{"id":1006,"date":"2020-11-17T04:48:39","date_gmt":"2020-11-16T19:48:39","guid":{"rendered":"https:\/\/mell0w-5phere.net\/jaded5phere\/?p=1006"},"modified":"2020-11-17T04:48:39","modified_gmt":"2020-11-16T19:48:39","slug":"coriolis","status":"publish","type":"post","link":"https:\/\/mell0w-5phere.net\/jaded5phere\/2020\/11\/17\/coriolis\/","title":{"rendered":"\u3051\u3063\u304d\u3087\u304f\u30b3\u30ea\u30aa\u30ea\u306e\u529b\u3063\u3066\u306a\u3093\u3084\u306d\u3093"},"content":{"rendered":"\n

\u8a66\u9a13\u671f\u9593\u304a\u306a\u3058\u307f\u306e\u9003\u907f\u8a18\u4e8b\u3002<\/p>\n

\u30fb\u524d\u63d0<\/h4>\n

\"\"
\n\u3000\u6163\u6027\u5ea7\u6a19\u7cfb\\(O-xyz\\)\u7cfb\u3068\u3001\u89d2\u901f\u5ea6\\(\\boldsymbol \\omega\\)\u3067\u56de\u308b\u56de\u8ee2\u5ea7\u6a19\u7cfb\\(O-x^\\prime y^\\prime z^\\prime\\)\u7cfb\u304c\u3001\u539f\u70b9\u3092\u5171\u6709\u3057\u3066\u3044\u308b\u3088\u3046\u306a\u72b6\u6cc1\u3092\u8003\u3048\u308b\u3002\u306a\u304a\u3001\u56f3\u3067\u306f\u56de\u8ee2\u5ea7\u6a19\u7cfb\u304c\\(z^\\prime\\)\u8ef8\u5468\u308a\u306b\u56de\u8ee2\u3057\u3066\u3044\u308b\u304c\u3001\u56de\u8ee2\u5ea7\u6a19\u7cfb\u306e\u56de\u8ee2\u8ef8\u306f\u5225\u306b\u5ea7\u6a19\u8ef8\u306b\u4e00\u81f4\u3057\u3066\u3044\u306a\u304f\u3066\u3082\u69cb\u308f\u306a\u3044\u3002
\n\u3000\u6163\u6027\u5ea7\u6a19\u7cfb\u306e\u57fa\u5e95\u3092\\(\\mathscr W=\\{\\boldsymbol i_0,\\boldsymbol j_0,\\boldsymbol k_0\\}\\)\u3068\u3057\u3001\u56de\u8ee2\u5ea7\u6a19\u7cfb\u306e\u57fa\u5e95\u3092\\(\\mathscr V=\\{\\boldsymbol i,\\boldsymbol j,\\boldsymbol k\\}\\)\u3068\u3059\u308b\u3002<\/p>\n

\u3000\u6163\u6027\u5ea7\u6a19\u7cfb\u304b\u3089\u7269\u4f53\u3092\u898b\u305f\u5ea7\u6a19\u30d9\u30af\u30c8\u30eb\u3092$$\\boldsymbol R=\\left[\\begin{array}{cc}x\\\\y\\\\z\\end{array}\\right]$$\u3068\u3057\u3001\u56de\u8ee2\u5ea7\u6a19\u7cfb\u304b\u3089\u7269\u4f53\u3092\u898b\u305f\u5ea7\u6a19\u30d9\u30af\u30c8\u30eb\u3092$$\\boldsymbol r=\\left[\\begin{array}{cc}x^\\prime\\\\y^\\prime\\\\z^\\prime\\end{array}\\right]$$\u3068\u3059\u308b\u3002<\/p>\n

\u30fb\u4f4d\u7f6e\u306b\u95a2\u3059\u308b\u95a2\u4fc2\u5f0f<\/h4>\n

\u3000\u6163\u6027\u5ea7\u6a19\u7cfb\u3068\u56de\u8ee2\u5ea7\u6a19\u7cfb\u3067\u3001\u898b\u304b\u3051\u306e\u5ea7\u6a19\u306f\u7570\u306a\u308b\u304c\u3001\u5b9f\u4f53\u306e\u4f4d\u7f6e\u306f\u540c\u3058\u3067\u3042\u308b\u3002\u3064\u307e\u308a
\n$$x\\boldsymbol i_0+y\\boldsymbol j_0+z\\boldsymbol k_0 = x^\\prime\\boldsymbol i+y^\\prime\\boldsymbol j+z^\\prime\\boldsymbol k$$
\n\u3067\u3042\u308b\u3002\u57fa\u5e95\u3068\u5ea7\u6a19\u30d9\u30af\u30c8\u30eb\u3092\u4f7f\u3048\u3070\u3001
\n$$\\mathscr W\\boldsymbol R=\\mathscr V\\boldsymbol r\\quad\\cdots(1)$$
\n\u3068\u66f8\u3051\u308b\u3002<\/p>\n

\u30fb\u901f\u5ea6\u306b\u95a2\u3059\u308b\u95a2\u4fc2\u5f0f<\/h4>\n

\u3000\u5148\u306e(1)\u5f0f\u306e\u4e21\u8fba\u3092\u5fae\u5206\u3059\u308b\u3002
\n$$\\begin{eqnarray}\\frac{\\mathrm d}{\\mathrm dt}(\\mathscr W\\boldsymbol R)&=&\\frac{\\mathrm d}{\\mathrm dt}(\\mathscr V\\boldsymbol r)\\\\
\n\\Leftrightarrow\\frac{\\mathrm d\\mathscr W}{\\mathrm dt}\\boldsymbol R+\\mathscr W\\frac{\\mathrm d\\boldsymbol R}{\\mathrm dt}&=&\\frac{\\mathrm d\\mathscr V}{\\mathrm dt}\\boldsymbol r+\\mathscr V\\frac{\\mathrm d\\boldsymbol r}{\\mathrm dt}\\quad\\cdots(2)
\n\\end{eqnarray}$$
\n\u3000\u3053\u3053\u3067\u3001\u6163\u6027\u5ea7\u6a19\u7cfb\u306e\u57fa\u5e95\u3067\u3042\u308b\\(\\mathscr W\\)\u306f\u6642\u9593\u5909\u5316\u3057\u306a\u3044\u306e\u3067\u3001(2)\u5f0f\u306e\u5de6\u8fba\u306e\u7b2c\u4e00\u9805\u306f0\u3067\u3042\u308b\u3002\u4e00\u65b9\u3067\u3001\u53f3\u8fba\u7b2c\u4e00\u9805\u306b\u3064\u3044\u3066\u306f\u3001\u56de\u8ee2\u5ea7\u6a19\u7cfb\u306e\u57fa\u5e95\\(\\mathscr V\\)\u306f\u89d2\u901f\u5ea6\\(\\boldsymbol \\omega\\)\u3067\u56de\u8ee2\u3057\u3066\u3044\u308b\u306e\u3067\u3001\u3053\u306e\u5f71\u97ff\u3092\u8003\u3048\u306a\u304f\u3066\u306f\u306a\u3089\u306a\u3044\u3002
\n\u3000\u89d2\u901f\u5ea6\u30d9\u30af\u30c8\u30eb\u306e\u5b9a\u7fa9\u3092\u8003\u3048\u308c\u3070\u3001
\n$$\\begin{eqnarray}\\frac{\\mathrm d\\boldsymbol i}{\\mathrm dt}&=&\\boldsymbol \\omega \\times\\boldsymbol i\\\\
\n\\frac{\\mathrm d\\boldsymbol j}{\\mathrm dt}&=&\\boldsymbol \\omega \\times\\boldsymbol j\\\\
\n\\frac{\\mathrm d\\boldsymbol k}{\\mathrm dt}&=&\\boldsymbol \\omega \\times\\boldsymbol k\\end{eqnarray}$$
\n\u3067\u3042\u308b\u306e\u3067\u3001$$\\frac{\\mathrm d\\mathscr V}{\\mathrm dt}\\boldsymbol r=\\boldsymbol \\omega\\times(\\mathscr V\\boldsymbol r)$$
\n\u304c\u6210\u308a\u7acb\u3064\u3002\u3053\u308c\u3088\u308a$$\\mathscr W\\frac{\\mathrm d\\boldsymbol R}{\\mathrm dt}=\\boldsymbol \\omega\\times(\\mathscr V\\boldsymbol r)+\\mathscr V\\frac{\\mathrm d\\boldsymbol r}{\\mathrm dt}\\quad\\cdots(3)$$\u3068\u306a\u308b\u3002
\n\u3000(3)\u5f0f\u306e\u5de6\u8fba\u306f\u6163\u6027\u5ea7\u6a19\u7cfb\u304b\u3089\u898b\u305f\u7269\u4f53\u306e\u901f\u5ea6\\(\\boldsymbol V\\)\u3067\u3042\u308b\u3002\u4e00\u65b9\u3067\u3001\u53f3\u8fba\u7b2c\u4e00\u9805\u306f\u56de\u8ee2\u5ea7\u6a19\u7cfb\u306e\u56de\u8ee2\u306b\u3088\u3063\u3066\u751f\u3058\u308b\u901f\u5ea6\u3067\u3042\u308a\u3001\u53f3\u8fba\u7b2c\u4e8c\u9805\u304c\u56de\u8ee2\u5ea7\u6a19\u7cfb\u304b\u3089\u898b\u305f\u7269\u4f53\u306e\u901f\u5ea6\\(\\boldsymbol v\\)\u3068\u306a\u308b\u3002\u3053\u3053\u307e\u3067\u306f\u76f4\u611f\u7684\u306b\u3082\u308f\u304b\u308a\u3084\u3059\u3044\u3002<\/p>\n

\u30fb\u52a0\u901f\u5ea6\u306b\u95a2\u3059\u308b\u95a2\u4fc2\u5f0f<\/h4>\n

\u3000(3)\u5f0f\u306e\u4e21\u8fba\u3092\u3055\u3089\u306b\u5fae\u5206\u3059\u308b\u3002
\n$$\\begin{eqnarray}
\n\\frac{\\mathrm d}{\\mathrm dt}\\left(\\mathscr W\\frac{\\mathrm d\\boldsymbol R}{\\mathrm dt}\\right)&=&\\frac{\\mathrm d}{\\mathrm dt}\\left(\\boldsymbol \\omega\\times(\\mathscr V\\boldsymbol r)+\\mathscr V\\frac{\\mathrm d\\boldsymbol r}{\\mathrm dt}\\right)\\\\
\n\\Leftrightarrow\\quad\\quad\\mathscr W\\frac{\\mathrm d^2\\boldsymbol R}{\\mathrm dt^2}&=&\\frac{\\mathrm d\\boldsymbol \\omega}{\\mathrm dt}\\times(\\mathscr V\\boldsymbol r)+\\boldsymbol \\omega\\times\\frac{\\mathrm d}{\\mathrm dt}(\\mathscr V\\boldsymbol r)+\\frac{\\mathrm d\\mathscr V}{\\mathrm dt}\\frac{\\mathrm d\\boldsymbol r}{\\mathrm dt}+\\mathscr V\\frac{\\mathrm d^2\\boldsymbol r}{\\mathrm dt^2}\\\\
\n\\Leftrightarrow\\quad\\quad\\mathscr W\\frac{\\mathrm d^2\\boldsymbol R}{\\mathrm dt^2}&=&\\frac{\\mathrm d\\boldsymbol \\omega}{\\mathrm dt}\\times(\\mathscr V\\boldsymbol r)+\\boldsymbol \\omega\\times\\left(\\boldsymbol \\omega\\times(\\mathscr V\\boldsymbol r)+\\mathscr V\\frac{\\mathrm d\\boldsymbol r}{\\mathrm dt}\\right)+\\boldsymbol\\omega\\times\\left(\\mathscr V\\frac{\\mathrm d\\boldsymbol r}{\\mathrm dt}\\right)+\\mathscr V\\frac{\\mathrm d^2\\boldsymbol r}{\\mathrm dt^2}\\\\
\n\\Leftrightarrow\\quad\\quad\\mathscr W\\frac{\\mathrm d^2\\boldsymbol R}{\\mathrm dt^2}&=&\\frac{\\mathrm d\\boldsymbol \\omega}{\\mathrm dt}\\times(\\mathscr V\\boldsymbol r)+\\boldsymbol \\omega\\times(\\boldsymbol \\omega\\times(\\mathscr V\\boldsymbol r))+2\\boldsymbol\\omega\\times\\left(\\mathscr V\\frac{\\mathrm d\\boldsymbol r}{\\mathrm dt}\\right)+\\mathscr V\\frac{\\mathrm d^2\\boldsymbol r}{\\mathrm dt^2}\\\\
\n\\end{eqnarray}$$
\n\u3000\u5de6\u8fba\u306f1\u9805\u3057\u304b\u306a\u3044\u306e\u306b\u5bfe\u3057\u53f3\u8fba\u306f\u9805\u304c4\u3064\u3082\u3042\u308b\u3002\u3082\u3046\u306a\u306b\u304c\u306a\u3093\u3060\u304b\u308f\u304b\u3089\u306a\u3044\u304c\u3001\u3055\u3089\u306b\u4e21\u8fba\u306b\u7269\u4f53\u306e\u8cea\u91cf\\(m\\)\u3092\u304b\u3051\u3066\u307f\u3088\u3046\u3002
\n$$\\mathscr Wm\\frac{\\mathrm d^2\\boldsymbol R}{\\mathrm dt^2}=m\\frac{\\mathrm d\\boldsymbol \\omega}{\\mathrm dt}\\times(\\mathscr V\\boldsymbol r)+m\\boldsymbol \\omega\\times(\\boldsymbol \\omega\\times(\\mathscr V\\boldsymbol r))+2m\\boldsymbol\\omega\\times\\left(\\mathscr V\\frac{\\mathrm d\\boldsymbol r}{\\mathrm dt}\\right)+\\mathscr Vm\\frac{\\mathrm d^2\\boldsymbol r}{\\mathrm dt^2}$$
\n\u3000\u3055\u3089\u306b\u56de\u8ee2\u5ea7\u6a19\u7cfb\u8996\u70b9\u3067\u8003\u3048\u308b\u305f\u3081\u306b\u30b4\u30c1\u30e3\u30b4\u30c1\u30e3\u3068\u79fb\u9805\u3059\u308b\u3002
\n$$\\mathscr Vm\\frac{\\mathrm d^2\\boldsymbol r}{\\mathrm dt^2}=\\mathscr Wm\\frac{\\mathrm d^2\\boldsymbol R}{\\mathrm dt^2}-m\\frac{\\mathrm d\\boldsymbol \\omega}{\\mathrm dt}\\times(\\mathscr V\\boldsymbol r)-m\\boldsymbol \\omega\\times(\\boldsymbol \\omega\\times(\\mathscr V\\boldsymbol r))-2m\\boldsymbol\\omega\\times\\left(\\mathscr V\\frac{\\mathrm d\\boldsymbol r}{\\mathrm dt}\\right)$$
\n\u3000\u307e\u305a\u5de6\u8fba\u306e\\(m\\frac{\\mathrm d^2\\boldsymbol r}{\\mathrm dt^2}\\)\u306f\u56de\u8ee2\u5ea7\u6a19\u7cfb\u3067\u89b3\u6e2c\u3055\u308c\u305f\u7269\u4f53\u306b\u304b\u304b\u308b\u529b\\(\\boldsymbol F\\)\u3067\u3042\u308b\u3002\u4e00\u65b9\u53f3\u8fba\u7b2c\u4e00\u9805\u306e\\(m\\frac{\\mathrm d^2\\boldsymbol R}{\\mathrm dt^2}\\)\u306f\u6163\u6027\u5ea7\u6a19\u7cfb\u3067\u89b3\u6e2c\u3055\u308c\u305f\u529b\u3067\u3042\u308b\u3053\u3068\u304b\u3089\u3001\u5916\u529b\\(\\boldsymbol f\\)\u3092\u8868\u3057\u3066\u3044\u308b\u3002\u3064\u307e\u308a
\n$$\\mathscr V\\boldsymbol F=\\mathscr W\\boldsymbol f-m\\frac{\\mathrm d\\boldsymbol \\omega}{\\mathrm dt}\\times(\\mathscr V\\boldsymbol r)-m\\boldsymbol \\omega\\times(\\boldsymbol \\omega\\times(\\mathscr V\\boldsymbol r))-2m\\boldsymbol\\omega\\times\\left(\\mathscr V\\frac{\\mathrm d\\boldsymbol r}{\\mathrm dt}\\right)\\quad\\cdots(4)$$
\n\u3067\u3042\u308a\u3001\u56de\u8ee2\u5ea7\u6a19\u7cfb\u3067\u7269\u4f53\u306b\u304b\u304b\u3063\u3066\u3044\u308b\u3088\u3046\u306b\u898b\u3048\u308b\u529b\\(\\boldsymbol F\\)\u306e\u3046\u3061\u3001\u5b9f\u969b\u306b\u304b\u304b\u3063\u3066\u3044\u308b\u306e\u306f\u53f3\u8fba\u7b2c\u4e00\u9805\u306e\\(\\boldsymbol f\\)\u3060\u3051\u3067\u3001\u6b8b\u308a\u306f\u898b\u304b\u3051\u306e\u529b\u3001\u3059\u306a\u308f\u3061\u6163\u6027\u529b\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002
\n\u3000(4)\u5f0f\u53f3\u8fba\u7b2c\u4e8c\u9805\u306f\u56de\u8ee2\u5ea7\u6a19\u7cfb\u306e\u89d2\u52a0\u901f\u5ea6\\(\\frac{\\mathrm d\\boldsymbol \\omega}{\\mathrm dt}\\)\u306b\u3088\u308b\u3082\u306e\u3067\u3001\u7279\u306b\u540d\u524d\u304c\u306a\u304f\u3001\u7b49\u89d2\u901f\u5ea6\u306e\u7cfb\u3092\u6271\u3046\u3068\u304d\u306f\u7121\u8996\u3055\u308c\u3066\u3057\u307e\u3046\u53ef\u54c0\u305d\u3046\u306a\u9805\u3067\u3042\u308b\u3002
\n\u3000(4)\u5f0f\u53f3\u8fba\u7b2c\u4e09\u9805\u306f\u9060\u5fc3\u529b$$-m\\boldsymbol \\omega\\times(\\boldsymbol \\omega\\times(\\mathscr V\\boldsymbol r))$$\u3067\u3042\u308b\u3002\u5927\u304d\u3055\u306f\\(mr_\\perp\\omega^2\\)\u3067\u3042\u308a\u3001\u5411\u304d\u306f\\(\\boldsymbol\\omega\\)\u306b\u5782\u76f4\u3067\\(\\boldsymbol r_\\perp\\)\u306b\u5e73\u884c\u3067\u3042\u308b\u3002
\n\u3000\u3053\u308c\u306f\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d\u306e\u516c\u5f0f\u3092\u4f7f\u3046\u3068\u7406\u89e3\u3067\u304d\u308b\u3002$$-m\\boldsymbol \\omega\\times(\\boldsymbol \\omega\\times(\\mathscr V\\boldsymbol r))=m(\\boldsymbol \\omega\\cdot\\boldsymbol \\omega)\\mathscr V\\boldsymbol r-m(\\boldsymbol \\omega\\cdot(\\mathscr V\\boldsymbol r))\\boldsymbol \\omega\\quad\\cdots(5)$$
\n\u3000(5)\u5f0f\u53f3\u8fba\u5404\u9805\u306e\\(\\boldsymbol\\omega\\)\u306b\u5e73\u884c\u306a\u6210\u5206\u306f\u5927\u304d\u3055\u304c\u540c\u3058\\(mr_{\/\/}\\omega^2\\)\u3067\u3042\u308b\u306e\u3067\u6253\u3061\u6d88\u3057\u3042\u3044\u3001\\(\\boldsymbol r_\\perp\\)\u306b\u5e73\u884c\u306a\u6210\u5206\u306e\u307f\u304c\u6b8b\u308b\u3002
\n\"\"
\n\u3000\u305d\u3057\u3066(4)\u5f0f\u53f3\u8fba\u7b2c\u56db\u9805\u304c\u304a\u5f85\u3061\u304b\u306d\u30b3\u30ea\u30aa\u30ea\u529b$$-2m\\boldsymbol\\omega\\times\\left(\\mathscr V\\frac{\\mathrm d\\boldsymbol r}{\\mathrm dt}\\right)$$\u3067\u3042\u308b\u3002\u5927\u304d\u3055\u306f\\(2m\\omega v_\\perp\\)\u3001\u5411\u304d\u306f\\(\\boldsymbol\\omega\\)\u304a\u3088\u3073\\(\\boldsymbol v_\\perp\\)\u306b\u5782\u76f4\u3067\u3042\u308b\u3002\u4e0b\u56f3\u3067\u3044\u3046\u3068\u753b\u9762\u306b\u5bfe\u3057\u3066\u5782\u76f4\u624b\u524d\u65b9\u5411\u306b\u529b\u304c\u304b\u304b\u308b\u3053\u3068\u306b\u306a\u308b\u3002\"\"<\/p>\n

\n

\u3000\u57fa\u5e95\u3092\u660e\u793a\u7684\u306b\u3059\u308b\u3068\u5ea7\u6a19\u7cfb\u306b\u4f9d\u308b\u8996\u70b9\u306e\u9055\u3044\u3092\u3059\u3093\u306a\u308a\u7406\u89e3\u3067\u304d\u308b\u306a\u3068\u601d\u3063\u3066\u3084\u3063\u3066\u307f\u305f\u304c\u3001\u898b\u305f\u76ee\u304c\u3088\u304f\u306a\u3044\u306e\u304c\u96e3\u70b9\u3067\u3042\u308b\u3002\u3053\u306e\u5f62\u306a\u3089\u6b63\u898f\u76f4\u4ea4\u57fa\u5e95\u3067\u306a\u304f\u3066\u3082\u6210\u308a\u7acb\u3064\u306f\u305a\u3002
\n\u3000<\/p>\n","protected":false},"excerpt":{"rendered":"

\u8a66\u9a13\u671f\u9593\u304a\u306a\u3058\u307f\u306e\u9003\u907f\u8a18\u4e8b\u3002 \u30fb\u524d\u63d0 \u3000\u6163\u6027\u5ea7\u6a19\u7cfb\\(O-xyz\\)\u7cfb\u3068\u3001\u89d2\u901f\u5ea6\\(\\boldsymbol \\omega\\)\u3067\u56de\u308b\u56de\u8ee2\u5ea7\u6a19\u7cfb\\(O-x^\\prime y^\\prime z^\\prime\\)\u7cfb\u304c\u3001\u539f\u70b9\u3092\u5171\u6709<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[83],"tags":[],"_links":{"self":[{"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts\/1006"}],"collection":[{"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/comments?post=1006"}],"version-history":[{"count":26,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts\/1006\/revisions"}],"predecessor-version":[{"id":1037,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts\/1006\/revisions\/1037"}],"wp:attachment":[{"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/media?parent=1006"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/categories?post=1006"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/tags?post=1006"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}