andatoshiki/toshiki-notebook
1
0
Fork 0
mirror of https://github.com/andatoshiki/toshiki-notebook.git synced 2026-06-06 08:16:42 +00:00
Code Issues Actions Packages Projects Releases Wiki Activity
6bf3b71981
toshiki-notebook/academic/physics/ipho-formulas-jpn/4.html

47 lines
155 KiB
HTML
Raw Blame History

This file contains invisible Unicode characters

This file contains invisible Unicode characters that are indistinguishable to humans but may be processed differently by a computer. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

<!DOCTYPE html>
<html lang="en-US" dir="ltr">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<title>Formulas for IPhO 日本語版: Section 4 | Toshiki's Note</title>
<meta name="description" content="Toshiki's web notebook served via Vitepress!">
<link rel="preload stylesheet" href="/assets/style.eaf38c7a.css" as="style">
<link rel="modulepreload" href="/assets/chunks/VPAlgoliaSearchBox.6d4ee094.js">
<link rel="modulepreload" href="/assets/app.b4df1b52.js">
<link rel="modulepreload" href="/assets/academic_physics_ipho-formulas-jpn_4.md.eeb6afee.lean.js">
<link rel="stylesheet" href="https://cdnjs.toshiki.dev/ajax/libs/KaTeX/0.16.0/katex.min.css">
<link rel="stylesheet" href="https://cdnjs.toshiki.dev/ajax/libs/font-awesome/6.3.0/css/all.min.css">
<link rel="icon" href="/favicon.ico">
<meta name="author" content="Anda Toshiki">
<meta name="keywords" content="Toshiki, Anda Toshiki, andatoshiki, GitHub, GitHub action, Vitepress, Vite, Notebook, Knowledge base, Programming, Programming Notes, Academic, Personal, Notebook, Productivity, Journal, Note-taking, Markdown, Notepad, Organization, Tutorial">
<meta name="HandheldFriendly" content="True">
<meta name="MobileOptimized" content="320">
<meta name="theme-color" content="#3c8772">
<meta property="og:type" content="website">
<meta property="og:locale" content="en-US">
<meta property="og:title" content="Toshiki&#39;s Note">
<meta property="og:description" content="Toshiki&#39;s web notebook served via Vitepress!">
<meta property="og:site" content="https://note.toshiki.dev">
<meta property="og:site_name" content="Toshiki&#39;s Note">
<meta property="og:image" content="https://note.toshiki.dev/og-cover.png">
<script>function siteruntime(){window.setTimeout("siteruntime()",1e3),X=new Date("8/24/2021 10:28:00"),Y=new Date,T=Y.getTime()-X.getTime(),M=24*60*60*1e3,a=T/M,A=Math.floor(a),b=(a-A)*24,B=Math.floor(b),c=(b-B)*60,C=Math.floor((b-B)*60),D=Math.floor((c-C)*60),siteruntime_span.innerHTML="This site has been running for: "+A+" day(s) "+B+" hour(s) "+C+" minute(s) "+D+" second(s)"}siteruntime();</script>
<script async="true" defer="true" data-website-id="86de8554-d4c9-4f2b-b62a-068b71241048" src="https://umami.toshiki.dev/umami.js"></script>
<script id="check-dark-light">(()=>{const e=localStorage.getItem("vitepress-theme-appearance")||"",a=window.matchMedia("(prefers-color-scheme: dark)").matches;(!e||e==="auto"?a:e==="dark")&&document.documentElement.classList.add("dark")})();</script>
</head>
<body>
<div id="app"><div class="Layout" data-v-93a960b4><!--[--><!--]--><!--[--><span tabindex="-1" data-v-151f2593></span><a href="#VPContent" class="VPSkipLink visually-hidden" data-v-151f2593> Skip to content </a><!--]--><!----><header class="VPNav" data-v-93a960b4 data-v-0fa0e57d><div class="VPNavBar has-sidebar" data-v-0fa0e57d data-v-be450ad9><div class="container" data-v-be450ad9><div class="title" data-v-be450ad9><div class="VPNavBarTitle has-sidebar" data-v-be450ad9 data-v-6d2fb2d9><a class="title" href="/" data-v-6d2fb2d9><!--[--><!--]--><!--[--><img class="VPImage logo" src="/logos/logo.png" alt data-v-6db2186b><!--]--><!--[-->Toshiki&#39;s Note<!--]--><!--[--><!--]--></a></div></div><div class="content" data-v-be450ad9><div class="curtain" data-v-be450ad9></div><div class="content-body" data-v-be450ad9><!--[--><!--]--><div class="VPNavBarSearch search" data-v-be450ad9 style="--636b0e38:&#39;Meta&#39;;"><div id="docsearch"><button type="button" class="DocSearch DocSearch-Button" aria-label="Search"><span class="DocSearch-Button-Container"><svg class="DocSearch-Search-Icon" width="20" height="20" viewBox="0 0 20 20"><path d="M14.386 14.386l4.0877 4.0877-4.0877-4.0877c-2.9418 2.9419-7.7115 2.9419-10.6533 0-2.9419-2.9418-2.9419-7.7115 0-10.6533 2.9418-2.9419 7.7115-2.9419 10.6533 0 2.9419 2.9418 2.9419 7.7115 0 10.6533z" stroke="currentColor" fill="none" fill-rule="evenodd" stroke-linecap="round" stroke-linejoin="round"></path></svg><span class="DocSearch-Button-Placeholder">Search</span></span><span class="DocSearch-Button-Keys"><kbd class="DocSearch-Button-Key"></kbd><kbd class="DocSearch-Button-Key">K</kbd></span></button></div></div><nav aria-labelledby="main-nav-aria-label" class="VPNavBarMenu menu" data-v-be450ad9 data-v-bdedfc22><span id="main-nav-aria-label" class="visually-hidden" data-v-bdedfc22>Main Navigation</span><!--[--><!--[--><a class="VPLink link VPNavBarMenuLink" href="/development/" data-v-bdedfc22 data-v-95f5d58b data-v-30c06bd3><!--[-->Development<!--]--><!----></a><!--]--><!--[--><div class="VPFlyout VPNavBarMenuGroup active" data-v-bdedfc22 data-v-96001b6b><button type="button" class="button" aria-haspopup="true" aria-expanded="false" data-v-96001b6b><span class="text" data-v-96001b6b><!----> Academic <svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="text-icon" data-v-96001b6b><path d="M12,16c-0.3,0-0.5-0.1-0.7-0.3l-6-6c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l5.3,5.3l5.3-5.3c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-6,6C12.5,15.9,12.3,16,12,16z"></path></svg></span></button><div class="menu" data-v-96001b6b><div class="VPMenu" data-v-96001b6b data-v-e7ea1737><div class="items" data-v-e7ea1737><!--[--><!--[--><div class="VPMenuGroup" data-v-e7ea1737 data-v-b66affaf><p class="title" data-v-b66affaf>K-12</p><!--[--><!--[--><div class="VPMenuLink" data-v-b66affaf data-v-a5bbb52c><a class="VPLink link" href="/academic/chemistry/index" data-v-a5bbb52c data-v-30c06bd3><!--[-->Chemistry<!--]--><!----></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-b66affaf data-v-a5bbb52c><a class="VPLink link" href="/discrete-math/index" data-v-a5bbb52c data-v-30c06bd3><!--[-->Discrete Math.<!--]--><!----></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-b66affaf data-v-a5bbb52c><a class="VPLink link" href="/academic/literature/index" data-v-a5bbb52c data-v-30c06bd3><!--[-->Literature<!--]--><!----></a></div><!--]--><!--]--></div><!--]--><!--[--><div class="VPMenuGroup" data-v-e7ea1737 data-v-b66affaf><p class="title" data-v-b66affaf>Tools</p><!--[--><!--[--><div class="VPMenuLink" data-v-b66affaf data-v-a5bbb52c><a class="VPLink link active" href="/academic/physics/ipho-formulas-jpn/1" data-v-a5bbb52c data-v-30c06bd3><!--[-->Formulas for IPhO JPN.<!--]--><!----></a></div><!--]--><!--]--></div><!--]--><!--[--><div class="VPMenuLink" data-v-e7ea1737 data-v-a5bbb52c><span class="VPLink" data-v-a5bbb52c data-v-30c06bd3><!--[--><!--]--><!----></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-e7ea1737 data-v-a5bbb52c><span class="VPLink" data-v-a5bbb52c data-v-30c06bd3><!--[--><!--]--><!----></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-e7ea1737 data-v-a5bbb52c><span class="VPLink" data-v-a5bbb52c data-v-30c06bd3><!--[--><!--]--><!----></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-e7ea1737 data-v-a5bbb52c><span class="VPLink" data-v-a5bbb52c data-v-30c06bd3><!--[--><!--]--><!----></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-e7ea1737 data-v-a5bbb52c><span class="VPLink" data-v-a5bbb52c data-v-30c06bd3><!--[--><!--]--><!----></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-e7ea1737 data-v-a5bbb52c><span class="VPLink" data-v-a5bbb52c data-v-30c06bd3><!--[--><!--]--><!----></span></div><!--]--><!--[--><div class="VPMenuLink" data-v-e7ea1737 data-v-a5bbb52c><span class="VPLink" data-v-a5bbb52c data-v-30c06bd3><!--[--><!--]--><!----></span></div><!--]--><!--]--></div><!--[--><!--]--></div></div></div><!--]--><!--[--><div class="VPFlyout VPNavBarMenuGroup" data-v-bdedfc22 data-v-96001b6b><button type="button" class="button" aria-haspopup="true" aria-expanded="false" data-v-96001b6b><span class="text" data-v-96001b6b><!----> Application <svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="text-icon" data-v-96001b6b><path d="M12,16c-0.3,0-0.5-0.1-0.7-0.3l-6-6c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l5.3,5.3l5.3-5.3c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-6,6C12.5,15.9,12.3,16,12,16z"></path></svg></span></button><div class="menu" data-v-96001b6b><div class="VPMenu" data-v-96001b6b data-v-e7ea1737><div class="items" data-v-e7ea1737><!--[--><!--[--><div class="VPMenuGroup" data-v-e7ea1737 data-v-b66affaf><p class="title" data-v-b66affaf>Personal projects</p><!--[--><!--[--><div class="VPMenuLink" data-v-b66affaf data-v-a5bbb52c><a class="VPLink link" href="/application/markdown-it-katex/how-to-use" data-v-a5bbb52c data-v-30c06bd3><!--[-->markdown-it-katex<!--]--><!----></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-b66affaf data-v-a5bbb52c><span class="VPLink" data-v-a5bbb52c data-v-30c06bd3><!--[--><!--]--><!----></span></div><!--]--><!--]--></div><!--]--><!--]--></div><!--[--><!--]--></div></div></div><!--]--><!--[--><div class="VPFlyout VPNavBarMenuGroup" data-v-bdedfc22 data-v-96001b6b><button type="button" class="button" aria-haspopup="true" aria-expanded="false" data-v-96001b6b><span class="text" data-v-96001b6b><!----> Save <svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="text-icon" data-v-96001b6b><path d="M12,16c-0.3,0-0.5-0.1-0.7-0.3l-6-6c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l5.3,5.3l5.3-5.3c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-6,6C12.5,15.9,12.3,16,12,16z"></path></svg></span></button><div class="menu" data-v-96001b6b><div class="VPMenu" data-v-96001b6b data-v-e7ea1737><div class="items" data-v-e7ea1737><!--[--><!--[--><div class="VPMenuLink" data-v-e7ea1737 data-v-a5bbb52c><a class="VPLink link" href="/save/reading/index" data-v-a5bbb52c data-v-30c06bd3><!--[-->Reading<!--]--><!----></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-e7ea1737 data-v-a5bbb52c><a class="VPLink link" href="/academic/vocabulary/index" data-v-a5bbb52c data-v-30c06bd3><!--[-->Vocabulary<!--]--><!----></a></div><!--]--><!--]--></div><!--[--><!--]--></div></div></div><!--]--><!--]--></nav><!----><div class="VPNavBarAppearance appearance" data-v-be450ad9 data-v-da3f667a><button class="VPSwitch VPSwitchAppearance" type="button" role="switch" aria-label="toggle dark mode" aria-checked="false" data-v-da3f667a data-v-0d529b6d data-v-f3c41672><span class="check" data-v-f3c41672><span class="icon" data-v-f3c41672><!--[--><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="sun" data-v-0d529b6d><path d="M12,18c-3.3,0-6-2.7-6-6s2.7-6,6-6s6,2.7,6,6S15.3,18,12,18zM12,8c-2.2,0-4,1.8-4,4c0,2.2,1.8,4,4,4c2.2,0,4-1.8,4-4C16,9.8,14.2,8,12,8z"></path><path d="M12,4c-0.6,0-1-0.4-1-1V1c0-0.6,0.4-1,1-1s1,0.4,1,1v2C13,3.6,12.6,4,12,4z"></path><path d="M12,24c-0.6,0-1-0.4-1-1v-2c0-0.6,0.4-1,1-1s1,0.4,1,1v2C13,23.6,12.6,24,12,24z"></path><path d="M5.6,6.6c-0.3,0-0.5-0.1-0.7-0.3L3.5,4.9c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l1.4,1.4c0.4,0.4,0.4,1,0,1.4C6.2,6.5,5.9,6.6,5.6,6.6z"></path><path d="M19.8,20.8c-0.3,0-0.5-0.1-0.7-0.3l-1.4-1.4c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l1.4,1.4c0.4,0.4,0.4,1,0,1.4C20.3,20.7,20,20.8,19.8,20.8z"></path><path d="M3,13H1c-0.6,0-1-0.4-1-1s0.4-1,1-1h2c0.6,0,1,0.4,1,1S3.6,13,3,13z"></path><path d="M23,13h-2c-0.6,0-1-0.4-1-1s0.4-1,1-1h2c0.6,0,1,0.4,1,1S23.6,13,23,13z"></path><path d="M4.2,20.8c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l1.4-1.4c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-1.4,1.4C4.7,20.7,4.5,20.8,4.2,20.8z"></path><path d="M18.4,6.6c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l1.4-1.4c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-1.4,1.4C18.9,6.5,18.6,6.6,18.4,6.6z"></path></svg><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="moon" data-v-0d529b6d><path d="M12.1,22c-0.3,0-0.6,0-0.9,0c-5.5-0.5-9.5-5.4-9-10.9c0.4-4.8,4.2-8.6,9-9c0.4,0,0.8,0.2,1,0.5c0.2,0.3,0.2,0.8-0.1,1.1c-2,2.7-1.4,6.4,1.3,8.4c2.1,1.6,5,1.6,7.1,0c0.3-0.2,0.7-0.3,1.1-0.1c0.3,0.2,0.5,0.6,0.5,1c-0.2,2.7-1.5,5.1-3.6,6.8C16.6,21.2,14.4,22,12.1,22zM9.3,4.4c-2.9,1-5,3.6-5.2,6.8c-0.4,4.4,2.8,8.3,7.2,8.7c2.1,0.2,4.2-0.4,5.8-1.8c1.1-0.9,1.9-2.1,2.4-3.4c-2.5,0.9-5.3,0.5-7.5-1.1C9.2,11.4,8.1,7.7,9.3,4.4z"></path></svg><!--]--></span></span></button></div><div class="VPSocialLinks VPNavBarSocialLinks social-links" data-v-be450ad9 data-v-2ab2a029 data-v-f6988cfb><!--[--><a class="VPSocialLink" href="https://github.com/andatoshiki" target="_blank" rel="noopener" data-v-f6988cfb data-v-e57698f6><svg role="img" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><title>GitHub</title><path d="M12 .297c-6.63 0-12 5.373-12 12 0 5.303 3.438 9.8 8.205 11.385.6.113.82-.258.82-.577 0-.285-.01-1.04-.015-2.04-3.338.724-4.042-1.61-4.042-1.61C4.422 18.07 3.633 17.7 3.633 17.7c-1.087-.744.084-.729.084-.729 1.205.084 1.838 1.236 1.838 1.236 1.07 1.835 2.809 1.305 3.495.998.108-.776.417-1.305.76-1.605-2.665-.3-5.466-1.332-5.466-5.93 0-1.31.465-2.38 1.235-3.22-.135-.303-.54-1.523.105-3.176 0 0 1.005-.322 3.3 1.23.96-.267 1.98-.399 3-.405 1.02.006 2.04.138 3 .405 2.28-1.552 3.285-1.23 3.285-1.23.645 1.653.24 2.873.12 3.176.765.84 1.23 1.91 1.23 3.22 0 4.61-2.805 5.625-5.475 5.92.42.36.81 1.096.81 2.22 0 1.606-.015 2.896-.015 3.286 0 .315.21.69.825.57C20.565 22.092 24 17.592 24 12.297c0-6.627-5.373-12-12-12"/></svg></a><a class="VPSocialLink" href="https://twitter.com/andatoshiki" target="_blank" rel="noopener" data-v-f6988cfb data-v-e57698f6><svg role="img" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><title>Twitter</title><path d="M23.953 4.57a10 10 0 01-2.825.775 4.958 4.958 0 002.163-2.723c-.951.555-2.005.959-3.127 1.184a4.92 4.92 0 00-8.384 4.482C7.69 8.095 4.067 6.13 1.64 3.162a4.822 4.822 0 00-.666 2.475c0 1.71.87 3.213 2.188 4.096a4.904 4.904 0 01-2.228-.616v.06a4.923 4.923 0 003.946 4.827 4.996 4.996 0 01-2.212.085 4.936 4.936 0 004.604 3.417 9.867 9.867 0 01-6.102 2.105c-.39 0-.779-.023-1.17-.067a13.995 13.995 0 007.557 2.209c9.053 0 13.998-7.496 13.998-13.985 0-.21 0-.42-.015-.63A9.935 9.935 0 0024 4.59z"/></svg></a><!--]--></div><div class="VPFlyout VPNavBarExtra extra" data-v-be450ad9 data-v-66bb1f24 data-v-96001b6b><button type="button" class="button" aria-haspopup="true" aria-expanded="false" aria-label="extra navigation" data-v-96001b6b><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="icon" data-v-96001b6b><circle cx="12" cy="12" r="2"></circle><circle cx="19" cy="12" r="2"></circle><circle cx="5" cy="12" r="2"></circle></svg></button><div class="menu" data-v-96001b6b><div class="VPMenu" data-v-96001b6b data-v-e7ea1737><!----><!--[--><!--[--><!----><div class="group" data-v-66bb1f24><div class="item appearance" data-v-66bb1f24><p class="label" data-v-66bb1f24>Appearance</p><div class="appearance-action" data-v-66bb1f24><button class="VPSwitch VPSwitchAppearance" type="button" role="switch" aria-label="toggle dark mode" aria-checked="false" data-v-66bb1f24 data-v-0d529b6d data-v-f3c41672><span class="check" data-v-f3c41672><span class="icon" data-v-f3c41672><!--[--><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="sun" data-v-0d529b6d><path d="M12,18c-3.3,0-6-2.7-6-6s2.7-6,6-6s6,2.7,6,6S15.3,18,12,18zM12,8c-2.2,0-4,1.8-4,4c0,2.2,1.8,4,4,4c2.2,0,4-1.8,4-4C16,9.8,14.2,8,12,8z"></path><path d="M12,4c-0.6,0-1-0.4-1-1V1c0-0.6,0.4-1,1-1s1,0.4,1,1v2C13,3.6,12.6,4,12,4z"></path><path d="M12,24c-0.6,0-1-0.4-1-1v-2c0-0.6,0.4-1,1-1s1,0.4,1,1v2C13,23.6,12.6,24,12,24z"></path><path d="M5.6,6.6c-0.3,0-0.5-0.1-0.7-0.3L3.5,4.9c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l1.4,1.4c0.4,0.4,0.4,1,0,1.4C6.2,6.5,5.9,6.6,5.6,6.6z"></path><path d="M19.8,20.8c-0.3,0-0.5-0.1-0.7-0.3l-1.4-1.4c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l1.4,1.4c0.4,0.4,0.4,1,0,1.4C20.3,20.7,20,20.8,19.8,20.8z"></path><path d="M3,13H1c-0.6,0-1-0.4-1-1s0.4-1,1-1h2c0.6,0,1,0.4,1,1S3.6,13,3,13z"></path><path d="M23,13h-2c-0.6,0-1-0.4-1-1s0.4-1,1-1h2c0.6,0,1,0.4,1,1S23.6,13,23,13z"></path><path d="M4.2,20.8c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l1.4-1.4c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-1.4,1.4C4.7,20.7,4.5,20.8,4.2,20.8z"></path><path d="M18.4,6.6c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l1.4-1.4c0.4-0.4,1-0.4,1.4,0s0.4,1,0,1.4l-1.4,1.4C18.9,6.5,18.6,6.6,18.4,6.6z"></path></svg><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="moon" data-v-0d529b6d><path d="M12.1,22c-0.3,0-0.6,0-0.9,0c-5.5-0.5-9.5-5.4-9-10.9c0.4-4.8,4.2-8.6,9-9c0.4,0,0.8,0.2,1,0.5c0.2,0.3,0.2,0.8-0.1,1.1c-2,2.7-1.4,6.4,1.3,8.4c2.1,1.6,5,1.6,7.1,0c0.3-0.2,0.7-0.3,1.1-0.1c0.3,0.2,0.5,0.6,0.5,1c-0.2,2.7-1.5,5.1-3.6,6.8C16.6,21.2,14.4,22,12.1,22zM9.3,4.4c-2.9,1-5,3.6-5.2,6.8c-0.4,4.4,2.8,8.3,7.2,8.7c2.1,0.2,4.2-0.4,5.8-1.8c1.1-0.9,1.9-2.1,2.4-3.4c-2.5,0.9-5.3,0.5-7.5-1.1C9.2,11.4,8.1,7.7,9.3,4.4z"></path></svg><!--]--></span></span></button></div></div></div><div class="group" data-v-66bb1f24><div class="item social-links" data-v-66bb1f24><div class="VPSocialLinks social-links-list" data-v-66bb1f24 data-v-f6988cfb><!--[--><a class="VPSocialLink" href="https://github.com/andatoshiki" target="_blank" rel="noopener" data-v-f6988cfb data-v-e57698f6><svg role="img" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><title>GitHub</title><path d="M12 .297c-6.63 0-12 5.373-12 12 0 5.303 3.438 9.8 8.205 11.385.6.113.82-.258.82-.577 0-.285-.01-1.04-.015-2.04-3.338.724-4.042-1.61-4.042-1.61C4.422 18.07 3.633 17.7 3.633 17.7c-1.087-.744.084-.729.084-.729 1.205.084 1.838 1.236 1.838 1.236 1.07 1.835 2.809 1.305 3.495.998.108-.776.417-1.305.76-1.605-2.665-.3-5.466-1.332-5.466-5.93 0-1.31.465-2.38 1.235-3.22-.135-.303-.54-1.523.105-3.176 0 0 1.005-.322 3.3 1.23.96-.267 1.98-.399 3-.405 1.02.006 2.04.138 3 .405 2.28-1.552 3.285-1.23 3.285-1.23.645 1.653.24 2.873.12 3.176.765.84 1.23 1.91 1.23 3.22 0 4.61-2.805 5.625-5.475 5.92.42.36.81 1.096.81 2.22 0 1.606-.015 2.896-.015 3.286 0 .315.21.69.825.57C20.565 22.092 24 17.592 24 12.297c0-6.627-5.373-12-12-12"/></svg></a><a class="VPSocialLink" href="https://twitter.com/andatoshiki" target="_blank" rel="noopener" data-v-f6988cfb data-v-e57698f6><svg role="img" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><title>Twitter</title><path d="M23.953 4.57a10 10 0 01-2.825.775 4.958 4.958 0 002.163-2.723c-.951.555-2.005.959-3.127 1.184a4.92 4.92 0 00-8.384 4.482C7.69 8.095 4.067 6.13 1.64 3.162a4.822 4.822 0 00-.666 2.475c0 1.71.87 3.213 2.188 4.096a4.904 4.904 0 01-2.228-.616v.06a4.923 4.923 0 003.946 4.827 4.996 4.996 0 01-2.212.085 4.936 4.936 0 004.604 3.417 9.867 9.867 0 01-6.102 2.105c-.39 0-.779-.023-1.17-.067a13.995 13.995 0 007.557 2.209c9.053 0 13.998-7.496 13.998-13.985 0-.21 0-.42-.015-.63A9.935 9.935 0 0024 4.59z"/></svg></a><!--]--></div></div></div><!--]--><!--]--></div></div></div><!--[--><!--]--><button type="button" class="VPNavBarHamburger hamburger" aria-label="mobile navigation" aria-expanded="false" aria-controls="VPNavScreen" data-v-be450ad9 data-v-e5dd9c1c><span class="container" data-v-e5dd9c1c><span class="top" data-v-e5dd9c1c></span><span class="middle" data-v-e5dd9c1c></span><span class="bottom" data-v-e5dd9c1c></span></span></button></div></div></div></div><!----></header><div class="VPLocalNav" data-v-93a960b4 data-v-2817d72e><button class="menu" aria-expanded="false" aria-controls="VPSidebarNav" data-v-2817d72e><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="menu-icon" data-v-2817d72e><path d="M17,11H3c-0.6,0-1-0.4-1-1s0.4-1,1-1h14c0.6,0,1,0.4,1,1S17.6,11,17,11z"></path><path d="M21,7H3C2.4,7,2,6.6,2,6s0.4-1,1-1h18c0.6,0,1,0.4,1,1S21.6,7,21,7z"></path><path d="M21,15H3c-0.6,0-1-0.4-1-1s0.4-1,1-1h18c0.6,0,1,0.4,1,1S21.6,15,21,15z"></path><path d="M17,19H3c-0.6,0-1-0.4-1-1s0.4-1,1-1h14c0.6,0,1,0.4,1,1S17.6,19,17,19z"></path></svg><span class="menu-text" data-v-2817d72e>Menu</span></button><a class="top-link" href="#" data-v-2817d72e>Return to top</a></div><aside class="VPSidebar" data-v-93a960b4 data-v-c79ccefa><div class="curtain" data-v-c79ccefa></div><nav class="nav" id="VPSidebarNav" aria-labelledby="sidebar-aria-label" tabindex="-1" data-v-c79ccefa><span class="visually-hidden" id="sidebar-aria-label" data-v-c79ccefa> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="group" data-v-c79ccefa><section class="VPSidebarItem level-0 collapsible has-active" data-v-c79ccefa data-v-b05232f3><div class="item" role="button" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link" data-v-b05232f3 data-v-30c06bd3><!--[--><h2 class="text" data-v-b05232f3>IPhO Formulas: JP Ver.</h2><!--]--><!----></a><div class="caret" role="button" data-v-b05232f3><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-b05232f3><path d="M9,19c-0.3,0-0.5-0.1-0.7-0.3c-0.4-0.4-0.4-1,0-1.4l5.3-5.3L8.3,6.7c-0.4-0.4-0.4-1,0-1.4s1-0.4,1.4,0l6,6c0.4,0.4,0.4,1,0,1.4l-6,6C9.5,18.9,9.3,19,9,19z"></path></svg></div></div><div class="items" data-v-b05232f3><!--[--><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/1" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>1: 数学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/2" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>2: 一般的な推奨事</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/3" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>3: 運動学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link is-active has-active" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/4" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>4: 力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/5" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>5: 振動と波</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/6" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>6: 幾何光学，測光</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/7" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>7: 波動光学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/8" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>8: 電気回路</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/9" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>9: 電磁気学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/10" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>10: 熱力</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/11" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>11: 量子力学</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/12" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>12: Keplerの法則</p><!--]--><!----></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-b05232f3 data-v-b05232f3><div class="item" data-v-b05232f3><div class="indicator" data-v-b05232f3></div><a class="VPLink link link" href="/academic/physics/ipho-formulas-jpn/13" data-v-b05232f3 data-v-30c06bd3><!--[--><p class="text" data-v-b05232f3>13: 相対性理論</p><!--]--><!----></a><!----></div><!----></div><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-93a960b4 data-v-0bd490fb><div class="VPDoc has-sidebar has-aside" data-v-0bd490fb data-v-c5936a1e><div class="container" data-v-c5936a1e><div class="aside" data-v-c5936a1e><div class="aside-curtain" data-v-c5936a1e></div><div class="aside-container" data-v-c5936a1e><div class="aside-content" data-v-c5936a1e><div class="VPDocAside" data-v-c5936a1e data-v-cdc66372><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" data-v-cdc66372 data-v-5dd9d5f6><div class="content" data-v-5dd9d5f6><div class="outline-marker" data-v-5dd9d5f6></div><div class="outline-title" data-v-5dd9d5f6>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-5dd9d5f6><span class="visually-hidden" id="doc-outline-aria-label" data-v-5dd9d5f6> Table of Contents for current page </span><ul class="root" data-v-5dd9d5f6 data-v-1188541a><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-cdc66372></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/wechat-color.jpg"></article></a></div><!--]--></div></section><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid medium"><!--[--><div class="vp-sponsor-grid-item"><a class="vp-sponsor-grid-link" target="_blank" rel="sponsored noopener"><article class="vp-sponsor-grid-box"><h4 class="visually-hidden"></h4><img class="vp-sponsor-grid-image" src="https://cdn.jsdelivr.net/gh/maomao1996/picture/sponsor/alipay-color.jpg"></article></a></div><!--]--></div></section><!--]--></div></div><!--]--><!--]--><!--]--><!--]--></div></div></div></div><div class="content" data-v-c5936a1e><div class="content-container" data-v-c5936a1e><!--[--><!--]--><main class="main" data-v-c5936a1e><div style="position:relative;" class="vp-doc _academic_physics_ipho-formulas-jpn_4" data-v-c5936a1e><div><h1 id="formulas-for-ipho-日本語版-section-4" tabindex="-1">Formulas for IPhO 日本語版: Section 4 <a class="header-anchor" href="#formulas-for-ipho-日本語版-section-4" aria-hidden="true">#</a></h1><h2 id="_4-力学" tabindex="-1">4: 力学 <a class="header-anchor" href="#_4-力学" aria-hidden="true">#</a></h2><h3 id="_4-1-剛体の二次元的な平衡" tabindex="-1">4.1: 剛体の二次元的な平衡 <a class="header-anchor" href="#_4-1-剛体の二次元的な平衡" aria-hidden="true">#</a></h3><ol><li>剛体の二次元的な平衡 : 力についての 2 つの式とトル クについての 1 つの式. 1 （又は 2 ）個の力についての 式は 1（又は 2）個のトルクについての式で代用でき る. トルクの方が良い場合が多く，原点を適切に選択 することで「退屈な」力を消すことができる. もし 2 点 のみに力がかかっているならば，（正味の）力がかかっ ている直線は一致する. 3 点であれば, 3 つの直線は 1 点で交わる.</li></ol><h3 id="_4-2-垂直抗力" tabindex="-1">4.2: 垂直抗力 <a class="header-anchor" href="#_4-2-垂直抗力" aria-hidden="true">#</a></h3><ol start="2"><li>垂直抗力と摩擦力は 1 つの力に合成でき, 垂直抗力に 対して <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arctan</mi><mo>⁡</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">\arctan \mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">arctan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span></span></span></span> の角度で接触点に加わる.</li></ol><h3 id="_4-3-並進運動と回転運動" tabindex="-1">4.3: 並進運動と回転運動 <a class="header-anchor" href="#_4-3-並進運動と回転運動" aria-hidden="true">#</a></h3><ol start="3"><li>並進運動と回転運動についての Newton の第二法則：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold-italic">F</mi><mo>=</mo><mi>m</mi><mi mathvariant="bold-italic">a</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">M</mi><mo>=</mo><mi>I</mi><mi mathvariant="bold-italic">ε</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi mathvariant="bold-italic">M</mi><mo>=</mo><mi mathvariant="bold-italic">r</mi><mo>×</mo><mi mathvariant="bold-italic">F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\boldsymbol{F}=m \boldsymbol{a}, \boldsymbol{M}=I \boldsymbol{\varepsilon} \quad(\boldsymbol{M}=\boldsymbol{r} \times \boldsymbol{F}) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord"><span class="mord boldsymbol">a</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.11424em;">M</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mord"><span class="mord"><span class="mord boldsymbol">ε</span></span></span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.11424em;">M</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="mclose">)</span></span></span></span></span></p> 二次元の場合には <span class="katex"><span class="katex-mathml"><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 class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ε</mi></mrow><annotation encoding="application/x-tex">\varepsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ε</span></span></span></span> は本質的にスカラーで, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mo>=</mo><mi>F</mi><mi>l</mi><mo>=</mo><msub><mi>F</mi><mi>t</mi></msub><mi>r</mi><mo stretchy="false">(</mo><mi>l</mi></mrow><annotation encoding="application/x-tex">M=F l=F_t r(l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">Fl</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span> は力のうでの長さ <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mclose">)</span></span></span></span></li></ol><h3 id="_4-4-一般化座標" tabindex="-1">4.4: 一般化座標 <a class="header-anchor" href="#_4-4-一般化座標" aria-hidden="true">#</a></h3><ol start="4"><li>一般化座標. 系の状態が 1 つの変数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ξ</mi></mrow><annotation encoding="application/x-tex">\xi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span></span></span> とその時間微分 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>ξ</mi><mo>˙</mo></mover></mrow><annotation encoding="application/x-tex">\dot{\xi}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1257em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0278em;"><span class="mord">˙</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span></span></span></span> で表され，ポテンシャルエネルギーが <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U=U(\xi)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span></span></span></span>, 運動エネルギーが <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo>=</mo><mi>μ</mi><msup><mi>ξ</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">K=\mu \xi^2 / 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal">μ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/2</span></span></span></span> であるならば, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mover accent="true"><mi>ξ</mi><mo>¨</mo></mover><mo>=</mo></mrow><annotation encoding="application/x-tex">\mu \ddot{\xi}=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1257em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1389em;"><span class="mord">¨</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi mathvariant="normal">d</mi><mi>U</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>ξ</mi></mrow><annotation encoding="application/x-tex">-\mathrm{d} U(\xi) / \mathrm{d} \xi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span></span></span>. (したがって並進運動では, 力はポテン シャルエネルギーの微分)</li></ol><h3 id="_4-5-系質点" tabindex="-1">4.5: 系質点 <a class="header-anchor" href="#_4-5-系質点" aria-hidden="true">#</a></h3><ol start="5"><li>系が質点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>m</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">m_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> で構成されているとき：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><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"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi mathvariant="bold-italic">r</mi><mi>c</mi></msub><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mi mathvariant="normal">/</mi><mo>∑</mo><msub><mi>m</mi><mi>j</mi></msub><mo separator="true">,</mo><mi mathvariant="bold-italic">P</mi><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="bold-italic">L</mi><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>×</mo><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub><mo separator="true">,</mo><mi>K</mi><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msubsup><mi>v</mi><mi>i</mi><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mn>2</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>I</mi><mi>z</mi></msub><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>y</mi><mi>i</mi><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mo>=</mo><mo>∫</mo><mrow><mo fence="true">(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo fence="true">)</mo></mrow><mi mathvariant="normal">d</mi><mi>m</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; \boldsymbol{r}_c=\sum m_i \boldsymbol{r}_i / \sum m_j, \boldsymbol{P}=\sum m_i \boldsymbol{v}_i \\ &amp; \boldsymbol{L}=\sum m_i \boldsymbol{r}_i \times \boldsymbol{v}_i, K=\sum m_i v_i^2 / 2 \\ &amp; I_z=\sum m_i\left(x_i^2+y_i^2\right)=\int\left(x^2+y^2\right) \mathrm{d} m \\ &amp; \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.8223em;vertical-align:-3.6611em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1611em;"><span style="top:-6.4711em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"></span></span><span style="top:-4.5711em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"></span></span><span style="top:-2.3611em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"></span></span><span style="top:-0.3589em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6611em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1611em;"><span style="top:-6.4711em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.5711em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/2</span></span></span><span style="top:-2.3611em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathrm">d</span><span class="mord mathnormal">m</span></span></span><span style="top:-0.3589em;"><span class="pstrut" style="height:3.36em;"></span><span class="mord"><span class="mord"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6611em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_4-6-質量中心の速度" tabindex="-1">4.6: 質量中心の速度 <a class="header-anchor" href="#_4-6-質量中心の速度" aria-hidden="true">#</a></h3><ol start="6"><li>質量中心の速度が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> であるような系 (添え字 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span> は質量 中心についての物理量であることを示す）:<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">L</mi><mo>=</mo><msub><mi mathvariant="bold-italic">L</mi><mi>c</mi></msub><mo>+</mo><msub><mi>M</mi><mi mathvariant="normal">Σ</mi></msub><msub><mi mathvariant="bold-italic">R</mi><mi>c</mi></msub><mo>×</mo><msub><mi mathvariant="bold-italic">v</mi><mi>c</mi></msub><mo separator="true">,</mo><mi>K</mi><mo>=</mo><msub><mi>K</mi><mi>c</mi></msub><mo>+</mo><msub><mi>M</mi><mi mathvariant="normal">Σ</mi></msub><msubsup><mi>v</mi><mi>c</mi><mn>2</mn></msubsup><mi mathvariant="normal">/</mi><mn>2</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">P</mi><mo>=</mo><msub><mi mathvariant="bold-italic">P</mi><mi>c</mi></msub><mo>+</mo><msub><mi>M</mi><mi mathvariant="normal">Σ</mi></msub><msub><mi mathvariant="bold-italic">v</mi><mi>c</mi></msub><mi mathvariant="normal">.</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{gathered} \boldsymbol{L}=\boldsymbol{L}_c+M_{\Sigma} \boldsymbol{R}_c \times \boldsymbol{v}_c, K=K_c+M_{\Sigma} v_c^2 / 2 \\ \boldsymbol{P}=\boldsymbol{P}_c+M_{\Sigma} \boldsymbol{v}_c . \end{gathered} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0241em;vertical-align:-1.2621em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.8979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.00421em;">R</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">/2</span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_4-7-steiner-定理" tabindex="-1">4.7: Steiner 定理 <a class="header-anchor" href="#_4-7-steiner-定理" aria-hidden="true">#</a></h3><ol start="7"><li>Steiner の定理（平行軸の定理）も同じような形で は質量中心の回転軸からの距離）:<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>I</mi><mo>=</mo><msub><mi>I</mi><mi>c</mi></msub><mo>+</mo><mi>m</mi><msup><mi>b</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">I=I_c+m b^2 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_4-8-ポイント-6" tabindex="-1">4.8: ポイント 6 <a class="header-anchor" href="#_4-8-ポイント-6" aria-hidden="true">#</a></h3><ol start="8"><li>ポイント 6 の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">P</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{P}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span></span></span></span> と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">L</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span></span></span></span> を用いて, Newton の第二法則 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="bold-italic">F</mi><mi mathvariant="normal">Σ</mi></msub><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">P</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">M</mi><mi mathvariant="normal">Σ</mi></msub><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="bold-italic">L</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{F}_{\Sigma}=\mathrm{d} \boldsymbol{P} / \mathrm{d} t, \boldsymbol{M}_{\Sigma}=\mathrm{d} \boldsymbol{L} / \mathrm{d} t </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8361em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">F</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.15972em;">P</span></span></span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.11424em;">M</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathrm">d</span><span class="mord"><span class="mord"><span class="mord boldsymbol">L</span></span></span><span class="mord">/</span><span class="mord mathrm">d</span><span class="mord mathnormal">t</span></span></span></span></span></p></li></ol><h3 id="_4-9-ポイント-5" tabindex="-1">4.9: ポイント 5 <a class="header-anchor" href="#_4-9-ポイント-5" aria-hidden="true">#</a></h3><ol start="9"><li>ポイント 5 にに加えて，質量中心を通る <span class="katex"><span class="katex-mathml"><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 class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span> 軸に対する慣性モ一メントは <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mrow><mi>z</mi><mn>0</mn></mrow></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">I_{z 0}=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mo>∑</mo><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><msub><mi>m</mi><mi>i</mi></msub><msub><mi>m</mi><mi>j</mi></msub><mrow><mo fence="true">[</mo><msup><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>x</mi><mi>j</mi></msub><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo fence="true">(</mo><msub><mi>y</mi><mi>i</mi></msub><mo>−</mo><msub><mi>y</mi><mi>j</mi></msub><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo fence="true">]</mo></mrow><mi mathvariant="normal">/</mi><mrow><mo fence="true">(</mo><mn>2</mn><msub><mi>M</mi><mi mathvariant="normal">Σ</mi></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\sum_{i, j} m_i m_j\left[\left(x_i-x_j\right)^2+\left(y_i-y_j\right)^2\right] /\left(2 M_{\Sigma}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8em;vertical-align:-0.65em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.162em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">[</span></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Σ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span></li></ol><h3 id="_4-10-原点に対する慣性" tabindex="-1">4.10: 原点に対する慣性 <a class="header-anchor" href="#_4-10-原点に対する慣性" aria-hidden="true">#</a></h3><ol start="10"><li>原点に対する慣性モ一メント <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi><mo>=</mo><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub><msubsup><mi>r</mi><mi>i</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\theta=\sum m_i r_i^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0728em;vertical-align:-0.2587em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4413em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> は, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>θ</mi><mo>=</mo><msub><mi>I</mi><mi>x</mi></msub><mo>+</mo><msub><mi>I</mi><mi>y</mi></msub><mo>+</mo><msub><mi>I</mi><mi>z</mi></msub></mrow><annotation encoding="application/x-tex">2 \theta=I_x+I_y+I_z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> を用いることで二次元物体や等 方性のある物体の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mi>z</mi></msub></mrow><annotation encoding="application/x-tex">I_z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> を計算するのに有用.</li></ol><h3 id="_4-11-相当単振子の長" tabindex="-1">4.11: 相当単振子の長 <a class="header-anchor" href="#_4-11-相当単振子の長" aria-hidden="true">#</a></h3><ol start="11"><li>相当単振子の長さが <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>l</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{l}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9313em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.6134em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span></span></span></span> である物理振子 :<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><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"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msup><mi>ω</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mi mathvariant="normal">/</mi><mrow><mo fence="true">(</mo><mi>l</mi><mo>+</mo><msub><mi>I</mi><mi>c</mi></msub><mi mathvariant="normal">/</mi><mi>m</mi><mi>l</mi><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><mi>ω</mi><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><mover accent="true"><mi>l</mi><mo>~</mo></mover><mo>−</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mi>g</mi><mi mathvariant="normal">/</mi><mover accent="true"><mi>l</mi><mo>~</mo></mover></mrow></msqrt><mo separator="true">,</mo><mspace width="1em"></mspace><mover accent="true"><mi>l</mi><mo>~</mo></mover><mo>=</mo><mi>l</mi><mo>+</mo><msub><mi>I</mi><mi>c</mi></msub><mi mathvariant="normal">/</mi><mi>m</mi><mi>l</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} &amp; \omega^2(l)=g /\left(l+I_c / m l\right) \\ &amp; \omega(l)=\omega(\tilde{l}-l)=\sqrt{g / \tilde{l}}, \quad \tilde{l}=l+I_c / m l \\ &amp; \end{aligned} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.1641em;vertical-align:-2.3321em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.8321em;"><span style="top:-5.3425em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"></span></span><span style="top:-3.3079em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"></span></span><span style="top:-1.7025em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.3321em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.8321em;"><span style="top:-5.3425em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span><span style="top:-3.3079em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.6134em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3745em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord">/</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.6134em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span></span></span><span style="top:-3.3345em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
l0 -0
c4,-6.7,10,-10,18,-10 H400000v40
H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7
s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744
c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30
c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722
c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5
c53.7,-170.3,84.5,-266.8,92.5,-289.5z
M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4655em;"><span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span><span style="top:-3.6134em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathnormal">m</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span><span style="top:-1.7025em;"><span class="pstrut" style="height:3.3745em;"></span><span class="mord"><span class="mord"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.3321em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ol><h3 id="_4-12-慣性モーメントの係数" tabindex="-1">4.12: 慣性モーメントの係数 <a class="header-anchor" href="#_4-12-慣性モーメントの係数" aria-hidden="true">#</a></h3><ol start="12"><li>慣性モーメントの係数 : 円柱 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 球 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>2</mn><mn>5</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{2}{5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 球殼 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{2}{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 棒 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>12</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{12}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> （端に対しては <span class="katex"><span class="katex-mathml"><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 class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> ), 正方形 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{6}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li></ol><h3 id="_4-13-よく使われる保存則" tabindex="-1">4.13: よく使われる保存則 <a class="header-anchor" href="#_4-13-よく使われる保存則" aria-hidden="true">#</a></h3><ol start="13"><li>よく使われる保存則：エネルギー（弾性衝突，摩擦な し), 運動量（正味の外力なし, 各方向について成立), 角運動量（正味の外トルクなし, 例えば, 外力のうでの 長さが 0 (これが 2 又は 3 点のまわりに成り立てば運 動量保存で代用できる))</li></ol><h3 id="_4-14-非慣性系における見かけの力" tabindex="-1">4.14: 非慣性系における見かけの力 <a class="header-anchor" href="#_4-14-非慣性系における見かけの力" aria-hidden="true">#</a></h3><ol start="14"><li>非慣性系における見かけの力 : 慣性力 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>m</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">-m a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord mathnormal">ma</span></span></span></span>, 遠心力 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><msup><mi>ω</mi><mn>2</mn></msup><mi mathvariant="bold-italic">R</mi></mrow><annotation encoding="application/x-tex">m \omega^2 \boldsymbol{R}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.00421em;">R</span></span></span></span></span></span>, Coriolis 力 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>m</mi><mi mathvariant="bold-italic">v</mi><mo>×</mo><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">2 m \boldsymbol{v} \times \Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">m</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span> (避けた方がよい. 速 度に垂直なので仕事はしない).</li></ol><h3 id="_4-15-傾いた座標" tabindex="-1">4.15: 傾いた座標 <a class="header-anchor" href="#_4-15-傾いた座標" aria-hidden="true">#</a></h3><ol start="15"><li>傾いた座標：斜面上での運動については, 斜面に平行 と垂直な方向に軸をとるのがよい。このとき重力加速 度は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 成分と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 成分をもつ. 軸は斜交することもある が, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">v</mi><mo>=</mo><msub><mi>v</mi><mi>x</mi></msub><msub><mi mathvariant="bold-italic">e</mi><mi>x</mi></msub><mo>+</mo><msub><mi>v</mi><mi>y</mi></msub><msub><mi mathvariant="bold-italic">e</mi><mi>y</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}=v_x \boldsymbol{e}_x+v_y \boldsymbol{e}_y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7305em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">e</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> のとき <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">v_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">v</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03704em;">v</span></span></span></span></span></span> の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 軸への射 影ではない.</li></ol><h3 id="_4-16-2-つの物体の衝突" tabindex="-1">4.16: 2 つの物体の衝突 <a class="header-anchor" href="#_4-16-2-つの物体の衝突" aria-hidden="true">#</a></h3><ol start="16"><li>2 つの物体の衝突 : 保存されるのは, a) 全運動量, b) 全角運動量，c) 一方の物体の衝突点に関する角運動量, d) 全エネルギー（弾性衝突の場合， 摩擦がある場合 は, 摩擦力に垂直な方向の運動エネルギーが保存される. e) 衝突中に滑りが止まったならば，接触点の最終 速度は接触面上にある. f) 滑りが止まらなかったなら ば, 一方の物体から他方に伝わる運動量は, 接触面の 法線と <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arctan</mi><mo>⁡</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">\arctan \mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">arctan</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">μ</span></span></span></span> の角度をなす.</li></ol><h3 id="_4-17-剛体のすべての運動" tabindex="-1">4.17: 剛体のすべての運動 <a class="header-anchor" href="#_4-17-剛体のすべての運動" aria-hidden="true">#</a></h3><ol start="17"><li>剛体のすべての運動は (物体の各点の速度を見ると) 瞬 間回転中心 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> まわりの回転として表せる. 物体上の点 <span class="katex"><span class="katex-mathml"><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 class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> の <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> からの距離は <span class="katex"><span class="katex-mathml"><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 class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> の軌跡の曲率半径とは異なる ことに注意せよ.</li></ol><h3 id="_4-18-紐の張力" tabindex="-1">4.18: 紐の張力 <a class="header-anchor" href="#_4-18-紐の張力" aria-hidden="true">#</a></h3><ol start="18"><li>紐の張力：重さのある吊り紐では, 張力の水平成分は 一定で垂直成分は下にある紐の重さにより変わる. 滑 らかな面の上の紐による（単位長さあたりの）力は，そ の曲率半径と張力で決まり, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mi>T</mi><mi mathvariant="normal">/</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">N=T / R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span>. 似た場合と して, 表面張力による圧力は <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn><mi>σ</mi><mi mathvariant="normal">/</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">p=2 \sigma / R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span>. 導出には直 径に沿った圧力を調ベる.</li></ol><h3 id="_4-19-液体の表面" tabindex="-1">4.19: 液体の表面 <a class="header-anchor" href="#_4-19-液体の表面" aria-hidden="true">#</a></h3><ol start="19"><li>液体の表面は（表面張力を無視すれば）等ポテンシャ ル面になる. 非圧縮性流体では, <span class="katex"><span class="katex-mathml"><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 class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span></span></span></span> をポテンシャルエ ネルギーの体積密度として, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><msub><mi>P</mi><mn>0</mn></msub><mo>−</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">p=P_0-w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span></span></span></span>.</li></ol><h3 id="_4-20-非圧縮性流体に対する-bernoulli-の法則" tabindex="-1">4.20: 非圧縮性流体に対する Bernoulli の法則 <a class="header-anchor" href="#_4-20-非圧縮性流体に対する-bernoulli-の法則" aria-hidden="true">#</a></h3><ol start="20"><li>非圧縮性流体に対する Bernoulli の法則：<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>ρ</mi><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mi>ρ</mi><mi>ϕ</mi><mo>=</mo><mtext> const. </mtext></mrow><annotation encoding="application/x-tex">p+\frac{1}{2} \rho v^2+\rho \phi=\text { const. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">ρ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> const. </span></span></span></span></span></span></p> 一様な重力場では <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo>=</mo><mi>g</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">\phi=g h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">h</span></span></span></span>. 比熱が <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>p</mi></msub><mo stretchy="false">[</mo><mrow><mtext> </mtext><mi mathvariant="normal">J</mi></mrow><mi mathvariant="normal">/</mi><mrow><mi mathvariant="normal">k</mi><mi mathvariant="normal">g</mi></mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">c_p[\mathrm{~J} / \mathrm{kg}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">J</span></span><span class="mord">/</span><span class="mord"><span class="mord mathrm" style="margin-right:0.01389em;">kg</span></span><span class="mclose">]</span></span></span></span> である気 体では,<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><msub><mi>c</mi><mi>p</mi></msub><mi>T</mi><mo>=</mo><mtext> const. </mtext></mrow><annotation encoding="application/x-tex">\frac{1}{2} v^2+c_p T=\text { const. } </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord text"><span class="mord"> const. </span></span></span></span></span></span></p></li></ol><h3 id="_4-21-直線的な流線" tabindex="-1">4.21: 直線的な流線 <!----> <a class="header-anchor" href="#_4-21-直線的な流線" aria-hidden="true">#</a></h3><ol start="21"><li>直線的な流線に沾う運動量の連続性 : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>+</mo><mi>ρ</mi><msup><mi>v</mi><mn>2</mn></msup><mo>=</mo></mrow><annotation encoding="application/x-tex">p+\rho v^2=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span></span> const.</li></ol><h3 id="_4-22-断熱不変量" tabindex="-1">4.22: 断熱不変量 <!----> <a class="header-anchor" href="#_4-22-断熱不変量" aria-hidden="true">#</a></h3><ol start="22"><li>断熱不変量 : 振動する系の 1 周期の間のパラメータの 相対的な変化が小さければ，位相空間（ <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>−</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">x-p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> 座標で表さ れる）上に書かれるループの面積は非常に高い精度で 保存される.</li></ol><h3 id="_4-23-安定性" tabindex="-1">4.23: 安定性 <a class="header-anchor" href="#_4-23-安定性" aria-hidden="true">#</a></h3><ol start="23"><li>安定性を調ベるには, a) ポテンシャルエネルギー最小 の原理，又は b) 仮想仕事の原理を用いる.</li></ol><h3 id="_4-24-空間的に有限な運動に対する-virial-定理" tabindex="-1">4.24: 空間的に有限な運動に対する Virial 定理 <!----> <a class="header-anchor" href="#_4-24-空間的に有限な運動に対する-virial-定理" aria-hidden="true">#</a></h3><ol start="24"><li>空間的に有限な運動に対する Virial 定理：a) もし <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>∝</mo><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">r</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">F \propto|\boldsymbol{r}|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mord">∣</span></span></span></span> ならば <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>K</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>U</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle K\rangle=\langle U\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mclose">⟩</span></span></span></span> (時間平均). b) もし <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>∝</mo><mi mathvariant="normal">∣</mi><mi mathvariant="bold-italic">r</mi><msup><mi mathvariant="normal">∣</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">F \propto|\boldsymbol{r}|^{-2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.03194em;">r</span></span></span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span> ならば <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo stretchy="false">⟨</mo><mi>K</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo>−</mo><mo stretchy="false">⟨</mo><mi>U</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">2\langle K\rangle=-\langle U\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mclose">⟩</span></span></span></span>.</li></ol><h3 id="_4-25-tsiolkovsky-公式" tabindex="-1">4.25: Tsiolkovsky 公式 <a class="header-anchor" href="#_4-25-tsiolkovsky-公式" aria-hidden="true">#</a></h3><ol start="25"><li>Tsiolkovsky の公式（ロケット）: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>v</mi><mo>=</mo><mi>u</mi><mi>ln</mi><mo>⁡</mo><mfrac><mi>M</mi><mi>m</mi></mfrac></mrow><annotation encoding="application/x-tex">\Delta v=u \ln \frac{M}{m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2173em;vertical-align:-0.345em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></li></ol></div></div></main><!--[--><!--[--><!--[--><!----><!--]--><!--]--><!--]--><footer class="VPDocFooter" data-v-c5936a1e data-v-e033cd21><div class="edit-info" data-v-e033cd21><div class="edit-link" data-v-e033cd21><a class="VPLink link edit-link-button" href="https://github.com/andatoshiki/toshiki-notebook/edit/master/docs/academic/physics/ipho-formulas-jpn/4.md" target="_blank" rel="noreferrer" data-v-e033cd21 data-v-30c06bd3><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 24 24" class="edit-link-icon" data-v-e033cd21><path d="M18,23H4c-1.7,0-3-1.3-3-3V6c0-1.7,1.3-3,3-3h7c0.6,0,1,0.4,1,1s-0.4,1-1,1H4C3.4,5,3,5.4,3,6v14c0,0.6,0.4,1,1,1h14c0.6,0,1-0.4,1-1v-7c0-0.6,0.4-1,1-1s1,0.4,1,1v7C21,21.7,19.7,23,18,23z"></path><path d="M8,17c-0.3,0-0.5-0.1-0.7-0.3C7,16.5,6.9,16.1,7,15.8l1-4c0-0.2,0.1-0.3,0.3-0.5l9.5-9.5c1.2-1.2,3.2-1.2,4.4,0c1.2,1.2,1.2,3.2,0,4.4l-9.5,9.5c-0.1,0.1-0.3,0.2-0.5,0.3l-4,1C8.2,17,8.1,17,8,17zM9.9,12.5l-0.5,2.1l2.1-0.5l9.3-9.3c0.4-0.4,0.4-1.1,0-1.6c-0.4-0.4-1.2-0.4-1.6,0l0,0L9.9,12.5z M18.5,2.5L18.5,2.5L18.5,2.5z"></path></svg> Edit this page on GitHub<!--]--><!----></a></div><div class="last-updated" data-v-e033cd21><p class="VPLastUpdated" data-v-e033cd21 data-v-355aa5ef>Last updated: <time datetime="2023-03-24T09:25:41.000Z" data-v-355aa5ef></time></p></div></div><div class="prev-next" data-v-e033cd21><div class="pager" data-v-e033cd21><a class="pager-link prev" href="/academic/physics/ipho-formulas-jpn/3" data-v-e033cd21><span class="desc" data-v-e033cd21>Previous page</span><span class="title" data-v-e033cd21>3: 運動学</span></a></div><div class="has-prev pager" data-v-e033cd21><a class="pager-link next" href="/academic/physics/ipho-formulas-jpn/5" data-v-e033cd21><span class="desc" data-v-e033cd21>Next page</span><span class="title" data-v-e033cd21>5: 振動と波</span></a></div></div></footer><!--[--><!--]--></div></div></div></div></div><footer class="VPFooter has-sidebar" data-v-93a960b4 data-v-d24360a6><div class="container" data-v-d24360a6><p class="message" data-v-d24360a6>Wrote with <i class="heart fa fa-heart fa-xs fa-beat"></i> and <i class="coffee fa fa-coffee fa-xs" aria-hidden="true"></i> by <a href="https://toshiki.dev">Anda Toshiki</a> at <code>root@andatoshiki:/~</code></p><p class="copyright" data-v-d24360a6>Copyright © 2023-2023 <a href="https://github.com/andatoshiki">Anda Toshiki</a>, <a href="https://github.com/lolilab">LoliLab</a> and <a href="https://github.com/toshikidev">Toshiki Dev</a> present <br /><span id="siteruntime_span"></span></p></div></footer><!--[--><!--]--></div></div>
<script>__VP_HASH_MAP__ = JSON.parse("{\"academic_chemistry_index.md\":\"fa53861e\",\"academic_chemistry_problems_03-02-2.md\":\"9190de7a\",\"academic_physics_ipho-formulas-jpn_3.md\":\"ebe05286\",\"academic_literature_writing_methods-of-development.md\":\"789cc26c\",\"academic_chemistry_problems_02-20.md\":\"34de7858\",\"academic_chemistry_problems_03-02-3.md\":\"c0d95fdc\",\"academic_chemistry_problems_03-02-1.md\":\"9ed41a38\",\"academic_physics_ipho-formulas-jpn_12.md\":\"b36f34f2\",\"academic_physics_ipho-formulas-jpn_2.md\":\"5f49d39f\",\"academic_physics_ipho-formulas-jpn_4.md\":\"eeb6afee\",\"academic_vocabulary_2023_02_2023-02-27.md\":\"67fda3b7\",\"academic_vocabulary_index.md\":\"bd7d9e30\",\"academic_literature_index.md\":\"cf2b2a11\",\"academic_physics_ipho-formulas-jpn_5.md\":\"8fecd163\",\"academic_physics_ipho-formulas-jpn_6.md\":\"0fa33499\",\"academic_physics_ipho-formulas-jpn_7.md\":\"4d700c2a\",\"academic_physics_ipho-formulas-jpn_9.md\":\"4790c4ee\",\"application_markdown-it-katex_how-to-use.md\":\"86b38519\",\"getting-started.md\":\"e767a449\",\"index.md\":\"cb83f397\",\"javascript_notes_1_1-1.md\":\"82d73735\",\"javascript_notes_1_1-2.md\":\"6a7421a9\",\"roadmap.md\":\"958d3e89\",\"save_reading_index.md\":\"3a052c6c\",\"save_reading_outliers_1.md\":\"bc528405\",\"save_reading_outliers_2.md\":\"93d4b0ec\",\"save_reading_outliers_3.md\":\"20bd5446\",\"save_reading_outliers_4.md\":\"509cd158\",\"academic_physics_ipho-formulas-jpn_11.md\":\"b8b41f81\",\"academic_chemistry_notes_12-5.md\":\"c1e408a4\",\"academic_physics_ipho-formulas-jpn_8.md\":\"6b85a478\",\"academic_physics_ipho-formulas-jpn_1.md\":\"00b662f0\",\"academic_physics_index.md\":\"735544d4\",\"academic_physics_ipho-formulas-jpn_10.md\":\"75467e68\",\"academic_physics_ipho-formulas-jpn_13.md\":\"ffe70365\",\"application_markdown-it-katex_support-function.md\":\"80d3ff38\",\"application_markdown-it-katex_support-table.md\":\"43ca6e38\"}")</script>
<script type="module" async src="/assets/app.b4df1b52.js"></script>
</body>
</html>
Reference in New Issue View Git Blame Copy Permalink