{"id":36786,"date":"2025-05-15T08:10:42","date_gmt":"2025-05-15T00:10:42","guid":{"rendered":"http:\/\/www.qiusir.com\/?p=36786"},"modified":"2025-05-15T08:10:42","modified_gmt":"2025-05-15T00:10:42","slug":"%e6%97%a0%e9%99%90%e9%95%bf%e6%91%86%e9%95%bf%e5%8d%95%e6%91%86%e7%9a%84%e5%91%a8%e6%9c%9f%c2%b7%c2%b7%c2%b7","status":"publish","type":"post","link":"https:\/\/www.qiusir.com\/?p=36786","title":{"rendered":"\u65e0\u9650\u957f\u6446\u957f\u5355\u6446\u7684\u5468\u671f\u00b7\u00b7\u00b7"},"content":{"rendered":"

JQX\/\u8fdb\u53d6\u82af \u5e2d\u660e\u7eb3\u7b2c7\u671f\uff082025.5.8\uff09
\n\"\"
\n\u65e0\u9650\u5e38\u6446\u957f\u5355\u6446\u5468\u671f\u95ee\u9898<\/p>\n

\u96f6\u3001\u5173\u4e8e\\( T = 2\\pi\\sqrt{\\frac{R}{g}} \\)
\n\u8fd9\u4e2a\u516c\u5f0f\u662fQiusir\u5728JQX\u7b2c\u56db\u671f\u4e0e\u65b0\u7586\u90e8\u5b66\u751f\u8fdb\u884c\u5934\u8111\u98ce\u66b4\u65f6\u7528\u5230\u7684\u516c\u5f0f\u3002
\n\u6839\u636e\u5929\u4f53\u8fd0\u52a8\u4e2d\u4e07\u6709\u5f15\u529b\u5145\u5f53\u5411\u5fc3\u529b\uff0c\u53ef\u4ee5\u63a8\u5bfc\u8fd1\u5730\u536b\u661f\uff08\u8d34\u5730\u536b\u661f\uff09\u7684\u5468\u671f\u516c\u5f0f\uff0c\u4e3a\\( T = 2\\pi\\sqrt{\\frac{R}{g}} \\)\uff0c\u8fd9\u4e2a\u516c\u5f0f\u5f88\u5bb9\u6613\u8ba9\u4eba\u60f3\u5230\u6446\u957f\u4e3aR\u5355\u6446\u7684\u5468\u671f\uff0c\u53ef\u4e8b\u5b9e\u4e0a\u662f\u8fd9\u6837\u5417\uff1f<\/p>\n

\u4e00\u3001\u5173\u4e8e\\( T = 2\\pi\\sqrt{\\frac{r}{g}} \\)
\n\u540e\u6765\u8ba8\u8bba\u7684\u8fc7\u7a0b\u4e2d\uff0c\u5b66\u751f\u63d0\u5230\u4ed6\u60f3\u8868\u8fbe\u7684\u516c\u5f0f\u662f\u8fd9\u4e2a\u3002\u8fd9\u4e2a\u516c\u5f0f\u53ef\u4ee5\u770b\u505a\u4e00\u4e2a\u5c0f\u7403\u5728\u5706\u8f68\u9053\u505a\u5c0f\u632f\u5e45\u5f80\u590d\u8fd0\u52a8\u7684\u5468\u671f\uff08\u7b49\u6548\u5355\u6446\uff09\u3002<\/p>\n

\u4e8c\u3001\u4ece\u80fd\u91cf\u89d2\u5ea6\u63a8\u5bfc\u7b80\u8c10\u632f\u52a8\u7684\u52a8\u529b\u5b66\u65b9\u7a0b\u548c\u5355\u6446\u5468\u671f\u516c\u5f0f
\n1. \u7b80\u8c10\u8fd0\u52a8
\n\u7b80\u8c10\u8fd0\u52a8\u6ee1\u8db3\u80fd\u91cf\u5b88\u6052\u65b9\u7a0b\uff1a\\( \\frac{1}{2}mv^{2}+\\frac{1}{2}kx^{2}=E \\)\uff0c
\n\u7b80\u5316\u8fd9\u4e2a\u65b9\u7a0b\uff0c\u5373\u5f53\u52a8\u529b\u5b66\u65b9\u7a0b\u6ee1\u8db3\\( \\frac{k}{m}x^{2}+\\dot{x}^{2}=C \\)\uff08\\( C \\)\u4e3a\u5e38\u6570\uff09\u65f6\uff0c\u53ef\u5224\u5b9a\u4e3a\u7b80\u8c10\u8fd0\u52a8\u3002\u5176\u4e2d\uff0c\\( \\frac{k}{m} \\)\u5b9a\u4e49\u4e3a\u89d2\u9891\u7387\\( \\omega \\)\u7684\u5e73\u65b9\u3002
\n2. \u5355\u6446
\n\u9009\u53d6\u5355\u6446\u6446\u52a8\u7684\u6700\u4f4e\u70b9\u4f5c\u4e3a\u91cd\u529b\u52bf\u80fd\u96f6\u70b9\u3002\u5bf9\u4e8e\u6446\u89d2\u4e3a\\( \\theta \\)\u7684\u4efb\u610f\u4f4d\u7f6e\uff0c\u6839\u636e\u673a\u68b0\u80fd\u5b88\u6052\u5b9a\u5f8b\uff0c\u5355\u6446\u7cfb\u7edf\u7684\u603b\u80fd\u91cf\\( E \\)\u6ee1\u8db3\u65b9\u7a0b\\( \\frac{1}{2}mv^{2}+mg(l – l\\cos\\theta)=E \\)\uff0c\u5176\u4e2d\\( m \\)\u4e3a\u6446\u7403\u8d28\u91cf\uff0c\\( v \\)\u4e3a\u6446\u7403\u7ebf\u901f\u5ea6\uff0c\\( l \\)\u4e3a\u6446\u957f\uff0c\\( g \\)\u4e3a\u91cd\u529b\u52a0\u901f\u5ea6\u3002
\n\u6839\u636e\u5706\u5468\u8fd0\u52a8\u7ebf\u901f\u5ea6\u4e0e\u89d2\u901f\u5ea6\u5173\u7cfb\\( v = \\omega l \\)\uff0c\u4e14\u89d2\u901f\u5ea6\\( \\omega \\)\u53ef\u8868\u793a\u4e3a\u6446\u89d2\\( \\theta \\)\u5bf9\u65f6\u95f4\u7684\u4e00\u9636\u5bfc\u6570\\( \\omega=\\dot{\\theta} \\)\uff0c\u540c\u65f6\uff0c\u5229\u7528\u5c0f\u89d2\u5ea6\u8fd1\u4f3c\u6761\u4ef6\uff08\u5f53\\( \\theta\\approx0 \\)\u65f6\uff09\uff0c\\( \\cos\\theta \\)\u53ef\u5c55\u5f00\u4e3a\\( \\cos\\theta\\approx1-\\frac{\\theta^{2}}{2} \\)\u3002
\n\u5c06\u4e0a\u8ff0\u5173\u7cfb\u4ee3\u5165\u80fd\u91cf\u65b9\u7a0b\uff0c\u53ef\u5f97\uff1a\\( \\frac{1}{2}m\\cdot\\dot{\\theta}^{2}\\cdot l^{2}+mg\\cdot l\\cdot\\frac{\\theta^{2}}{2}=E \\)<\/p>\n

\u7531\u4e8e\u5728\u5c0f\u89d2\u5ea6\u4e0b\\( \\theta\\approx\\frac{x}{l} \\)\uff08\\( x \\)\u4e3a\u6446\u7403\u504f\u79bb\u5e73\u8861\u4f4d\u7f6e\u7684\u6c34\u5e73\u4f4d\u79fb\uff09\uff0c\u5c06\u5176\u4ee3\u5165\u4e0a\u5f0f\uff0c\u6574\u7406\u540e\u5f97\u5230\uff1a\\( \\dot{x}^{2}+\\frac{g}{l}x^{2}=C \\)\u3002\u5bf9\u6bd4\u7b80\u8c10\u8fd0\u52a8\u7684\u52a8\u529b\u5b66\u65b9\u7a0b\uff0c\u53ef\u5f97\u51fa\u5355\u6446\u8fd0\u52a8\u7684\u89d2\u9891\u7387\\( \\omega^{2}=\\frac{g}{l} \\)\u3002<\/p>\n

\u53ef\u5f97\u5230\u5355\u6446\u7684\u5468\u671f\u516c\u5f0f\uff1a\\( T = 2\\pi\\sqrt{\\frac{l}{g}} \\)<\/p>\n

\u4e09\u3001\u65e0\u9650\u6446\u957f\u95ee\u9898\u7684\u63d0\u51fa
\n\u6211\u4eec\u77e5\u9053\u5355\u6446\u7684\u5468\u671f\u516c\u5f0f\u662f\\( T = 2\\pi\\sqrt{\\frac{l}{g}} \\)\uff0c\u73b0\u5728\u6211\u4eec\u6765\u601d\u8003\u4e00\u4e0b\uff0c\u5f53\u6446\u957f\\( l \\)\u8d8b\u4e8e\u65e0\u7a77\u5927\uff0c\u8fd8\u6709\u5f53\u6446\u957f\\( l \\)\u7b49\u4e8e\u5730\u7403\u534a\u5f84\u7684\u65f6\u5019\uff0c\u8fd9\u4e2a\u5468\u671f\u4f1a\u662f\u591a\u5c11\u5462\uff1f
\n\u6211\u4eec\u5148\u6765\u63a8\u5bfc\u4e00\u4e0b\u5355\u6446\u5468\u671f\u516c\u5f0f\u3002
\n\u5f53\u5355\u6446\u7684\u6446\u89d2\\( \\theta \\)\u6781\u5c0f\u65f6\uff0c\u91cd\u529b\u6cbf\u5706\u5f27\u5207\u7ebf\u65b9\u5411\u7684\u5206\u529b\\( F = mg\\sin\\theta \\)\u4f5c\u4e3a\u56de\u590d\u529b\u3002<\/p>\n

\u7531\u4e8e\u6446\u89d2\\( \\theta \\)\u6781\u5c0f\uff0c\u6b64\u65f6\\( \\sin\\theta\\approx\\theta \\)\uff0c\u4e14\\( \\theta=\\frac{\\overset{\\frown}{OP}}{l}\\approx\\frac{x}{l} \\)\uff0c\u6709\u56de\u590d\u529b\\( F = – \\frac{mg}{l}x \\)\u3002\u8bbe\\( k = \\frac{mg}{l} \\)\uff0c\u5219\u56de\u590d\u529b\\( F=-kx \\)\uff0c\u8fd9\u610f\u5473\u7740\u5355\u6446\u5728\u6446\u89d2\u5f88\u5c0f\u65f6\u505a\u7b80\u8c10\u8fd0\u52a8\u3002\u628a\\( k \\)\u4ee3\u5165\u7b80\u8c10\u8fd0\u52a8\u5468\u671f\u516c\u5f0f\\( T = 2\\pi\\sqrt{\\frac{m}{k}} \\)\uff0c\u4fbf\u80fd\u63a8\u5bfc\u5f97\u51fa\u76f8\u5e94\u7ed3\u679c\u3002
\n\u90a3\u4e48\uff0c\u5f53\u6446\u957f\u65e0\u9650\u5927\u65f6\uff0c\u5468\u671f\u5e94\u8be5\u4e3a\u591a\u5c11\u5462\uff1f\u662f\u65e0\u7a77\u5927\u5417\uff1f\u6211\u4eec\u6162\u6162\u9053\u6765\uff0c\u5148\u6765\u770b\u4e00\u4e0b\u8fd9\u4e2a\u60c5\u51b5\uff1a<\/p>\n

\u56db\u3001\u8fd1\u5730\u536b\u661f\u5468\u671f
\n\u4e07\u6709\u5f15\u529b\u5145\u5f53\u5411\u5fc3\u529b\uff0c\u53ef\u4ee5\u63a8\u5f97\\( T = 2\\pi\\sqrt{\\frac{R}{g}} \\)\uff0c\u53e6\u5916\uff0c\u5982\u679c\u4f7f\u7528\u8fd1\u5730\u536b\u661f\u5468\u671f\u53ef\u4ee5\u6c42\u5bc6\u5ea6\u7684\u7ed3\u8bba\uff1a\\( \\rho = \\frac{3\\pi}{GT^{2}} \\)\uff0c\u5c06\u5bc6\u5ea6\u53cd\u5411\u5e26\u56de\uff0c\u4e5f\u53ef\u4ee5\u63a8\u5bfc\u5468\u671f\u3002<\/p>\n

\u4e94\u3001”\u5730\u4e0b\u94c1”\u5468\u671f
\n\u6211\u4eec\u5047\u8bbe\u5728\u5730\u7403\u8d64\u9053\u5bf9\u79f0\u4e24\u7ea7\u6316\u901a\u4e00\u4e2a\u96a7\u9053\uff0c\u8bbe\u5730\u7403\u534a\u5f84\u4e3a\\( R \\)\uff0c\u8d28\u91cf\u4e3a\\( M \\)\u3002\u5728\u96a7\u9053\u4e00\u7aef\u7531\u9759\u6b62\u91ca\u653e\u4e00\u4e2a\u7269\u4f53\uff0c\u8d28\u91cf\u4e3a\\( m \\)\uff0c\u4e0e\u5730\u5fc3\u7684\u8ddd\u79bb\u8bb0\u4e3a\\( x \\)\u3002\u6211\u4eec\u6765\u5206\u6790\u4e00\u4e0b\u8fd9\u4e2a\u7269\u4f53\u7684\u8fd0\u52a8\u3002<\/p>\n

\u6211\u4eec\u77e5\u9053\u7406\u60f3\u7403\u58f3\u5bf9\u5185\u90e8\u7269\u4f53\u7684\u5f15\u529b\u5408\u529b\u4e3a\u96f6\uff0c\u5728\u8fd9\u4e2a\u7269\u4f53\u5728\u96a7\u9053\u91cc\u9762\u8fd0\u52a8\u4e2d\u8ddd\u79bb\u5730\u5fc3\\( x \\)\u5904\u65f6\uff0c\u6240\u53d7\u5f15\u529b\u7b49\u6548\u4e8e\u534a\u5f84\u4e3a\\( x \\)\u7684\u90e8\u5206\u5730\u7403\u5bf9\u5176\u65bd\u52a0\u7684\u5f15\u529b\u3002
\n\u7403\u4f53\u8d28\u91cf\u516c\u5f0f\\( M = \\rho\\times\\frac{4}{3}\\pi R^{3} \\)\uff08\\( \\rho \\)\u4e3a\u5730\u7403\u5e73\u5747\u5bc6\u5ea6\uff09\uff0c\u53ef\u63a8\u5bfc\u51fa\u534a\u5f84\u4e3a\\( x \\)\u7684\u7403\u4f53\u8d28\u91cf\\( M_{x}=\\rho\\times\\frac{4}{3}\\pi x^{3} \\)\u3002\u7ed3\u5408\\( F = G\\frac{Mm}{r^2} \\)\u3001\\( mg = G\\frac{Mm}{R^{2}} \\)\u3001\\( M = \\rho\\times\\frac{4}{3}\\pi R^{3} \\)\uff0c\u53ef\u5f97\u5230\\( F = -\\frac{mg}{R}x \\)\uff0c\u56fe\u50cf\u8bf7\u89c1\u9ed1\u677f\u3002\u5e26\u5165\u7b80\u8c10\u5468\u671f\u516c\u5f0f\uff0c\u5bb9\u6613\u7b97\u5f97\\( T = 2\\pi\\sqrt{\\frac{R}{g}} \\)\u3002
\n\u53ef\u4ee5\u53d1\u73b0\uff0c”\u5730\u4e0b\u94c1”\u5468\u671f\u7684\u8ba1\u7b97\u7ed3\u679c\uff0c\u4e0e\u8fd1\u5730\u536b\u661f\u7684\u5468\u671f\u6570\u503c\u662f\u76f8\u7b49\u7684\uff0c\u5982\u679c\u5728\u5730\u4e0b\u94c1\u91ca\u653e\u7269\u4f53\u7684\u540c\u65f6\uff0c\u5728\u5730\u9762\u540c\u65f6\u53d1\u5c04\u4e00\u9897\u8fd1\u5730\u536b\u661f\uff0c\u4f1a\u53d1\u73b0\u4e8c\u8005\u5728\u5404\u81ea\u8f68\u9053\u4e0a\u7684\u8fd0\u52a8\u5177\u6709\u540c\u6b65\u6027\uff08\u6295\u5f71\u5171\u7ebf\uff09\uff0c\u8fd9\u4e00\u73b0\u8c61\u80cc\u540e\u5176\u5b9e\u8574\u542b\u7740\u7b80\u8c10\u8fd0\u52a8\u662f\u5300\u901f\u5706\u5468\u8fd0\u52a8\u5206\u8fd0\u52a8\u7684\u7269\u7406\u89c4\u5f8b\u3002<\/p>\n

\u5982\u679c\u6211\u4eec\u5c06\u5730\u4e0b\u94c1\u7684\u8f68\u9053\u6362\u5230\u66f4\u9ad8\u7eac\u5ea6\uff0c\u7ecf\u8fc7\u8ba1\u7b97\uff08\u6d89\u53ca\u5230\u53d7\u529b\u5206\u89e3\u7b49\uff0c\u8fc7\u7a0b\u8bf7\u5c1d\u8bd5\u81ea\u884c\u63a8\u5bfc\uff09\u4f9d\u7136\u53ef\u4ee5\u7b97\u5f97\\( T = 2\\pi\\sqrt{\\frac{R}{g}} \\)\uff0c\u5982\u679c\u6211\u4eec\u8fdb\u4e00\u6b65\u63d0\u9ad8\u7eac\u5ea6\uff0c\u5c06\u7eac\u5ea6\u6781\u9650\u5230\u5317\u6781\u9644\u8fd1\u975e\u5e38\u5c0f\u7684\u4e00\u90e8\u5206\uff0c\u90a3\u4e48\u8fd9\u4e2a\u5f80\u590d\u8fd0\u52a8\uff0c\u5e94\u8be5\u53ef\u4ee5\u89c6\u4e3a\u65e0\u9650\u6446\u957f\u7684\u5355\u6446\u8fd0\u52a8\uff08\u6446\u957f\u65e0\u9650\uff0c\u8fd1\u4f3c\u4e8e\u76f4\u7ebf\u8fd0\u52a8\uff09\uff0c\u90a3\u4e48\uff0c\u8fd9\u4e2a\u65e0\u9650\u6446\u957f\u7684\u5355\u6446\u5468\u671f\uff0c\u7adf\u7136\u4e0d\u662f\u65e0\u7a77\u5927\uff0c\u800c\u662f\\( T = 2\\pi\\sqrt{\\frac{R}{g}} \\)\u3002<\/p>\n

\u516d\u3001\u5355\u6446\u5468\u671f\u516c\u5f0f\u7684\u4fee\u6b63
\n\u901a\u8fc7\u524d\u9762\u7684\u8ba8\u8bba\u6211\u4eec\u53d1\u73b0\uff0c\u4f20\u7edf\u5355\u6446\u5468\u671f\u516c\u5f0f\u5728\u7279\u6b8a\u60c5\u51b5\u4e0b\u9700\u8981\u4fee\u6b63\u3002\u5728\u63a8\u5bfc\u5355\u6446\u516c\u5f0f\u65f6\uff0c\u6211\u4eec\u9ed8\u8ba4\u91cd\u529b\u573a\u7c7b\u4f3c\u5300\u5f3a\u7535\u573a\u4e00\u6837\u65b9\u5411\u4e0d\u53d8\uff0c\u4f46\u5b9e\u9645\u5c06\u89c6\u89d2\u6269\u5927\u5230\u5730\u7403\u534a\u5f84\u5c3a\u5ea6\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u53d1\u73b0\u5176\u5b9e\u91cd\u529b\u65b9\u5411\u662f\u6307\u5411\u5730\u5fc3\uff08\u5ffd\u7565\u81ea\u8f6c\uff09\uff0c\u90a3\u4e48\u4fee\u6b63\u540e\u7684\u5355\u6446\u5468\u671f\u516c\u5f0f\u662f\u4ec0\u4e48\u5462\uff1f
\n\u5982\u9ed1\u677f\u4e03\u56fe\u6240\u793a\uff0c\u8bbe\u6446\u957f\u4e3a\\( l \\)\uff0c\u5730\u7403\u534a\u5f84\u4e3a\\( R \\)\u3002\u4e07\u6709\u5f15\u529b\u5373\u4e3a\u91cd\u529b\uff0c\\( G \\frac{Mm}{R^2}=mg \\)
\n\u56de\u590d\u529b\\( F = mg \\sin(\\beta + \\alpha) \\)
\n\u57fa\u4e8e\u5c0f\u89d2\u5ea6\u8fd1\u4f3c\u5173\u7cfb\\( \\sin\\theta \\approx \\theta \\)\uff0c\u6709\\( \\sin(\\beta + \\alpha) \\approx \\beta + \\alpha \\)
\n\u518d\u6839\u636e\u5c0f\u89d2\u5ea6\u51e0\u4f55\u5173\u7cfb\uff0c\u6709\\( \\beta = \\frac{x}{R}, \\quad \\alpha = \\frac{x}{l} \\)
\n\u8003\u8651\u5230\u56de\u590d\u529b\u65b9\u5411\u4e0e\u4f4d\u79fb\u65b9\u5411\u7684\u53cd\u5411\u5173\u7cfb\uff0c\u8fdb\u800c\u5f97\u5230\uff1a\\( F = -mg \\left( \\frac{1}{R} + \\frac{1}{l} \\right) x \\)
\n\u56de\u590d\u529b\u7cfb\u6570\u4e3a\\( k = mg \\left( \\frac{1}{R} + \\frac{1}{l} \\right) \\)\uff0c\u4ee3\u5165\u7b80\u8c10\u632f\u52a8\u5468\u671f\u516c\u5f0f \\( T = 2\\pi\\sqrt{\\frac{m}{k}} \\)\uff0c
\n\u7efc\u4e0a\uff0c\u5355\u6446\u5468\u671f\u7684\u6700\u7ec8\u8868\u8fbe\u5f0f\u4e3a\uff1a\\( T = 2\\pi\\sqrt{\\frac{Rl}{g(R + l)}} \\)
\n\u8fd9\u4e2a\u516c\u5f0f\u4e2d\u7684\u6446\u957f\uff0c\u53ef\u4ee5\u7b49\u6548\u4e3a\u4e24\u4e2a\u6446\u957f\u5e76\u8054\u7684\u7ed3\u679c\uff0c\u6211\u4eec\u524d\u51e0\u671f\u63a2\u8ba8\u4e86\u7ea6\u5316\u8d28\u91cf\u4e5f\u662f\u8fd9\u4e2a\u6a21\u5f0f\uff0c\u8fd9\u4e2a\u96be\u9053\u662f”\u7ea6\u5316\u957f\u5ea6”\uff1f<\/p>\n

\u4e03\u3001\u5355\u6446\u516c\u5f0f\u4fee\u6b63\u540e\u7684\u7ed3\u8bba
\n\u6839\u636e\u4fee\u6b63\u516c\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u53d1\u73b0\u4ee5\u4e0b\u7ed3\u8bba\uff1a
\n\u5f53l\u76f8\u5bf9R\u8db3\u591f\u5c0f\u65f6\uff0c\u8be5\u516c\u5f0f\u5c31\u8fd1\u4f3c\u4e8e\u4f20\u7edf\u5355\u6446\u5468\u671f\u516c\u5f0f\uff0c\u8fd9\u4e5f\u89e3\u91ca\u4e86\u4e3a\u4ec0\u4e48\u5728\u65e5\u5e38\u4f7f\u7528\u4f20\u7edf\u516c\u5f0f\u8ba1\u7b97\u666e\u901a\u5355\u6446\u5468\u671f\u65f6\u80fd\u5f97\u5230\u8f83\u4e3a\u51c6\u786e\u7684\u7ed3\u679c\u3002
\n\u800c\u5f53l\u4e0eR\u5904\u4e8e\u76f8\u8fd1\u6570\u91cf\u7ea7\u65f6\uff0c\u4e8c\u8005\u5dee\u5f02\u663e\u8457\uff0c\u5fc5\u987b\u4f7f\u7528\u4fee\u6b63\u516c\u5f0f\u8fdb\u884c\u8ba1\u7b97\uff0c
\n\u5f53\\( l = R \\)\u65f6\uff0c\\( T = 2\\pi\\sqrt{\\frac{R}{2g}} \\)\u3002
\n\u5f53 \\( l \\)\u65e0\u9650\u5927\u65f6\uff0c\u5355\u6446\u7684\u5468\u671f \\( T = 2\\pi\\sqrt{\\frac{R}{g}} \\)\u3002\u8be5\u7ed3\u679c\u4e0e\u6b64\u524d\u63a8\u5bfc\u7684\u8fd1\u5730\u536b\u661f\u5468\u671f\u3001”\u5730\u4e0b\u94c1” \u5468\u671f\u6570\u503c\u543b\u5408\u3002<\/p>\n

\u516b\u3001\u9759\u7535\u573a\u4e2d\u7684\u7b80\u8c10\u8fd0\u52a8
\n\u6211\u4eec\u518d\u6765\u8bf4\u8bf4\u5173\u4e8e\u7535\u573a\u7684\u95ee\u9898\u3002\u5982\u9ed1\u677f8\u56fe\uff0c\u5047\u8bbe\u6709\u4e00\u4e2a\u6b63\u7535\u8377\\( 4q \\)\u548c\u4e00\u4e2a\u8d1f\u7535\u8377\\( q \\)\uff0c\u56fa\u5b9a\u5728\u8ddd\u79bb\u4e3a\\( l \\)\u7684\u4f4d\u7f6e\u4e0a\uff0c\u5ef6\u957f\u7ebf\u6709\u70b9\\( O \\)\uff0c\u4e0e\u8d1f\u7535\u8377q \u7684\u8ddd\u79bb\u4e5f\u662f\\( l \\)\u3002
\n\u6211\u4eec\u6765\u5206\u6790\u4e00\u4e0b\u5728\\( O \\)\u70b9\u4e24\u4fa7\u6781\u77ed\u7684\u8ddd\u79bb\u5185\uff0c\u6709\u4e00\u4e2a\u6b63\u7535\u8377\\( q_{0} \\)\u91ca\u653e\u4e4b\u540e\u7684\u8fd0\u52a8\u60c5\u51b5\uff1a
\n\u6839\u636e\u5e93\u4ed1\u5b9a\u5f8b\u53ef\u5f97\u7535\u8377\u6240\u53d7\u5408\u529b\\( F = \\frac{4k q q_0}{(2l + x)^2} – \\frac{k q q_0}{(l + x)^2} \\)\uff08\u5176\u4e2d\\( q_{0} \\)\u662f\u91ca\u653e\u7684\u7535\u8377\uff09
\n\u6211\u4eec\u4f7f\u7528AI\u8f6f\u4ef6\u5e2e\u6211\u4eec\u8fdb\u884c\u4e00\u4e9b\u4f53\u529b\u64cd\u4f5c\uff0c\u628a\u8fd9\u4e2a\u529b\\( F \\)\u6cf0\u52d2\u5c55\u5f00\u5c55\u5f00\u4e00\u4e0b\uff0c\\( F = \\frac{k q q_0}{l^3}x – \\frac{9k q q_0}{4l^4}x^2 + \\frac{7k q q_0}{2l^5}x^3 + \\cdots \\)
\n\u56e0\u4e3a\\( x \\)\u8fdc\u8fdc\u5c0f\u4e8e\\( l \\)\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u628a\u4e00\u4e9b\u9ad8\u9636\u65e0\u7a77\u5c0f\u9879\u5ffd\u7565\u6389\uff0c\u6709\\( F = – \\frac{q_{0}}{l^{3}}x \\)\uff0c\u6ee1\u8db3\u7b80\u8c10\u8fd0\u52a8\u56de\u590d\u529b\u5173\u7cfb\u3002
\n\u4ece\u52bf\u80fd\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u6211\u4eec\u77e5\u9053\u52bf\u80fd\u548c\u529f\u7684\u5173\u7cfb\u662f\\( W = – \\Delta E_{p} \\)\uff0c\u6211\u4eec\u5bf9\u52bf\u80fd\u6c42\u5bfc\u5c31\u53ef\u4ee5\u5f97\u5230\u529b\u3002\u540c\u6837\u7684\uff0c\u6211\u4eec\u5982\u679c\u5199\u51fa\u52bf\u80fd\u4e0e\u4f4d\u79fb\u7684\u8868\u8fbe\u5f0f\uff0c\u8fdb\u884c\u6c42\u5bfc\uff0c\u540c\u6837\u4e5f\u53ef\u4ee5\u5f97\u5230\u529b\u7684\u8868\u8fbe\u5f0f\u3002<\/p>\n

\u4e5d\u3001\u52bf\u80fd\u7684\u6781\u503c\u9644\u8fd1\uff0c\u7269\u7406\u505a\u7b80\u8c10\u8fd0\u52a8
\n\u8096\u8001\u5e08\u8865\u5145\uff1a\u5176\u5b9e\u5bf9\u4e8e\u52bf\u80fd\u6781\u503c\u7684\u4f4d\u7f6e\uff0c\u5728\u9644\u8fd1\u505a\u5f80\u590d\u8fd0\u52a8\u5927\u591a\uff08\u5e76\u4e0d\u662f\u6240\u6709\uff09\u53ef\u4ee5\u8ba4\u4e3a\u662f\u4e00\u79cd\u7b80\u8c10\u8fd0\u52a8\u3002
\n\u5982\u679c\u5c06\u52bf\u80fd\u7684\u8868\u8fbe\u5f0f\u8fdb\u884c\u6c42\u5bfc\uff0c\u6cf0\u52d2\u5c55\u5f00\u540e\u5ffd\u7565\u9ad8\u9636\u5c0f\u91cf\uff0c\u53ef\u4ee5\u5f97\u5230\u529b\u4e0e\u4f4d\u79fb\u7684\u5173\u7cfb\uff08\u6ee1\u8db3\u56de\u590d\u529b\u5f62\u5f0f\uff09\uff0c\u518d\u5c06\u529b\u8fdb\u884c\u6c42\u5bfc\uff0c\u5f97\u5230\u7684\u6570\u503c\u5373\u4e3a\u56de\u590d\u529b\u5e38\u91cfk\uff0c\u8fdb\u800c\u53ef\u4ee5\u6c42\u89e3\u5468\u671f\u7b49\u3002
\n\u5176\u5b9e\uff0c\u6211\u4eec\u5728\u7b2c\u4e8c\u90e8\u5206\u7528\u80fd\u91cf\u8fdb\u884c\u6c42\u89e3\u5355\u6446\u5468\u671f\u7684\u8fc7\u7a0b\uff0c\u5b8c\u5168\u53ef\u4ee5\u53ea\u5206\u6790\u52bf\u80fd\uff0c\u63a8\u5bfc\u8fc7\u7a0b\u53ef\u4ee5\u975e\u5e38\u7b80\u5316\u5982\u4e0b\uff1a
\n\u5355\u6446\u7684\u91cd\u529b\u52bf\u80fd\u8868\u8fbe\u5f0f\u4e3a\uff1a\\( E_{p}=mgl(1 – \\cos\\theta) \\)
\n\u5f53\u6446\u89d2\\( \\theta \\)\u5f88\u5c0f\u65f6\uff08\u6ee1\u8db3\\( \\cos\\theta \\approx 1 – \\frac{\\theta^{2}}{2} \\)\uff09\uff0c\u91cd\u529b\u52bf\u80fd\u53ef\u8fd1\u4f3c\u4e3a\uff1a\\( E_{p}=mgl\\cdot\\frac{\\theta^{2}}{2} \\)
\n\u7ed3\u5408\u5355\u6446\u5c0f\u89d2\u5ea6\u6446\u52a8\u65f6\uff0c\u6446\u7403\u4f4d\u79fb\\( x \\)\u4e0e\u6446\u957f\\( l \\)\u3001\u6446\u89d2\\( \\theta \\)\u7684\u5173\u7cfb\\( \\theta \\approx \\frac{x}{l} \\)\uff0c\u8fdb\u4e00\u6b65\u63a8\u5bfc\u53ef\u5f97\uff1a\\( E_{p}=\\frac{mg}{2l}\\cdot x^{2} \\)
\n\u6c42\u4e8c\u9636\u5bfc\u6570\uff0c\u5f97\u5230\\( E_{p}”(x)=\\frac{mg}{l} \\)\uff0c\u5373\u4e3ak\u503c\u3002<\/p>\n

\u5341\u3001\u5229\u7528\u6cf0\u52d2\u5c55\u5f00\u8fdb\u4e00\u6b65\u5206\u6790
\n\u52bf\u80fd\u7684\u4e00\u9636\u5bfc\u4e3a\u8be5\u52bf\u80fd\u6240\u5bf9\u5e94\u7684\u4fdd\u5b88\u529b\uff1a$$F(x) = -\\frac{dU}{dx}$$\uff0c\u5728\u5e73\u8861\u4f4d\u7f6e\u9644\u8fd1\uff0c\u529b\u5173\u4e8e\u4f4d\u7f6e\u8fd1\u4f3c\u5448\u7ebf\u6027\u5173\u7cfb\uff1a$$F(x) \\approx -k(x – x_0)$$\uff0c\u8fd9\u4e0e\u7b80\u8c10\u632f\u52a8\u7684\u56de\u590d\u529b\u5f62\u5f0f\u76f8\u540c\u3002\u56e0\u6b64\u7269\u4f53\u8fd1\u4f3c\u505a\u7b80\u8c10\u632f\u52a8\u3002\u5176\u4e2d$$k = \\left. \\frac{d^2U}{dx^2} \\right|_{x_0}$$\uff0c\u662f\u7b49\u6548\u52b2\u5ea6\u7cfb\u6570\u3002
\n\u6cf0\u52d2\u5c55\u5f00\u7684\u4e00\u822c\u5f62\u5f0f\u4e3a\uff1a$$f(x) = f(a) + f'(a)(x – a) + \\frac{f”(a)}{2!}(x – a)^2 + \\frac{f^{(3)}(a)}{3!}(x – a)^3 + \\cdots + \\frac{f^{(n)}(a)}{n!}(x – a)^n + \\cdots$$
\n\u6211\u4eec\u628a\u52bf\u80fdU(x)\u5728\u5e73\u8861\u4f4d\u7f6e\u9644\u8fd1\u5c55\u5f00\u5f97\u5230\uff1a$$U(x) = U(x_0) + (x – x_0) \\left. \\frac{dU}{dx} \\right|_{x_0} + \\frac{1}{2}(x – x_0)^2 \\left. \\frac{d^2U}{dx^2} \\right|_{x_0} + \\cdots$$\uff0c\u7531\u4e8e\u52bf\u80fd\u5728\u5e73\u8861\u4f4d\u7f6e\u7684\u4e00\u9636\u5bfc\u6570\u4e3a\u96f6\uff0c\u5bf9\u4e8e\u8db3\u591f\u5c0f\u7684\u4f4d\u79fb\uff0c\u7565\u53bb\u4e09\u9636\u4ee5\u540e\u7684\u9879\uff0c\u53ef\u8fd1\u4f3c\u5f97\u5230\uff1a$$U(x) \\approx U(x_0) + \\frac{1}{2}(x – x_0)^2 \\left. \\frac{d^2U}{dx^2} \\right|_{x_0}$$\u3002
\n\u7b80\u8c10\u632f\u5b50\u52bf\u80fd\u7684\u6807\u51c6\u5f62\u5f0f\u4e3a\uff1a$$U(x) = \\text{C} + \\frac{1}{2}k(x – x_0)^2$$\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u770b\u51fa\u7b49\u6548\u52b2\u5ea6\u7cfb\u6570k\u4e3a\uff1a$$k = \\left. \\frac{d^2U}{dx^2} \\right|_{x_0}$$\u3002
\n


\nJQX|Jin<\/center>
\n\u672c\u671f\u7814\u8ba8Qiusir\u7531\u5355\u6446\u5165\u624b\uff0c\u901a\u8fc7\u63ed\u793a\u4f20\u7edf\u6a21\u578b\u5728\u65e0\u9650\u6446\u957f\u6761\u4ef6\u4e0b\u7684\u77db\u76fe\uff0c\u4fee\u6b63\u4e86\u8003\u8651\u91cd\u529b\u65b9\u5411\u53d8\u5316\u7684\u5355\u6446\u5468\u671f\u516c\u5f0f\uff0c\u5e76\u9a8c\u8bc1\u4e86\u5728\u5e38\u89c4\u5355\u6446\u3001\u65e0\u9650\u6446\u957f\u4e0b\u7684\u81ea\u6d3d\u6027\u3002\u66f4\u5e05\u7684\u662f\uff0c\u8fdb\u4e00\u6b65\u901a\u8fc7\u52bf\u80fd\u6cf0\u52d2\u5c55\u5f00\u6cd5\uff0c\u9610\u660e\u7b80\u8c10\u8fd0\u52a8\u7684\u666e\u9002\u6027\u2014\u2014\u5e73\u8861\u4f4d\u7f6e\u9644\u8fd1\u7684\u52bf\u80fd\u4e8c\u9636\u5c55\u5f00\u51b3\u5b9a\u56de\u590d\u529b\u7cfb\u6570\uff0c\u7edf\u4e00\u89e3\u91ca\u4e86\u5355\u6446\u3001\u7535\u573a\u632f\u52a8\u7b49\u5468\u671f\u6027\u73b0\u8c61\u3002
\n\u8fd9\u4e00\u8fc7\u7a0b\u903b\u8f91\u4e25\u8c28\u53c8\u524d\u540e\u547c\u5e94\uff0c\u4e0d\u4ec5\u4f18\u5316\u4e86\u5355\u6446\u6a21\u578b\uff0c\u66f4\u51f8\u663e\u7269\u7406\u4e2d\u901a\u8fc7\u6253\u7834\u7406\u60f3\u5047\u8bbe\u3001\u63a2\u5bfb\u771f\u5b9e\u7cfb\u7edf\u7684\u7814\u7a76\uff1a\u4ece\u77db\u76fe\u51fa\u53d1\uff0c\u4ee5\u6570\u5b66\u91cd\u6784\u6a21\u578b\uff0c\u6700\u7ec8\u5b9e\u73b0\u89c4\u5f8b\u7684\u672c\u8d28\u6027\u7edf\u4e00\u3002<\/p>\n

\u3010\u4e0b\u671f\u9884\u544a\u3011\u52a0\u901f\u5ea6\u7684\u5173\u8054\u00b7\u00b7\u00b7
\n\u901f\u5ea6\u5173\u8054\u95ee\u9898\u662f\u9ad8\u4e2d\u7269\u7406\u8fd0\u52a8\u5408\u6210\u4e2d\u7684\u96be\u70b9\u4e4b\u4e00\u3002\u901f\u5ea6\u53ef\u4ee5\u5173\u8054\uff0c\u52a0\u901f\u5ea6\u4e5f\u53ef\u4ee5\u5173\u8054\u5417\uff1f
\n\u4e0b\u6b21\u5e2d\u660e\u7eb3\uff0c\u91d1\u8001\u5e08\u5c06\u4ece\u4e00\u9053\u7ecf\u5178\u4e60\u9898\u5165\u624b\uff0c\u63a2\u5bfb\u4e09\u79cd\u5e38\u89c1\u6a21\u578b\u7684\u901f\u5ea6\u3001\u52a0\u901f\u5ea6\u5173\u8054\u7684\u672c\u8d28\uff0c\u656c\u8bf7\u671f\u5f85\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

JQX\/\u8fdb\u53d6\u82af \u5e2d\u660e\u7eb3\u7b2c7\u671f\uff082025.5.8\uff09 \u65e0\u9650\u5e38\u6446\u957f\u5355\u6446\u5468\u671f\u95ee\u9898 \u96f6\u3001\u5173\u4e8e\\( T = 2\\pi\\sq … \u9605\u8bfb\u66f4\u591a<\/a><\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[9],"tags":[460],"class_list":["post-36786","post","type-post","status-publish","format-standard","hentry","category-jqx","tag-jqx"],"_links":{"self":[{"href":"https:\/\/www.qiusir.com\/index.php?rest_route=\/wp\/v2\/posts\/36786","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.qiusir.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.qiusir.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.qiusir.com\/index.php?rest_route=\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/www.qiusir.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=36786"}],"version-history":[{"count":0,"href":"https:\/\/www.qiusir.com\/index.php?rest_route=\/wp\/v2\/posts\/36786\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.qiusir.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=36786"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.qiusir.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=36786"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.qiusir.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=36786"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}