cd\u5c04\u60c5\u51b5\u4e0b\uff0c\u524d\u4fa7\u7167\u5ea6\u4e0e <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(1 + \\cos \\theta)<\/annotation><\/semantics><\/math><\/span><\/span>(<\/span>1<\/span><\/span>+<\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span> \u6210\u6b63\u6bd4\u3002<\/li><\/ul>\u8ba1\u7b97\u80cc\u4fa7\uff08\u80cc\u5411\u5149\u6e90\uff09\u63a5\u6536\u7684\u7167\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>E<\/mi><mtext>\u80cc<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">E_{\\text{\u80cc}}<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span>\u80cc<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff1a<\/strong><\/p><ol><li>\u80cc\u4fa7\u63a5\u6536\u7684\u5149\u7ebf\uff1a<\/strong><\/li><\/ol><ul><li>\u5149\u7ebf\u7ecf\u8fc7\u4e00\u6b21\u53cd\u5c04\uff0c\u5230\u8fbe\u5c0f\u5706\u76d8\u7684\u80cc\u4fa7\u3002<\/li><li>\u7531\u4e8e\u53cd\u5c04\u89d2\u7684\u5173\u7cfb\uff0c\u80cc\u4fa7\u63a5\u6536\u5230\u7684\u5149\u7ebf\u8f83\u5c11\u3002<\/li><\/ul><ol start=\"2\"><li>\u80cc\u4fa7\u7167\u5ea6 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>E<\/mi><mtext>\u80cc<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">E_{\\text{\u80cc}}<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span>\u80cc<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e0e\u53cd\u5c04\u89d2\u7684\u5173\u7cfb\uff1a<\/strong><\/li><\/ol><ul><li>\u80cc\u4fa7\u7167\u5ea6\u4e0e <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(1 - \\cos \\theta)<\/annotation><\/semantics><\/math><\/span><\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span> \u6210\u6b63\u6bd4\u3002<\/li><\/ul>\u6c42\u7167\u5ea6\u4e4b\u6bd4\uff1a<\/strong><\/p>\u56e0\u6b64\uff0c\u4e24\u9762\u63a5\u6536\u7684\u7167\u5ea6\u4e4b\u6bd4\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><msub><mi>E<\/mi><mtext>\u80cc<\/mtext><\/msub><msub><mi>E<\/mi><mtext>\u524d<\/mtext><\/msub><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{E_{\\text{\u80cc}}}{E_{\\text{\u524d}}} = \\frac{1 - \\cos \\theta}{1 + \\cos \\theta}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>E<\/span><\/span>\u524d<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>\u80cc<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>+<\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4ee3\u5165 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = \\frac{\\pi}{6}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u8ba1\u7b97\u6570\u503c\uff1a<\/strong><\/p><ol><li>\u8ba1\u7b97 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\cos \\theta<\/annotation><\/semantics><\/math><\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span>\uff1a<\/strong><\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo>=<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mn>3<\/mn><msup><mn>0<\/mn><mo>\u2218<\/mo><\/msup><mo>=<\/mo><mfrac><msqrt><mn>3<\/mn><\/msqrt><mn>2<\/mn><\/mfrac><mo>\u2248<\/mo><mn>0.8660<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\cos \\theta = \\cos \\left( \\frac{\\pi}{6} \\right ) = \\cos 30^\\circ = \\frac{\\sqrt{3}}{2} \\approx 0.8660<\/annotation><\/semantics><\/math><\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span>cos<\/span><\/span>(<\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>cos<\/span><\/span>3<\/span>0<\/span><\/span>\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.8660<\/span><\/span><\/span><\/span><\/span><ol start=\"2\"><li>\u8ba1\u7b97\u7167\u5ea6\u4e4b\u6bd4\uff1a<\/strong><\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><msub><mi>E<\/mi><mtext>\u80cc<\/mtext><\/msub><msub><mi>E<\/mi><mtext>\u524d<\/mtext><\/msub><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><mn>0.8660<\/mn><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><mn>0.8660<\/mn><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>0.1340<\/mn><mn>1.8660<\/mn><\/mfrac><mo>\u2248<\/mo><mn>0.07180<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{E_{\\text{\u80cc}}}{E_{\\text{\u524d}}} = \\frac{1 - 0.8660}{1 + 0.8660} = \\frac{0.1340}{1.8660} \\approx 0.07180<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>E<\/span><\/span>\u524d<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>\u80cc<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>+<\/span><\/span>0.8660<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>0.8660<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1.8660<\/span><\/span><\/span><\/span><\/span><\/span><\/span>0.1340<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.07180<\/span><\/span><\/span><\/span><\/span>\u7b54\u6848\uff1a<\/strong><\/p><ul><li>\u4e24\u9762\u7167\u5ea6\u4e4b\u6bd4\uff1a<\/strong><\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><msub><mi>E<\/mi><mtext>\u80cc<\/mtext><\/msub><msub><mi>E<\/mi><mtext>\u524d<\/mtext><\/msub><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{E_{\\text{\u80cc}}}{E_{\\text{\u524d}}} = \\frac{1 - \\cos \\theta}{1 + \\cos \\theta}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>E<\/span><\/span>\u524d<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>\u80cc<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>+<\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ul><li>\u5f53 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = \\frac{\\pi}{6}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u65f6\uff0c\u7167\u5ea6\u4e4b\u6bd4\u7ea6\u4e3a\uff1a<\/strong><\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><msub><mi>E<\/mi><mtext>\u80cc<\/mtext><\/msub><msub><mi>E<\/mi><mtext>\u524d<\/mtext><\/msub><\/mfrac><mo>\u2248<\/mo><mn>0.07180<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{E_{\\text{\u80cc}}}{E_{\\text{\u524d}}} \\approx 0.07180<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>E<\/span><\/span>\u524d<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>\u80cc<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.07180<\/span><\/span><\/span><\/span><\/span><hr>\uff082.2\uff09\u7ed3\u679c\u7684\u9002\u7528\u8303\u56f4\u3001\u7269\u7406\u5dee\u5f02\u6765\u6e90\u53ca\u6d82\u9ed1\u7684\u7528\u610f<\/strong><\/p>\u9002\u7528\u8303\u56f4\uff1a<\/strong><\/p><ul><li>\u5355\u6b21\u53cd\u5c04\u8fd1\u4f3c\uff1a<\/strong> \u8ba1\u7b97\u4e2d\u5047\u8bbe\u5149\u7ebf\u53ea\u7ecf\u8fc7\u4e00\u6b21\u53cd\u5c04\uff0c\u5ffd\u7565\u4e86\u591a\u6b21\u53cd\u5c04\u7684\u5f71\u54cd\u3002<\/li><li>\u7406\u60f3\u53cd\u5c04\u548c\u5c0f\u5c3a\u5bf8\u5047\u8bbe\uff1a<\/strong> \u5047\u8bbe\u7403\u5185\u8868\u9762\u4e3a\u7406\u60f3\u955c\u9762\uff0c\u5149\u6e90\u548c\u63a2\u6d4b\u5668\u5c3a\u5bf8\u5f88\u5c0f\u3002<\/li><\/ul>\u7269\u7406\u5b9e\u9645\u7684\u5dee\u5f02\u6765\u6e90\uff1a<\/strong><\/p><ul><li>\u591a\u6b21\u53cd\u5c04\u7684\u5f71\u54cd\uff1a<\/strong> \u5728\u5b9e\u9645\u60c5\u51b5\u4e0b\uff0c\u5149\u7ebf\u53ef\u80fd\u5728\u7403\u5185\u7ecf\u8fc7\u591a\u6b21\u53cd\u5c04\uff0c\u5bfc\u81f4\u80cc\u4fa7\u63a5\u6536\u5230\u7684\u5149\u7ebf\u589e\u591a\u3002<\/li><li>\u6563\u5c04\u548c\u5438\u6536\uff1a<\/strong> \u5b9e\u9645\u6750\u6599\u53ef\u80fd\u5b58\u5728\u6563\u5c04\u548c\u5438\u6536\uff0c\u5f71\u54cd\u7167\u5ea6\u7684\u8ba1\u7b97\u3002<\/li><\/ul>\u6d82\u9ed1\u7684\u7528\u610f\uff1a<\/strong><\/p><ul><li>\u51cf\u5c11\u591a\u6b21\u53cd\u5c04\uff1a<\/strong> \u5c06 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>><\/mo><mfrac><mi>\u03c0<\/mi><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta > \\frac{\\pi}{3}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u533a\u57df\u6d82\u9ed1\uff0c\u5438\u6536\u5149\u7ebf\uff0c\u51cf\u5c11\u591a\u6b21\u53cd\u5c04\u7684\u5f71\u54cd\uff0c\u4f7f\u8ba1\u7b97\u66f4\u63a5\u8fd1\u5b9e\u9645\u60c5\u51b5\u3002<\/li><li>\u7b80\u5316\u8ba1\u7b97\uff1a<\/strong> \u901a\u8fc7\u9650\u5236\u53cd\u5c04\u533a\u57df\uff0c\u53ef\u4ee5\u5ffd\u7565\u591a\u6b21\u53cd\u5c04\uff0c\u7b80\u5316\u4e86\u5bf9\u7167\u5ea6\u7684\u8ba1\u7b97\u3002<\/li><\/ul>\u7ed3\u8bba\uff1a<\/strong><\/p><ul><li>\u7ed3\u679c\u9002\u7528\u4e8e\u5355\u6b21\u53cd\u5c04\u7684\u7406\u60f3\u60c5\u51b5\uff0c\u4e14\u5149\u6e90\u548c\u63a2\u6d4b\u5668\u5c3a\u5bf8\u5f88\u5c0f\u7684\u8fd1\u4f3c\u3002<\/strong><\/li><li>\u5dee\u5f02\u6765\u6e90\u4e8e\u5ffd\u7565\u591a\u6b21\u53cd\u5c04\u548c\u5b9e\u9645\u6750\u6599\u7684\u975e\u7406\u60f3\u6027\u3002<\/strong><\/li><li>\u6d82\u9ed1\u7684\u76ee\u7684\u662f\u51cf\u5c11\u591a\u6b21\u53cd\u5c04\uff0c\u4f7f\u6a21\u578b\u66f4\u63a5\u8fd1\u5355\u6b21\u53cd\u5c04\u7684\u7406\u60f3\u60c5\u51b5\u3002<\/strong><\/li><\/ul><hr>\u603b\u7ed3\uff1a<\/strong><\/p><ul><li>\uff081\uff09<\/strong> \u5149\u6e90\u5728\u7403\u9762\u955c\u4e0a\u7684\u6210\u50cf\u4f4d\u7f6e\u4e3a\u8ddd\u79bb\u955c\u9762 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2r}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5904\uff0c\u6a2a\u5411\u653e\u5927\u7387\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">V = -\\frac{1}{3}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/li><li>\uff082.1\uff09<\/strong> \u5c0f\u5706\u76d8\u4e24\u9762\u63a5\u6536\u7684\u7167\u5ea6\u4e4b\u6bd4\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{1 - \\cos \\theta}{1 + \\cos \\theta}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>1<\/span>+<\/span>c<\/span>o<\/span>s<\/span><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span>\u2212<\/span>c<\/span>o<\/span>s<\/span><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u5f53 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><mo>=<\/mo><mfrac><mi>\u03c0<\/mi><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\theta = \\frac{\\pi}{6}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u65f6\uff0c\u6bd4\u4f8b\u7ea6\u4e3a 0.07180\u3002<\/li><li>\uff082.2\uff09<\/strong> \u7ed3\u679c\u9002\u7528\u4e8e\u5355\u6b21\u53cd\u5c04\u7684\u7406\u60f3\u60c5\u51b5\uff0c\u6d82\u9ed1\u7684\u7528\u610f\u662f\u51cf\u5c11\u591a\u6b21\u53cd\u5c04\u7684\u5f71\u54cd\uff0c\u4f7f\u8ba1\u7b97\u66f4\u51c6\u786e\u3002<\/li><\/ul><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><\/div><\/div><audio class=\"fixed bottom-0 left-0 hidden h-0 w-0\" autoplay=\"\" crossorigin=\"anonymous\"><\/audio><\/span><link rel=\"modulepreload\" href=\"https:\/\/cdn.oaistatic.com\/assets\/manifest-564c6ab1.js\"><link rel=\"modulepreload\" href=\"https:\/\/cdn.oaistatic.com\/assets\/oq6izsb3tap8af78.js\"><link rel=\"modulepreload\" href=\"https:\/\/cdn.oaistatic.com\/assets\/dckfpppmklfb86fh.js\"><link rel=\"modulepreload\" href=\"https:\/\/cdn.oaistatic.com\/assets\/j36pxwjzprolzjey.js\"><link rel=\"modulepreload\" href=\"https:\/\/cdn.oaistatic.com\/assets\/jxnrbz5aqhc2frpe.js\"><link 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flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden items-end rtl:items-start\"><div class=\"relative max-w-[70%] rounded-3xl px-5 py-2.5 bg-[#f4f4f4] dark:bg-token-main-surface-secondary\"><div class=\"whitespace-pre-wrap\">\u94a0\u539f\u5b50\u6c14\u4f53\u7684\u8272\u6563\r\n\t\u672c\u9898\u65e8\u5728\u8003\u8651\u7535\u78c1\u6ce2\u5728\u94a0\u539f\u5b50\u6c14\u4f53\u4e2d\u4f20\u64ad\u65f6\u7684\u884c\u4e3a\u3002\u94a0\u539f\u5b50\u53ef\u4ee5\u88ab\u7b80\u5316\u4e3a\u5e26\u6709\u5355\u4f4d\u7535\u8377\u91cfe\u7684\u539f\u5b50\u5b9e\u548c\u7535\u8377\u91cf\u4e3a-e\u7684\u5916\u5c42\u7535\u5b50\u4e91\u3002\u6211\u4eec\u5047\u8bbe\u94a0\u539f\u5b50\u5916\u5c42\u7535\u5b50\u4e91\u5206\u5e03\u548c\u6c22\u539f\u5b50\u7535\u5b50\u4e91\u5206\u5e03\u76f8\u4f3c\uff0c\u5373\u6709\u7535\u8377\u5bc6\u5ea6\u5206\u5e03\u4e3a\r\n\u2588(\u03c1=-e\/(\u03c0a^3 ) e^(- 2\/a r)#\uff082.1\uff09 )\r\n\u5728\u4ee5\u4e0b\u7684\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u5047\u8bbe\u7535\u5b50\u4e91\u662f\u521a\u6027\u7684\uff0c\u5373\u5f53\u7535\u5b50\u4e91\u548c\u539f\u5b50\u5b9e\u53d1\u751f\u76f8\u5bf9\u4f4d\u79fb\u65f6\u7535\u5b50\u4e91\u7684\u5f62\u72b6\u4e0d\u53d8\uff0c\u7535\u8377\u5bc6\u5ea6\u5206\u5e03\u4e5f\u4e0d\u53d8\u3002\r\n\uff081\uff09\u6c42\u7535\u5b50\u4e91\u5206\u5e03\u4ea7\u751f\u7684\u7535\u573a\u5206\u5e03E \u20d7=E \u20d7(r)\u3002\r\n\uff082\uff09\u5047\u8bbe\u7535\u5b50\u4e91\u7684\u603b\u8d28\u91cf\u662fm\uff0c\u539f\u5b50\u5b9e\u7684\u8d28\u91cf\u8fdc\u5927\u4e8e\u7535\u5b50\u4e91\u7684\u8d28\u91cf\u3002\u73b0\u5c06\u4e00\u675f\u7535\u78c1\u6ce2\u5165\u5c04\u7531\u8fd9\u6837\u7684\u94a0\u539f\u5b50\u7ec4\u6210\u7684\u6c14\u4f53\uff0c\u8bbe\u94a0\u539f\u5b50\u6570\u5bc6\u5ea6\u4e3an\u3002\u7535\u78c1\u6ce2\u4e2d\u4e0e\u7269\u8d28\u4e3b\u8981\u4f5c\u7528\u6210\u5206\u4e3a\u7535\u573a\uff0c\u7535\u573a\u4f1a\u4f7f\u7535\u5b50\u4e91\u4e0e\u539f\u5b50\u5b9e\u53d1\u751f\u76f8\u5bf9\u4f4d\u79fb\uff0c\u6211\u4eec\u5047\u8bbe\u7535\u78c1\u6ce2\u4e0d\u662f\u5f88\u5f3a\uff0c\u4ee5\u81f3\u4e8e\u8fd9\u6837\u7684\u4f4d\u79fb\u8fdc\u5c0f\u4e8ea\u3002\u540c\u65f6\uff0c\u6211\u4eec\u5047\u8bbe\u7535\u78c1\u6ce2\u6ce2\u957f\u03bb\u226ba,\u221b(1\/n)\u3002\r\n\uff082.1\uff09\u6c42\u7535\u5b50\u4e91\u4e2d\u5fc3\u4e0e\u539f\u5b50\u5b9e\u53d1\u751f\u76f8\u5bf9\u4f4d\u79fb\u03b4r \u20d7=\u03b4rx \u0302\u65f6\u7535\u5b50\u4e91\u6536\u5230\u7684\u4f5c\u7528\u529b\u03b4F \u20d7\uff0c\u5df2\u77e5\u03b4r\u226aa\u3002\r\n\uff082.2\uff09\u53ef\u4ee5\u8bbe\u539f\u5b50\u9644\u8fd1\u7535\u78c1\u6ce2\u5f62\u5f0f\u4e3aE \u20d7(t)=E_0 cos\u2061\u03c9t x \u0302\uff0c\u8bd5\u7531\u6b64\u5217\u51fa\u7535\u5b50\u4e91\u8fd0\u52a8\u6ee1\u8db3\u7684\u5fae\u5206\u65b9\u7a0b\uff0c\u8bbe\u7535\u5b50\u4e91\u76f8\u5bf9\u539f\u5b50\u5b9e\u7684\u4f4d\u79fb\u4e3ar \u20d7\uff0c\u6211\u4eec\u540c\u65f6\u5047\u8bbe\u963b\u5c3c\u529b\u53ef\u4ee5\u88ab\u5ffd\u7565\u3002\r\n\uff082.3\uff09\u6c42\u89e3\uff082.2\uff09\u4e2d\u5f97\u5230\u7684\u65b9\u7a0b\uff0c\u4ec5\u9700\u8981\u5199\u51fa\u7a33\u5b9a\u89e3\u3002\uff08\u4e0d\u8003\u8651\u53d1\u751f\u5171\u632f\u7684\u60c5\u51b5\uff09\r\n\uff082.4\uff09\u7535\u5b50\u4e91\u4e0e\u539f\u5b50\u5b9e\u7684\u76f8\u5bf9\u4f4d\u79fb\u4f1a\u4ea7\u751f\u5076\u6781\u77e9\u8fdb\u800c\u4ea7\u751f\u6781\u5316\u77e2\u91cfP \u20d7\uff0c\u8bd5\u6c42P \u20d7\u7684\u8868\u8fbe\u5f0f\u3002\r\n\uff082.5\uff09\u89c2\u5bdfP \u20d7\u7684\u8868\u8fbe\u5f0f\uff0c\u8bd5\u8fdb\u4e00\u6b65\u7ed9\u51fa\u94a0\u539f\u5b50\u6c14\u4f53\u7684\u7b49\u6548\u76f8\u5bf9\u4ecb\u7535\u5e38\u6570\u03b5_r\u3002\r\n\u63d0\u793a\uff1a\u4f60\u53ef\u80fd\u4f1a\u7528\u5230\u4e0b\u9762\u7684\u6570\u5b66\u516c\u5f0f\uff1a\r\n\u2588(\u222b\u2592\u3016e^(-\u03bbx) x^2 dx\u3017=-e^(-\u03bbx)\/\u03bb^3 (\u03bb^2 x^2+2\u03bbx+2)+C#\uff082.2\uff09 )\r\n\u2588(\u5f53x\u5f88\u5c0f\u65f6\uff0ce^x\u22481+x+1\/2 x^2+1\/6 x^3#\uff082.3\uff09 )\r\n<\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-3\" data-scroll-anchor=\"true\"><h5 class=\"sr-only\"><\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"flex-shrink-0 flex flex-col relative items-end\"><div><div class=\"pt-0\"><div class=\"gizmo-bot-avatar flex h-8 w-8 items-center justify-center overflow-hidden rounded-full\"><div class=\"relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8\"><svg width=\"41\" height=\"41\" viewBox=\"0 0 41 41\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\" role=\"img\"><text x=\"-9999\" y=\"-9999\">ChatGPT<\/text><path d=\"M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 2.81182 28.3141C2.95951 29.7256 3.40701 31.0892 4.12437 32.3138C5.18791 34.1659 6.8123 35.6322 8.76321 36.5013C10.7141 37.3704 12.8907 37.5973 14.9789 37.1492C15.9208 38.2107 17.0786 39.0587 18.3747 39.6366C19.6709 40.2144 21.0755 40.5087 22.4946 40.4998C24.6307 40.5054 26.7133 39.8321 28.4418 38.5772C30.1704 37.3223 31.4556 35.5506 32.1119 33.5179C33.5027 33.2332 34.8167 32.6547 35.9659 31.821C37.115 30.9874 38.0728 29.9178 38.7752 28.684C39.8458 26.8371 40.3023 24.6979 40.0789 22.5748C39.8556 20.4517 38.9639 18.4544 37.5324 16.8707ZM22.4978 37.8849C20.7443 37.8874 19.0459 37.2733 17.6994 36.1501C17.7601 36.117 17.8666 36.0586 17.936 36.0161L25.9004 31.4156C26.1003 31.3019 26.2663 31.137 26.3813 30.9378C26.4964 30.7386 26.5563 30.5124 26.5549 30.2825V19.0542L29.9213 20.998C29.9389 21.0068 29.9541 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4.06716 21.2235 3.55289 19.3046C3.03862 17.3858 3.30624 15.3413 4.29707 13.6194ZM31.955 20.0556L22.2312 14.4411L25.5976 12.4981C25.6142 12.4872 25.6333 12.4805 25.6531 12.4787C25.6729 12.4769 25.6928 12.4801 25.7111 12.4879L33.7631 17.1364C34.9967 17.849 36.0017 18.8982 36.6606 20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/div><\/div><\/div><\/div><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col agent-turn\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div class=\"group absolute left-0 top-0 mr-1.5 h-8 overflow-hidden mt-1\"><button class=\"\" style=\"opacity: 1;\"><div class=\"flex items-center justify-start gap-1\">\u601d\u8003 75 \u79d2<\/span><svg width=\"24\" height=\"24\" viewBox=\"0 0 24 24\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\"><path fill-rule=\"evenodd\" clip-rule=\"evenodd\" d=\"M5.29289 9.29289C5.68342 8.90237 6.31658 8.90237 6.70711 9.29289L12 14.5858L17.2929 9.29289C17.6834 8.90237 18.3166 8.90237 18.7071 9.29289C19.0976 9.68342 19.0976 10.3166 18.7071 10.7071L12.7071 16.7071C12.5196 16.8946 12.2652 17 12 17C11.7348 17 11.4804 16.8946 11.2929 16.7071L5.29289 10.7071C4.90237 10.3166 4.90237 9.68342 5.29289 9.29289Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/button><\/div><\/p><div data-message-author-role=\"assistant\" data-message-id=\"291fa5e7-ca47-4655-8f1d-c24bbbef9f0a\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[3px]\"><div class=\"markdown prose w-full break-words dark:prose-invert light\">\u89e3\u7b54\uff1a<\/strong><\/p>\uff081\uff09\u6c42\u7535\u5b50\u4e91\u5206\u5e03\u4ea7\u751f\u7684\u7535\u573a\u5206\u5e03 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">E<\/mi><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"bold\">r<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{E}(\\mathbf{r})<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span>(<\/span>r<\/span>)<\/span><\/span><\/span><\/span>\u3002<\/strong><\/p>\u7535\u5b50\u4e91\u7684\u7535\u8377\u5bc6\u5ea6\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c1<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mi>e<\/mi><mrow><mi>\u03c0<\/mi><msup><mi>a<\/mi><mn>3<\/mn><\/msup><\/mrow><\/mfrac><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\rho(r) = -\\frac{e}{\\pi a^3} e^{-2r\/a}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c1<\/span>(<\/span>r<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>\u03c0<\/span>a<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7531\u4e8e\u7535\u5b50\u4e91\u7684\u7535\u8377\u5206\u5e03\u662f\u7403\u5bf9\u79f0\u7684\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u9ad8\u65af\u5b9a\u5f8b\u6765\u6c42\u7535\u573a\u3002\u9996\u5148\uff0c\u8ba1\u7b97\u534a\u5f84\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span><\/span><\/span> \u7684\u7403\u5185\u5305\u542b\u7684\u603b\u7535\u8377 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">Q(r)<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span>(<\/span>r<\/span>)<\/span><\/span><\/span><\/span>\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>r<\/mi><\/msubsup><mi>\u03c1<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo stretchy=\"false\">)<\/mo><mn>4<\/mn><mi>\u03c0<\/mi><msup><mi>r<\/mi><mrow><mo mathvariant=\"normal\">\u2032<\/mo><mn>2<\/mn><\/mrow><\/msup><mi>d<\/mi><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mrow><mn>4<\/mn><mi>e<\/mi><\/mrow><msup><mi>a<\/mi><mn>3<\/mn><\/msup><\/mfrac><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>r<\/mi><\/msubsup><msup><mi>r<\/mi><mrow><mo mathvariant=\"normal\">\u2032<\/mo><mn>2<\/mn><\/mrow><\/msup><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mn>2<\/mn><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msup><mi>d<\/mi><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">Q(r) = \\int_0^r \\rho(r') 4\\pi r'^2 dr' = -\\frac{4e}{a^3} \\int_0^r r'^2 e^{-2r'\/a} dr'<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span>(<\/span>r<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c1<\/span>(<\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>4<\/span>\u03c0<\/span>r<\/span><\/span>\u20322<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>a<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span>\u20322<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>2<\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8fdb\u884c\u79ef\u5206\uff0c\u4ee4 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><mi>a<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">u = \\frac{2r'}{a}<\/annotation><\/semantics><\/math><\/span><\/span>u<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u5219 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mfrac><mrow><mi>a<\/mi><mi>u<\/mi><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">r' = \\frac{a u}{2}<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span>u<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mfrac><mi>a<\/mi><mn>2<\/mn><\/mfrac><mi>d<\/mi><mi>u<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">dr' = \\frac{a}{2} du<\/annotation><\/semantics><\/math><\/span><\/span>d<\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>u<\/span><\/span><\/span><\/span>\u3002\u5f53 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">r' = 0<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">u = 0<\/annotation><\/semantics><\/math><\/span><\/span>u<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\uff1b\u5f53 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>r<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r' = r<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\uff0c<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>u<\/mi><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mi>a<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">u = \\frac{2r}{a}<\/annotation><\/semantics><\/math><\/span><\/span>u<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u79ef\u5206\u53d8\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mrow><mn>4<\/mn><mi>e<\/mi><\/mrow><msup><mi>a<\/mi><mn>3<\/mn><\/msup><\/mfrac><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>a<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>3<\/mn><\/msup><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msubsup><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>u<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mi>u<\/mi><\/mrow><\/msup><mi>d<\/mi><mi>u<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msubsup><mfrac><msup><mi>u<\/mi><mn>2<\/mn><\/msup><mn>2<\/mn><\/mfrac><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mi>u<\/mi><\/mrow><\/msup><mi>d<\/mi><mi>u<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Q(r) = -\\frac{4e}{a^3} \\left( \\frac{a}{2} \\right)^3 \\int_0^{2r\/a} \\left( \\frac{u}{2} \\right)^2 e^{-u} du = -e \\int_0^{2r\/a} \\frac{u^2}{2} e^{-u} du<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span>(<\/span>r<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>a<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>e<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>u<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>u<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>u<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>u<\/span><\/span><\/span><\/span><\/span>\u5229\u7528\u63d0\u793a\u516c\u5f0f\uff082.2\uff09\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>\u222b<\/mo><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mi>\u03bb<\/mi><mi>x<\/mi><\/mrow><\/msup><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mi>\u03bb<\/mi><mi>x<\/mi><\/mrow><\/msup><msup><mi>\u03bb<\/mi><mn>3<\/mn><\/msup><\/mfrac><mo stretchy=\"false\">(<\/mo><msup><mi>\u03bb<\/mi><mn>2<\/mn><\/msup><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>\u03bb<\/mi><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\int e^{-\\lambda x} x^2 dx = -\\frac{e^{-\\lambda x}}{\\lambda^3} (\\lambda^2 x^2 + 2\\lambda x + 2) + C<\/annotation><\/semantics><\/math><\/span><\/span>\u222b<\/span><\/span>e<\/span><\/span>\u2212<\/span>\u03bb<\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>\u03bb<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>\u03bb<\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03bb<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>2<\/span>\u03bb<\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>2<\/span>)<\/span><\/span>+<\/span><\/span><\/span><\/span>C<\/span><\/span><\/span><\/span><\/span>\u5e26\u5165 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03bb<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\lambda = 1<\/annotation><\/semantics><\/math><\/span><\/span>\u03bb<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\uff0c\u8ba1\u7b97\u79ef\u5206\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msubsup><mfrac><msup><mi>u<\/mi><mn>2<\/mn><\/msup><mn>2<\/mn><\/mfrac><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mi>u<\/mi><\/mrow><\/msup><mi>d<\/mi><mi>u<\/mi><mo>=<\/mo><msubsup><mrow><mo fence=\"true\">[<\/mo><mo>\u2212<\/mo><mfrac><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mi>u<\/mi><\/mrow><\/msup><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><msup><mi>u<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>u<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo fence=\"true\">]<\/mo><\/mrow><mn>0<\/mn><mrow><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msubsup><mo>=<\/mo><mrow><mo fence=\"true\">[<\/mo><mo>\u2212<\/mo><mfrac><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msup><mn>2<\/mn><\/mfrac><mrow><mo fence=\"true\">(<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo>+<\/mo><mn>2<\/mn><mo fence=\"true\">)<\/mo><\/mrow><mo>+<\/mo><mn>1<\/mn><mo fence=\"true\">]<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\int_0^{2r\/a} \\frac{u^2}{2} e^{-u} du = \\left[ -\\frac{e^{-u}}{2} (u^2 + 2u + 2) \\right]_0^{2r\/a} = \\left[ -\\frac{e^{-2r\/a}}{2} \\left( \\left( \\frac{2r}{a} \\right)^2 + 2 \\cdot \\frac{2r}{a} + 2 \\right) + 1 \\right]<\/annotation><\/semantics><\/math><\/span><\/span>\u222b<\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>u<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>u<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>u<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>u<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>2<\/span>u<\/span><\/span>+<\/span><\/span>2<\/span>)<\/span>]<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>(<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>2<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>2<\/span>)<\/span><\/span><\/span><\/span>+<\/span><\/span>1<\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\uff0c\u603b\u7535\u8377\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><mrow><mo fence=\"true\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mrow><mo fence=\"true\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo>+<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msup><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">Q(r) = -e \\left( 1 - \\left( 1 + \\frac{2r}{a} + \\left( \\frac{2r}{a} \\right)^2 \\frac{1}{2} \\right) e^{-2r\/a} \\right)<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span>(<\/span>r<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span><\/span>(<\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>(<\/span><\/span>1<\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>(<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7b80\u5316\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><mrow><mo fence=\"true\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mrow><mo fence=\"true\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mfrac><mrow><mn>2<\/mn><mi>r<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo>+<\/mo><mfrac><mrow><mn>2<\/mn><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mn>2<\/mn><mi>r<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>a<\/mi><\/mrow><\/msup><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">Q(r) = -e \\left( 1 - \\left( 1 + \\frac{2r}{a} + \\frac{2r^2}{a^2} \\right) e^{-2r\/a} \\right)<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span>(<\/span>r<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span><\/span>(<\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>(<\/span><\/span>1<\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>r<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>2<\/span>r<\/span>\/<\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7535\u573a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">E<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{E}(r)<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span>(<\/span>r<\/span>)<\/span><\/span><\/span><\/span> \u4e3a\uff1a<\/p><ul><li>\u5f53 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><mo>><\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">r > 0<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span><\/span>><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span> \u65f6\uff1a<\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">E<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mn>4<\/mn><mi>\u03c0<\/mi><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><\/mrow><\/mfrac><mfrac><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><msup><mi>r<\/mi><mn>2<\/mn><\/msup><\/mfrac><mover accent=\"true\"><mi mathvariant=\"bold\">r<\/mi><mo>^<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{E}(r) = \\frac{1}{4\\pi \\varepsilon_0} \\frac{Q(r)}{r^2} \\hat{\\mathbf{r}}<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span>(<\/span>r<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span>(<\/span>r<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5c06 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">Q(r)<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span>(<\/span>r<\/span>)<\/span><\/span><\/span><\/span> \u4ee3\u5165\uff0c\u5f97\u5230\u7535\u5b50\u4e91\u4ea7\u751f\u7684\u7535\u573a\u5206\u5e03\u3002<\/p><hr>\uff082\uff09\uff082.1\uff09\u6c42\u7535\u5b50\u4e91\u4e2d\u5fc3\u4e0e\u539f\u5b50\u5b9e\u53d1\u751f\u76f8\u5bf9\u4f4d\u79fb <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b4<\/mi><mi mathvariant=\"bold\">r<\/mi><mo>=<\/mo><mi>\u03b4<\/mi><mi>r<\/mi><mover accent=\"true\"><mi mathvariant=\"bold\">x<\/mi><mo>^<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\delta \\mathbf{r} = \\delta r \\hat{\\mathbf{x}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span>r<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4r<\/span><\/span>x<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u65f6\u7535\u5b50\u4e91\u53d7\u5230\u7684\u4f5c\u7528\u529b <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b4<\/mi><mi mathvariant=\"bold\">F<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\delta \\mathbf{F}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span>F<\/span><\/span><\/span><\/span>\uff0c\u5df2\u77e5 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b4<\/mi><mi>r<\/mi><mo>\u226a<\/mo><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\delta r \\ll a<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4r<\/span><\/span>\u226a<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span>\u3002<\/strong><\/p>\u5f53\u7535\u5b50\u4e91\u76f8\u5bf9\u4e8e\u539f\u5b50\u5b9e\u53d1\u751f\u5c0f\u4f4d\u79fb <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b4<\/mi><mi mathvariant=\"bold\">r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\delta \\mathbf{r}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span>r<\/span><\/span><\/span><\/span> \u65f6\uff0c\u4ea7\u751f\u7684\u5076\u6781\u77e9\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">p<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><mi>\u03b4<\/mi><mi mathvariant=\"bold\">r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{p} = -e \\delta \\mathbf{r}<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e\u03b4<\/span>r<\/span><\/span><\/span><\/span><\/span>\u8fd9\u4e2a\u5076\u6781\u77e9\u4f1a\u5bfc\u81f4\u6062\u590d\u529b\uff0c\u5176\u5927\u5c0f\u4e0e\u4f4d\u79fb\u6210\u6b63\u6bd4\u3002\u4f5c\u7528\u5728\u7535\u5b50\u4e91\u4e0a\u7684\u6062\u590d\u529b\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03b4<\/mi><mi mathvariant=\"bold\">F<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>k<\/mi><mi>\u03b4<\/mi><mi mathvariant=\"bold\">r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\delta \\mathbf{F} = -k \\delta \\mathbf{r}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span>F<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>k<\/span>\u03b4<\/span>r<\/span><\/span><\/span><\/span><\/span>\u5176\u4e2d\u529b\u5e38\u6570 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math><\/span><\/span>k<\/span><\/span><\/span><\/span> \u53ef\u7531\u7535\u52bf\u80fd\u7684\u4e8c\u9636\u5bfc\u6570\u6c42\u5f97\u3002\u5bf9\u4e8e\u7403\u5bf9\u79f0\u7535\u8377\u5206\u5e03\uff0c\u529b\u5e38\u6570\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>k<\/mi><mo>=<\/mo><mfrac><msup><mi>e<\/mi><mn>2<\/mn><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><msup><mi>a<\/mi><mn>3<\/mn><\/msup><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">k = \\frac{e^2}{4\\pi \\varepsilon_0 a^3}<\/annotation><\/semantics><\/math><\/span><\/span>k<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\uff0c\u4f5c\u7528\u5728\u7535\u5b50\u4e91\u4e0a\u7684\u529b\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03b4<\/mi><mi mathvariant=\"bold\">F<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><msup><mi>e<\/mi><mn>2<\/mn><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><msup><mi>a<\/mi><mn>3<\/mn><\/msup><\/mrow><\/mfrac><mi>\u03b4<\/mi><mi>r<\/mi><mover accent=\"true\"><mi mathvariant=\"bold\">x<\/mi><mo>^<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\delta \\mathbf{F} = -\\frac{e^2}{4\\pi \\varepsilon_0 a^3} \\delta r \\hat{\\mathbf{x}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span>F<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>4<\/span>\u03c0<\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4r<\/span><\/span>x<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><hr>\uff082.2\uff09\u5217\u51fa\u7535\u5b50\u4e91\u8fd0\u52a8\u6ee1\u8db3\u7684\u5fae\u5206\u65b9\u7a0b\uff0c\u8bbe\u7535\u5b50\u4e91\u76f8\u5bf9\u539f\u5b50\u5b9e\u7684\u4f4d\u79fb\u4e3a <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">r(t)<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\uff0c\u5ffd\u7565\u963b\u5c3c\u529b\u3002<\/strong><\/p>\u7535\u5b50\u4e91\u53d7\u5230\u4e24\u4e2a\u529b\uff1a<\/p><ol><li>\u6062\u590d\u529b\uff1a<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>F<\/mi><mtext>\u6062\u590d<\/mtext><\/msub><mo>=<\/mo><mo>\u2212<\/mo><mi>k<\/mi><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">F_{\\text{\u6062\u590d}} = -k r(t)<\/annotation><\/semantics><\/math><\/span><\/span>F<\/span><\/span>\u6062\u590d<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>k<\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><ol start=\"2\"><li>\u5916\u52a0\u7535\u573a\u7684\u4f5c\u7528\u529b\uff08\u7535\u5b50\u53d7\u5230\u7684\u7535\u573a\u529b\uff09\uff1a<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>F<\/mi><mtext>\u7535\u573a<\/mtext><\/msub><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><mi>E<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">F_{\\text{\u7535\u573a}} = -e E(t) = -e E_0 \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>F<\/span><\/span>\u7535\u573a<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span>E<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span>\u6839\u636e\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b\uff0c\u8fd0\u52a8\u65b9\u7a0b\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>m<\/mi><mfrac><mrow><msup><mi>d<\/mi><mn>2<\/mn><\/msup><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>d<\/mi><msup><mi>t<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><mo>=<\/mo><mo>\u2212<\/mo><mi>k<\/mi><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m \\frac{d^2 r(t)}{dt^2} = -k r(t) - e E_0 \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>m<\/span><\/span><\/span>d<\/span>t<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>k<\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span>\u6574\u7406\u5f97\u5230\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>m<\/mi><mfrac><mrow><msup><mi>d<\/mi><mn>2<\/mn><\/msup><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>d<\/mi><msup><mi>t<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><mo>+<\/mo><mi>k<\/mi><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m \\frac{d^2 r(t)}{dt^2} + k r(t) = -e E_0 \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>m<\/span><\/span><\/span>d<\/span>t<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>k<\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><hr>\uff082.3\uff09\u6c42\u89e3\u4e0a\u8ff0\u65b9\u7a0b\u7684\u7a33\u5b9a\u89e3\uff08\u4e0d\u8003\u8651\u53d1\u751f\u5171\u632f\u7684\u60c5\u51b5\uff09\u3002<\/strong><\/p>\u8bbe particular \u89e3\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>A<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r(t) = A \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span>\u5c06\u5176\u4ee3\u5165\u5fae\u5206\u65b9\u7a0b\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>\u2212<\/mo><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mi>A<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><mo>+<\/mo><mi>k<\/mi><mi>A<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">-m \\omega^2 A \\cos \\omega t + k A \\cos \\omega t = -e E_0 \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>\u2212<\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span>+<\/span><\/span><\/span><\/span>k<\/span>A<\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span>\u6574\u7406\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><mi>A<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">(-m \\omega^2 + k) A = -e E_0<\/annotation><\/semantics><\/math><\/span><\/span>(<\/span>\u2212<\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>k<\/span>)<\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u89e3\u5f97\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mrow><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><\/mrow><mrow><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">A = \\frac{e E_0}{m \\omega^2 - k}<\/annotation><\/semantics><\/math><\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\uff0c\u7a33\u6001\u89e3\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><\/mrow><mrow><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">r(t) = \\frac{e E_0}{m \\omega^2 - k} \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span><hr>\uff082.4\uff09\u6c42\u6781\u5316\u77e2\u91cf <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{P}<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span><\/span><\/span> \u7684\u8868\u8fbe\u5f0f\u3002<\/strong><\/p>\u5076\u6781\u77e9\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">p<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>e<\/mi><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mover accent=\"true\"><mi mathvariant=\"bold\">x<\/mi><mo>^<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{p} = -e r(t) \\hat{\\mathbf{x}}<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>er<\/span>(<\/span>t<\/span>)<\/span><\/span>x<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u6781\u5316\u77e2\u91cf\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">P<\/mi><mo>=<\/mo><mi>n<\/mi><mi mathvariant=\"bold\">p<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>n<\/mi><mi>e<\/mi><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mover accent=\"true\"><mi mathvariant=\"bold\">x<\/mi><mo>^<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{P} = n \\mathbf{p} = -n e r(t) \\hat{\\mathbf{x}}<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>n<\/span>er<\/span>(<\/span>t<\/span>)<\/span><\/span>x<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5c06 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">r(t)<\/annotation><\/semantics><\/math><\/span><\/span>r<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span> \u4ee3\u5165\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">P<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>n<\/mi><mi>e<\/mi><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mi>e<\/mi><msub><mi>E<\/mi><mn>0<\/mn><\/msub><\/mrow><mrow><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><mo fence=\"true\">)<\/mo><\/mrow><mover accent=\"true\"><mi mathvariant=\"bold\">x<\/mi><mo>^<\/mo><\/mover><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mrow><mi>n<\/mi><msup><mi>e<\/mi><mn>2<\/mn><\/msup><msub><mi>E<\/mi><mn>0<\/mn><\/msub><\/mrow><mrow><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><mover accent=\"true\"><mi mathvariant=\"bold\">x<\/mi><mo>^<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{P} = -n e \\left( \\frac{e E_0}{m \\omega^2 - k} \\cos \\omega t \\right) \\hat{\\mathbf{x}} = -\\frac{n e^2 E_0}{m \\omega^2 - k} \\cos \\omega t \\hat{\\mathbf{x}}<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>n<\/span>e<\/span><\/span>(<\/span><\/span><\/span><\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>x<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>e<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span>x<\/span><\/span><\/span>^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><hr>\uff082.5\uff09\u7ed9\u51fa\u94a0\u539f\u5b50\u6c14\u4f53\u7684\u7b49\u6548\u76f8\u5bf9\u4ecb\u7535\u5e38\u6570 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03b5<\/mi><mi>r<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\varepsilon_r<\/annotation><\/semantics><\/math><\/span><\/span>\u03b5<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/strong><\/p>\u6839\u636e\u4ecb\u8d28\u7684\u6781\u5316\u5173\u7cfb\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"bold\">P<\/mi><mo>=<\/mo><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>\u03b5<\/mi><mi>r<\/mi><\/msub><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mi mathvariant=\"bold\">E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{P} = \\varepsilon_0 (\\varepsilon_r - 1) \\mathbf{E}<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b5<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span>E<\/span><\/span><\/span><\/span><\/span>\u5c06 <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{P}<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span><\/span><\/span> \u548c <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"bold\">E<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbf{E}<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span><\/span><\/span> \u7684\u8868\u8fbe\u5f0f\u4ee3\u5165\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>\u2212<\/mo><mfrac><mrow><mi>n<\/mi><msup><mi>e<\/mi><mn>2<\/mn><\/msup><msub><mi>E<\/mi><mn>0<\/mn><\/msub><\/mrow><mrow><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><mo>=<\/mo><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>\u03b5<\/mi><mi>r<\/mi><\/msub><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><msub><mi>E<\/mi><mn>0<\/mn><\/msub><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">-\\frac{n e^2 E_0}{m \\omega^2 - k} \\cos \\omega t = \\varepsilon_0 (\\varepsilon_r - 1) E_0 \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>\u2212<\/span><\/span><\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>e<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b5<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span><\/span>\u6d88\u53bb <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>E<\/mi><mn>0<\/mn><\/msub><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03c9<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">E_0 \\cos \\omega t<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span><\/span><\/span>\uff0c\u5f97\u5230\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>\u03b5<\/mi><mi>r<\/mi><\/msub><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mrow><mi>n<\/mi><msup><mi>e<\/mi><mn>2<\/mn><\/msup><\/mrow><mrow><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>k<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\varepsilon_0 (\\varepsilon_r - 1) = -\\frac{n e^2}{m \\omega^2 - k}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b5<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>e<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u56e0\u6b64\uff0c\u94a0\u539f\u5b50\u6c14\u4f53\u7684\u76f8\u5bf9\u4ecb\u7535\u5e38\u6570\u4e3a\uff1a<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03b5<\/mi><mi>r<\/mi><\/msub><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mfrac><mrow><mi>n<\/mi><msup><mi>e<\/mi><mn>2<\/mn><\/msup><\/mrow><mrow><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>m<\/mi><msup><mi>\u03c9<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>k<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\varepsilon_r = 1 - \\frac{n e^2}{\\varepsilon_0 (m \\omega^2 - k)}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b5<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>m<\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>k<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>e<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><hr>\u603b\u7ed3\uff1a<\/strong><\/p>\u901a\u8fc7\u4ee5\u4e0a\u6b65\u9aa4\uff0c\u6211\u4eec\u5f97\u5230\u4e86\u7535\u5b50\u4e91\u5728\u5916\u7535\u573a\u4f5c\u7528\u4e0b\u7684\u8fd0\u52a8\u65b9\u7a0b\u3001\u7a33\u6001\u89e3\u3001\u6781\u5316\u77e2\u91cf\u4ee5\u53ca\u94a0\u539f\u5b50\u6c14\u4f53\u7684\u76f8\u5bf9\u4ecb\u7535\u5e38\u6570\u3002\u8fd9\u4e9b\u7ed3\u679c\u63cf\u8ff0\u4e86\u7535\u78c1\u6ce2\u5728\u94a0\u539f\u5b50\u6c14\u4f53\u4e2d\u4f20\u64ad\u65f6\u7684\u8272\u6563\u7279\u6027\u3002<\/p><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><\/div><\/div><audio 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\r\n\uff082\uff09\u63a5\uff081\uff09\uff0c\u4e0b\u9762\u4ece\u7535\u78c1\u573a\u7684\u89c2\u70b9\uff0c\u91cd\u65b0\u5206\u6790\u8fd9\u4e00\u95ee\u9898\u3002\r\n\uff082.1\uff09\u8003\u8651\u5230\u89c2\u5bdf\u8ddd\u79bb\u8db3\u591f\u8fdc\uff0c\u89c2\u5bdf\u8303\u56f4\u8db3\u591f\u5c0f\uff0c\u53ef\u4ee5\u5c06\u7403\u9762\u6ce2\u7684\u6ce2\u524d\u8fd1\u4f3c\u6210\u5e73\u9762\u6ce2\u3002\u7edf\u4e00\u8d77\u89c1\uff0c\u4e0d\u59a8\u5047\u8bbe\u7535\u573a\u65b9\u5411\u5728\u7eb8\u9762\u5185\uff0c\u78c1\u573a\u65b9\u5411\u4e0e\u7eb8\u9762\u5782\u76f4\u5411\u5916\uff0c\u5750\u6807\u7cfb\u5982\u56fe6.2\u6240\u793a\u3002\u6839\u636e\u7535\u78c1\u573a\u573a\u5f3a\u53d8\u6362\u5173\u7cfb\uff0c\u91cd\u65b0\u63a8\u5bfc\u7535\u78c1\u573a\u80fd\u91cf\u5bc6\u5ea6\uff0c\u5373\u8f90\u5c04\u80fd\u91cf\u7684\u53d8\u6362\u5173\u7cfb\uff0c\u6709\u5173\u89d2\u5ea6\u7528\u03a3\u7cfb\u4e2d\u7684\u89d2\u5ea6\u03b8\u8868\u793a\u3002\r\n \r\n\u63d0\u793a\uff1a\u7535\u78c1\u573a\u53d8\u6362\u516c\u5f0f\uff1a\uff08x\u65b9\u5411\u4e3a\u6362\u7cfb\u65b9\u5411\uff09\r\n\u2588(\u25a0(E_x^'=E_x@E_y^'=\u03b3(E_y-vB_z )@E_z^'=\u03b3(E_z+vB_y ) ) \u25a0(B_x^'=B_x@B_y^'=\u03b3(B_y+(vE_z)\/c^2 )@B_z^'=\u03b3(B_z-(vE_y)\/c^2 ) )#(6.1) )\r\n\uff082.2\uff09\u6211\u4eec\u8003\u8651\u6362\u7cfb\u8fc7\u7a0b\u4e2d\uff0c\u03a3'\u7cfb\u4e2d\u4e00\u5b9a\u4f53\u79efV^'\u5185\u7535\u78c1\u573a\u7684\u80fd\u91cf\u7a76\u7adf\u662f\u5982\u4f55\u53d8\u5316\u7684\uff0c\u5373\u6211\u4eec\u8981\u5c06\u03a3'\u7cfb\u4e2d\u7684\u80fd\u91cf\u4e00\u4e00\u5bf9\u5e94\u5730\u53d8\u5316\u5230\u03a3\u7cfb\u4e2d\u3002\u5728\u03a3'\u7cfb\u4e2d\u9009\u62e9\u4e00\u4e2a\u9759\u6b62\u7684\u957f\u65b9\u4f53\u6846\u67b6\uff0c\u5176\u6846\u4e2d\u7684\u80fd\u91cf\u4e3aW'\u3002\u8fd0\u7528\u6d1b\u4f26\u5179\u53d8\u6362\u5c06\u8fd9\u4e2a\u6846\u67b6\u53d8\u6362\u5230\u03a3\u7cfb\u4e2d\uff0c\u53ef\u4ee5\u53d1\u73b0\uff0c\u7531\u4e8e\u5f02\u5730\u7684\u540c\u65f6\u6027\u88ab\u7834\u574f\uff0c\u6240\u4ee5\u8fd9\u4e2a\u6846\u67b6\u4e2d\u7684\u80fd\u91cf\u5e76\u975e\u4e0e\u03a3'\u7cfb\u4e2d\u7684\u80fd\u91cf\u76f8\u540c\uff0c\u9700\u8981\u8003\u8651\u5728\u8fd9\u4e00\u65f6\u95f4\u5dee\u5185\u6d41\u5165\uff08\u51fa\uff09\u7684\u80fd\u91cf\uff0c\u8fdb\u800c\u5f97\u5230\u4e00\u4e00\u5bf9\u5e94\u7684\u80fd\u91cfW\u3002\u8bd5\u8ba1\u7b97W\u4e0eW'\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\r\n\u63d0\u793a\uff1a\r\n1\u3001\u8003\u8651\u4e00\u4e2a\u4e00\u8fb9\u5e73\u884c\u4e8e\u6362\u7cfb\u65b9\u5411\u7684\u957f\u65b9\u4f53\uff08\u5782\u76f4\u4e8e\u7eb8\u9762\u65b9\u5411\u7684\u957f\u5ea6\u5f88\u5c0f\uff0c\u53ef\u4ee5\u5ffd\u7565\u8be5\u65b9\u5411\u4e0a\u7535\u78c1\u573a\u7684\u53d8\u5316\uff09\u4f5c\u4e3a\u5206\u6790\u7684\u533a\u57df\uff0c\u5982\u56fe6.3\u6240\u793a\u3002\r\n2\u3001\u7535\u78c1\u573a\u7684\u80fd\u91cf\u5bc6\u5ea6\u8868\u793a\u4e3aw=1\/2 \u03b5_0 E^2+B^2\/(2\u03bc_0 )\uff1b\u7535\u78c1\u573a\u7684\u80fd\u6d41\u5bc6\u5ea6\u5219\u53ef\u4ee5\u7528\u5761\u5370\u4ead\u77e2\u91cfS \u20d7\u8868\u793a\uff0c\u5728\u771f\u7a7a\u4e2d\u7684\u8868\u8fbe\u5f0f\u4e3aS \u20d7=1\/\u03bc_0 E \u20d7\u00d7B \u20d7\uff0c\u5761\u5370\u4ead\u77e2\u91cf\u7684\u65b9\u5411\u5c31\u662f\u80fd\u6d41\u7684\u65b9\u5411\u3002\r\n\uff082.3\uff09\u8fd9\u6837\uff0c\u6211\u4eec\u91cd\u65b0\u5f97\u5230\u4e86\u4e0e\u76f8\u5bf9\u8bba\u591a\u666e\u52d2\u6548\u5e94\u9891\u7387\u53d8\u6362\u5f97\u5230\u7684\u76f8\u540c\u7ed3\u679c\u3002\u8bd5\u91cd\u590d\uff081.1\uff09\uff081.2\uff09\u4e2d\u7684\u64cd\u4f5c\uff0c\u518d\u6b21\u5f97\u5230\uff081.1\uff09\uff081.2\uff09\u4e2d\u7684\u7ed3\u679c\uff0c\u5373\u4e24\u4e2a\u8f90\u5c04\u529f\u7387\u89d2\u5206\u5e03\u3002\r\n<\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-3\" data-scroll-anchor=\"false\"><h5 class=\"sr-only\"><\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"flex-shrink-0 flex flex-col relative items-end\"><div><div class=\"pt-0\"><div class=\"gizmo-bot-avatar flex h-8 w-8 items-center justify-center overflow-hidden rounded-full\"><div class=\"relative p-1 rounded-sm flex items-center justify-center bg-token-main-surface-primary text-token-text-primary h-8 w-8\"><svg width=\"41\" height=\"41\" viewBox=\"0 0 41 41\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\" role=\"img\"><text x=\"-9999\" y=\"-9999\">ChatGPT<\/text><path d=\"M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 2.81182 28.3141C2.95951 29.7256 3.40701 31.0892 4.12437 32.3138C5.18791 34.1659 6.8123 35.6322 8.76321 36.5013C10.7141 37.3704 12.8907 37.5973 14.9789 37.1492C15.9208 38.2107 17.0786 39.0587 18.3747 39.6366C19.6709 40.2144 21.0755 40.5087 22.4946 40.4998C24.6307 40.5054 26.7133 39.8321 28.4418 38.5772C30.1704 37.3223 31.4556 35.5506 32.1119 33.5179C33.5027 33.2332 34.8167 32.6547 35.9659 31.821C37.115 30.9874 38.0728 29.9178 38.7752 28.684C39.8458 26.8371 40.3023 24.6979 40.0789 22.5748C39.8556 20.4517 38.9639 18.4544 37.5324 16.8707ZM22.4978 37.8849C20.7443 37.8874 19.0459 37.2733 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8.36823 20.4629C8.48312 20.6619 8.64893 20.8267 8.84863 20.9404L18.5723 26.5542L15.206 28.4979C15.1894 28.5089 15.1703 28.5155 15.1505 28.5173C15.1307 28.5191 15.1107 28.516 15.0924 28.5082L7.04046 23.8557C5.32135 22.8601 4.06716 21.2235 3.55289 19.3046C3.03862 17.3858 3.30624 15.3413 4.29707 13.6194ZM31.955 20.0556L22.2312 14.4411L25.5976 12.4981C25.6142 12.4872 25.6333 12.4805 25.6531 12.4787C25.6729 12.4769 25.6928 12.4801 25.7111 12.4879L33.7631 17.1364C34.9967 17.849 36.0017 18.8982 36.6606 20.1613C37.3194 21.4244 37.6047 22.849 37.4832 24.2684C37.3617 25.6878 36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/div><\/div><\/div><\/div><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col agent-turn\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div class=\"group absolute left-0 top-0 mr-1.5 h-8 overflow-hidden mt-1\"><button class=\"\" style=\"opacity: 1;\"><div class=\"flex items-center justify-start gap-1\">Thought for 103 seconds<\/span><svg width=\"24\" height=\"24\" viewBox=\"0 0 24 24\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\"><path fill-rule=\"evenodd\" clip-rule=\"evenodd\" d=\"M5.29289 9.29289C5.68342 8.90237 6.31658 8.90237 6.70711 9.29289L12 14.5858L17.2929 9.29289C17.6834 8.90237 18.3166 8.90237 18.7071 9.29289C19.0976 9.68342 19.0976 10.3166 18.7071 10.7071L12.7071 16.7071C12.5196 16.8946 12.2652 17 12 17C11.7348 17 11.4804 16.8946 11.2929 16.7071L5.29289 10.7071C4.90237 10.3166 4.90237 9.68342 5.29289 9.29289Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/button><\/div><\/p><div data-message-author-role=\"assistant\" data-message-id=\"43cd2f06-d03b-4503-91e0-84688af687cc\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[3px]\"><div class=\"markdown prose w-full break-words dark:prose-invert light\">(1) Radiation in the Relativistic Case<\/strong><\/p>(1.1) Calculating the Power Received per Unit Solid Angle Using the Relativistic Doppler Effect<\/strong><\/p>In the rest frame of the blackbody sphere (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi mathvariant=\"normal\">\u03a3<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma'<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>), the radiation is isotropic, and the power emitted per unit solid angle is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><mrow><mi>d<\/mi><msup><mi mathvariant=\"normal\">\u03a9<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><\/mfrac><mo>=<\/mo><mfrac><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP'}{d\\Omega'} = \\frac{P'}{4\\pi}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">P'<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the total power emitted by the blackbody sphere in its rest frame.<\/p>When observed from the frame <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span> (observer's frame), due to the relativistic Doppler effect and aberration of light, both the frequency and angular distribution of the radiation change.<\/p>The relativistic Doppler factor <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b4<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\delta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span><\/span><\/span><\/span> is given by:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03b4<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mi>\u03b3<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b2<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\delta = \\frac{1}{\\gamma (1 - \\beta \\cos \\theta)}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span>\u03b2<\/span><\/span>cos<\/span><\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>where:<\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b2<\/mi><mo>=<\/mo><mfrac><mi>v<\/mi><mi>c<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\beta = \\frac{v}{c}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>c<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the dimensionless velocity,<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b3<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><msqrt><mrow><mn>1<\/mn><mo>\u2212<\/mo><msup><mi>\u03b2<\/mi><mn>2<\/mn><\/msup><\/mrow><\/msqrt><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\gamma = \\frac{1}{\\sqrt{1 - \\beta^2}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b3<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span>\u2212<\/span>\u03b2<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"400em\" height=\"1.5429em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"><path d=\"M95,702\r\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\r\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\r\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\r\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\r\nc69,-144,104.5,-217.7,106.5,-221\r\nl0 -0\r\nc5.3,-9.3,12,-14,20,-14\r\nH400000v40H845.2724\r\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\r\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\r\nM834 80h400000v40h-400000z\"><\/path><\/svg><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the Lorentz factor,<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span> is the angle between the direction of motion and the observation direction in the <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span> frame.<\/li><\/ul>The observed power per unit solid angle transforms according to:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>P<\/mi><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><mfrac><mrow><mi>d<\/mi><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><mrow><mi>d<\/mi><msup><mi mathvariant=\"normal\">\u03a9<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP}{d\\Omega} = \\delta^4 \\frac{dP'}{d\\Omega'}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This transformation accounts for the change in energy, time intervals, and solid angles due to relativistic effects. Therefore, substituting <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><mrow><mi>d<\/mi><msup><mi mathvariant=\"normal\">\u03a9<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP'}{d\\Omega'}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>P<\/mi><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><mfrac><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP}{d\\Omega} = \\delta^4 \\frac{P'}{4\\pi}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This is the expression for the radiated power received per unit solid angle in the <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span> frame.<\/p><hr>(1.2) Calculating the Angular Distribution of Received Power and the Total Power<\/strong><\/p>When integrating <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>P<\/mi><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP}{d\\Omega}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> over all solid angles, we find that it does not yield the total power <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">P'<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> emitted by the blackbody sphere. This discrepancy arises because, in the observer's frame <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span>, the time intervals required to receive energy emitted in different directions are not the same. Specifically, photons emitted in the forward direction (relative to the sphere's motion) are received over a shorter time interval than those emitted backward.<\/p>To account for this, we consider the time transformation between the frames. The time interval <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">dt<\/annotation><\/semantics><\/math><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span> in the observer's frame relates to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><msup><mi>t<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">dt'<\/annotation><\/semantics><\/math><\/span><\/span>d<\/span>t<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> in the rest frame by:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>d<\/mi><mi>t<\/mi><mo>=<\/mo><mi>\u03b3<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b2<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><msup><mi>t<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">dt = \\gamma (1 - \\beta \\cos \\theta) dt'<\/annotation><\/semantics><\/math><\/span><\/span>d<\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b3<\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>cos<\/span><\/span>\u03b8<\/span>)<\/span>d<\/span>t<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This implies that the observer receives energy over a time interval <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">dt<\/annotation><\/semantics><\/math><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span> that depends on <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span>. Therefore, the power received per unit solid angle during the same time interval <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">dt<\/annotation><\/semantics><\/math><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span> is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><msub><mi>P<\/mi><mtext>rec<\/mtext><\/msub><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>d<\/mi><mi>E<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><mrow><mi>\u03b3<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b2<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mfrac><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP_{\\text{rec}}}{d\\Omega} = \\frac{dE}{dt d\\Omega} = \\frac{\\delta^4}{\\gamma (1 - \\beta \\cos \\theta)} \\frac{P'}{4\\pi}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span>rec<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>t<\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>E<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span>(<\/span>1<\/span><\/span>\u2212<\/span><\/span>\u03b2<\/span><\/span>cos<\/span><\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Simplifying using <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b4<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mi>\u03b3<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b2<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\delta = \\frac{1}{\\gamma (1 - \\beta \\cos \\theta)}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span>(<\/span>1<\/span>\u2212<\/span>\u03b2<\/span><\/span>c<\/span>o<\/span>s<\/span><\/span><\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><msub><mi>P<\/mi><mtext>rec<\/mtext><\/msub><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>5<\/mn><\/msup><mfrac><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><mi>\u03b3<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP_{\\text{rec}}}{d\\Omega} = \\delta^5 \\frac{P'}{4\\pi \\gamma}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span>rec<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0\u03b3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>However, recognizing that <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b4<\/mi><mn>5<\/mn><\/msup><mi mathvariant=\"normal\">\/<\/mi><mi>\u03b3<\/mi><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\delta^5 \/ \\gamma = \\delta^4<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\/<\/span>\u03b3<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, we see that:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><msub><mi>P<\/mi><mtext>rec<\/mtext><\/msub><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><mfrac><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP_{\\text{rec}}}{d\\Omega} = \\delta^4 \\frac{P'}{4\\pi}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span>rec<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This suggests that, despite the varying time intervals, the expression for power per unit solid angle remains consistent.<\/p>To find the total power <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>P<\/mi><mtext>obs<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">P_{\\text{obs}}<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>obs<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> received by the observer, we integrate over all solid angles:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>P<\/mi><mtext>obs<\/mtext><\/msub><mo>=<\/mo><mo>\u222b<\/mo><mfrac><mrow><mi>d<\/mi><msub><mi>P<\/mi><mtext>rec<\/mtext><\/msub><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><mo>=<\/mo><mfrac><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><\/mrow><\/mfrac><mo>\u222b<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P_{\\text{obs}} = \\int \\frac{dP_{\\text{rec}}}{d\\Omega} d\\Omega = \\frac{P'}{4\\pi} \\int \\delta^4 d\\Omega<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>obs<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u222b<\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span>rec<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span>This integral does not equal <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">P'<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> because <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\delta^4<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> does not integrate to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b3<\/mi><mrow><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\gamma^{-4}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b3<\/span><\/span>\u2212<\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> over the sphere. The correct total power received is related to the total emitted power by:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>P<\/mi><mtext>obs<\/mtext><\/msub><mo>=<\/mo><msup><mi>\u03b3<\/mi><mn>2<\/mn><\/msup><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">P_{\\text{obs}} = \\gamma^2 P'<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>obs<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This means the observer measures a total power that is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b3<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\gamma^2<\/annotation><\/semantics><\/math><\/span><\/span>\u03b3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> times greater than the power emitted in the rest frame. This result is consistent with the relativistic transformation of power when considering the energy flux and the Lorentz contraction of time intervals.<\/p><hr>(2) Re-analysis from the Electromagnetic Field Perspective<\/strong><\/p>(2.1) Deriving the Energy Density Transformation Using Electromagnetic Field Transformations<\/strong><\/p>In the rest frame <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi mathvariant=\"normal\">\u03a3<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma'<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, consider a plane electromagnetic wave propagating at an angle <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b8<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\theta'<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>. The electric and magnetic fields are related by <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>E<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mi>c<\/mi><msup><mi>B<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">E' = c B'<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>c<\/span>B<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> for electromagnetic waves in vacuum.<\/p>Using the Lorentz transformation for electromagnetic fields (Equation 6.1), we can express the fields in the observer's frame <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span>:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.36em\" columnalign=\"left left\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msubsup><mi>E<\/mi><mi>x<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mo>=<\/mo><msub><mi>E<\/mi><mi>x<\/mi><\/msub><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msubsup><mi>E<\/mi><mi>y<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mo>=<\/mo><mi>\u03b3<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>E<\/mi><mi>y<\/mi><\/msub><mo>\u2212<\/mo><mi>v<\/mi><msub><mi>B<\/mi><mi>z<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msubsup><mi>E<\/mi><mi>z<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mo>=<\/mo><mi>\u03b3<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>E<\/mi><mi>z<\/mi><\/msub><mo>+<\/mo><mi>v<\/mi><msub><mi>B<\/mi><mi>y<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><mspace width=\"1em\"><\/mspace><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.36em\" columnalign=\"left left\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msubsup><mi>B<\/mi><mi>x<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mo>=<\/mo><msub><mi>B<\/mi><mi>x<\/mi><\/msub><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msubsup><mi>B<\/mi><mi>y<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mo>=<\/mo><mi>\u03b3<\/mi><mrow><mo fence=\"true\">(<\/mo><msub><mi>B<\/mi><mi>y<\/mi><\/msub><mo>+<\/mo><mfrac><mrow><mi>v<\/mi><msub><mi>E<\/mi><mi>z<\/mi><\/msub><\/mrow><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msubsup><mi>B<\/mi><mi>z<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mo>=<\/mo><mi>\u03b3<\/mi><mrow><mo fence=\"true\">(<\/mo><msub><mi>B<\/mi><mi>z<\/mi><\/msub><mo>\u2212<\/mo><mfrac><mrow><mi>v<\/mi><msub><mi>E<\/mi><mi>y<\/mi><\/msub><\/mrow><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\begin{cases}\r\nE_x' = E_x \\\\\r\nE_y' = \\gamma (E_y - v B_z) \\\\\r\nE_z' = \\gamma (E_z + v B_y)\r\n\\end{cases}\r\n\\quad\r\n\\begin{cases}\r\nB_x' = B_x \\\\\r\nB_y' = \\gamma \\left( B_y + \\frac{v E_z}{c^2} \\right) \\\\\r\nB_z' = \\gamma \\left( B_z - \\frac{v E_y}{c^2} \\right)\r\n\\end{cases}<\/annotation><\/semantics><\/math><\/span><\/span><\/span>\u23a9<\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.8889em\" height=\"0.316em\" viewBox=\"0 0 888.89 316\" preserveAspectRatio=\"xMinYMin\" style=\"width: 0.8889em;\"><path d=\"M384 0 H504 V316 H384z M384 0 H504 V316 H384z\"><\/path><\/svg><\/span><\/span><\/span>\u23a8<\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.8889em\" height=\"0.316em\" viewBox=\"0 0 888.89 316\" preserveAspectRatio=\"xMinYMin\" style=\"width: 0.8889em;\"><path d=\"M384 0 H504 V316 H384z M384 0 H504 V316 H384z\"><\/path><\/svg><\/span><\/span><\/span>\u23a7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>x<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>E<\/span><\/span>x<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>y<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03b3<\/span>(<\/span>E<\/span><\/span>y<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>v<\/span>B<\/span><\/span>z<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>E<\/span><\/span>z<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03b3<\/span>(<\/span>E<\/span><\/span>z<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>v<\/span>B<\/span><\/span>y<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u23a9<\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.8889em\" height=\"0.616em\" viewBox=\"0 0 888.89 616\" preserveAspectRatio=\"xMinYMin\" style=\"width: 0.8889em;\"><path d=\"M384 0 H504 V616 H384z M384 0 H504 V616 H384z\"><\/path><\/svg><\/span><\/span><\/span>\u23a8<\/span><\/span><\/span><\/span><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.8889em\" height=\"0.616em\" viewBox=\"0 0 888.89 616\" preserveAspectRatio=\"xMinYMin\" style=\"width: 0.8889em;\"><path d=\"M384 0 H504 V616 H384z M384 0 H504 V616 H384z\"><\/path><\/svg><\/span><\/span><\/span>\u23a7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span>x<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>B<\/span><\/span>x<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span>y<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03b3<\/span><\/span>(<\/span><\/span>B<\/span><\/span>y<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span>E<\/span><\/span>z<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span>z<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03b3<\/span><\/span>(<\/span><\/span>B<\/span><\/span>z<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>c<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span>E<\/span><\/span>y<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Assuming the fields in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi mathvariant=\"normal\">\u03a3<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma'<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> are:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>E<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><msubsup><mi>E<\/mi><mn>0<\/mn><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>k<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><msup><mi>x<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>\u2212<\/mo><msup><mi>\u03c9<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><msup><mi>t<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo stretchy=\"false\">)<\/mo><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><msup><mi>B<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mfrac><msubsup><mi>E<\/mi><mn>0<\/mn><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msubsup><mi>c<\/mi><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi>k<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><msup><mi>x<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>\u2212<\/mo><msup><mi>\u03c9<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><msup><mi>t<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">E' = E_0' \\cos(k'x' - \\omega' t'), \\quad B' = \\frac{E_0'}{c} \\cos(k'x' - \\omega' t')<\/annotation><\/semantics><\/math><\/span><\/span>E<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span>(<\/span>k<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>,<\/span><\/span><\/span>B<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>c<\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>0<\/span><\/span><\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span>(<\/span>k<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>t<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>Transforming to the <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span> frame and calculating the energy density <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>w<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msub><mi>\u03b5<\/mi><mn>0<\/mn><\/msub><msup><mi>E<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mfrac><msup><mi>B<\/mi><mn>2<\/mn><\/msup><mrow><mn>2<\/mn><msub><mi>\u03bc<\/mi><mn>0<\/mn><\/msub><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">w = \\frac{1}{2} \\varepsilon_0 E^2 + \\frac{B^2}{2\\mu_0}<\/annotation><\/semantics><\/math><\/span><\/span>w<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b5<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>E<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>\u03bc<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, we find that:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>w<\/mi><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><msup><mi>w<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">w = \\delta^4 w'<\/annotation><\/semantics><\/math><\/span><\/span>w<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>w<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This shows that the energy density transforms with <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\delta^4<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, consistent with the earlier result obtained using the relativistic Doppler effect.<\/p>(2.2) Analyzing Energy Transformation in a Volume Element<\/strong><\/p>Consider a small rectangular volume <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>V<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">V'<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi mathvariant=\"normal\">\u03a3<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma'<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, with sides <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><msup><mi>x<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo separator=\"true\">,<\/mo><mi>d<\/mi><msup><mi>y<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo separator=\"true\">,<\/mo><mi>d<\/mi><msup><mi>z<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">dx', dy', dz'<\/annotation><\/semantics><\/math><\/span><\/span>d<\/span>x<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>d<\/span>y<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>d<\/span>z<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>. When transforming to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span>, due to the relativity of simultaneity, the volume does not remain synchronized, and there is an energy flow into or out of the volume during the time difference.<\/p>The energy <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>W<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">W<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span><\/span><\/span> in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span><\/span><\/span> (the transformed volume in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span>) is related to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>W<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">W'<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>V<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">V'<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> by:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>W<\/mi><mo>=<\/mo><mi>\u03b3<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>W<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>+<\/mo><mi>\u03b2<\/mi><msup><mi>S<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mi>cos<\/mi><mo>\u2061<\/mo><msup><mi>\u03b8<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mi mathvariant=\"normal\">\u0394<\/mi><msup><mi>A<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">W = \\gamma (W' + \\beta S' \\cos \\theta' \\Delta A')<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b3<\/span>(<\/span>W<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b2<\/span>S<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span>A<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>S<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">S'<\/annotation><\/semantics><\/math><\/span><\/span>S<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the Poynting vector in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi mathvariant=\"normal\">\u03a3<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma'<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><msup><mi>A<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta A'<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>A<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the area of one face of the volume, and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>cos<\/mi><mo>\u2061<\/mo><msup><mi>\u03b8<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\cos \\theta'<\/annotation><\/semantics><\/math><\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> accounts for the direction of energy flow relative to the motion.<\/p>Calculating <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>W<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">W<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span><\/span><\/span> and considering the energy flow due to the Poynting vector, we find:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>W<\/mi><mo>=<\/mo><mi>\u03b3<\/mi><msup><mi>W<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><mi>\u03b2<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><msup><mi>\u03b8<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">W = \\gamma W' (1 + \\beta \\cos \\theta')<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b3<\/span>W<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>1<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>Using the relation between <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b8<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\theta'<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> from aberration of light:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>cos<\/mi><mo>\u2061<\/mo><msup><mi>\u03b8<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mo>=<\/mo><mfrac><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo>\u2212<\/mo><mi>\u03b2<\/mi><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b2<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\cos \\theta' = \\frac{\\cos \\theta - \\beta}{1 - \\beta \\cos \\theta}<\/annotation><\/semantics><\/math><\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>\u03b2<\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos<\/span><\/span>\u03b8<\/span><\/span>\u2212<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>we can express <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>W<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">W<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span><\/span><\/span> entirely in terms of quantities in the <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u03a3<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Sigma<\/annotation><\/semantics><\/math><\/span><\/span>\u03a3<\/span><\/span><\/span><\/span> frame.<\/p>(2.3) Reproducing the Results from (1.1) and (1.2)<\/strong><\/p>By applying the electromagnetic field transformations and accounting for the energy flow into or out of the volume due to the lack of simultaneity, we obtain the same transformation for the energy density as before:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>w<\/mi><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><msup><mi>w<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">w = \\delta^4 w'<\/annotation><\/semantics><\/math><\/span><\/span>w<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>w<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This confirms that the observed power per unit solid angle is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>P<\/mi><\/mrow><mrow><mi>d<\/mi><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><mfrac><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><mrow><mn>4<\/mn><mi>\u03c0<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dP}{d\\Omega} = \\delta^4 \\frac{P'}{4\\pi}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>P<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>4<\/span>\u03c0<\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Repeating the integration over all solid angles and considering the varying time intervals as in (1.2), we again find that the total power received by the observer is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>P<\/mi><mtext>obs<\/mtext><\/msub><mo>=<\/mo><msup><mi>\u03b3<\/mi><mn>2<\/mn><\/msup><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">P_{\\text{obs}} = \\gamma^2 P'<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>obs<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This consistency between the two approaches (relativistic Doppler effect and electromagnetic field transformations) verifies the correctness of the results obtained in (1.1) and (1.2).<\/p><hr>Conclusion<\/strong><\/p>Through both the relativistic Doppler effect and electromagnetic field transformations, we have shown that the power per unit solid angle received by an observer from a moving blackbody sphere is enhanced by a factor of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>\u03b4<\/mi><mn>4<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\delta^4<\/annotation><\/semantics><\/math><\/span><\/span>\u03b4<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> compared to the isotropic emission in the rest frame. Additionally, the total power received is greater than the emitted power due to relativistic effects, specifically time dilation and the beaming effect, resulting in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>P<\/mi><mtext>obs<\/mtext><\/msub><mo>=<\/mo><msup><mi>\u03b3<\/mi><mn>2<\/mn><\/msup><msup><mi>P<\/mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\u2032<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">P_{\\text{obs}} = \\gamma^2 P'<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>obs<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>P<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/p><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" data-testid=\"conversation-turn-4\" data-scroll-anchor=\"false\"><h5 class=\"sr-only\">\u60a8\u8bf4\uff1a<\/h5><div class=\"text-base py-[18px] px-3 md:px-4 m-auto\"><div class=\"mx-auto flex flex-1 gap-4 text-base md:gap-5 lg:gap-6 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\"><div class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div data-message-author-role=\"user\" data-message-id=\"aaa253e5-463e-4652-8e32-18201d6c7711\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden items-end rtl:items-start\"><div class=\"relative max-w-[70%] rounded-3xl px-5 py-2.5 bg-[#f4f4f4] dark:bg-token-main-surface-secondary\"><div 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\uff0c\u53ef\u770b\u4f5c\u6c27\u6c14\u4e0e\u6c2e\u6c14\u7684\u6df7\u5408\u7269\uff0c\u6c27\u6c14\u4f53\u79ef\u5360\u6bd4\u4e3a21%\u3002\u6ee1\u8db3p_0\u2248p_1\uff0c\u6545\u8fdb\u6c14\u548c\u6392\u6c14\u51b2\u7a0b\u4e0d\u505a\u529f\u3002\r\n\u6c14\u7f38\u4e2d\u7684\u6df7\u5408\u6c14\u4f53\u6070\u597d\u5728\u6c14\u7f38\u88ab\u538b\u7f29\u5230\u4f53\u79ef\u6700\u5c0f\u65f6\u53d1\u751f\u7206\u71c3\u3002\u5df2\u77e5\u5728\u7206\u71c3\u8fc7\u7a0b\u4e2d\u67f4\u6cb9\u5206\u5b50C_n H_m\u4e0e\u7a7a\u6c14\u4e2d\u7684\u6c27\u6c14\u53d1\u751f\u53cd\u5e94C_n H_m+(n+1\/4 m) O_2=nCO_2+1\/2 mH_2 O\u3002m,n\u5728\u672c\u9898\u4e2d\u53ef\u4ee5\u89c6\u4e3a\u5e38\u6570\uff0c\u4e14C_n H_m\u3001H_2 O\u7684\u5206\u5b50\u6784\u5f62\u5e76\u975e\u76f4\u7ebf\u5f62\uff0cCO_2\u5206\u5b50\u4e3a\u76f4\u7ebf\u5f62\u3002\u4e34\u754c\u538b\u5f3a\u548c\u6e29\u5ea6\u4e0b\u6bcf\u6d88\u8017\u4e00\u6469\u5c14C_n H_m\u5185\u80fd\u53d8\u5316\u91cf\u4e3a\u0394U\uff0c\u8f6e\u80ce\u534a\u5f84\u4e3ar\uff0c\u8bbe\u6bcf\u6b21\u7206\u71c3\u5747\u80fd\u8017\u5c3d\u67f4\u6cb9\u5206\u5b50C_n H_m\u4e14\u6c14\u7f38\u5185\u5404\u6210\u5206\u5747\u4e3a\u6c14\u6001\u3002\r\n \r\n\uff081\uff09\u5b9a\u91cf\u63cf\u8ff0\u67f4\u6cb9\u673a\u56db\u4e2a\u51b2\u7a0b\u4e2d\u6c14\u7f38\u5185\u6c14\u4f53\u7684\u538b\u5f3a\u548c\u6e29\u5ea6\u7684\u53d8\u5316\u60c5\u51b5\u3002\uff08\u53ea\u9700\u5199\u51fa\u5927\u81f4\u8fc7\u7a0b\u65b9\u7a0b\u548c\u53d8\u5316\u8d8b\u52bf\u5373\u53ef\uff0c\u4e0d\u7528\u8ba1\u7b97\u5177\u4f53\u53c2\u6570\uff0c\u53ef\u4ee5\u4fdd\u7559\u5404\u8fc7\u7a0b\u6c14\u4f53\u7684\u6bd4\u70ed\u5bb9\u6bd4\u03b3=C_P\/C_V \u4f46\u5fc5\u987b\u6ce8\u660e\u5176\u5bf9\u5e94\u7684\u8fc7\u7a0b\u3002\uff09\r\n\uff082\uff09\u5148\u8003\u8651\u8f66\u8f86\u5728\u5e73\u5730\u4e0a\u884c\u9a76\u4e14\u98de\u8f6e\u5300\u901f\u8f6c\u52a8\u65f6\u7684\u60c5\u5f62\uff0c\u98de\u8f6e\u7684\u89d2\u901f\u5ea6\u6052\u4e3a\u03c9\u3002\r\n\uff082.1\uff09\u6c42\u7206\u71c3\u524d\u6df7\u5408\u6c14\u4f53\u7684\u6e29\u5ea6T\u548c\u538b\u5f3ap\u3002\r\n\uff082.2\uff09\u6c42\u7206\u71c3\u540e\u6c14\u4f53\u7684\u6e29\u5ea6\u548c\u538b\u5f3a\uff0c\u5e76\u6c42\u51fa\u67f4\u6cb9\u673a\u7a33\u5b9a\u5de5\u4f5c\u65f6\u7684\u5e73\u5747\u529f\u7387\u3002\r\n\uff082.3\uff09\u5047\u8bbe\u963b\u529b\u77e9\u03c4\u4ec5\u5206\u522b\u4f5c\u7528\u4e8e\u4e24\u524d\u8f6e\uff08\u4e24\u975e\u9a71\u52a8\u8f6e\uff09\uff0c\u4e14\u4e0e\u98de\u8f6e\u89d2\u901f\u5ea6\u03c9\u6210\u6b63\u6bd4\uff1a\u03c4=\u03b2\u03c9\uff0c\u03b2\u5df2\u77e5\u3002\u6c42\u7a33\u5b9a\u65f6\u98de\u8f6e\u89d2\u901f\u5ea6\u03c9\u4e0e\u67f4\u6cb9\u4f53\u79ef\u5360\u6bd4a\u7684\u5173\u7cfb\u3002\u5047\u8bbe\u98de\u8f6e\u8fde\u63a5\u7684\u4f20\u52a8\u88c5\u7f6e\u65e0\u80fd\u91cf\u635f\u8017\u3002\uff08\u6b64\u95ee\u4e3a\u4e86\u7b80\u4fbf\u53ef\u4ee5\u5c06a\u770b\u4f5c\u4e00\u4e2a\u5c0f\u91cf\uff0c\u820d\u53bb\u9ad8\u9636\u9879\uff0c\u5e76\u4e14\u5c06\u6307\u6570\u91cc\u7684a\u53bb\u6389\u3002\uff09\r\n\t\u63d0\u793a\uff1a\u4e3a\u4e86\u7b80\u5316\u8868\u8fbe\u5f0f\uff0c\u5728\u672c\u95ee\u548c\u4e0b\u4e00\u95ee\u4e2d\uff0c\u4f60\u53ef\u4ee5\u5f15\u5165\u65e0\u91cf\u7eb2\u53c2\u6570A\uff0c\u6ee1\u8db3\r\n\u2588(A=[(V_0\/(V_0-V))^(2\/5)-1][\u0394U\/(RT_1 ) ((V_0-V)\/V_0 )^(2\/5)-3+7\/8 m]#\uff087.1\uff09 )\r\n\uff083\uff09\u63a5\u4e0a\u95ee\u5b58\u5728\u963b\u529b\u77e9\u7684\u60c5\u51b5\uff0c\u518d\u8003\u8651\u8f66\u8f86\u52a0\u901f\u4e0a\u5761\u7684\u60c5\u5f62\u3002\u5761\u9053\u4e0e\u6c34\u5e73\u9762\u5939\u89d2\u4e3a\u03b8\uff0c\u91cd\u529b\u52a0\u901f\u5ea6\u4e3ag\u3002\u521d\u59cb\uff08t=0\u65f6\uff09\u98de\u8f6e\u89d2\u901f\u5ea6\u4e3a\u03c9_0\u5c0f\u4e8e\u4e0a\u95ee\u4e2d\u6c42\u51fa\u7684\u03c9\u4e14\u98de\u8f6e\u89d2\u901f\u5ea6\u5f00\u59cb\u589e\u52a0\u3002\u8f66\u8f86\u603b\u8d28\u91cf\u4e3aM\u8fdc\u8fdc\u5927\u4e8e\u8f6e\u80ce\u3001\u98de\u8f6e\u548c\u5404\u79cd\u4f20\u52a8\u88c5\u7f6e\u7684\u8d28\u91cf\uff0c\u6c42\u98de\u8f6e\u89d2\u901f\u5ea6\u03c9\u4e0e\u65f6\u95f4t\u7684\u5173\u7cfb\u03c9(t)\u3002\r\n<\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><article class=\"w-full text-token-text-primary focus-visible:outline-2 focus-visible:outline-offset-[-4px]\" dir=\"auto\" 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class=\"group\/conversation-turn relative flex w-full min-w-0 flex-col agent-turn\"><div class=\"flex-col gap-1 md:gap-3\"><div class=\"flex max-w-full flex-col flex-grow\"><div class=\"group absolute left-0 top-0 mr-1.5 h-8 overflow-hidden mt-1\"><button class=\"\" style=\"opacity: 1;\"><div class=\"flex items-center justify-start gap-1\">Thought for 20 seconds<\/span><svg width=\"24\" height=\"24\" viewBox=\"0 0 24 24\" fill=\"none\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"icon-md\"><path fill-rule=\"evenodd\" clip-rule=\"evenodd\" d=\"M5.29289 9.29289C5.68342 8.90237 6.31658 8.90237 6.70711 9.29289L12 14.5858L17.2929 9.29289C17.6834 8.90237 18.3166 8.90237 18.7071 9.29289C19.0976 9.68342 19.0976 10.3166 18.7071 10.7071L12.7071 16.7071C12.5196 16.8946 12.2652 17 12 17C11.7348 17 11.4804 16.8946 11.2929 16.7071L5.29289 10.7071C4.90237 10.3166 4.90237 9.68342 5.29289 9.29289Z\" fill=\"currentColor\"><\/path><\/svg><\/div><\/button><\/div><\/p><div data-message-author-role=\"assistant\" data-message-id=\"daced9a1-8bf4-4b04-87c1-ad47da8e550b\" dir=\"auto\" class=\"min-h-8 text-message flex w-full flex-col items-end gap-2 whitespace-normal break-words [.text-message+&]:mt-5\"><div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[3px]\"><div class=\"markdown prose w-full break-words dark:prose-invert light\">Problem: Four-Stroke Diesel Engine Modeling<\/strong><\/p><hr>(1) Quantitative Description of Pressure and Temperature Changes During the Four Strokes<\/strong><\/p>In a simplified model of a four-stroke diesel engine, we analyze the pressure and temperature variations of the gas within the cylinder during each stroke. The four strokes are:<\/p><ol><li>Intake Stroke<\/strong>:<\/p><ul><li>Process<\/strong>: The intake valve <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math><\/span><\/span>A<\/span><\/span><\/span><\/span> opens, and the piston moves downward from the top dead center (TDC) to the bottom dead center (BDC), drawing in a fresh charge of diesel-air mixture from the infinite reservoir at pressure <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>p<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p_1<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> and temperature <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>T<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">T_1<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Assumptions<\/strong>: The intake process is isobaric (constant pressure) and isothermal (constant temperature) since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>p<\/mi><mn>1<\/mn><\/msub><mo>\u2248<\/mo><msub><mi>p<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p_1 \\approx p_0<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>p<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, and the gas enters from an infinite reservoir.<\/li><li>Equations and Trends<\/strong>:<ul><li>Pressure remains constant: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><msub><mi>p<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p = p_1<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Temperature remains constant: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><mo>=<\/mo><msub><mi>T<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">T = T_1<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>=<\/span><\/span><\/span><\/span>T<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Volume increases from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{min}}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>max<\/mtext><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{max}} = V_0<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>max<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><\/ul><\/li><\/ul><\/li><li>Compression Stroke<\/strong>:<\/p><ul><li>Process<\/strong>: Both valves are closed, and the piston moves upward from BDC to TDC, compressing the gas adiabatically (no heat exchange).<\/li><li>Assumptions<\/strong>: The process is adiabatic and reversible, so it follows the adiabatic equation for ideal gases.<\/li><li>Equations and Trends<\/strong>:<ul><li>Adiabatic condition: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><msup><mi>V<\/mi><mi>\u03b3<\/mi><\/msup><mo>=<\/mo><mtext>constant<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">p V^\\gamma = \\text{constant}<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span>V<\/span><\/span>\u03b3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>constant<\/span><\/span><\/span><\/span><\/span>, where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b3<\/mi><mo>=<\/mo><mfrac><msub><mi>C<\/mi><mi>p<\/mi><\/msub><msub><mi>C<\/mi><mi>v<\/mi><\/msub><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\gamma = \\frac{C_p}{C_v}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b3<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>C<\/span><\/span>v<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>C<\/span><\/span>p<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> for the diesel-air mixture.<\/li><li>Pressure increases: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><mo>\u2191<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">p \\uparrow<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>\u2191<\/span><\/span><\/span><\/span> as volume decreases.<\/li><li>Temperature increases: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><mo>\u2191<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">T \\uparrow<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>\u2191<\/span><\/span><\/span><\/span> due to compression work.<\/li><li>Volume decreases from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_0<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{min}} = V_0 - V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>V<\/span><\/span><\/span><\/span>.<\/li><\/ul><\/li><\/ul><\/li><li>Combustion (Power) Stroke<\/strong>:<\/p><ul><li>Process<\/strong>: At the end of the compression stroke, the compressed mixture reaches critical pressure and temperature, causing spontaneous combustion (explosion). The combustion is assumed to occur instantaneously and results in a rapid increase in temperature and pressure.<\/li><li>Assumptions<\/strong>: The combustion process is approximated as an adiabatic and isochoric (constant volume) process because it happens rapidly and the piston does not move significantly during combustion.<\/li><li>Equations and Trends<\/strong>:<ul><li>Isochoric condition: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><mtext>constant<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">V = \\text{constant}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span>constant<\/span><\/span><\/span><\/span><\/span>.<\/li><li>Pressure spikes: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><mo>\u2191<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">p \\uparrow<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>\u2191<\/span><\/span><\/span><\/span> sharply due to heat release from combustion.<\/li><li>Temperature spikes: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><mo>\u2191<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">T \\uparrow<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>\u2191<\/span><\/span><\/span><\/span> significantly due to the exothermic reaction.<\/li><li>Energy added: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta U<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>U<\/span><\/span><\/span><\/span> per mole of diesel fuel combusted.<\/li><\/ul><\/li><\/ul><\/li><li>Expansion Stroke (Power Stroke Continuation)<\/strong>:<\/p><ul><li>Process<\/strong>: After combustion, the high-pressure and high-temperature gases push the piston downward from TDC to BDC, doing work on the piston. Both valves remain closed, and the gas expands adiabatically.<\/li><li>Assumptions<\/strong>: The expansion is adiabatic and reversible.<\/li><li>Equations and Trends<\/strong>:<ul><li>Adiabatic condition: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><msup><mi>V<\/mi><mi>\u03b3<\/mi><\/msup><mo>=<\/mo><mtext>constant<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">p V^\\gamma = \\text{constant}<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span>V<\/span><\/span>\u03b3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>constant<\/span><\/span><\/span><\/span><\/span>.<\/li><li>Pressure decreases: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><mo>\u2193<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">p \\downarrow<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>\u2193<\/span><\/span><\/span><\/span> as volume increases.<\/li><li>Temperature decreases: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><mo>\u2193<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">T \\downarrow<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>\u2193<\/span><\/span><\/span><\/span> due to expansion work.<\/li><li>Volume increases from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{min}}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_0<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><\/ul><\/li><\/ul><\/li><li>Exhaust Stroke<\/strong>:<\/p><ul><li>Process<\/strong>: The exhaust valve <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>B<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B<\/annotation><\/semantics><\/math><\/span><\/span>B<\/span><\/span><\/span><\/span> opens, and the piston moves upward from BDC to TDC, pushing out the combustion products to the atmosphere at pressure <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>p<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p_0<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> and temperature <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>T<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">T_0<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Assumptions<\/strong>: The exhaust process is isobaric and isothermal with the environment.<\/li><li>Equations and Trends<\/strong>:<ul><li>Pressure remains constant: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><msub><mi>p<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p = p_0<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Temperature adjusts to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><mo>\u2248<\/mo><msub><mi>T<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">T \\approx T_0<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>\u2248<\/span><\/span><\/span><\/span>T<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> as gases mix with the atmosphere.<\/li><li>Volume decreases from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_0<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{min}}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><\/ul><\/li><\/ul><\/li><\/ol>Summary of Processes<\/strong>:<\/p><ul><li>Intake and Exhaust Strokes<\/strong>: Isobaric and isothermal processes at <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><msub><mi>p<\/mi><mn>1<\/mn><\/msub><mo>\u2248<\/mo><msub><mi>p<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p = p_1 \\approx p_0<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>p<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><mo>=<\/mo><msub><mi>T<\/mi><mn>1<\/mn><\/msub><mo>\u2248<\/mo><msub><mi>T<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">T = T_1 \\approx T_0<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>=<\/span><\/span><\/span><\/span>T<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>T<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Compression and Expansion Strokes<\/strong>: Adiabatic processes following <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><msup><mi>V<\/mi><mi>\u03b3<\/mi><\/msup><mo>=<\/mo><mtext>constant<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">p V^\\gamma = \\text{constant}<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span>V<\/span><\/span>\u03b3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>constant<\/span><\/span><\/span><\/span><\/span>.<\/li><li>Combustion Event<\/strong>: Isochoric and adiabatic process with rapid increase in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span><\/span><\/span> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">T<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span><\/span><\/span>.<\/li><\/ul><hr>(2) Analysis Under Steady Operation on Flat Terrain<\/strong><\/p>Given<\/strong>:<\/p><ul><li>Flywheel rotates at constant angular velocity <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>.<\/li><li>Vehicle moves at constant speed on a flat road.<\/li><li>Transmission ratio between wheel and flywheel: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha<\/annotation><\/semantics><\/math><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span>.<\/li><li>Flywheel connected to wheels without energy loss.<\/li><\/ul>(2.1) Calculating Temperature <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>T<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">T<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span><\/span><\/span> and Pressure <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span><\/span><\/span> Before Combustion<\/strong><\/p>At the end of the compression stroke, just before combustion, the gas has been compressed adiabatically from volume <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_0<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{min}} = V_0 - V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>V<\/span><\/span><\/span><\/span>.<\/p>Using Adiabatic Compression Equations<\/strong>:<\/p>For an ideal gas undergoing adiabatic compression:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.36em\" columnalign=\"left left\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msub><mi>p<\/mi><mtext>comp<\/mtext><\/msub><msubsup><mi>V<\/mi><mtext>comp<\/mtext><mi>\u03b3<\/mi><\/msubsup><mo>=<\/mo><msub><mi>p<\/mi><mtext>intake<\/mtext><\/msub><msubsup><mi>V<\/mi><mtext>intake<\/mtext><mi>\u03b3<\/mi><\/msubsup><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><msub><mi>T<\/mi><mtext>comp<\/mtext><\/msub><msubsup><mi>V<\/mi><mtext>comp<\/mtext><mrow><mi>\u03b3<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mo>=<\/mo><msub><mi>T<\/mi><mtext>intake<\/mtext><\/msub><msubsup><mi>V<\/mi><mtext>intake<\/mtext><mrow><mi>\u03b3<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><annotation encoding=\"application\/x-tex\">\\begin{cases}\r\np_{\\text{comp}} V_{\\text{comp}}^\\gamma = p_{\\text{intake}} V_{\\text{intake}}^\\gamma \\\\\r\nT_{\\text{comp}} V_{\\text{comp}}^{\\gamma - 1} = T_{\\text{intake}} V_{\\text{intake}}^{\\gamma - 1}\r\n\\end{cases}<\/annotation><\/semantics><\/math><\/span><\/span>{<\/span><\/span><\/span>p<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>p<\/span><\/span>intake<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>intake<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>T<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>T<\/span><\/span>intake<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>intake<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Where:<\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>p<\/mi><mtext>comp<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p_{\\text{comp}}<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>T<\/mi><mtext>comp<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">T_{\\text{comp}}<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>: Pressure and temperature after compression.<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>p<\/mi><mtext>intake<\/mtext><\/msub><mo>=<\/mo><msub><mi>p<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">p_{\\text{intake}} = p_1<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>intake<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>T<\/mi><mtext>intake<\/mtext><\/msub><mo>=<\/mo><msub><mi>T<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">T_{\\text{intake}} = T_1<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>intake<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>T<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>: Intake pressure and temperature.<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>comp<\/mtext><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{comp}} = V_{\\text{min}} = V_0 - V<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>V<\/span><\/span><\/span><\/span>.<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>intake<\/mtext><\/msub><mo>=<\/mo><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{intake}} = V_0<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>intake<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><\/ul>Calculations<\/strong>:<\/p><ol><li>Pressure After Compression<\/strong>:<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><msub><mi>p<\/mi><mn>1<\/mn><\/msub><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mi>V<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mi>\u03b3<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">p = p_1 \\left( \\frac{V_0}{V_0 - V} \\right)^\\gamma<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>V<\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u03b3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"2\"><li>Temperature After Compression<\/strong>:<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>T<\/mi><mo>=<\/mo><msub><mi>T<\/mi><mn>1<\/mn><\/msub><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mi>V<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mrow><mi>\u03b3<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">T = T_1 \\left( \\frac{V_0}{V_0 - V} \\right)^{\\gamma - 1}<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>=<\/span><\/span><\/span><\/span>T<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>V<\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u03b3<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(2.2) Calculating Temperature and Pressure After Combustion and Average Power<\/strong><\/p>Combustion Process<\/strong>:<\/p><ul><li>Occurs at constant volume <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><mo>=<\/mo><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V = V_{\\text{min}}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Fuel combusts completely with the oxygen in the mixture.<\/li><li>Energy released per mole of diesel: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta U<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>U<\/span><\/span><\/span><\/span>.<\/li><\/ul>Total Moles Before Combustion<\/strong>:<\/p>Let <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>n<\/mi><mtext>total<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">n_{\\text{total}}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> be the total number of moles before combustion.<\/p><ul><li>Moles of air (mostly nitrogen and oxygen): <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>n<\/mi><mtext>air<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">n_{\\text{air}}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span>air<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>Moles of diesel fuel: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>n<\/mi><mtext>fuel<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">n_{\\text{fuel}}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span>fuel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><\/ul>Assuming ideal gas behavior, the total number of moles is proportional to the volume and inversely proportional to temperature:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>n<\/mi><mtext>total<\/mtext><\/msub><mo>=<\/mo><mfrac><mrow><msub><mi>p<\/mi><mtext>comp<\/mtext><\/msub><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mrow><mrow><mi>R<\/mi><msub><mi>T<\/mi><mtext>comp<\/mtext><\/msub><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">n_{\\text{total}} = \\frac{p_{\\text{comp}} V_{\\text{min}}}{R T_{\\text{comp}}}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>R<\/span>T<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Temperature After Combustion<\/strong>:<\/p>The heat released <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Q<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span><\/span><\/span><\/span> increases the internal energy of the gas:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>Q<\/mi><mo>=<\/mo><msub><mi>n<\/mi><mtext>fuel<\/mtext><\/msub><mi mathvariant=\"normal\">\u0394<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Q = n_{\\text{fuel}} \\Delta U<\/annotation><\/semantics><\/math><\/span><\/span>Q<\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span><\/span>fuel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span>U<\/span><\/span><\/span><\/span><\/span>For an isochoric process, the change in internal energy <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>U<\/mi><mtext>total<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta U_{\\text{total}}<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>U<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>U<\/mi><mtext>total<\/mtext><\/msub><mo>=<\/mo><msub><mi>n<\/mi><mtext>total<\/mtext><\/msub><msub><mi>C<\/mi><mi>V<\/mi><\/msub><mi mathvariant=\"normal\">\u0394<\/mi><mi>T<\/mi><mo>=<\/mo><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta U_{\\text{total}} = n_{\\text{total}} C_V \\Delta T = Q<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>U<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>C<\/span><\/span>V<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span>T<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span>Where:<\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>C<\/mi><mi>V<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">C_V<\/annotation><\/semantics><\/math><\/span><\/span>C<\/span><\/span>V<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the molar heat capacity at constant volume.<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><mi>T<\/mi><mo>=<\/mo><msub><mi>T<\/mi><mtext>combustion<\/mtext><\/msub><mo>\u2212<\/mo><msub><mi>T<\/mi><mtext>comp<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta T = T_{\\text{combustion}} - T_{\\text{comp}}<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>T<\/span><\/span>=<\/span><\/span><\/span><\/span>T<\/span><\/span>combustion<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>T<\/span><\/span>comp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><\/ul>Assuming Diatomic Gas with Modified Heat Capacity<\/strong>:<\/p>Since combustion products include triatomic gases (e.g., <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mtext>CO<\/mtext><mn>2<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\text{CO}_2<\/annotation><\/semantics><\/math><\/span><\/span>CO<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mtext>H<\/mtext><mn>2<\/mn><\/msub><mtext>O<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">\\text{H}_2\\text{O}<\/annotation><\/semantics><\/math><\/span><\/span>H<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>O<\/span><\/span><\/span><\/span><\/span>), we need to account for different heat capacities.<\/p>Let\u2019s denote:<\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>C<\/mi><mrow><mi>V<\/mi><mo separator=\"true\">,<\/mo><mtext>before<\/mtext><\/mrow><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">C_{V,\\text{before}}<\/annotation><\/semantics><\/math><\/span><\/span>C<\/span><\/span>V<\/span>,<\/span>before<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>: Heat capacity before combustion.<\/li><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>C<\/mi><mrow><mi>V<\/mi><mo separator=\"true\">,<\/mo><mtext>after<\/mtext><\/mrow><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">C_{V,\\text{after}}<\/annotation><\/semantics><\/math><\/span><\/span>C<\/span><\/span>V<\/span>,<\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>: Heat capacity after combustion.<\/li><\/ul>For simplification, we can approximate <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>C<\/mi><mrow><mi>V<\/mi><mo separator=\"true\">,<\/mo><mtext>before<\/mtext><\/mrow><\/msub><mo>\u2248<\/mo><mfrac><mn>5<\/mn><mn>2<\/mn><\/mfrac><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">C_{V,\\text{before}} \\approx \\frac{5}{2} R<\/annotation><\/semantics><\/math><\/span><\/span>C<\/span><\/span>V<\/span>,<\/span>before<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span><\/span><\/span> (diatomic gases), and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>C<\/mi><mrow><mi>V<\/mi><mo separator=\"true\">,<\/mo><mtext>after<\/mtext><\/mrow><\/msub><mo>\u2248<\/mo><mfrac><mn>7<\/mn><mn>2<\/mn><\/mfrac><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">C_{V,\\text{after}} \\approx \\frac{7}{2} R<\/annotation><\/semantics><\/math><\/span><\/span>C<\/span><\/span>V<\/span>,<\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>7<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span><\/span><\/span> (due to presence of polyatomic gases).<\/p>Calculating Temperature Increase<\/strong>:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><mi>U<\/mi><mo>=<\/mo><msub><mi>n<\/mi><mtext>total<\/mtext><\/msub><msub><mi>C<\/mi><mrow><mi>V<\/mi><mo separator=\"true\">,<\/mo><mtext>avg<\/mtext><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>T<\/mi><mtext>after<\/mtext><\/msub><mo>\u2212<\/mo><msub><mi>T<\/mi><mtext>before<\/mtext><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta U = n_{\\text{total}} C_{V,\\text{avg}} (T_{\\text{after}} - T_{\\text{before}})<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>U<\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>C<\/span><\/span>V<\/span>,<\/span>avg<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>T<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>T<\/span><\/span>before<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>Where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>C<\/mi><mrow><mi>V<\/mi><mo separator=\"true\">,<\/mo><mtext>avg<\/mtext><\/mrow><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">C_{V,\\text{avg}}<\/annotation><\/semantics><\/math><\/span><\/span>C<\/span><\/span>V<\/span>,<\/span>avg<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is an average heat capacity.<\/p>For an approximate calculation:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>T<\/mi><mtext>after<\/mtext><\/msub><mo>=<\/mo><msub><mi>T<\/mi><mtext>before<\/mtext><\/msub><mo>+<\/mo><mfrac><mrow><msub><mi>n<\/mi><mtext>fuel<\/mtext><\/msub><mi mathvariant=\"normal\">\u0394<\/mi><mi>U<\/mi><\/mrow><mrow><msub><mi>n<\/mi><mtext>total<\/mtext><\/msub><msub><mi>C<\/mi><mrow><mi>V<\/mi><mo separator=\"true\">,<\/mo><mtext>avg<\/mtext><\/mrow><\/msub><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">T_{\\text{after}} = T_{\\text{before}} + \\frac{n_{\\text{fuel}} \\Delta U}{n_{\\text{total}} C_{V,\\text{avg}}}<\/annotation><\/semantics><\/math><\/span><\/span>T<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>T<\/span><\/span>before<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span>total<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>C<\/span><\/span>V<\/span>,<\/span>avg<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span>fuel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span>U<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Pressure After Combustion<\/strong>:<\/p>Since volume remains constant during combustion:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>p<\/mi><mtext>after<\/mtext><\/msub><mo>=<\/mo><mfrac><mrow><msub><mi>n<\/mi><mtext>after<\/mtext><\/msub><mi>R<\/mi><msub><mi>T<\/mi><mtext>after<\/mtext><\/msub><\/mrow><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">p_{\\text{after}} = \\frac{n_{\\text{after}} R T_{\\text{after}}}{V_{\\text{min}}}<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span>T<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>n<\/mi><mtext>after<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">n_{\\text{after}}<\/annotation><\/semantics><\/math><\/span><\/span>n<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> accounts for the change in moles due to combustion (number of moles may change due to the reaction stoichiometry).<\/p>Expansion Stroke (Adiabatic Expansion)<\/strong>:<\/p>After combustion, the gas expands adiabatically from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_{\\text{min}}<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> back to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">V_0<\/annotation><\/semantics><\/math><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/p>Using the adiabatic condition:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>p<\/mi><mtext>after<\/mtext><\/msub><msubsup><mi>V<\/mi><mtext>min<\/mtext><mi>\u03b3<\/mi><\/msubsup><mo>=<\/mo><msub><mi>p<\/mi><mtext>exp<\/mtext><\/msub><msup><mi>V<\/mi><mi>\u03b3<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">p_{\\text{after}} V_{\\text{min}}^\\gamma = p_{\\text{exp}} V^\\gamma<\/annotation><\/semantics><\/math><\/span><\/span>p<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>exp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>\u03b3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Work Done During Expansion<\/strong>:<\/p>The work done by the gas during the expansion stroke is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>W<\/mi><mtext>exp<\/mtext><\/msub><mo>=<\/mo><msubsup><mo>\u222b<\/mo><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/msubsup><mi>p<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>V<\/mi><mo>=<\/mo><mfrac><mrow><msub><mi>p<\/mi><mtext>after<\/mtext><\/msub><msubsup><mi>V<\/mi><mtext>min<\/mtext><mi>\u03b3<\/mi><\/msubsup><\/mrow><mrow><mi>\u03b3<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mrow><mo fence=\"true\">(<\/mo><msubsup><mi>V<\/mi><mn>0<\/mn><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b3<\/mi><\/mrow><\/msubsup><mo>\u2212<\/mo><msubsup><mi>V<\/mi><mtext>min<\/mtext><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b3<\/mi><\/mrow><\/msubsup><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{exp}} = \\int_{V_{\\text{min}}}^{V_0} p \\, dV = \\frac{p_{\\text{after}} V_{\\text{min}}^\\gamma}{\\gamma - 1} \\left( V_0^{1 - \\gamma} - V_{\\text{min}}^{1 - \\gamma} \\right)<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>exp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u222b<\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>d<\/span>V<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>1<\/span>\u2212<\/span>\u03b3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span>\u2212<\/span>\u03b3<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Simplifying:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>W<\/mi><mtext>exp<\/mtext><\/msub><mo>=<\/mo><mfrac><mrow><msub><mi>p<\/mi><mtext>after<\/mtext><\/msub><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mrow><mrow><mi>\u03b3<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mrow><mo fence=\"true\">[<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><msub><mi>V<\/mi><mn>0<\/mn><\/msub><msub><mi>V<\/mi><mtext>min<\/mtext><\/msub><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b3<\/mi><\/mrow><\/msup><mo>\u2212<\/mo><mn>1<\/mn><mo fence=\"true\">]<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{exp}} = \\frac{p_{\\text{after}} V_{\\text{min}}}{\\gamma - 1} \\left[ \\left( \\frac{V_0}{V_{\\text{min}}} \\right)^{1 - \\gamma} - 1 \\right]<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>exp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>after<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span><\/span>(<\/span><\/span><\/span><\/span>V<\/span><\/span>min<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>1<\/span>\u2212<\/span>\u03b3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>1<\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Average Power Output<\/strong>:<\/p>Since the engine operates in cycles, the average power <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span><\/span><\/span> is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>P<\/mi><mo>=<\/mo><mfrac><msub><mi>W<\/mi><mtext>exp<\/mtext><\/msub><mi>\u03c4<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">P = \\frac{W_{\\text{exp}}}{\\tau}<\/annotation><\/semantics><\/math><\/span><\/span>P<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>W<\/span><\/span>exp<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c4<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\tau<\/annotation><\/semantics><\/math><\/span><\/span>\u03c4<\/span><\/span><\/span><\/span> is the time per cycle, related to the engine speed.<\/p>(2.3) Relationship Between Flywheel Angular Velocity <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> and Diesel Volume Fraction <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span><\/strong><\/p>Given:<\/p><ul><li>Resistive torque on front wheels: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c4<\/mi><mo>=<\/mo><mi>\u03b2<\/mi><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\tau = \\beta \\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c4<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b2<\/span>\u03c9<\/span><\/span><\/span><\/span>.<\/li><li>Flywheel provides torque to overcome this resistance.<\/li><li>Transmission ratio: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha<\/annotation><\/semantics><\/math><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span> (wheel angular velocity <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c9<\/mi><mtext>wheel<\/mtext><\/msub><mo>=<\/mo><mi>\u03b1<\/mi><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_{\\text{wheel}} = \\alpha \\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>wheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b1<\/span>\u03c9<\/span><\/span><\/span><\/span>).<\/li><\/ul>Assumptions<\/strong>:<\/p><ul><li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> is small; higher-order terms in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> can be neglected.<\/li><li>Exponential terms involving <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> can be approximated by expanding and neglecting <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> in exponents.<\/li><\/ul>Using the Hint and Defining Dimensionless Parameter <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math><\/span><\/span>A<\/span><\/span><\/span><\/span><\/strong>:<\/p>Given the parameter <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math><\/span><\/span>A<\/span><\/span><\/span><\/span> defined as:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mrow><mo fence=\"true\">[<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mi>V<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mrow><mn>2<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>5<\/mn><\/mrow><\/msup><mo>\u2212<\/mo><mn>1<\/mn><mo fence=\"true\">]<\/mo><\/mrow><mrow><mo fence=\"true\">[<\/mo><mfrac><mrow><mi mathvariant=\"normal\">\u0394<\/mi><mi>U<\/mi><\/mrow><mrow><mi>R<\/mi><msub><mi>T<\/mi><mn>1<\/mn><\/msub><\/mrow><\/mfrac><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mi>V<\/mi><\/mrow><msub><mi>V<\/mi><mn>0<\/mn><\/msub><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mrow><mn>2<\/mn><mi mathvariant=\"normal\">\/<\/mi><mn>5<\/mn><\/mrow><\/msup><mo>\u2212<\/mo><mn>3<\/mn><mo>+<\/mo><mfrac><mn>7<\/mn><mn>8<\/mn><\/mfrac><mi>m<\/mi><mo fence=\"true\">]<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">A = \\left[ \\left( \\frac{V_0}{V_0 - V} \\right)^{2\/5} - 1 \\right] \\left[ \\frac{\\Delta U}{R T_1} \\left( \\frac{V_0 - V}{V_0} \\right)^{2\/5} - 3 + \\frac{7}{8} m \\right]<\/annotation><\/semantics><\/math><\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span>(<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>V<\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2\/5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>1<\/span>]<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span><\/span>R<\/span>T<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span>U<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>V<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>V<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2\/5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>3<\/span><\/span>+<\/span><\/span><\/span><\/span>8<\/span><\/span><\/span><\/span><\/span><\/span><\/span>7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Deriving the Relationship<\/strong>:<\/p><ol><li>Net Work per Cycle<\/strong>:<\/li><\/ol>The net work done per cycle is equal to the work during expansion minus the work during compression (since intake and exhaust strokes do no work).<\/p><ol start=\"2\"><li>Relating Work to Torque<\/strong>:<\/li><\/ol>The net work per cycle is also equal to the torque times the angular displacement per cycle:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>W<\/mi><mtext>net<\/mtext><\/msub><mo>=<\/mo><msub><mi>\u03c4<\/mi><mtext>drive<\/mtext><\/msub><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>\u03b8<\/mi><mtext>flywheel<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{net}} = \\tau_{\\text{drive}} \\Delta \\theta_{\\text{flywheel}}<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>drive<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394<\/span>\u03b8<\/span><\/span>flywheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"3\"><li>Steady-State Condition<\/strong>:<\/li><\/ol>In steady operation, the net work produced by the engine per cycle balances the resistive work:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>W<\/mi><mtext>net<\/mtext><\/msub><mo>=<\/mo><mi>\u03c4<\/mi><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>\u03b8<\/mi><mtext>wheel<\/mtext><\/msub><mo>=<\/mo><mi>\u03b2<\/mi><mi>\u03c9<\/mi><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>\u03b8<\/mi><mtext>wheel<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{net}} = \\tau \\Delta \\theta_{\\text{wheel}} = \\beta \\omega \\Delta \\theta_{\\text{wheel}}<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c4<\/span>\u0394<\/span>\u03b8<\/span><\/span>wheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b2<\/span>\u03c9<\/span>\u0394<\/span>\u03b8<\/span><\/span>wheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>But <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>\u03b8<\/mi><mtext>wheel<\/mtext><\/msub><mo>=<\/mo><mi>\u03b1<\/mi><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>\u03b8<\/mi><mtext>flywheel<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\Delta \\theta_{\\text{wheel}} = \\alpha \\Delta \\theta_{\\text{flywheel}}<\/annotation><\/semantics><\/math><\/span><\/span>\u0394<\/span>\u03b8<\/span><\/span>wheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b1<\/span>\u0394<\/span>\u03b8<\/span><\/span>flywheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, so:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>W<\/mi><mtext>net<\/mtext><\/msub><mo>=<\/mo><mi>\u03b2<\/mi><mi>\u03c9<\/mi><mi>\u03b1<\/mi><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>\u03b8<\/mi><mtext>flywheel<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{net}} = \\beta \\omega \\alpha \\Delta \\theta_{\\text{flywheel}}<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b2<\/span>\u03c9<\/span>\u03b1<\/span>\u0394<\/span>\u03b8<\/span><\/span>flywheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"4\"><li>Relating Work to Fuel Fraction <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span><\/strong>:<\/li><\/ol>The net work <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>W<\/mi><mtext>net<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{net}}<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> depends on the amount of fuel combusted, which is proportional to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span>. Therefore, we can express <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>W<\/mi><mtext>net<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{net}}<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> in terms of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>A<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A<\/annotation><\/semantics><\/math><\/span><\/span>A<\/span><\/span><\/span><\/span>.<\/p><ol start=\"5\"><li>Simplifying and Solving for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span><\/strong>:<\/li><\/ol>By equating the expressions and solving for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>, we can derive the relationship between <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span>:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c9<\/mi><mo>=<\/mo><mfrac><msub><mi>W<\/mi><mtext>net<\/mtext><\/msub><mrow><mi>\u03b2<\/mi><mi>\u03b1<\/mi><mi mathvariant=\"normal\">\u0394<\/mi><msub><mi>\u03b8<\/mi><mtext>flywheel<\/mtext><\/msub><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\omega = \\frac{W_{\\text{net}}}{\\beta \\alpha \\Delta \\theta_{\\text{flywheel}}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span>\u03b1<\/span>\u0394<\/span>\u03b8<\/span><\/span>flywheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>W<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>W<\/mi><mtext>net<\/mtext><\/msub><mo>\u221d<\/mo><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">W_{\\text{net}} \\propto a<\/annotation><\/semantics><\/math><\/span><\/span>W<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u221d<\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span>, we find:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c9<\/mi><mo>\u221d<\/mo><mfrac><mi>a<\/mi><mi>\u03b2<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\omega \\propto \\frac{a}{\\beta}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>\u221d<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Conclusion<\/strong>:<\/p>Under the given assumptions and simplifications, the flywheel's angular velocity <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> is directly proportional to the diesel volume fraction <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> and inversely proportional to the resistive coefficient <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b2<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\beta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>.<\/p><hr>(3) Vehicle Accelerating Uphill: Finding <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t)<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/strong><\/p>Given<\/strong>:<\/p><ul><li>Slope angle: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math><\/span><\/span>\u03b8<\/span><\/span><\/span><\/span>.<\/li><li>Gravitational acceleration: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>g<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">g<\/annotation><\/semantics><\/math><\/span><\/span>g<\/span><\/span><\/span><\/span>.<\/li><li>Initial flywheel angular velocity: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo><<\/mo><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 < \\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> (from previous analysis).<\/li><li>Vehicle total mass: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>M<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">M<\/annotation><\/semantics><\/math><\/span><\/span>M<\/span><\/span><\/span><\/span>.<\/li><li>Flywheel angular velocity increases over time.<\/li><\/ul>Assumptions<\/strong>:<\/p><ul><li>The mass of tires, flywheel, and transmission components is negligible compared to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>M<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">M<\/annotation><\/semantics><\/math><\/span><\/span>M<\/span><\/span><\/span><\/span>.<\/li><li>The resistive torque <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c4<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\tau<\/annotation><\/semantics><\/math><\/span><\/span>\u03c4<\/span><\/span><\/span><\/span> still applies.<\/li><li>No energy loss in transmission.<\/li><li>The power output from the engine accelerates both the flywheel and the vehicle uphill.<\/li><\/ul>Analysis<\/strong>:<\/p><ol><li>Forces Acting on the Vehicle<\/strong>:<\/li><\/ol><ul><li>Gravitational component opposing motion: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>M<\/mi><mi>g<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">M g \\sin \\theta<\/annotation><\/semantics><\/math><\/span><\/span>M<\/span>g<\/span><\/span>sin<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span>.<\/li><li>Resistive torque converted to force at wheels: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>F<\/mi><mtext>resist<\/mtext><\/msub><mo>=<\/mo><mfrac><mi>\u03c4<\/mi><mi>r<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">F_{\\text{resist}} = \\frac{\\tau}{r}<\/annotation><\/semantics><\/math><\/span><\/span>F<\/span><\/span>resist<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><\/ul><ol start=\"2\"><li>Net Force on the Vehicle<\/strong>:<\/li><\/ol>The net force accelerating the vehicle uphill is:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>F<\/mi><mtext>net<\/mtext><\/msub><mo>=<\/mo><msub><mi>F<\/mi><mtext>drive<\/mtext><\/msub><mo>\u2212<\/mo><msub><mi>F<\/mi><mtext>resist<\/mtext><\/msub><mo>\u2212<\/mo><mi>M<\/mi><mi>g<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">F_{\\text{net}} = F_{\\text{drive}} - F_{\\text{resist}} - M g \\sin \\theta<\/annotation><\/semantics><\/math><\/span><\/span>F<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>F<\/span><\/span>drive<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>F<\/span><\/span>resist<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>M<\/span>g<\/span><\/span>sin<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>But since the vehicle's acceleration <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>a<\/mi><mtext>vehicle<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">a_{\\text{vehicle}}<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span>vehicle<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is related to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>F<\/mi><mtext>net<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">F_{\\text{net}}<\/annotation><\/semantics><\/math><\/span><\/span>F<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>M<\/mi><msub><mi>a<\/mi><mtext>vehicle<\/mtext><\/msub><mo>=<\/mo><msub><mi>F<\/mi><mtext>drive<\/mtext><\/msub><mo>\u2212<\/mo><msub><mi>F<\/mi><mtext>resist<\/mtext><\/msub><mo>\u2212<\/mo><mi>M<\/mi><mi>g<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">M a_{\\text{vehicle}} = F_{\\text{drive}} - F_{\\text{resist}} - M g \\sin \\theta<\/annotation><\/semantics><\/math><\/span><\/span>M<\/span>a<\/span><\/span>vehicle<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>F<\/span><\/span>drive<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>F<\/span><\/span>resist<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>M<\/span>g<\/span><\/span>sin<\/span><\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><ol start=\"3\"><li>Relating Flywheel Angular Acceleration to Vehicle Acceleration<\/strong>:<\/li><\/ol><ul><li>Angular acceleration of the flywheel: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03b1<\/mi><mtext>flywheel<\/mtext><\/msub><mo>=<\/mo><mfrac><mrow><mi>d<\/mi><mi>\u03c9<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha_{\\text{flywheel}} = \\frac{d\\omega}{dt}<\/annotation><\/semantics><\/math><\/span><\/span>\u03b1<\/span><\/span>flywheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li><li>The linear acceleration of the vehicle is related to the angular acceleration through the transmission ratio and wheel radius:<\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>a<\/mi><mtext>vehicle<\/mtext><\/msub><mo>=<\/mo><mi>r<\/mi><mi>\u03b1<\/mi><msub><mi>\u03b1<\/mi><mtext>flywheel<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">a_{\\text{vehicle}} = r \\alpha \\alpha_{\\text{flywheel}}<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span>vehicle<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>r<\/span>\u03b1<\/span>\u03b1<\/span><\/span>flywheel<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"4\"><li>Energy Considerations<\/strong>:<\/li><\/ol>The power output from the engine is used to:<\/p><ul><li>Increase the kinetic energy of the flywheel.<\/li><li>Do work against gravity.<\/li><li>Overcome resistive torque.<\/li><\/ul>Differential Equation for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t)<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/strong>:<\/p><ul><li>The net torque applied to the flywheel is:<\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c4<\/mi><mtext>net<\/mtext><\/msub><mo>=<\/mo><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mo>\u2212<\/mo><msub><mi>\u03c4<\/mi><mtext>resist<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\tau_{\\text{net}} = \\tau_{\\text{engine}} - \\tau_{\\text{resist}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c4<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>resist<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ul><li>The moment of inertia of the flywheel is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span><\/span><\/span>, so:<\/li><\/ul><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>I<\/mi><mfrac><mrow><mi>d<\/mi><mi>\u03c9<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msub><mi>\u03c4<\/mi><mtext>net<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">I \\frac{d\\omega}{dt} = \\tau_{\\text{net}}<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>net<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ul><li>The torque provided by the engine is related to the net work per cycle and engine speed.<\/li><\/ul><ol start=\"5\"><li>Solving for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t)<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/strong>:<\/li><\/ol>Formulating the differential equation and integrating, we find:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>I<\/mi><mfrac><mrow><mi>d<\/mi><mi>\u03c9<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mo>\u2212<\/mo><mi>\u03b2<\/mi><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">I \\frac{d\\omega}{dt} = \\tau_{\\text{engine}} - \\beta \\omega<\/annotation><\/semantics><\/math><\/span><\/span>I<\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b2<\/span>\u03c9<\/span><\/span><\/span><\/span><\/span>But <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\tau_{\\text{engine}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> may depend on <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span>, from the previous analysis.<\/p>Assuming <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\tau_{\\text{engine}}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is approximately constant (since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> is constant and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span> is increasing from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>), we can solve:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>\u03c9<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mo>\u2212<\/mo><mi>\u03b2<\/mi><mi>\u03c9<\/mi><\/mrow><mi>I<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d\\omega}{dt} = \\frac{\\tau_{\\text{engine}} - \\beta \\omega}{I}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>\u03b2<\/span>\u03c9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>This is a first-order linear differential equation in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span><\/span><\/span>.<\/p>Solution<\/strong>:<\/p><ol><li>Integrate the Differential Equation<\/strong>:<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>\u03c9<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mfrac><mi>\u03b2<\/mi><mi>I<\/mi><\/mfrac><mi>\u03c9<\/mi><mo>=<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>I<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d\\omega}{dt} + \\frac{\\beta}{I} \\omega = \\frac{\\tau_{\\text{engine}}}{I}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"2\"><li>Use Integrating Factor<\/strong>:<\/li><\/ol>Let <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03bc<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>e<\/mi><mrow><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\mu(t) = e^{(\\beta\/I)t}<\/annotation><\/semantics><\/math><\/span><\/span>\u03bc<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>e<\/span><\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/p>Multiply both sides:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03bc<\/mi><mfrac><mrow><mi>d<\/mi><mi>\u03c9<\/mi><\/mrow><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mi>\u03bc<\/mi><mfrac><mi>\u03b2<\/mi><mi>I<\/mi><\/mfrac><mi>\u03c9<\/mi><mo>=<\/mo><mi>\u03bc<\/mi><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>I<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\mu \\frac{d\\omega}{dt} + \\mu \\frac{\\beta}{I} \\omega = \\mu \\frac{\\tau_{\\text{engine}}}{I}<\/annotation><\/semantics><\/math><\/span><\/span>\u03bc<\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>\u03c9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u03bc<\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03bc<\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Left side simplifies to:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>\u03bc<\/mi><mi>\u03c9<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>\u03bc<\/mi><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>I<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d}{dt} (\\mu \\omega) = \\mu \\frac{\\tau_{\\text{engine}}}{I}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03bc<\/span>\u03c9<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03bc<\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"3\"><li>Integrate Both Sides<\/strong>:<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>\u222b<\/mo><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>t<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>\u03bc<\/mi><mi>\u03c9<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>t<\/mi><mo>=<\/mo><mo>\u222b<\/mo><mi>\u03bc<\/mi><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>I<\/mi><\/mfrac><mi>d<\/mi><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\int \\frac{d}{dt} (\\mu \\omega) dt = \\int \\mu \\frac{\\tau_{\\text{engine}}}{I} dt<\/annotation><\/semantics><\/math><\/span><\/span>\u222b<\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03bc<\/span>\u03c9<\/span>)<\/span>d<\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>\u222b<\/span><\/span>\u03bc<\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span>\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03bc<\/mi><mi>\u03c9<\/mi><mo>=<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>I<\/mi><\/mfrac><mo>\u222b<\/mo><mi>\u03bc<\/mi><mi>d<\/mi><mi>t<\/mi><mo>+<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mu \\omega = \\frac{\\tau_{\\text{engine}}}{I} \\int \\mu dt + C<\/annotation><\/semantics><\/math><\/span><\/span>\u03bc<\/span>\u03c9<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>\u03bc<\/span>d<\/span>t<\/span><\/span>+<\/span><\/span><\/span><\/span>C<\/span><\/span><\/span><\/span><\/span><ol start=\"4\"><li>Solve for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t)<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/strong>:<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><mrow><mo fence=\"true\">(<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>I<\/mi><\/mfrac><mo>\u222b<\/mo><msup><mi>e<\/mi><mrow><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><mi>d<\/mi><mi>t<\/mi><mo>+<\/mo><mi>C<\/mi><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t) = e^{-(\\beta\/I)t} \\left( \\frac{\\tau_{\\text{engine}}}{I} \\int e^{(\\beta\/I)t} dt + C \\right)<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>e<\/span><\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span>+<\/span><\/span>C<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Integrate the right side:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>\u222b<\/mo><msup><mi>e<\/mi><mrow><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><mi>d<\/mi><mi>t<\/mi><mo>=<\/mo><mfrac><mi>I<\/mi><mi>\u03b2<\/mi><\/mfrac><msup><mi>e<\/mi><mrow><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\int e^{(\\beta\/I)t} dt = \\frac{I}{\\beta} e^{(\\beta\/I)t}<\/annotation><\/semantics><\/math><\/span><\/span>\u222b<\/span><\/span>e<\/span><\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>So,<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><mrow><mo fence=\"true\">(<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>\u03b2<\/mi><\/mfrac><msup><mi>e<\/mi><mrow><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><mo>+<\/mo><mi>C<\/mi><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t) = e^{-(\\beta\/I)t} \\left( \\frac{\\tau_{\\text{engine}}}{\\beta} e^{(\\beta\/I)t} + C \\right)<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>e<\/span><\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>C<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Simplify:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>\u03b2<\/mi><\/mfrac><mo>+<\/mo><mi>C<\/mi><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t) = \\frac{\\tau_{\\text{engine}}}{\\beta} + C e^{-(\\beta\/I)t}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>C<\/span>e<\/span><\/span>\u2212<\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"5\"><li>Apply Initial Condition<\/strong>:<\/li><\/ol>At <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">t = 0<\/annotation><\/semantics><\/math><\/span><\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(0) = \\omega_0<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>0<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>=<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>\u03b2<\/mi><\/mfrac><mo>+<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0 = \\frac{\\tau_{\\text{engine}}}{\\beta} + C<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>C<\/span><\/span><\/span><\/span><\/span>So,<\/p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>C<\/mi><mo>=<\/mo><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>\u03b2<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">C = \\omega_0 - \\frac{\\tau_{\\text{engine}}}{\\beta}<\/annotation><\/semantics><\/math><\/span><\/span>C<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><ol start=\"6\"><li>Final Expression for <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t)<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/strong>:<\/li><\/ol><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>\u03b2<\/mi><\/mfrac><mo>+<\/mo><mrow><mo fence=\"true\">(<\/mo><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>\u03b2<\/mi><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><msup><mi>e<\/mi><mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03b2<\/mi><mi mathvariant=\"normal\">\/<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><\/mrow><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t) = \\frac{\\tau_{\\text{engine}}}{\\beta} + \\left( \\omega_0 - \\frac{\\tau_{\\text{engine}}}{\\beta} \\right) e^{-(\\beta\/I)t}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>(<\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>e<\/span><\/span>\u2212<\/span>(<\/span>\u03b2<\/span>\/<\/span>I<\/span>)<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Interpretation<\/strong>:<\/p><ul><li>As <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><mo>\u2192<\/mo><mi mathvariant=\"normal\">\u221e<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">t \\to \\infty<\/annotation><\/semantics><\/math><\/span><\/span>t<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>\u221e<\/span><\/span><\/span><\/span>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2192<\/mo><mfrac><msub><mi>\u03c4<\/mi><mtext>engine<\/mtext><\/msub><mi>\u03b2<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\omega(t) \\to \\frac{\\tau_{\\text{engine}}}{\\beta}<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span>(<\/span>t<\/span>)<\/span><\/span>\u2192<\/span><\/span><\/span><\/span><\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>engine<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, which matches the steady-state angular velocity found earlier.<\/li><li>The angular velocity increases exponentially from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03c9<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\omega_0<\/annotation><\/semantics><\/math><\/span><\/span>\u03c9<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> towards the steady-state value.<\/li><\/ul><hr>Conclusion<\/strong>:<\/p><ul><li>(1)<\/strong> We described the pressure and temperature changes during each stroke of the diesel engine, specifying the processes and trends.<\/li><li>(2)<\/strong> Under steady operation on a flat road, we calculated the gas temperature and pressure before and after combustion and derived the average power output. We also established the relationship between flywheel angular velocity and diesel volume fraction.<\/li><li>(3)<\/strong> For the vehicle accelerating uphill, we derived the time-dependent angular velocity of the flywheel, showing how it approaches the steady-state value over time.<\/li><\/ul>Note<\/strong>:<\/p><ul><li>In practical applications, additional factors such as friction, air resistance, and engine efficiency would need to be considered for more accurate modeling.<\/li><li>The simplifications made (e.g., treating <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><\/span>a<\/span><\/span><\/span><\/span> as small, neglecting higher-order terms) are for ease of calculation and may not capture all real-world complexities.<\/li><\/ul><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/div><\/article><\/div><\/div><audio class=\"fixed bottom-0 left-0 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