{"id":10236,"date":"2025-05-20T13:51:58","date_gmt":"2025-05-20T04:51:58","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=10236"},"modified":"2025-05-21T17:34:54","modified_gmt":"2025-05-21T08:34:54","slug":"%e5%bc%b7%e5%88%b6%e6%8c%af%e5%8b%95%e3%81%a8%e5%85%b1%e9%b3%b4","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/%e5%b8%b8%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%bc%8f\/%e5%ae%9a%e6%95%b0%e4%bf%82%e6%95%b02%e9%9a%8e%e7%b7%9a%e5%bd%a2%e9%9d%9e%e5%90%8c%e6%ac%a1%e6%96%b9%e7%a8%8b%e5%bc%8f\/%e5%bc%b7%e5%88%b6%e6%8c%af%e5%8b%95%e3%81%a8%e5%85%b1%e9%b3%b4\/","title":{"rendered":"\u5f37\u5236\u632f\u52d5\u3068\u5171\u9cf4"},"content":{"rendered":"

\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u304c\u4eba\u751f\u306e\u3069\u3046\u3044\u3046\u3068\u304d\u306b\u5fc5\u8981\u306b\u306a\u308b\u304b\u3068\u3044\u3046\u3068\uff0c\u529b\u5b66\u3067\u3084\u3063\u305f\uff08\u3053\u308c\u304b\u3089\u3084\u308b\uff1f\uff09\u3088\u3046\u306a\u5f37\u5236\u632f\u52d5\u3068\u5171\u9cf4\u3092\u7406\u89e3\u3059\u308b\u305f\u3081\u306b\u5fc5\u8981\u306a\u3093\u3067\u3059\u3088\u3002\u5ff5\u306e\u305f\u3081\uff0c\u5f37\u5236\u632f\u52d5\u3068\u5171\u9cf4\u306b\u95a2\u3059\u308b\u304a\u3055\u3089\u3044\u3092\u3002<\/p>\n

<\/p>\n

\u8abf\u548c\u632f\u52d5<\/h3>\n

\u7c21\u5358\u306e\u305f\u3081\u306b1\u6b21\u5143\uff08$x$ \u65b9\u5411\uff09\u306e\u904b\u52d5\u3092\u8003\u3048\u308b\u3002 \u30d5\u30c3\u30af\u306e\u6cd5\u5247\u306b\u3088\u3063\u3066\uff0c\uff08\u3064\u308a\u3042\u3044\u306e\u4f4d\u7f6e\u304b\u3089\u306e\uff09\u5909\u4f4d $x$ \u306b\u6bd4\u4f8b\u3059\u308b\u5fa9\u5143\u529b $F = – k\\, x$ \u3092\u53d7\u3051\u308b\u8cea\u91cf $m$ \u306e\u8cea\u70b9\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n

$$ m \\frac{d^2 x}{dt^2} = -k\\, x$$<\/p>\n

\u30d0\u30cd\u5b9a\u6570 $k$ \u3092 $k \\equiv m\\, \\omega_{0}^{\\,2} = m \\times (\\omega_{0})^2$ \u3068\u304a\u304f\u3068<\/p>\n

$$\\frac{d^2 x}{dt^2} = -\\omega_0^{\\,2}\\, x$$<\/p>\n

\u3053\u306e\u304b\u305f\u3061\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\u300c\u6700\u3082\u7c21\u5358\u306a\u5b9a\u6570\u4fc2\u65702\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\uff1a\u7d9a\u304d<\/a>\u300d\u3067\u8aac\u660e\u3057\u3066\u3044\u3066\uff0c\u305f\u3060\u3061\u306b\u57fa\u672c\u89e3\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u6c42\u3081\u3089\u308c\u308b\u3002<\/p>\n

\\begin{eqnarray}
\nx_1(t) &=& \\cos \\omega_0\\, t \\\\
\nx_2(t) &=& \\sin \\omega_0\\, t
\n\\end{eqnarray}<\/p>\n

\u8abf\u548c\u632f\u52d5\u5b50\u306e\u4e00\u822c\u89e3\u3068\u521d\u671f\u6761\u4ef6<\/h4>\n

\u8abf\u548c\u632f\u52d5\u5b50\u306e\u4e00\u822c\u89e3\u306f\uff0c2\u3064\u306e\u57fa\u672c\u89e3\u306e\u7dda\u5f62\u7d50\u5408\u3067\u66f8\u304b\u308c\u3066<\/p>\n

\\begin{eqnarray}
\nx(t) &=& A\\, x_1(t) + B\\, x_2(t) \\\\
\n&=& A\u00a0 \\cos \\omega_0\\, t + B \\sin \\omega_0\\, t
\n\\end{eqnarray}<\/p>\n

2\u3064\u306e\u7a4d\u5206\u5b9a\u6570 $A,\\\u00a0 B$ \u306f\u521d\u671f\u6761\u4ef6\u304b\u3089\u3002\u4f8b\u3048\u3070\uff0c\u7c21\u5358\u306e\u305f\u3081\u306b $t = 0$ \u3067 $x(0) = a,\\\u00a0 \\dot{x}(0) = 0$ \u3068\u3059\u308b\u3068\uff0c<\/p>\n

$$ A = a, \\quad B = 0$$<\/p>\n

\u3068\u6c7a\u307e\u308a\uff0c\u3057\u305f\u304c\u3063\u3066<\/p>\n

$$x(t) =\u00a0 a \\cos \\omega_0\\, t$$<\/p>\n

\u5f37\u5236\u632f\u52d5<\/h3>\n

\u30d5\u30c3\u30af\u306e\u6cd5\u5247\u306b\u3088\u308b\u5fa9\u5143\u529b\u4ee5\u5916\u306b\uff0c\u6642\u9593\u7684\u306b\u5909\u52d5\u3059\u308b\u5916\u529b<\/strong><\/span> $F(t)$ \u304c\u50cd\u304f\u5834\u5408\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\uff0c<\/p>\n

$$\\frac{d^2 x}{dt^2}\u00a0 + \\omega_0^{\\,2}\\, x = \\frac{F(t)}{m}$$<\/p>\n

\u7279\u306b\uff0c\u5916\u529b $F(t)$ \u304c\u6642\u9593\u306e\u5468\u671f\u95a2\u6570<\/p>\n

$$F(t) = F_0 \\cos \\omega\\, t \\equiv m f_0 \\cos \\omega\\, t $$<\/p>\n

\u3067\u3042\u308b\u5834\u5408\u306f\uff0c<\/p>\n

$$\\frac{d^2 x}{dt^2}\u00a0 + \\omega_0^{\\,2}\\, x = f_0 \\cos \\omega\\, t$$<\/p>\n

\u3053\u306e\u5f62\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\uff0c\uff08\u672a\u77e5\u95a2\u6570 $x(t)$ \u3092\u542b\u3080\u9805\u3092\u5168\u3066\u5de6\u8fba\u306b\u3082\u3063\u3066\u3044\u3063\u3066\u3082\u53f3\u8fba\u306b\u6b8b\u3063\u3066\u3044\u308b\uff09\u975e\u540c\u6b21\u9805<\/strong><\/span>\u304c\u3042\u308b\u306e\u3067\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f<\/strong><\/span>\u3068\u547c\u3070\u308c\u308b\u3002<\/p>\n

\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u3092\u5f62\u6c7a\u3081\u6253\u3061\u3067\u6c42\u3081\u308b<\/h3>\n

\u5f37\u5236\u632f\u52d5\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\uff08\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\uff09\u306e\u7279\u6b8a\u89e3 $x_s$ \u306e\u5f62\u3092<\/p>\n

$$x_s(t) = C \\cos\\omega\\, t$$<\/p>\n

\u3068\u6c7a\u3081\u6253\u3061\u306b\u3057\u3066\u89e3\u3044\u3066\u307f\u308b\u3002\uff08\u306a\u3093\u3067\u3053\u3046\u7f6e\u304f\u304b\u3063\u3066\uff1f\u3053\u3046\u3059\u308b\u3068\u89e3\u3051\u308b\u304b\u3089\u3067\u3059\u3088\u3002\u306f\u3058\u3081\u304b\u3089\u7b54\u3048\u3092\u77e5\u3063\u3066\u3044\u308b\u304b\u306e\u5982\u304f\uff0c\u89e3\u306e\u5f62\u3092\u6c7a\u3081\u6253\u3061\u3059\u308b\u306e\u304c\u3057\u3063\u304f\u308a\u3053\u306a\u3044\u3042\u306a\u305f\u306e\u305f\u3081\u306b\uff0c\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3\u3092\u4f7f\u3063\u305f\u516c\u5f0f\u3067\u6c42\u3081\u308b\u4e00\u822c\u7684\u65b9\u6cd5\u3082\u4ee5\u4e0b\u306b\u8aac\u660e\u3057\u3066\u3044\u307e\u3059\u304b\u3089\uff0c\u305d\u3061\u3089\u3082\u3069\u3046\u305e\u3002\uff09<\/p>\n

\\begin{eqnarray}
\n\\frac{d^2 x_s}{dt^2}\u00a0 + \\omega_0^{\\,2}\\, x_s &=& f_0 \\cos \\omega\\, t \\\\
\n\\left( -\\omega^2 + \\omega_0^{\\, 2}\\right) \\, C\u00a0 \\cos\\omega\\, t &=& f_0 \\cos \\omega\\, t \\\\
\n\\therefore\\ \\ C &=& \\frac{f_0}{\\omega_0^{\\, 2} -\\omega^2} \\\\
\n\\therefore\\ \\ x_s(t) &=& \\frac{f_0}{\\omega_0^{\\, 2} -\\omega^2} \\cos\\omega\\, t
\n\\end{eqnarray}<\/p>\n

\u5f37\u5236\u632f\u52d5\u306e\u4e00\u822c\u89e3\u3068\u521d\u671f\u6761\u4ef6<\/h4>\n

\u3057\u305f\u304c\u3063\u3066\uff0c\u5f37\u5236\u632f\u52d5\u306e\u4e00\u822c\u89e3\u306f<\/p>\n

\\begin{eqnarray}
\nx(t) &=& A\\, x_1(t) + B\\, x_2(t) + x_s(t) \\\\
\n&=& A \\cos\\omega_0\\, t + B \\sin \\omega_0\\, t + \\frac{f_0}{\\omega_0^{\\, 2} -\\omega^2} \\cos\\omega\\, t
\n\\end{eqnarray}<\/p>\n

2\u3064\u306e\u7a4d\u5206\u5b9a\u6570 $A, B$ \u306f\u521d\u671f\u6761\u4ef6\u304b\u3089\u3002\u521d\u671f\u6761\u4ef6\u3092\u7c21\u5358\u306e\u305f\u3081\u306b $t = 0$ \u3067 $x = a, \\ \\dot{x} = 0$ \u3068\u3059\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\nA &=& a -\\frac{f_0}{\\omega_0^{\\, 2} -\\omega^2} \\\\
\nB &=& 0 \\\\
\n\\therefore\\ \\ x(t) &=& \\left\\{ a -\\frac{f_0}{\\omega_0^{\\, 2} -\\omega^2} \\right\\} \\cos\\omega_0\\, t
\n+ \\frac{f_0}{\\omega_0^{\\, 2} -\\omega^2} \\cos\\omega\\, t \\\\
\n&=& a \\cos\\omega_0\\, t
\n+ f_0 \\frac{ \\cos\\omega\\, t -\\cos\\omega_0\\, t}{\\omega_0^{\\, 2} -\\omega^2}
\n\\end{eqnarray}<\/p>\n

\u5171\u9cf4<\/h4>\n

\u53c2\u8003\u307e\u3067\u306b\uff0c\u30ed\u30d4\u30bf\u30eb\u306e\u5b9a\u7406\u3068\u306f\uff08\uff11\u5e74\u751f\u306e\u6388\u696d\u3067\u3084\u308a\u307e\u3057\u305f\uff0c\u3053\u3053<\/a><\/strong><\/span>\u3068\u304b\u3092\u53c2\u7167\uff09<\/p>\n

$f(a) = 0, \\ g(a) = 0$ \u306e\u3068\u304d\uff0c<\/p>\n

$$\\lim_{x \\rightarrow a} \\frac{f(x)}{g(x)}\u00a0 =\\frac{f'(a)}{g'(a)} \\ \\left( = \\lim_{x \\rightarrow a} \\frac{f'(x)}{g'(x)}\\right)$$<\/p>\n

\u3068\u306a\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

$\\omega \\rightarrow \\omega_0$ \u306e\u3068\u304d\uff0c$f_0$ \u3092\u4fc2\u6570\u3068\u3059\u308b\u9805\u306f $\\displaystyle \\frac{0}{0}$ \u306e\u4e0d\u5b9a\u5f62\u3068\u306a\u308b\u304c\uff0c\uff08\u30e9\u30f3\u30c0\u30a6-\u30ea\u30d5\u30b7\u30c3\u30c4\u300c\u529b\u5b66\u300d\u7b2c22\u7bc0\u3092\u53c2\u7167\uff09\u30ed\u30d4\u30bf\u30eb\u306e\u5b9a\u7406\u3092\u4f7f\u3046\u3068\uff08\u5206\u6bcd\u30fb\u5206\u5b50\u3092 $\\omega$ \u3067\u5fae\u5206\u3057\u3066… \uff09<\/p>\n

\\begin{eqnarray}
\n\\lim_{\\omega \\rightarrow \\omega_0} \\frac{ \\cos\\omega\\, t -\\cos\\omega_0\\, t}{\\omega_0^{\\, 2} -\\omega^2}
\n&=& \\lim_{\\omega \\rightarrow \\omega_0} \\frac{\\frac{d}{d\\omega}\\left\\{ \\cos\\omega\\, t -\\cos\\omega_0\\, t\\right\\}}{\\frac{d}{d\\omega}\\left\\{\\omega_0^{\\, 2} -\\omega^2\\right\\}}
\n&=& \\lim_{\\omega \\rightarrow \\omega_0} \\frac{-t\\,\\sin \\omega\\, t}{-2 \\omega} \\\\
\n&=& \\frac{t}{2 \\omega_0} \\sin \\omega_0\\, t
\n\\end{eqnarray}<\/p>\n

$$\\therefore\\ \\ x(t) = a \\cos\\omega_0\\, t + \\frac{f_0}{2 \\omega_0}\\, {\\color{red}{t}}\\, \\sin \\omega_0\\, t$$<\/p>\n

\u8abf\u548c\u632f\u52d5\u5b50\u306e\u56fa\u6709\u632f\u52d5\u6570 $\\omega_0$ \u3068\u540c\u3058\u632f\u52d5\u6570\u306e\u5468\u671f\u7684\u5916\u529b\u3092\u52a0\u3048\u308b\u3068\uff0c\u632f\u5e45\u304c\u6642\u9593 ${\\color{red}{t}}$ \u306b\u6bd4\u4f8b\u3057\u3066\u5358\u8abf\u5897\u52a0\u3059\u308b\u9805\u304c\u73fe\u308c\u308b\u3002\u3053\u308c\u3092\u5171\u9cf4<\/strong><\/span>\u3068\u3044\u3046\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\"\"<\/p>\n

\u3082\u3061\u308d\u3093\uff0c\u73fe\u5b9f\u4e16\u754c\u3067\u306f\u632f\u5e45\u304c $t$ \u306b\u6bd4\u4f8b\u3057\u3066\u7121\u9650\u306b\u5897\u5927\u3057\u3064\u3065\u3051\u308b\u3053\u3068\u306f\u306a\u304f\u3066\uff0c\u3044\u305a\u308c\u7834\u7dbb\u3059\u308b\u3053\u3068\u306b\u306a\u308b\u3002\u4f8b\u3048\u3070\uff0c\u30d5\u30c3\u30af\u306e\u6cd5\u5247\u304c\u3084\u3076\u308c\u308b\u3068\u304b\uff0c\u30d0\u30cd\u304c\u3072\u304d\u3061\u304e\u308c\u3066\u58ca\u308c\u3066\u3057\u307e\u3046\u3068\u304b…<\/p>\n

\u306a\u304a\uff0c\u4e00\u90e8\u306e\u30c6\u30ad\u30b9\u30c8\u306b\u306f\uff08\u4f8b\u3048\u3070\u30d5\u30a1\u30a4\u30f3\u30de\u30f3\u7269\u7406\u5b66\u2160\u529b\u5b66 \u7b2c21-5\u7bc0\uff09\u5468\u671f\u7684\u5916\u529b\u304c\u3042\u308b\u5834\u5408\u306e\u5f37\u5236\u632f\u52d5\u306e\u89e3\u306f<\/p>\n

$$x(t) = A \\cos\\omega_0\\, t + \\frac{f_0}{\\omega_0^{\\, 2} -\\omega^2} \\cos\\omega \\, t$$<\/p>\n

\u3067\u3042\u308a\uff0c$\\omega = \\omega_0$ \u306e\u5834\u5408\u306f\u5206\u6bcd\u304c\u30bc\u30ed\u306b\u306a\u308a\uff0c\u632f\u5e45\u304c\u767a\u6563\u3059\u308b\uff01\uff1f<\/strong><\/span>\u306a\u3069\u3068\u66f8\u3044\u3066\u3042\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u305d\u308c\u306f\u3061\u3068\u8a00\u3044\u904e\u304e\u3067\u3042\u308a\uff0c\u4e0a\u306b\u307e\u3068\u3081\u305f\u3088\u3046\u306b\u9069\u5207\u306b\u521d\u671f\u6761\u4ef6\u3092\u8a2d\u5b9a\u3057\u3066\u3084\u308a\uff0c\u3066\u3044\u306d\u3044\u306b $\\omega \\rightarrow \\omega_0$ \u306e\u6975\u9650\u3092\u3068\u3063\u3066\u3084\u308c\u3070\uff0c\u5171\u9cf4\u73fe\u8c61\u306e\u89e3\u304c\u5f97\u3089\u308c\u308b\u3053\u3068\u3092\u3053\u3053\u3067\u793a\u3057\u3066\u3044\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u3092\u516c\u5f0f\u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b<\/h3>\n

\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u3092\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3\u3092\u4f7f\u3063\u305f\u516c\u5f0f\u3067\u6c42\u3081\u308b\u4e00\u822c\u7684\u65b9\u6cd5\u306f\uff0c\u3053\u306e\u6388\u696d\u3067\u8aac\u660e\u3057\u3066\u3044\u308b\u3002\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u3092\u53c2\u7167\uff1a<\/p>\n