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<span>创建账号</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E5%8A%9F" title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-简介" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#简介"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>简介</span> </div> </a> <ul id="toc-简介-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-单位" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#单位"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>单位</span> </div> </a> <ul id="toc-单位-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-功與能" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#功與能"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>功與能</span> </div> </a> <button aria-controls="toc-功與能-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关功與能子章节</span> </button> <ul id="toc-功與能-sublist" class="vector-toc-list"> <li id="toc-只有机械能时的情况" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#只有机械能时的情况"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>只有机械能时的情况</span> </div> </a> <ul id="toc-只有机械能时的情况-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-約束力" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#約束力"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>約束力</span> </div> </a> <ul id="toc-約束力-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-数学计算" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#数学计算"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>数学计算</span> </div> </a> <button aria-controls="toc-数学计算-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关数学计算子章节</span> </button> <ul id="toc-数学计算-sublist" class="vector-toc-list"> <li id="toc-轉矩和轉動" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#轉矩和轉動"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>轉矩和轉動</span> </div> </a> <ul id="toc-轉矩和轉動-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-力与位移" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#力与位移"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>力与位移</span> </div> </a> <ul id="toc-力与位移-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-力矩" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#力矩"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>力矩</span> </div> </a> <ul id="toc-力矩-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-功與動能" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#功與動能"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>功與動能</span> </div> </a> <button aria-controls="toc-功與動能-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关功與動能子章节</span> </button> <ul id="toc-功與動能-sublist" class="vector-toc-list"> <li id="toc-概述" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#概述"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>概述</span> </div> </a> <ul id="toc-概述-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-注释" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#注释"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>注释</span> </div> </a> <ul id="toc-注释-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>参考</span> </div> </a> <ul id="toc-参考-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目录" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="开关目录" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">开关目录</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">功</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="前往另一种语言写成的文章。102种语言可用" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-102" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">102种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Arbeid" title="Arbeid – 南非荷兰语" lang="af" hreflang="af" data-title="Arbeid" data-language-autonym="Afrikaans" data-language-local-name="南非荷兰语" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Arbeit_(Physik)" title="Arbeit (Physik) – 瑞士德语" lang="gsw" hreflang="gsw" data-title="Arbeit (Physik)" data-language-autonym="Alemannisch" data-language-local-name="瑞士德语" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%A5%E1%88%AB" title="ሥራ – 阿姆哈拉语" lang="am" hreflang="am" data-title="ሥራ" data-language-autonym="አማርኛ" data-language-local-name="阿姆哈拉语" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Treballo_mecanico" title="Treballo mecanico – 阿拉贡语" lang="an" hreflang="an" data-title="Treballo mecanico" data-language-autonym="Aragonés" data-language-local-name="阿拉贡语" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B4%D8%BA%D9%84_(%D9%81%D9%8A%D8%B2%D9%8A%D8%A7%D8%A1)" title="شغل (فيزياء) – 阿拉伯语" lang="ar" hreflang="ar" data-title="شغل (فيزياء)" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%95%E0%A6%BE%E0%A7%B0%E0%A7%8D%E0%A6%AF%E0%A7%8D%E0%A6%AF" title="কাৰ্য্য – 阿萨姆语" lang="as" hreflang="as" data-title="কাৰ্য্য" data-language-autonym="অসমীয়া" data-language-local-name="阿萨姆语" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Trabayu_(f%C3%ADsica)" title="Trabayu (física) – 阿斯图里亚斯语" lang="ast" hreflang="ast" data-title="Trabayu (física)" data-language-autonym="Asturianu" data-language-local-name="阿斯图里亚斯语" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Mexaniki_i%C5%9F" title="Mexaniki iş – 阿塞拜疆语" lang="az" hreflang="az" data-title="Mexaniki iş" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D1%80%D0%B0%D0%B1%D0%BE%D1%82%D0%B0" title="Механічная работа – 白俄罗斯语" lang="be" hreflang="be" data-title="Механічная работа" data-language-autonym="Беларуская" data-language-local-name="白俄罗斯语" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D1%8D%D1%85%D0%B0%D0%BD%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D0%BF%D1%80%D0%B0%D1%86%D0%B0" title="Мэханічная праца – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Мэханічная праца" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D1%80%D0%B0%D0%B1%D0%BE%D1%82%D0%B0" title="Механична работа – 保加利亚语" lang="bg" hreflang="bg" data-title="Механична работа" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A6%BE%E0%A6%9C_(%E0%A6%AA%E0%A6%A6%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%A5%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8)" title="কাজ (পদার্থবিজ্ঞান) – 孟加拉语" lang="bn" hreflang="bn" data-title="কাজ (পদার্থবিজ্ঞান)" data-language-autonym="বাংলা" data-language-local-name="孟加拉语" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Labour_un_nerzh" title="Labour un nerzh – 布列塔尼语" lang="br" hreflang="br" data-title="Labour un nerzh" data-language-autonym="Brezhoneg" data-language-local-name="布列塔尼语" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Rad_(fizika)" title="Rad (fizika) – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Rad (fizika)" data-language-autonym="Bosanski" data-language-local-name="波斯尼亚语" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Treball_(f%C3%ADsica)" title="Treball (física) – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Treball (física)" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%A9%D8%A7%D8%B1_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9)" title="کار (فیزیک) – 中库尔德语" lang="ckb" hreflang="ckb" data-title="کار (فیزیک)" data-language-autonym="کوردی" data-language-local-name="中库尔德语" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Pr%C3%A1ce_(fyzika)" title="Práce (fyzika) – 捷克语" lang="cs" hreflang="cs" data-title="Práce (fyzika)" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-csb mw-list-item"><a href="https://csb.wikipedia.org/wiki/Rob%C3%B2ta_(fizyka)" title="Robòta (fizyka) – 卡舒比语" lang="csb" hreflang="csb" data-title="Robòta (fizyka)" data-language-autonym="Kaszëbsczi" data-language-local-name="卡舒比语" class="interlanguage-link-target"><span>Kaszëbsczi</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%C4%95%C3%A7" title="Механикăлла ĕç – 楚瓦什语" lang="cv" hreflang="cv" data-title="Механикăлла ĕç" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什语" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Arbejde_(fysik)" title="Arbejde (fysik) – 丹麦语" lang="da" hreflang="da" data-title="Arbejde (fysik)" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Arbeit_(Physik)" title="Arbeit (Physik) – 德语" lang="de" hreflang="de" data-title="Arbeit (Physik)" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%88%CF%81%CE%B3%CE%BF_(%CF%86%CF%85%CF%83%CE%B9%CE%BA%CE%AE)" title="Έργο (φυσική) – 希腊语" lang="el" hreflang="el" data-title="Έργο (φυσική)" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Work_(physics)" title="Work (physics) – 英语" lang="en" hreflang="en" data-title="Work (physics)" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Laboro_(fiziko)" title="Laboro (fiziko) – 世界语" lang="eo" hreflang="eo" data-title="Laboro (fiziko)" data-language-autonym="Esperanto" data-language-local-name="世界语" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Trabajo_(f%C3%ADsica)" title="Trabajo (física) – 西班牙语" lang="es" hreflang="es" data-title="Trabajo (física)" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Mehaaniline_t%C3%B6%C3%B6" title="Mehaaniline töö – 爱沙尼亚语" lang="et" hreflang="et" data-title="Mehaaniline töö" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Lan_(fisika)" title="Lan (fisika) – 巴斯克语" lang="eu" hreflang="eu" data-title="Lan (fisika)" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%A9%D8%A7%D8%B1_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9)" title="کار (فیزیک) – 波斯语" lang="fa" hreflang="fa" data-title="کار (فیزیک)" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ty%C3%B6_(fysiikka)" title="Työ (fysiikka) – 芬兰语" lang="fi" hreflang="fi" data-title="Työ (fysiikka)" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Travail_d%27une_force" title="Travail d'une force – 法语" lang="fr" hreflang="fr" data-title="Travail d'une force" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Werk" title="Werk – 北弗里西亚语" lang="frr" hreflang="frr" data-title="Werk" data-language-autonym="Nordfriisk" data-language-local-name="北弗里西亚语" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Obair_(fisic)" title="Obair (fisic) – 爱尔兰语" lang="ga" hreflang="ga" data-title="Obair (fisic)" data-language-autonym="Gaeilge" data-language-local-name="爱尔兰语" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Obair_(fiosaigs)" title="Obair (fiosaigs) – 苏格兰盖尔语" lang="gd" hreflang="gd" data-title="Obair (fiosaigs)" data-language-autonym="Gàidhlig" data-language-local-name="苏格兰盖尔语" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Traballo_(f%C3%ADsica)" title="Traballo (física) – 加利西亚语" lang="gl" hreflang="gl" data-title="Traballo (física)" data-language-autonym="Galego" data-language-local-name="加利西亚语" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Obbyr_(fishig)" title="Obbyr (fishig) – 马恩语" lang="gv" hreflang="gv" data-title="Obbyr (fishig)" data-language-autonym="Gaelg" data-language-local-name="马恩语" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%91%D7%95%D7%93%D7%94_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)" title="עבודה (פיזיקה) – 希伯来语" lang="he" hreflang="he" data-title="עבודה (פיזיקה)" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80)" title="कार्य (भौतिकी) – 印地语" lang="hi" hreflang="hi" data-title="कार्य (भौतिकी)" data-language-autonym="हिन्दी" data-language-local-name="印地语" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Rad_(fizika)" title="Rad (fizika) – 克罗地亚语" lang="hr" hreflang="hr" data-title="Rad (fizika)" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Travay_(fizik)" title="Travay (fizik) – 海地克里奥尔语" lang="ht" hreflang="ht" data-title="Travay (fizik)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="海地克里奥尔语" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Mechanikai_munka" title="Mechanikai munka – 匈牙利语" lang="hu" hreflang="hu" data-title="Mechanikai munka" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%B7%D5%AD%D5%A1%D5%BF%D5%A1%D5%B6%D6%84_(%D6%86%D5%AB%D5%A6%D5%AB%D5%AF%D5%A1)" title="Աշխատանք (ֆիզիկա) – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Աշխատանք (ֆիզիկա)" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Usaha_(fisika)" title="Usaha (fisika) – 印度尼西亚语" lang="id" hreflang="id" data-title="Usaha (fisika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Vinna_(e%C3%B0lisfr%C3%A6%C3%B0i)" title="Vinna (eðlisfræði) – 冰岛语" lang="is" hreflang="is" data-title="Vinna (eðlisfræði)" data-language-autonym="Íslenska" data-language-local-name="冰岛语" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Lavoro_(fisica)" title="Lavoro (fisica) – 意大利语" lang="it" hreflang="it" data-title="Lavoro (fisica)" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BB%95%E4%BA%8B_(%E7%89%A9%E7%90%86%E5%AD%A6)" title="仕事 (物理学) – 日语" lang="ja" hreflang="ja" data-title="仕事 (物理学)" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%A3%E1%83%A8%E1%83%90%E1%83%9D%E1%83%91%E1%83%90" title="მუშაობა – 格鲁吉亚语" lang="ka" hreflang="ka" data-title="მუშაობა" data-language-autonym="ქართული" data-language-local-name="格鲁吉亚语" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D0%B6%D2%B1%D0%BC%D1%8B%D1%81" title="Механикалық жұмыс – 哈萨克语" lang="kk" hreflang="kk" data-title="Механикалық жұмыс" data-language-autonym="Қазақша" data-language-local-name="哈萨克语" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%80%E1%9E%98%E1%9F%92%E1%9E%98%E1%9E%93%E1%9F%92%E1%9E%8F_(%E1%9E%9A%E1%9E%BC%E1%9E%94%E1%9E%9C%E1%9E%B7%E1%9E%91%E1%9F%92%E1%9E%99%E1%9E%B6)" title="កម្មន្ត (រូបវិទ្យា) – 高棉语" lang="km" hreflang="km" data-title="កម្មន្ត (រូបវិទ្យា)" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="高棉语" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%95%E0%B3%86%E0%B2%B2%E0%B2%B8" title="ಕೆಲಸ – 卡纳达语" lang="kn" hreflang="kn" data-title="ಕೆಲಸ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="卡纳达语" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%BC_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="일 (물리학) – 韩语" lang="ko" hreflang="ko" data-title="일 (물리학)" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Kar_(fiz%C3%AEk)" title="Kar (fizîk) – 库尔德语" lang="ku" hreflang="ku" data-title="Kar (fizîk)" data-language-autonym="Kurdî" data-language-local-name="库尔德语" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D0%B6%D1%83%D0%BC%D1%83%D1%88" title="Механикалык жумуш – 柯尔克孜语" lang="ky" hreflang="ky" data-title="Механикалык жумуш" data-language-autonym="Кыргызча" data-language-local-name="柯尔克孜语" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Labor_(physica)" title="Labor (physica) – 拉丁语" lang="la" hreflang="la" data-title="Labor (physica)" data-language-autonym="Latina" data-language-local-name="拉丁语" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Labora_(fisica)" title="Labora (fisica) – 新共同語言" lang="lfn" hreflang="lfn" data-title="Labora (fisica)" data-language-autonym="Lingua Franca Nova" data-language-local-name="新共同語言" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/La%C3%B4_(fisica)" title="Laô (fisica) – 倫巴底文" lang="lmo" hreflang="lmo" data-title="Laô (fisica)" data-language-autonym="Lombard" data-language-local-name="倫巴底文" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Mechaninis_darbas" title="Mechaninis darbas – 立陶宛语" lang="lt" hreflang="lt" data-title="Mechaninis darbas" data-language-autonym="Lietuvių" data-language-local-name="立陶宛语" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Darbs_(fizika)" title="Darbs (fizika) – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Darbs (fizika)" data-language-autonym="Latviešu" data-language-local-name="拉脱维亚语" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B1%D0%BE%D1%82%D0%B0_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Работа (физика) – 马其顿语" lang="mk" hreflang="mk" data-title="Работа (физика)" data-language-autonym="Македонски" data-language-local-name="马其顿语" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%B5%E0%B5%83%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%BF" title="പ്രവൃത്തി – 马拉雅拉姆语" lang="ml" hreflang="ml" data-title="പ്രവൃത്തി" data-language-autonym="മലയാളം" data-language-local-name="马拉雅拉姆语" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%90%D0%B6%D0%B8%D0%BB_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA)" title="Ажил (физик) – 蒙古语" lang="mn" hreflang="mn" data-title="Ажил (физик)" data-language-autonym="Монгол" data-language-local-name="蒙古语" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0)" title="कार्य (भौतिकशास्त्र) – 马拉地语" lang="mr" hreflang="mr" data-title="कार्य (भौतिकशास्त्र)" data-language-autonym="मराठी" data-language-local-name="马拉地语" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kerja_(fizik)" title="Kerja (fizik) – 马来语" lang="ms" hreflang="ms" data-title="Kerja (fizik)" data-language-autonym="Bahasa Melayu" data-language-local-name="马来语" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%9C%E1%80%AF%E1%80%95%E1%80%BA_(%E1%80%9B%E1%80%B0%E1%80%95%E1%80%97%E1%80%B1%E1%80%92)" title="အလုပ် (ရူပဗေဒ) – 缅甸语" lang="my" hreflang="my" data-title="အလုပ် (ရူပဗေဒ)" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="缅甸语" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0)" title="कार्य (भौतिकशास्त्र) – 尼泊尔语" lang="ne" hreflang="ne" data-title="कार्य (भौतिकशास्त्र)" data-language-autonym="नेपाली" data-language-local-name="尼泊尔语" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%B5%E0%A5%87%E0%A4%B2%E0%A5%88_(%E0%A4%B8%E0%A4%A8%E0%A5%8D_%E0%A5%A7%E0%A5%AF%E0%A5%AF%E0%A5%AE%E0%A4%AF%E0%A4%BE_%E0%A4%B8%E0%A4%82%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE)" title="वेलै (सन् १९९८या संकिपा) – 尼瓦尔语" lang="new" hreflang="new" data-title="वेलै (सन् १९९८या संकिपा)" data-language-autonym="नेपाल भाषा" data-language-local-name="尼瓦尔语" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Arbeid_(natuurkunde)" title="Arbeid (natuurkunde) – 荷兰语" lang="nl" hreflang="nl" data-title="Arbeid (natuurkunde)" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Arbeid_i_fysikk" title="Arbeid i fysikk – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Arbeid i fysikk" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Arbeid_(fysikk)" title="Arbeid (fysikk) – 书面挪威语" lang="nb" hreflang="nb" data-title="Arbeid (fysikk)" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Trabalh_(fisica)" title="Trabalh (fisica) – 奥克语" lang="oc" hreflang="oc" data-title="Trabalh (fisica)" data-language-autonym="Occitan" data-language-local-name="奥克语" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Dalagaa" title="Dalagaa – 奥罗莫语" lang="om" hreflang="om" data-title="Dalagaa" data-language-autonym="Oromoo" data-language-local-name="奥罗莫语" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%B0%E0%A8%AE_(%E0%A8%AD%E0%A9%8C%E0%A8%A4%E0%A8%BF%E0%A8%95_%E0%A8%B5%E0%A8%BF%E0%A8%97%E0%A8%BF%E0%A8%86%E0%A8%A8)" title="ਕੰਮ (ਭੌਤਿਕ ਵਿਗਿਆਨ) – 旁遮普语" lang="pa" hreflang="pa" data-title="ਕੰਮ (ਭੌਤਿਕ ਵਿਗਿਆਨ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="旁遮普语" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Praca_(fizyka)" title="Praca (fizyka) – 波兰语" lang="pl" hreflang="pl" data-title="Praca (fizyka)" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Travaj" title="Travaj – 皮埃蒙特文" lang="pms" hreflang="pms" data-title="Travaj" data-language-autonym="Piemontèis" data-language-local-name="皮埃蒙特文" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Trabalho_(f%C3%ADsica)" title="Trabalho (física) – 葡萄牙语" lang="pt" hreflang="pt" data-title="Trabalho (física)" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Ruray" title="Ruray – 克丘亚语" lang="qu" hreflang="qu" data-title="Ruray" data-language-autonym="Runa Simi" data-language-local-name="克丘亚语" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Lucru_mecanic" title="Lucru mecanic – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Lucru mecanic" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%80%D0%B0%D0%B1%D0%BE%D1%82%D0%B0" title="Механическая работа – 俄语" lang="ru" hreflang="ru" data-title="Механическая работа" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%A0%E1%B1%9F%E1%B1%B9%E1%B1%A2%E1%B1%A4" title="ᱠᱟᱹᱢᱤ – 桑塔利语" lang="sat" hreflang="sat" data-title="ᱠᱟᱹᱢᱤ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="桑塔利语" class="interlanguage-link-target"><span>ᱥᱟᱱᱛᱟᱲᱤ</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%DA%AA%D9%85_(%D8%B7%D8%A8%D8%B9%D9%8A%D8%A7%D8%AA)" title="ڪم (طبعيات) – 信德语" lang="sd" hreflang="sd" data-title="ڪم (طبعيات)" data-language-autonym="سنڌي" data-language-local-name="信德语" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Rad_(fizika)" title="Rad (fizika) – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Rad (fizika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞尔维亚-克罗地亚语" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9A%E0%B7%8F%E0%B6%BB%E0%B7%8A%E0%B6%BA%E0%B6%BA_(%E0%B6%B7%E0%B7%9E%E0%B6%AD%E0%B7%92%E0%B6%9A_%E0%B7%80%E0%B7%92%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B7%80)" title="කාර්යය (භෞතික විද්‍යාව) – 僧伽罗语" lang="si" hreflang="si" data-title="කාර්යය (භෞතික විද්‍යාව)" data-language-autonym="සිංහල" data-language-local-name="僧伽罗语" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Work_(physics)" title="Work (physics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Work (physics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Pr%C3%A1ca_(fyzika)" title="Práca (fyzika) – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Práca (fyzika)" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Delo_(fizika)" title="Delo (fizika) – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Delo (fizika)" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Basa_(fundoyetsimba)" title="Basa (fundoyetsimba) – 绍纳语" lang="sn" hreflang="sn" data-title="Basa (fundoyetsimba)" data-language-autonym="ChiShona" data-language-local-name="绍纳语" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Puna_(fizik%C3%AB)" title="Puna (fizikë) – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Puna (fizikë)" data-language-autonym="Shqip" data-language-local-name="阿尔巴尼亚语" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D1%87%D0%BA%D0%B8_%D1%80%D0%B0%D0%B4" title="Механички рад – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Механички рад" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Usaha_m%C3%A9kanik" title="Usaha mékanik – 巽他语" lang="su" hreflang="su" data-title="Usaha mékanik" data-language-autonym="Sunda" data-language-local-name="巽他语" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Arbete_(fysik)" title="Arbete (fysik) – 瑞典语" lang="sv" hreflang="sv" data-title="Arbete (fysik)" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AF%87%E0%AE%B2%E0%AF%88_(%E0%AE%87%E0%AE%AF%E0%AE%B1%E0%AF%8D%E0%AE%AA%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D)" title="வேலை (இயற்பியல்) – 泰米尔语" lang="ta" hreflang="ta" data-title="வேலை (இயற்பியல்)" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AA%E0%B0%A8%E0%B0%BF" title="పని – 泰卢固语" lang="te" hreflang="te" data-title="పని" data-language-autonym="తెలుగు" data-language-local-name="泰卢固语" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%87%E0%B8%B2%E0%B8%99_(%E0%B8%9F%E0%B8%B4%E0%B8%AA%E0%B8%B4%E0%B8%81%E0%B8%AA%E0%B9%8C)" title="งาน (ฟิสิกส์) – 泰语" lang="th" hreflang="th" data-title="งาน (ฟิสิกส์)" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0%C5%9F_(fizik)" title="İş (fizik) – 土耳其语" lang="tr" hreflang="tr" data-title="İş (fizik)" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B1%D0%BE%D1%82%D0%B0_(%D1%84%D1%96%D0%B7%D0%B8%D0%BA%D0%B0)" title="Робота (фізика) – 乌克兰语" lang="uk" hreflang="uk" data-title="Робота (фізика)" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DA%A9%D8%A7%D9%85_(%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C%D8%A7%D8%AA)" title="کام (طبیعیات) – 乌尔都语" lang="ur" hreflang="ur" data-title="کام (طبیعیات)" data-language-autonym="اردو" data-language-local-name="乌尔都语" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C3%B4ng_(v%E1%BA%ADt_l%C3%BD_h%E1%BB%8Dc)" title="Công (vật lý học) – 越南语" lang="vi" hreflang="vi" data-title="Công (vật lý học)" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wo mw-list-item"><a href="https://wo.wikipedia.org/wiki/Ligg%C3%A9ey_(j%C3%ABmm)" title="Liggéey (jëmm) – 沃洛夫语" lang="wo" hreflang="wo" data-title="Liggéey (jëmm)" data-language-autonym="Wolof" data-language-local-name="沃洛夫语" class="interlanguage-link-target"><span>Wolof</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8A%9F" title="功 – 吴语" lang="wuu" hreflang="wuu" data-title="功" data-language-autonym="吴语" data-language-local-name="吴语" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%90%D7%A8%D7%91%D7%A2%D7%98_(%D7%A4%D7%99%D7%96%D7%99%D7%A7)" title="ארבעט (פיזיק) – 意第绪语" lang="yi" hreflang="yi" data-title="ארבעט (פיזיק)" data-language-autonym="ייִדיש" data-language-local-name="意第绪语" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%8A%9F" title="功 – 文言文" lang="lzh" hreflang="lzh" data-title="功" data-language-autonym="文言" 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.ambox-protection{border-left-color:#a2a9b1!important}}</style><table class="box-Expand_language plainlinks metadata ambox ambox-notice" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span class="skin-invert" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Translation_to_zh_arrow.svg/50px-Translation_to_zh_arrow.svg.png" decoding="async" width="50" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Translation_to_zh_arrow.svg/75px-Translation_to_zh_arrow.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Translation_to_zh_arrow.svg/100px-Translation_to_zh_arrow.svg.png 2x" data-file-width="60" data-file-height="20" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">此條目<b>可参照<a href="https://en.wikipedia.org/wiki/Work_(physics)" class="extiw" title="en:Work (physics)">英語維基百科</a>相應條目来扩充</b>。<span 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href="/wiki/File:Wiki_letter_w.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/44px-Wiki_letter_w.svg.png" decoding="async" width="44" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/66px-Wiki_letter_w.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/88px-Wiki_letter_w.svg.png 2x" data-file-width="44" data-file-height="44" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">此條目<b>缺少有關</b><u><a href="/w/index.php?title=%E5%8A%9F_(%E7%94%B5%E5%9C%BA)&action=edit&redlink=1" class="new" title="功 (电场)（页面不存在）">功 (电场)</a>和<a href="/w/index.php?title=%E5%8A%9F_(%E7%83%AD%E5%8A%9B%E5%AD%A6)&action=edit&redlink=1" class="new" title="功 (热力学)（页面不存在）">功 (热力学)</a></u>的信息。<span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2023年11月16日</span>)</i></small><span class="hide-when-compact"><br /><small><span class="skin-invert" style="font-family:sans-serif;font-size:105%;;background-color:transparent;;">請<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E5%8A%9F&action=edit">擴充此條目</a>相關信息。<a href="/wiki/Talk:%E5%8A%9F" title="Talk:功">討論頁</a>可能有詳細細節。</span></small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83732972"><table class="box-Refimprove plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/File:Tango-nosources.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/45px-Tango-nosources.svg.png" decoding="async" width="45" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/68px-Tango-nosources.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/90px-Tango-nosources.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">此條目<b>需要补充更多<a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源">来源</a></b>。<span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2023年11月16日</span>)</i></small><span class="hide-when-compact"><br /><small>请协助補充多方面<a href="/wiki/Wikipedia:%E5%8F%AF%E9%9D%A0%E6%9D%A5%E6%BA%90" title="Wikipedia:可靠来源">可靠来源</a>以<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E5%8A%9F&action=edit">改善这篇条目</a>，<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AF%81" class="mw-redirect" title="Wikipedia:可供查证">无法查证</a>的内容可能會因為<a href="/wiki/Template:Fact" class="mw-redirect" title="Template:Fact">异议提出</a>而被移除。<br />致使用者：请搜索一下条目的标题（来源搜索：<span class="plainlinks"><a rel="nofollow" class="external text" href="//www.google.com/search?&as_eq=wikipedia&q=%22%E5%8A%9F%22">"功"</a> — <a rel="nofollow" class="external text" href="//www.google.com/search?q=%22%E5%8A%9F%22">网页</a>、<a rel="nofollow" class="external text" href="//www.google.com/search?tbm=nws&q=&as_src=-newswire+-wire+-presswire+-PR+-press+-release+-wikipedia&q=%22%E5%8A%9F%22">新闻</a>、<a rel="nofollow" class="external text" href="//books.google.com/books?&as_brr=0&as_pub=-icon&q=%22%E5%8A%9F%22">书籍</a>、<a rel="nofollow" class="external text" href="//scholar.google.com/scholar?&q=%22%E5%8A%9F%22">学术</a>、<a rel="nofollow" class="external text" href="//www.google.com/search?tbm=isch&safe=off&q=%22%E5%8A%9F%22">图像</a></span>），以检查网络上是否存在该主题的更多可靠来源（<a href="/wiki/Wikipedia:%E5%8F%AF%E9%9D%A0%E6%9D%A5%E6%BA%90" title="Wikipedia:可靠来源">判定指引</a>）。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <p><b>功</b>（英語：<span lang="en">work</span>），也叫<b>机械功</b>，是<a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%A6" title="物理学">物理学</a>中表示當外界沿物體<a href="/wiki/%E4%BD%8D%E7%A7%BB" title="位移">位移</a>的方向施加<a href="/wiki/%E5%8A%9B" title="力">力</a>時，外界的<a href="/wiki/%E8%83%BD%E9%87%8F" title="能量">能量</a>被轉移到移動的物體身上，因為根據<a href="/wiki/%E8%83%BD%E9%87%8F%E5%AE%88%E6%81%86%E5%AE%9A%E5%BE%8B" class="mw-redirect" title="能量守恆定律">能量守恆定律</a>，兩個物體的能量總和不會增加。<sup id="cite_ref-ri_1-0" class="reference"><a href="#cite_note-ri-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>与<a href="/wiki/%E6%9C%BA%E6%A2%B0%E8%83%BD" title="机械能">机械能</a>相似的是，功也是<a href="/wiki/%E6%A0%87%E9%87%8F_(%E7%89%A9%E7%90%86%E5%AD%A6)" title="标量 (物理学)">标量</a>，意味著有多少能量被轉移<sup id="cite_ref-ri_1-1" class="reference"><a href="#cite_note-ri-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>，<a href="/wiki/%E5%9B%BD%E9%99%85%E5%8D%95%E4%BD%8D%E5%88%B6" title="国际单位制">国际单位制</a>单位为<a href="/wiki/%E7%84%A6%E8%80%B3_(%E5%8D%95%E4%BD%8D)" class="mw-redirect" title="焦耳 (单位)">焦耳</a>。 </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Baseball_pitching_motion_2004.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Baseball_pitching_motion_2004.jpg/300px-Baseball_pitching_motion_2004.jpg" decoding="async" width="300" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Baseball_pitching_motion_2004.jpg/450px-Baseball_pitching_motion_2004.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/25/Baseball_pitching_motion_2004.jpg/600px-Baseball_pitching_motion_2004.jpg 2x" data-file-width="749" data-file-height="262" /></a><figcaption>球员对球作正功，使其动能增加。 <br />通用符號：W <br />SI單位：J（焦耳）<br />基本公式：W = F×D，W = τθ</figcaption></figure> <p>“功”一词最初是法国数学家<a href="/wiki/%E8%B4%BE%E6%96%AF%E5%B8%95-%E5%8F%A4%E6%96%AF%E5%A1%94%E5%A4%AB%C2%B7%E7%A7%91%E9%87%8C%E5%A5%A5%E5%88%A9" class="mw-redirect" title="贾斯帕-古斯塔夫·科里奥利">贾斯帕-古斯塔夫·科里奥利</a>创造的<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>。 </p><p>由<a href="/wiki/%E5%8A%A8%E8%83%BD%E5%AE%9A%E7%90%86" class="mw-redirect" title="动能定理">动能定理</a>，若一个外力作用于一物体使之<a href="/wiki/%E5%8A%A8%E8%83%BD" title="动能">动能</a>从<i>E<sub>k0</sub></i>增至<i>E<sub>k</sub></i>，那么，此力所做的机械功为： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\Delta E_{\rm {k}}=E_{\rm {k}}-E_{\mathrm {k} 0}={\tfrac {1}{2}}m(v^{2}-v_{0}^{2})\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <mo stretchy="false">(</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\Delta E_{\rm {k}}=E_{\rm {k}}-E_{\mathrm {k} 0}={\tfrac {1}{2}}m(v^{2}-v_{0}^{2})\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cfddc217eb607a28f441aa6ec5e0faa60521b76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-right: -0.387ex; width:38.874ex; height:3.509ex;" alt="{\displaystyle W=\Delta E_{\rm {k}}=E_{\rm {k}}-E_{\mathrm {k} 0}={\tfrac {1}{2}}m(v^{2}-v_{0}^{2})\,\!}"></span><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></dd></dl> <p>其中<i>m</i>是物体的质量，<i>v</i>是物体的速度。 </p><p>机械功就是力与位移的<a href="/wiki/%E5%85%A7%E7%A9%8D" class="mw-redirect mw-disambig" title="內積">內積</a>： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W={\vec {F}}\cdot {\vec {d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W={\vec {F}}\cdot {\vec {d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c68528fcde091a89e77192144476aac72657d4d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.531ex; height:2.843ex;" alt="{\displaystyle W={\vec {F}}\cdot {\vec {d}}}"></span><sup id="cite_ref-R&H7-2_5-0" class="reference"><a href="#cite_note-R&H7-2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></dd></dl> <p>若力与位移的夹角小于直角，则机械功为正，亦称为力作正功。若力与位移的夹角大于直角，则机械功为负，或力做负功，或物体克服力作功。 </p><p>若力的方向与位移方向垂直，则此力不作功：<sup id="cite_ref-R&H7-2_5-1" class="reference"><a href="#cite_note-R&H7-2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ec3e6bbd4113161b63114707f5b9048bcda0c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.696ex; height:2.176ex;" alt="{\displaystyle W=0}"></span></dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="简介"><span id=".E7.AE.80.E4.BB.8B"></span>简介</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%9F&action=edit&section=1" title="编辑章节：简介"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>即使存在力，也可能没有作功。例如，在<a href="/wiki/%E5%8C%80%E9%80%9F%E5%9C%86%E5%91%A8%E8%BF%90%E5%8A%A8" class="mw-redirect" title="匀速圆周运动">匀速圆周运动</a>中，<a href="/wiki/%E5%90%91%E5%BF%83%E5%8A%9B" title="向心力">向心力</a>没有作功，因为做圆周运动的物体的动能没有发生变化。同样的，桌上的一本书，尽管桌对书有支持力，但因没有位移而没有作功。 </p> <div class="mw-heading mw-heading2"><h2 id="单位"><span id=".E5.8D.95.E4.BD.8D"></span>单位</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%9F&action=edit&section=2" title="编辑章节：单位"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E5%9B%BD%E9%99%85%E5%8D%95%E4%BD%8D%E5%88%B6" title="国际单位制">国际单位制</a>中功的单位为<a href="/wiki/%E7%84%A6%E8%80%B3" title="焦耳">焦耳</a>（J）。焦耳被定义为用1牛顿的力对一物体使其发生1米的位移所做的机械功的大小。<a href="/wiki/%E9%87%8F%E7%BA%B2" title="量纲">量纲</a>相同的单位牛·米有时也使用，但是一般牛·米用于<a href="/wiki/%E5%8A%9B%E7%9F%A9" title="力矩">力矩</a>，使其跟功和能区别开。 </p><p>非国际单位制单位包括<a href="/wiki/%E7%88%BE%E6%A0%BC" title="爾格">尔格</a>、英尺·磅、千瓦时（kW·h）、大氣壓力、马力时（HP·h）。而由於具有相同的物理量─熱能，偶爾會見到以熱量熱能形式表示的測量單位，如：卡路里（cal）、BTU等。 </p> <div class="mw-heading mw-heading2"><h2 id="功與能"><span id=".E5.8A.9F.E8.88.87.E8.83.BD"></span>功與能</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%9F&action=edit&section=3" title="编辑章节：功與能"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>功與能息息相關，根據系統能量的守恆，內部總能量的變化等於添加的熱能加上环境对系統所作的功。見<a href="/wiki/%E7%86%B1%E5%8A%9B%E5%AD%B8%E7%AC%AC%E4%B8%80%E5%AE%9A%E5%BE%8B" class="mw-redirect" title="熱力學第一定律">熱力學第一定律</a>。 </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} E=\delta Q+\delta W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>E</mi> <mo>=</mo> <mi>δ</mi> <mi>Q</mi> <mo>+</mo> <mi>δ</mi> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} E=\delta Q+\delta W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3193b93f1629c23792183555b5cd916d818d6a0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.378ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} E=\delta Q+\delta W}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="只有机械能时的情况"><span id=".E5.8F.AA.E6.9C.89.E6.9C.BA.E6.A2.B0.E8.83.BD.E6.97.B6.E7.9A.84.E6.83.85.E5.86.B5"></span>只有机械能时的情况</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%9F&action=edit&section=4" title="编辑章节：只有机械能时的情况"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>1.<a href="/wiki/%E4%BF%9D%E5%AE%88%E5%8A%9B" title="保守力">保守力</a>作功使「存」在物體中的<a href="/wiki/%E5%8A%BF%E8%83%BD" title="势能">位能</a>釋放出來，亦即<b>保守力作功等於負的位能變化</b>： </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=-\Delta U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=-\Delta U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/817c77fc4aa36cac8441f88e2646c8a82f7dbad8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.06ex; height:2.343ex;" alt="{\displaystyle W=-\Delta U}"></span> </p><p>2.非保守力作功時，<b>若有保守力作負功則優先化為位能，剩下的功才化為物體的<a href="/wiki/%E5%8B%95%E8%83%BD" class="mw-redirect" title="動能">動能</a></b>，即<b>非保守力作功等於總<a href="/wiki/%E6%A9%9F%E6%A2%B0%E8%83%BD" class="mw-redirect" title="機械能">力學能</a>（動能+位能）變化</b>： </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\Delta E=\Delta E_{\rm {k}}+\Delta U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>E</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\Delta E=\Delta E_{\rm {k}}+\Delta U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cddf4d237febf50c3ddb0fa9980c83c4543aa467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.654ex; height:2.509ex;" alt="{\displaystyle W=\Delta E=\Delta E_{\rm {k}}+\Delta U}"></span> </p><p>3.綜合以上兩點，一物體所受的合力包含了保守力與非保守力，非保守力使總力學能變，而保守力將之部份化為位能，二者相加，即<b>合力作功等於動能變化</b>： </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\Delta E_{\rm {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\Delta E_{\rm {k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/498ccffd25514ee50912d304c115fed055976e54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.285ex; height:2.509ex;" alt="{\displaystyle W=\Delta E_{\rm {k}}}"></span> </p><p>根據這些公式證明功是與作用力相關的能量，所以作功是能被測量的，是一種具有物理單位的能量。 </p><p>上面所討論的作功、能量原理也适用于非机械能，例如電器和能源等，其原理是相同的。 </p> <div class="mw-heading mw-heading2"><h2 id="約束力"><span id=".E7.B4.84.E6.9D.9F.E5.8A.9B"></span>約束力</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%9F&action=edit&section=5" title="编辑章节：約束力"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>约束力决定了系统中物件的位移，將其限制在範圍內（以斜面加重力為例，當物體受到無法再伸長的緊繩約束使其不能再下滑，物體就會<i>卡</i>在斜面上）。它消除了在該方向上所有的位移，即物體平行此力的速度被約束為0，因此<b>約束力不對系統作功</b>。 </p><p>例如：用一根绳子系上一个小球做匀速圆周运动，小球会受到来自绳子，方向指向圆心的一个向心力。这个力的方向和球速度的方向垂直，所以这个力不作功（W=0）。又如桌上有一本书，施加外力会使书在桌面上移动。如果再对书施加一个垂直的力（实际上书受到的重力和支持力就属于这个力），和其欲移动之方向垂直，则此约束力（施加的垂直力）不作功。 </p><p>磁场中的带电粒子受到磁力（洛伦兹力）的大小为F = q<b>v×B</b>，其中q为电荷，v是粒子速度，而B为磁场强度。<a href="/wiki/%E5%A4%96%E7%A7%AF" class="mw-disambig" title="外积">外积</a>结果恆垂直于兩原向量，因此F⊥v。而两垂直向量的<a href="/wiki/%E5%85%A7%E7%A9%8D" class="mw-redirect mw-disambig" title="內積">內积</a>恆零，因此磁力作的功W = 0。磁力可改变此粒子运动的方向，但是永远无法改变运动速率（磁力不作功，動能不變）。帶電粒子在均強磁場中受到與速度方向垂直，固定大小的力，因此粒子會以原來的速率作圓周運動。(質譜儀的原理) </p> <div class="mw-heading mw-heading2"><h2 id="数学计算"><span id=".E6.95.B0.E5.AD.A6.E8.AE.A1.E7.AE.97"></span>数学计算</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%9F&action=edit&section=6" title="编辑章节：数学计算"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>對於一移動的物體而言，作功量/時間可以從距離/時間（即速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span>）來計算。因此，在任何時刻，力作功的<a href="/wiki/%E5%8A%9F%E7%8E%87" title="功率">功率</a>（焦耳/秒、瓦），其值為力的純量積（矢量）和作用點上的速度矢量。力的純量積和速度被歸類為瞬時功率。 </p><p>而正如速度可能會隨著時間的推移以獲得更長的距離，同一條路徑上的總功率也同樣是作用點沿著同一條路徑上之瞬時功率的時間積分的總和。 </p><p>功是指質點受外力作用位移而產生的量，當質點移動時，它沿著曲線<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>和速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span>在所有的時間<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>。少量的功<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/973e6b089758466f338610e76ab50fd4093efbc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.484ex; height:2.343ex;" alt="{\displaystyle \delta W}"></span>發生在瞬時時間<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/588a981eb3c6f32c01153f8710a7f70029b5e553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.132ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} t}"></span>能夠寫成： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\mathbf {F} \cdot \mathrm {d} \mathbf {s} =\mathbf {F} \cdot \mathbf {v} \,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\mathbf {F} \cdot \mathrm {d} \mathbf {s} =\mathbf {F} \cdot \mathbf {v} \,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/579bb78bbc68aef087758b473ca3b4ef0072ad68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.683ex; height:2.343ex;" alt="{\displaystyle \delta W=\mathbf {F} \cdot \mathrm {d} \mathbf {s} =\mathbf {F} \cdot \mathbf {v} \,\mathrm {d} t}"></span></dd></dl> <p>其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} \cdot \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} \cdot \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4badf4af56c391dc8ee7cc299a1a900d104f300d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.773ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} \cdot \mathbf {v} }"></span>是在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/588a981eb3c6f32c01153f8710a7f70029b5e553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.132ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} t}"></span>內的瞬時功率，這些少量功的總和超過該質點運動位移所產生的功量。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} \,\mathrm {d} t=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot {\frac {\mathrm {d} \mathbf {s} }{\mathrm {d} t}}\,\mathrm {d} t=\int _{C}\mathbf {F} \cdot \mathrm {d} \mathbf {s} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} \,\mathrm {d} t=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot {\frac {\mathrm {d} \mathbf {s} }{\mathrm {d} t}}\,\mathrm {d} t=\int _{C}\mathbf {F} \cdot \mathrm {d} \mathbf {s} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b56d266040654f20a2cb8020f0cbc2a7407d8d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.976ex; height:6.509ex;" alt="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} \,\mathrm {d} t=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot {\frac {\mathrm {d} \mathbf {s} }{\mathrm {d} t}}\,\mathrm {d} t=\int _{C}\mathbf {F} \cdot \mathrm {d} \mathbf {s} }"></span></dd></dl> <p>其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>的位移是從<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} (t_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (t_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3787293467cf91fa7742a1a32d07d018671c8ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.114ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} (t_{1})}"></span>到<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} (t_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (t_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1e2640276cdb0888d722a32e57916fd7f4eec38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.114ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} (t_{2})}"></span>，計算質點位移的積分。 </p><p>如果力的方向總是沿著這條線，力的大小為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>，那麼此積分可簡化為： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{C}F\,\mathrm {d} s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mi>F</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{C}F\,\mathrm {d} s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f01cf7e1dda57bd44ca70a40842301b0d7b9d60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.205ex; height:5.676ex;" alt="{\displaystyle W=\int _{C}F\,\mathrm {d} s}"></span></dd></dl> <p>其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>是沿著直線的位移，假設<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>固定，且沿著此直線，則此積分可進一步簡化成： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{C}F\,\mathrm {d} s=F\int _{C}\mathrm {d} s=Fs}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mi>F</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> <mo>=</mo> <mi>F</mi> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> <mo>=</mo> <mi>F</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{C}F\,\mathrm {d} s=F\int _{C}\mathrm {d} s=Fs}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12d474eb27e866314533cdfa4b29633a9dc7d58d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.905ex; height:5.676ex;" alt="{\displaystyle W=\int _{C}F\,\mathrm {d} s=F\int _{C}\mathrm {d} s=Fs}"></span></dd></dl> <p>其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>是質點沿著直線前進的距離。 </p><p>此計算可歸納為恆定力並非沿著線而是沿著質點。在此情況下<a href="/wiki/%E5%85%A7%E7%A9%8D" class="mw-redirect mw-disambig" title="內積">內積</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} \cdot \mathrm {d} \mathbf {s} =F\cos \theta \,\mathrm {d} s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo>=</mo> <mi>F</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} \cdot \mathrm {d} \mathbf {s} =F\cos \theta \,\mathrm {d} s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a43e2b30dd5747b0ede32094f5304539de0e9de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.295ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} \cdot \mathrm {d} \mathbf {s} =F\cos \theta \,\mathrm {d} s}"></span>，其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>是力矢量和運動方向之間的角度。即： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{C}\mathbf {F} \cdot \mathrm {d} \mathbf {s} =Fs\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo>=</mo> <mi>F</mi> <mi>s</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{C}\mathbf {F} \cdot \mathrm {d} \mathbf {s} =Fs\cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d51b77796dcded72feb424b8329f087648fddaaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.31ex; height:5.676ex;" alt="{\displaystyle W=\int _{C}\mathbf {F} \cdot \mathrm {d} \mathbf {s} =Fs\cos \theta }"></span></dd></dl> <p>一般常見的情況，施加的力和速度矢量對物體成<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c326d317eddef3ad3e6625e018a708e290a039f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}"></span>（向心力朝下物體繞一圓圈運動），由於<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos 90^{\circ }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos 90^{\circ }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4282c3ff0ada7f2d9fce6581bc9dddd92e187337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.138ex; height:2.343ex;" alt="{\displaystyle \cos 90^{\circ }=0}"></span>，所以不作功。因此可以延伸至重力對於星球在圓形軌道上運動不作功（此為理想情況，一般情況下軌道略呈橢圓形）。 </p><p>此外物體作等速圓周運動受到機械外力作用時，作的功也為0，就像在一理想情況之無摩擦力的離心機中作等速圓周運動一般。 </p><p>計算功在時間和力作用在一直線路徑上的數值只適用在最簡單的情況下，如上文所述。如果力會變化，或物體沿曲線方向移動，物體可能轉動甚至並非剛性物體，那麼其所作的功只和作用力的角度、路徑有關，並且只有部分的力平行在作用點上形成的速度才作功（相同方向為正，反方向為負值），此處的力可以被描述為純量或是切線分量的純量。（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22de614bc8f13a6865e8ee420419b393029f0ee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.717ex; height:2.176ex;" alt="{\displaystyle F\cos \theta }"></span>，其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>是力和速度之間的夾角）。 </p><p>至於功最普遍的定義如下：力作功是其沿著作用點上的路徑之切線分量的純量也就是線性積分。 </p> <div class="mw-heading mw-heading3"><h3 id="轉矩和轉動"><span id=".E8.BD.89.E7.9F.A9.E5.92.8C.E8.BD.89.E5.8B.95"></span>轉矩和轉動</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%9F&action=edit&section=7" title="编辑章节：轉矩和轉動"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>轉矩是從相等但方向相反的力作用於剛性體上兩個不同的點所形成。這些力總和為零，但它會對物體影響形成轉矩<b>Τ</b>，計算作功形成的轉矩公式為： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\mathbf {T} \cdot {\vec {\omega }}\delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\mathbf {T} \cdot {\vec {\omega }}\delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c2fe3d8222cbe6a0423de6e9d84a34fe44f14f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.455ex; height:2.343ex;" alt="{\displaystyle \delta W=\mathbf {T} \cdot {\vec {\omega }}\delta t}"></span>，其中<b>T.ω</b>是作用在時間點δ<i>t</i>上。這些少量的功之和大於剛性體運動軌跡所產生的功。</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot {\vec {\omega }}dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot {\vec {\omega }}dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bf2bc025e4b14aef26c36e2ac986f6a3d9b5693" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.082ex; height:6.509ex;" alt="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot {\vec {\omega }}dt}"></span>，此積分是計算剛體沿軌跡運動與時間變化的角速度ω，可以說與運動的路徑息息相關。</dd></dl> <p>如果角速度矢量保持恆定的方向，那麼可以寫成： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\omega }}={\dot {\phi }}\mathbf {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}={\dot {\phi }}\mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ed502c9baf0e4b5bb1d5c6f456c56c9766c7da7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.497ex; height:3.009ex;" alt="{\displaystyle {\vec {\omega }}={\dot {\phi }}\mathbf {S} }"></span>，其中φ為轉動角度，單位矢量<b>S</b>。在此情況下，功的轉矩可寫成：</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot {\vec {\omega }}dt=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot \mathbf {S} {\frac {d\phi }{dt}}dt=\int _{C}\mathbf {T} \cdot \mathbf {S} d\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mi>d</mi> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot {\vec {\omega }}dt=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot \mathbf {S} {\frac {d\phi }{dt}}dt=\int _{C}\mathbf {T} \cdot \mathbf {S} d\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3edcbdd1d1284ea8702a3c2f32031a5139f46352" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.09ex; height:6.509ex;" alt="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot {\vec {\omega }}dt=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot \mathbf {S} {\frac {d\phi }{dt}}dt=\int _{C}\mathbf {T} \cdot \mathbf {S} d\phi }"></span>，其中<i>C</i>是從<i>φ（t<sub>1</sub>）</i>到<i>φ（t<sub>2</sub>）</i>的運動軌跡。此積分取決於<i>φ（t）</i>的值，因此與路徑相關。</dd></dl> <p>如果轉矩<b>T</b>與角速度矢量一致，那麼可寫成： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} =\tau \mathbf {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} =\tau \mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2d441df60b1bbd49975fbb60012b9664870d4c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.645ex; height:2.176ex;" alt="{\displaystyle \mathbf {T} =\tau \mathbf {S} }"></span></dd></dl> <p>而且若轉矩和角速度是恆定的，那麼功可寫成這個形式： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{t_{1}}^{t_{2}}\tau {\dot {\phi }}dt=\tau (\phi _{2}-\phi _{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{t_{1}}^{t_{2}}\tau {\dot {\phi }}dt=\tau (\phi _{2}-\phi _{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291128feff0a1d86c137a46d2457d55b728c327a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.597ex; height:6.509ex;" alt="{\displaystyle W=\int _{t_{1}}^{t_{2}}\tau {\dot {\phi }}dt=\tau (\phi _{2}-\phi _{1})}"></span>。</dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Work_on_the_lever_arm.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Work_on_the_lever_arm.svg/220px-Work_on_the_lever_arm.svg.png" decoding="async" width="220" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Work_on_the_lever_arm.svg/330px-Work_on_the_lever_arm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Work_on_the_lever_arm.svg/440px-Work_on_the_lever_arm.svg.png 2x" data-file-width="211" data-file-height="184" /></a><figcaption>一个大小不变且垂直于杠杆臂的力</figcaption></figure> <p>此結果可以更簡單的理解，如圖所示。這股力將通過圓弧的距離<i>s=rφ</i>，所作的功即是： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=Fs=Fr\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>F</mi> <mi>s</mi> <mo>=</mo> <mi>F</mi> <mi>r</mi> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=Fs=Fr\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a74ce0cb6cd92a262fef7d62ef77aded1f2aae0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.638ex; height:2.509ex;" alt="{\displaystyle W=Fs=Fr\phi }"></span>，導出轉矩<i>τ=Fr</i>，得：</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=Fr\phi =\tau \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>F</mi> <mi>r</mi> <mi>ϕ</mi> <mo>=</mo> <mi>τ</mi> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=Fr\phi =\tau \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de9a722431277130e7d998a58d12240966cb7010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.394ex; height:2.509ex;" alt="{\displaystyle W=Fr\phi =\tau \phi }"></span></dd></dl> <p>以上，請注意只有轉矩在角速度矢量方向的部分才有作功。 </p> <div class="mw-heading mw-heading3"><h3 id="力与位移"><span id=".E5.8A.9B.E4.B8.8E.E4.BD.8D.E7.A7.BB"></span>力与位移</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%9F&action=edit&section=8" title="编辑章节：力与位移"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>力与位移都是<a href="/wiki/%E7%9F%A2%E9%87%8F" class="mw-redirect" title="矢量">矢量</a>。功是力与位移的<a href="/wiki/%E5%86%85%E7%A7%AF" class="mw-disambig" title="内积">内积</a>，為<a href="/wiki/%E6%A0%87%E9%87%8F_(%E7%89%A9%E7%90%86%E5%AD%A6)" title="标量 (物理学)">标量</a>。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W={\vec {F}}\cdot {\vec {d}}=Fd\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>F</mi> <mi>d</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W={\vec {F}}\cdot {\vec {d}}=Fd\cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6342ae6708e99a43605bd4fa17ed1ea5d653d7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.562ex; height:2.843ex;" alt="{\displaystyle W={\vec {F}}\cdot {\vec {d}}=Fd\cos \theta }"></span>           （1）</dd></dl> <p>其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>是力矢量和位移矢量的<a href="/wiki/%E5%A4%BE%E8%A7%92" class="mw-redirect" title="夾角">夹角</a>。 </p><p>为使此式正确，力须为常矢量，路径须为一条直线。 </p><p>如力随时间变化或路径不为直线，上式不再适用，此时需使用<a href="/wiki/%E6%9B%B2%E7%BA%BF%E7%A7%AF%E5%88%86" title="曲线积分">曲线积分</a>。故功的一般公式为： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{C}{\vec {F}}\cdot \mathrm {d} {\vec {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{C}{\vec {F}}\cdot \mathrm {d} {\vec {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f3fb69f293e1f4ffcfb2e32d508ada042975bd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.767ex; height:5.676ex;" alt="{\displaystyle W=\int _{C}{\vec {F}}\cdot \mathrm {d} {\vec {x}}}"></span>            （2）</dd></dl> <p>其中 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle _{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle _{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9949182464d7aa62382333aa85d7ea29051867c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.481ex; height:1.676ex;" alt="{\displaystyle \textstyle _{C}}"></span>是路径</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef40edff397a115ecdce7d3518001dfcc7f37d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.843ex;" alt="{\displaystyle {\vec {F}}}"></span>是力矢量</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}}"></span>是位移矢量</dd></dl> <p>表达式<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W={\vec {F}}\cdot \mathrm {d} {\vec {s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W={\vec {F}}\cdot \mathrm {d} {\vec {s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/257ee7e20c7a7a50218c7fd356a66ae94a8c8d9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.548ex; height:2.843ex;" alt="{\displaystyle \delta W={\vec {F}}\cdot \mathrm {d} {\vec {s}}}"></span>是一个<a href="/w/index.php?title=%E9%9D%9E%E6%81%B0%E5%BD%93%E5%BE%AE%E5%88%86&action=edit&redlink=1" class="new" title="非恰当微分（页面不存在）">非恰当微分</a>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {W_{C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {W_{C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f864a1cacdad459d637e00e69f76fca92a29c2a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.675ex; height:2.509ex;" alt="{\displaystyle \textstyle {W_{C}}}"></span>与路径有关，求微分后不能得到<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}\cdot \mathrm {d} {\vec {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}\cdot \mathrm {d} {\vec {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a453c60cd4d65ecc208075387126e8859a7afecc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle {\vec {F}}\cdot \mathrm {d} {\vec {x}}}"></span>。 </p><p>非零力可以不作功，这一点与<a href="/wiki/%E5%86%B2%E9%87%8F" title="冲量">冲量</a>不同。冲量是力对时间的累积。冲量是<a href="/wiki/%E7%9F%A2%E9%87%8F" class="mw-redirect" title="矢量">矢量</a>，所以圆周运动时虽向心力不作功，但产生了对物体的非零冲量。 </p> <div class="mw-heading mw-heading3"><h3 id="力矩"><span id=".E5.8A.9B.E7.9F.A9"></span>力矩</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%9F&action=edit&section=9" title="编辑章节：力矩"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E5%8A%9B%E7%9F%A9" title="力矩">力矩</a>所作功可由下式计算得到： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\tau \theta \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>τ</mi> <mi>θ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\tau \theta \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8592ae1786b7fa5852fb263e142df00caa012da9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.407ex; height:2.176ex;" alt="{\displaystyle W=\tau \theta \ }"></span></dd></dl> <p>其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span>为力矩。 </p> <div class="mw-heading mw-heading2"><h2 id="功與動能"><span id=".E5.8A.9F.E8.88.87.E5.8B.95.E8.83.BD"></span>功與動能</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%9F&action=edit&section=10" title="编辑章节：功與動能"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>（功）動能定理</b>（The work–kinetic energy theorem）或稱功能定理（The work–energy theorem）、功能原理（The work–energy principle），意指<a href="/wiki/%E5%90%88%E5%8A%9B" class="mw-redirect" title="合力">合力</a>作用在<a href="/wiki/%E7%89%A9%E8%B3%AA" class="mw-redirect" title="物質">物質</a>上（合力作功）的功等於物質的<a href="/wiki/%E5%8B%95%E8%83%BD" class="mw-redirect" title="動能">動能</a>變化量。 </p><p>合力<i>W</i>在<a href="/wiki/%E8%B3%AA%E9%BB%9E" title="質點">質點</a>上所作的功等於其動能的變化量<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7587849b44d775263271e89499f4327eeac5dc81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.804ex; height:2.509ex;" alt="{\displaystyle E_{k}}"></span>,<sup id="cite_ref-Young_6-0" class="reference"><a href="#cite_note-Young-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\Delta E_{\rm {k}}={\tfrac {1}{2}}mv_{2}^{2}-{\tfrac {1}{2}}mv_{1}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\Delta E_{\rm {k}}={\tfrac {1}{2}}mv_{2}^{2}-{\tfrac {1}{2}}mv_{1}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6888e9a7322280f8a5fe596be95fd3a84e83ca34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.985ex; height:3.509ex;" alt="{\displaystyle W=\Delta E_{\rm {k}}={\tfrac {1}{2}}mv_{2}^{2}-{\tfrac {1}{2}}mv_{1}^{2}}"></span>,</dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d33f5d498d528bd8c10edc8ac8c34347f32b3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{1}}"></span> 和 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb04c423c2cb809c30cac725befa14ffbf4c85f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{2}}"></span>分別是質點的初<a href="/wiki/%E9%80%9F%E5%BA%A6" title="速度">速度</a>和末速度，m則是<a href="/wiki/%E8%B3%AA%E9%87%8F" class="mw-redirect" title="質量">質量</a>。 </p> <div class="mw-heading mw-heading3"><h3 id="概述"><span id=".E6.A6.82.E8.BF.B0"></span>概述</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%9F&action=edit&section=11" title="编辑章节：概述"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>功與能的原理由<a href="/wiki/%E7%89%9B%E9%A0%93%E7%AC%AC%E4%BA%8C%E9%81%8B%E5%8B%95%E5%AE%9A%E5%BE%8B" title="牛頓第二運動定律">牛頓第二運動定律</a>推導，其中包括作用在質點上的合力和<a href="/wiki/%E7%B4%84%E6%9D%9F%E5%8A%9B" class="mw-redirect" title="約束力">約束反力</a>對質點造成的<a href="/wiki/%E4%BD%8D%E7%A7%BB" title="位移">位移</a>量。 </p><p>对于<a href="/wiki/%E5%8C%80%E5%8F%98%E9%80%9F%E7%9B%B4%E7%BA%BF%E8%BF%90%E5%8A%A8" class="mw-redirect" title="匀变速直线运动">匀变速直线运动</a>的情形，<a href="/wiki/%E6%8E%A8%E5%AF%BC" class="mw-disambig" title="推导">推导</a>如下式。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=Fd=mad=ma\left({\frac {v_{2}^{2}-v_{1}^{2}}{2a}}\right)={\frac {mv_{2}^{2}}{2}}-{\frac {mv_{1}^{2}}{2}}=\Delta {E_{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>F</mi> <mi>d</mi> <mo>=</mo> <mi>m</mi> <mi>a</mi> <mi>d</mi> <mo>=</mo> <mi>m</mi> <mi>a</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=Fd=mad=ma\left({\frac {v_{2}^{2}-v_{1}^{2}}{2a}}\right)={\frac {mv_{2}^{2}}{2}}-{\frac {mv_{1}^{2}}{2}}=\Delta {E_{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/541085175c582376075625bd6056406a5e5778dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:58.446ex; height:7.509ex;" alt="{\displaystyle W=Fd=mad=ma\left({\frac {v_{2}^{2}-v_{1}^{2}}{2a}}\right)={\frac {mv_{2}^{2}}{2}}-{\frac {mv_{1}^{2}}{2}}=\Delta {E_{k}}}"></span>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\frac {v_{2}^{2}-v_{1}^{2}}{2a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\frac {v_{2}^{2}-v_{1}^{2}}{2a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6de7592a33b72c1543c87c2d360607492cea1fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.355ex; height:5.843ex;" alt="{\displaystyle d={\frac {v_{2}^{2}-v_{1}^{2}}{2a}}}"></span>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{2}^{2}=v_{1}^{2}+2ad}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{2}^{2}=v_{1}^{2}+2ad}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49206d391fad434cef2b76607e5911af24b3c5e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.911ex; height:3.176ex;" alt="{\displaystyle v_{2}^{2}=v_{1}^{2}+2ad}"></span></dd></dl> <p>一般情况下的推导则如下式。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} dt=\int _{t_{1}}^{t_{2}}F\,vdt=\int _{t_{1}}^{t_{2}}ma\,vdt=m\int _{t_{1}}^{t_{2}}v\,{dv \over dt}\,dt=m\int _{v_{1}}^{v_{2}}v\,dv={\tfrac {1}{2}}m(v_{2}^{2}-v_{1}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>F</mi> <mspace width="thinmathspace" /> <mi>v</mi> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>m</mi> <mi>a</mi> <mspace width="thinmathspace" /> <mi>v</mi> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>m</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>v</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>v</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>m</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>m</mi> <mo stretchy="false">(</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} dt=\int _{t_{1}}^{t_{2}}F\,vdt=\int _{t_{1}}^{t_{2}}ma\,vdt=m\int _{t_{1}}^{t_{2}}v\,{dv \over dt}\,dt=m\int _{v_{1}}^{v_{2}}v\,dv={\tfrac {1}{2}}m(v_{2}^{2}-v_{1}^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee0a92fe5bc53d54687028235d44ada5d65870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:91.315ex; height:6.509ex;" alt="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} dt=\int _{t_{1}}^{t_{2}}F\,vdt=\int _{t_{1}}^{t_{2}}ma\,vdt=m\int _{t_{1}}^{t_{2}}v\,{dv \over dt}\,dt=m\int _{v_{1}}^{v_{2}}v\,dv={\tfrac {1}{2}}m(v_{2}^{2}-v_{1}^{2})}"></span>。</dd></dl> <div class="mw-heading mw-heading2"><h2 id="注释"><span id=".E6.B3.A8.E9.87.8A"></span>注释</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%9F&action=edit&section=12" title="编辑章节：注释"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-ri-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-ri_1-0"><sup><b>1.0</b></sup></a> <a href="#cite_ref-ri_1-1"><sup><b>1.1</b></sup></a></span> <span class="reference-text">（日）小幡彩贵,1分钟懂科学图解100个科学基本名词,海峡书局,2021.09,第15页</span> </li> <li id="cite_note-:1-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-:1_2-0">^</a></b></span> <span class="reference-text"><cite class="citation web"><a href="/w/index.php?title=National_Council_of_Educational_Research_and_Training&action=edit&redlink=1" class="new" title="National Council of Educational Research and Training（页面不存在）">NCERT</a>. <a rel="nofollow" class="external text" href="https://www.ncert.nic.in/ncerts/l/keph106.pdf">Physics Book</a> <span style="font-size:85%;">(PDF)</span>. ncert.nic.in. 2020 <span class="reference-accessdate"> [<span class="nowrap">24 November</span> 2021]</span>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20231003232127/https://ncert.nic.in/ncerts/l/keph106.pdf">存档</a> <span style="font-size:85%;">(PDF)</span>于2023-10-03）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%9F&rft.atitle=Physics+Book&rft.au=NCERT&rft.date=2020&rft.genre=unknown&rft.jtitle=ncert.nic.in&rft_id=https%3A%2F%2Fwww.ncert.nic.in%2Fncerts%2Fl%2Fkeph106.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation book">Jammer, Max. Concepts of Force. Dover Publications, Inc. 1957. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-486-40689-X" title="Special:网络书源/0-486-40689-X"><span title="国际标准书号">ISBN</span> 0-486-40689-X</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%9F&rft.aufirst=Max&rft.aulast=Jammer&rft.btitle=Concepts+of+Force&rft.date=1957&rft.genre=book&rft.isbn=0-486-40689-X&rft.pub=Dover+Publications%2C+Inc.&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation book">Tipler. 1991: 138.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%9F&rft.btitle=Tipler&rft.date=1991&rft.genre=book&rft.pages=138&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-R&H7-2-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-R&H7-2_5-0"><sup><b>5.0</b></sup></a> <a href="#cite_ref-R&H7-2_5-1"><sup><b>5.1</b></sup></a></span> <span class="reference-text"><cite class="citation book">Resnick, Robert and Halliday, David. <span></span><i>Physics</i>, Section 7-2 (Vol I and II, Combined edition). Wiley International Edition, Library of Congress Catalog Card No. 66-11527. 1966.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%9F&rft.au=Resnick%2C+Robert+and+Halliday%2C+David&rft.btitle=Physics%2C+Section+7-2+%28Vol+I+and+II%2C+Combined+edition%29&rft.date=1966&rft.genre=book&rft.pub=Wiley+International+Edition%2C+Library+of+Congress+Catalog+Card+No.+66-11527&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Young-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Young_6-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Hugh D. Young and Roger A. Freedman. <a rel="nofollow" class="external text" href="https://archive.org/details/universityphysic00youn_715">University Physics</a> 12th. Addison-Wesley. 2008: <a rel="nofollow" class="external text" href="https://archive.org/details/universityphysic00youn_715/page/n349">329</a>. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-321-50130-1" title="Special:网络书源/978-0-321-50130-1"><span title="国际标准书号">ISBN</span> 978-0-321-50130-1</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%9F&rft.au=Hugh+D.+Young+and+Roger+A.+Freedman&rft.btitle=University+Physics&rft.date=2008&rft.edition=12th&rft.genre=book&rft.isbn=978-0-321-50130-1&rft.pages=329&rft.pub=Addison-Wesley&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Funiversityphysic00youn_715&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="参考"><span id=".E5.8F.82.E8.80.83"></span>参考</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%9F&action=edit&section=13" title="编辑章节：参考"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation book">Serway, Raymond A.; Jewett, John W. Physics for Scientists and Engineers 6th ed. Brooks/Cole. 2004. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-534-40842-7" title="Special:网络书源/0-534-40842-7"><span title="国际标准书号">ISBN</span> 0-534-40842-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%9F&rft.au=Serway%2C+Raymond+A.%3B+Jewett%2C+John+W.&rft.btitle=Physics+for+Scientists+and+Engineers&rft.date=2004&rft.edition=6th+ed.&rft.genre=book&rft.isbn=0-534-40842-7&rft.pub=Brooks%2FCole&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33"> 引文格式1维护：冗余文本 (<a href="/wiki/Category:%E5%BC%95%E6%96%87%E6%A0%BC%E5%BC%8F1%E7%BB%B4%E6%8A%A4%EF%BC%9A%E5%86%97%E4%BD%99%E6%96%87%E6%9C%AC" title="Category:引文格式1维护：冗余文本">link</a>)</span></li> <li><cite class="citation book">Tipler, Paul. Physics for Scientists and Engineers: Mechanics 3rd ed., extended version. W. H. Freeman. 1991. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-87901-432-6" title="Special:网络书源/0-87901-432-6"><span title="国际标准书号">ISBN</span> 0-87901-432-6</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%9F&rft.au=Tipler%2C+Paul&rft.btitle=Physics+for+Scientists+and+Engineers%3A+Mechanics&rft.date=1991&rft.edition=3rd+ed.%2C+extended+version&rft.genre=book&rft.isbn=0-87901-432-6&rft.pub=W.+H.+Freeman&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output 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href="/wiki/Template_talk:%E7%B6%93%E5%85%B8%E5%8A%9B%E5%AD%B8" title="Template talk:經典力學"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E7%B6%93%E5%85%B8%E5%8A%9B%E5%AD%B8" title="Special:编辑页面/Template:經典力學"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="经典力学" style="font-size:110%;margin:0 5em"><a href="/wiki/%E7%BB%8F%E5%85%B8%E5%8A%9B%E5%AD%A6" title="经典力学">经典力学</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">表述形式</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%BB%8F%E5%85%B8%E5%8A%9B%E5%AD%A6" title="经典力学">矢量力学</a></li> <li><a href="/wiki/%E5%88%86%E6%9E%90%E5%8A%9B%E5%AD%A6" title="分析力学">分析力学</a>（<a href="/wiki/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E5%8A%9B%E5%AD%A6" title="拉格朗日力学">拉格朗日力学</a></li> <li><a href="/wiki/%E5%93%88%E5%AF%86%E9%A1%BF%E5%8A%9B%E5%AD%A6" title="哈密顿力学">哈密頓力學</a>）</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">基础概念</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%A9%BA%E9%96%93" title="空間">空间</a></li> <li><a href="/wiki/%E6%97%B6%E9%97%B4" title="时间">时间</a></li> <li><a href="/wiki/%E9%80%9F%E5%BA%A6" title="速度">速度</a></li> <li><a href="/wiki/%E5%8A%A0%E9%80%9F%E5%BA%A6" title="加速度">加速度</a></li> <li><a href="/wiki/%E8%B4%A8%E9%87%8F" title="质量">质量</a></li> <li><a href="/wiki/%E5%BC%95%E5%8A%9B" title="引力">引力</a></li> <li><a href="/wiki/%E5%8A%9B%E7%9F%A9" title="力矩">力矩</a></li> <li><a href="/wiki/%E5%8F%82%E8%80%83%E7%B3%BB" title="参考系">參考系</a></li> <li><a href="/wiki/%E5%8A%9B" title="力">力</a></li> <li><a href="/wiki/%E5%8A%9B%E5%81%B6" title="力偶">力偶</a></li> <li><a href="/wiki/%E5%86%B2%E9%87%8F" title="冲量">冲量</a></li> <li><a href="/wiki/%E5%8A%A8%E9%87%8F" title="动量">动量</a></li> <li><a href="/wiki/%E5%88%9A%E4%BD%93" title="刚体">刚体</a></li> <li><a href="/wiki/%E8%A7%92%E5%8A%A8%E9%87%8F" title="角动量">角动量</a></li> <li><a href="/wiki/%E6%85%A3%E6%80%A7" title="慣性">慣性</a></li> <li><a href="/wiki/%E8%BD%89%E5%8B%95%E6%85%A3%E9%87%8F" title="轉動慣量">轉動慣量</a></li> <li><a href="/wiki/%E8%83%BD%E9%87%8F" title="能量">能量</a></li> <li><a href="/wiki/%E5%8A%A8%E8%83%BD" title="动能">动能</a></li> <li><a href="/wiki/%E5%8A%BF%E8%83%BD" title="势能">位能</a></li> <li><a href="/wiki/%E8%99%9B%E5%8A%9F" title="虛功">虛功</a></li> <li><a href="/wiki/%E4%BD%9C%E7%94%A8%E9%87%8F" title="作用量">作用量</a></li> <li><a href="/wiki/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E9%87%8F" title="拉格朗日量">拉格朗日量</a></li> <li><a href="/wiki/%E5%93%88%E5%AF%86%E9%A1%BF%E5%8A%9B%E5%AD%A6" title="哈密顿力学">哈密頓量</a></li> <li><a class="mw-selflink selflink">功</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">重要理论</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%89%9B%E9%A1%BF%E8%BF%90%E5%8A%A8%E5%AE%9A%E5%BE%8B" title="牛顿运动定律">牛顿运动定律</a></li> <li><a href="/wiki/%E8%83%A1%E5%85%8B%E5%AE%9A%E5%BE%8B" title="胡克定律">胡克定律</a></li> <li><a href="/wiki/%E4%B8%87%E6%9C%89%E5%BC%95%E5%8A%9B%E5%AE%9A%E5%BE%8B" title="万有引力定律">牛顿万有引力定律</a></li> <li><a href="/wiki/%E7%B0%A1%E8%AB%A7%E9%81%8B%E5%8B%95" title="簡諧運動">簡諧運動</a></li> <li><a href="/wiki/%E9%81%94%E6%9C%97%E8%B2%9D%E7%88%BE%E5%8E%9F%E7%90%86" title="達朗貝爾原理">達朗貝爾原理</a></li> <li><a href="/wiki/%E6%AC%A7%E6%8B%89%E6%96%B9%E7%A8%8B_(%E5%88%9A%E4%BD%93%E8%BF%90%E5%8A%A8)" title="欧拉方程 (刚体运动)">歐拉方程式</a></li> <li><a href="/wiki/%E5%93%88%E5%AF%86%E9%A0%93%E5%8E%9F%E7%90%86" title="哈密頓原理">哈密頓原理</a></li> <li><a href="/wiki/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E6%96%B9%E7%A8%8B%E5%BC%8F" title="拉格朗日方程式">拉格朗日方程式</a></li> <li><a href="/wiki/%E6%9C%80%E5%B0%8F%E4%BD%9C%E7%94%A8%E9%87%8F%E5%8E%9F%E7%90%86" title="最小作用量原理">最小作用量原理</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">应用</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%AE%80%E5%8D%95%E6%9C%BA%E6%A2%B0" title="简单机械">简单机械</a></li> <li><a href="/wiki/%E6%96%9C%E9%9D%A2" title="斜面">斜面</a></li> <li><a href="/wiki/%E6%9D%A0%E6%9D%86" title="杠杆">杠杆</a></li> <li><a href="/wiki/%E6%BB%91%E8%BD%AE" title="滑轮">滑轮</a></li> <li><a href="/wiki/%E8%9E%BA%E6%97%8B_(%E7%B0%A1%E5%96%AE%E6%A9%9F%E6%A2%B0)" title="螺旋 (簡單機械)">螺旋</a></li> <li><a href="/wiki/%E6%A5%94%E5%AD%90" title="楔子">楔子</a></li> <li><a href="/wiki/%E8%BC%AA%E8%BB%B8" title="輪軸">輪軸</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">科学史</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%A6%E5%8F%B2#力学的发展史" title="物理学史">发展史</a></li> <li><a href="/wiki/%E7%BA%A6%E7%BF%B0%E5%86%85%E6%96%AF%C2%B7%E5%BC%80%E6%99%AE%E5%8B%92" title="约翰内斯·开普勒">开普勒</a></li> <li><a href="/wiki/%E8%89%BE%E8%96%A9%E5%85%8B%C2%B7%E7%89%9B%E9%A0%93" class="mw-redirect" title="艾薩克·牛頓">牛頓</a></li> <li><a href="/wiki/%E8%90%8A%E6%98%82%E5%93%88%E5%BE%B7%C2%B7%E6%AD%90%E6%8B%89" title="萊昂哈德·歐拉">歐拉</a></li> <li><a href="/wiki/%E8%AE%A9%C2%B7%E5%8B%92%E6%9C%97%C2%B7%E8%BE%BE%E6%9C%97%E8%B4%9D%E5%B0%94" title="让·勒朗·达朗贝尔">達朗貝爾</a></li> <li><a href="/wiki/%E5%A8%81%E5%BB%89%C2%B7%E5%93%88%E5%AF%86%E9%A0%93" title="威廉·哈密頓">哈密頓</a></li> <li><a href="/wiki/%E6%B5%B7%E5%9B%A0%E9%87%8C%E5%B8%8C%C2%B7%E8%B5%AB%E5%85%B9" title="海因里希·赫兹">赫茲</a></li> <li><a href="/wiki/%E7%BA%A6%E7%91%9F%E5%A4%AB%C2%B7%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5" title="约瑟夫·拉格朗日">拉格朗日</a></li> <li><a href="/wiki/%E7%9A%AE%E5%9F%83%E5%B0%94-%E8%A5%BF%E8%92%99%C2%B7%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF" title="皮埃尔-西蒙·拉普拉斯">拉普拉斯</a></li> <li><a href="/wiki/%E4%BC%BD%E5%88%A9%E7%95%A5%C2%B7%E4%BC%BD%E5%88%A9%E8%8E%B1" title="伽利略·伽利莱">伽利略</a></li> <li><a href="/wiki/%E5%8D%A1%E7%88%BE%C2%B7%E9%9B%85%E5%8F%AF%E6%AF%94" title="卡爾·雅可比">雅可比</a></li> <li><a href="/wiki/%E5%9F%83%E7%B1%B3%C2%B7%E8%AF%BA%E7%89%B9" title="埃米·诺特">諾特</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">分支</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E9%9D%99%E5%8A%9B%E5%AD%A6" title="静力学">静力学</a></li> <li><a href="/wiki/%E5%8B%95%E5%8A%9B%E5%AD%B8" title="動力學">動力學</a></li> <li><a href="/wiki/%E8%BF%90%E5%8A%A8%E5%AD%A6" title="运动学">运动学</a></li> <li><a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%8A%9B%E5%AD%A6" title="工程力学">工程力學</a></li> <li><a href="/wiki/%E5%A4%A9%E9%AB%94%E5%8A%9B%E5%AD%B8" title="天體力學">天體力學</a></li> <li><a href="/wiki/%E9%80%A3%E7%BA%8C%E4%BB%8B%E8%B3%AA%E5%8A%9B%E5%AD%B8" class="mw-redirect" title="連續介質力學">連續介質力學</a></li> <li><a href="/wiki/%E7%B5%B1%E8%A8%88%E5%8A%9B%E5%AD%B8" class="mw-redirect" title="統計力學">統計力學</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84261037"><style data-mw-deduplicate="TemplateStyles:r58896141">.mw-parser-output .serif{font-family:Times,serif}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link 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href="/w/index.php?title=Template_talk:%E7%BB%8F%E5%85%B8%E5%8A%9B%E5%AD%A6%E5%9B%BD%E9%99%85%E5%8D%95%E4%BD%8D&action=edit&redlink=1" class="new" title="Template talk:经典力学国际单位（页面不存在）"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E7%BB%8F%E5%85%B8%E5%8A%9B%E5%AD%A6%E5%9B%BD%E9%99%85%E5%8D%95%E4%BD%8D" title="Special:编辑页面/Template:经典力学国际单位"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="经典力学国际单位" style="font-size:110%;margin:0 5em"><a href="/wiki/%E7%BB%8F%E5%85%B8%E5%8A%9B%E5%AD%A6" title="经典力学">经典力学</a><a href="/wiki/%E5%9B%BD%E9%99%85%E5%8D%95%E4%BD%8D%E5%88%B6" title="国际单位制">国际单位</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0;"><table class="wikitable" style="text-align:center;line-height:0.9;border-collapse:collapse;margin:auto;border:none;background:none;"> <tbody><tr> <th colspan="4">线性（平动）的量</th> <td rowspan="12" style="border:none;backgound:none;"></td> <th colspan="4">角度（转动）的量</th> </tr> <tr> <th style="font-weight:normal;font-size:80%;">量纲</th> <th style="font-weight:normal;">—</th> <th style="font-weight:normal;">L</th> <th style="font-weight:normal;">L<sup>2</sup></th> <th style="font-weight:normal;font-size:80%;">量纲</th> <th style="font-weight:normal;">—</th> <th style="font-weight:normal;">—</th> <th style="font-weight:normal;">—</th> </tr> <tr> <th style="font-weight:normal;">T</th> <td><a href="/wiki/%E6%97%B6%E9%97%B4" title="时间">时间</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">t</span></span><br /><a href="/wiki/%E7%A7%92" title="秒">s</a></td> <td><a href="/wiki/%E4%BD%8D%E7%A7%BB%E7%A7%AF%E5%88%86" title="位移积分">位移积分</a>: <span class="serif"><span class="texhtml"><b>A</b></span></span><br /><span class="ilh-all" data-orig-title="m s" data-lang-code="en" data-lang-name="英语" data-foreign-title="meter second"><span class="ilh-page"><a href="/w/index.php?title=M_s&action=edit&redlink=1" class="new" title="M s（页面不存在）">m s</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/meter_second" class="extiw" title="en:meter second"><span lang="en" dir="auto">meter second</span></a></span>）</span></span></td> <td></td> <th style="font-weight:normal;">T</th> <td><a href="/wiki/%E6%97%B6%E9%97%B4" title="时间">时间</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">t</span></span><br /><a href="/wiki/%E7%A7%92" title="秒">s</a></td> <td></td> <td></td> </tr> <tr> <th style="font-weight:normal;">—</th> <td></td> <td><a href="/wiki/%E8%B7%9D%E7%A6%BB" title="距离">距离</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">d</span></span>, <span class="nowrap"><a href="/wiki/%E4%BD%8D%E7%BD%AE%E5%90%91%E9%87%8F" title="位置向量">位矢</a>: <span class="serif"><span class="texhtml"><b>r</b></span></span>, <span class="serif"><span class="texhtml"><b>s</b></span></span>, <span class="serif"><span class="texhtml"><b>x</b></span></span></span>, <a href="/wiki/%E4%BD%8D%E7%A7%BB" title="位移">位移</a><br /><a href="/wiki/%E7%B1%B3_(%E5%8D%95%E4%BD%8D)" title="米 (单位)">m</a></td> <td><a href="/wiki/%E9%9D%A2%E7%A7%AF" title="面积">面积</a>: <span class="serif"><span class="texhtml"><i>A</i></span></span><br /><a href="/wiki/%E5%B9%B3%E6%96%B9%E7%B1%B3" title="平方米">m<sup>2</sup></a></td> <th style="font-weight:normal;">—</th> <td></td> <td><a href="/wiki/%E8%A7%92" title="角">角度</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">θ</span></span>, <a href="/wiki/%E8%A7%92%E7%A7%BB" title="角移">角移</a>: <span class="serif"><span class="texhtml"><b>θ</b></span></span><br /><a href="/wiki/%E5%BC%A7%E5%BA%A6" title="弧度">rad</a></td> <td><span class="nowrap"><a href="/wiki/%E7%AB%8B%E9%AB%94%E8%A7%92" title="立體角">立體角</a>: <span class="serif"><span class="texhtml">Ω</span></span><br /><a href="/wiki/%E7%90%83%E9%9D%A2%E5%BA%A6" title="球面度">rad<sup>2</sup>, sr</a></span></td> </tr> <tr> <th style="font-weight:normal;">T<sup>−1</sup></th> <td><span class="nowrap"><a href="/wiki/%E9%A0%BB%E7%8E%87_(%E7%89%A9%E7%90%86%E5%AD%B8)" title="頻率 (物理學)">頻率</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">f</span></span></span><br /><span class="ilh-all" data-orig-title="每秒" data-lang-code="en" data-lang-name="英语" data-foreign-title="inverse second"><span class="ilh-page"><a href="/w/index.php?title=%E6%AF%8F%E7%A7%92&action=edit&redlink=1" class="new" title="每秒（页面不存在）">s<sup>−1</sup></a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/inverse_second" class="extiw" title="en:inverse second"><span lang="en" dir="auto">inverse second</span></a></span>）</span></span>, <a href="/wiki/%E8%B5%AB%E5%85%B9" title="赫兹">Hz</a></td> <td><a href="/wiki/%E9%80%9F%E7%8E%87" title="速率">速率</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">v</span></span>, <a href="/wiki/%E9%80%9F%E5%BA%A6" title="速度">速度</a>: <span class="serif"><span class="texhtml"><b>v</b></span></span><br /><a href="/wiki/%E7%B1%B3%E6%AF%8F%E7%A7%92" title="米每秒">m s<sup>−1</sup></a></td> <td><a href="/wiki/%E9%BB%8F%E5%BA%A6" title="黏度">面積速率</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">ν</span></span><br /><a href="https://en.wikipedia.org/wiki/metre_squared_per_second" class="extiw" title="en:metre squared per second">m<sup>2</sup> s<sup>−1</sup></a></td> <th style="font-weight:normal;">T<sup>−1</sup></th> <td><span class="nowrap"><a href="/wiki/%E9%A0%BB%E7%8E%87_(%E7%89%A9%E7%90%86%E5%AD%B8)" title="頻率 (物理學)">頻率</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">f</span></span></span><br /><span class="ilh-all" data-orig-title="每秒" data-lang-code="en" data-lang-name="英语" data-foreign-title="inverse second"><span class="ilh-page"><a href="/w/index.php?title=%E6%AF%8F%E7%A7%92&action=edit&redlink=1" class="new" title="每秒（页面不存在）">s<sup>−1</sup></a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/inverse_second" class="extiw" title="en:inverse second"><span lang="en" dir="auto">inverse second</span></a></span>）</span></span>, <a href="/wiki/%E8%B5%AB%E5%85%B9" title="赫兹">Hz</a></td> <td><a href="/wiki/%E8%A7%92%E9%80%9F%E5%BA%A6" title="角速度">角速率</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">ω</span></span>, <a href="/wiki/%E8%A7%92%E9%80%9F%E5%BA%A6" title="角速度">角速度</a>: <span class="serif"><span class="texhtml"><b>ω</b></span></span><br /><a href="/wiki/%E5%BC%A7%E5%BA%A6%E6%AF%8F%E7%A7%92" title="弧度每秒">rad<span style="letter-spacing:0.1em"> </span>s<sup>−1</sup></a></td> <td></td> </tr> <tr> <th style="font-weight:normal;">T<sup>−2</sup></th> <td></td> <td><a href="/wiki/%E5%8A%A0%E9%80%9F%E5%BA%A6" title="加速度">加速度</a>: <span class="serif"><span class="texhtml"><b>a</b></span></span><br /><a href="/wiki/%E7%B1%B3%E6%AF%8F%E4%BA%8C%E6%AC%A1%E6%96%B9%E7%A7%92" title="米每二次方秒">m s<sup>−2</sup></a></td> <td></td> <th style="font-weight:normal;">T<sup>−2</sup></th> <td></td> <td><a href="/wiki/%E8%A7%92%E5%8A%A0%E9%80%9F%E5%BA%A6" title="角加速度">角加速度</a>: <span class="serif"><span class="texhtml"><b>α</b></span></span><br />rad<span style="letter-spacing:0.1em"> </span>s<sup>−2</sup></td> <td></td> </tr> <tr> <th style="font-weight:normal;">T<sup>−3</sup></th> <td></td> <td><a href="/wiki/%E5%8A%A0%E5%8A%A0%E9%80%9F%E5%BA%A6" title="加加速度">加加速度</a>: <span class="serif"><span class="texhtml"><b>j</b></span></span><br />m s<sup>−3</sup></td> <td></td> <th style="font-weight:normal;">T<sup>−3</sup></th> <td></td> <td><a href="/wiki/%E5%8A%A0%E5%8A%A0%E9%80%9F%E5%BA%A6" title="加加速度">角加加速度</a>: <span class="serif"><span class="texhtml"><b>ζ</b></span></span><br />rad<span style="letter-spacing:0.1em"> </span>s<sup>−3</sup></td> <td></td> </tr> <tr> <td colspan="4" style="padding-top:0;"></td> <td colspan="4" style="padding-top:0;"></td> </tr> <tr><th style="font-weight:normal;">M</th> <td><a href="/wiki/%E8%B4%A8%E9%87%8F" title="质量">质量</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">m</span></span><br /><a href="/wiki/%E5%8D%83%E5%85%8B" title="千克">kg</a></td> <td></td> <td></td> <th style="font-weight:normal;">ML<sup>2</sup></th> <td><a href="/wiki/%E8%BD%89%E5%8B%95%E6%85%A3%E9%87%8F" title="轉動慣量">轉動慣量</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">I</span></span><br />kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup></td> <td></td> <td></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−1</sup></th> <td></td> <td><a href="/wiki/%E5%8A%A8%E9%87%8F" title="动量">动量</a>: <span class="serif"><span class="texhtml"><b>p</b></span></span>, <a href="/wiki/%E5%86%B2%E9%87%8F" title="冲量">冲量</a>: <span class="serif"><span class="texhtml"><b>J</b></span></span><br />kg<span style="letter-spacing:0.1em"> </span>m s<sup>−1</sup>, <span class="ilh-all" data-orig-title="N s" data-lang-code="en" data-lang-name="英语" data-foreign-title="newton second"><span class="ilh-page"><a href="/w/index.php?title=N_s&action=edit&redlink=1" class="new" title="N s（页面不存在）">N s</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/newton_second" class="extiw" title="en:newton second"><span lang="en" dir="auto">newton second</span></a></span>）</span></span></td> <td><a href="/wiki/%E4%BD%9C%E7%94%A8%E9%87%8F" title="作用量">作用量</a>: <span class="serif"><span class="texhtml">𝒮</span></span>, <a href="/wiki/Actergy" class="mw-redirect" title="Actergy">actergy</a>: <span class="serif"><span class="texhtml">ℵ</span></span><br />kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup> s<sup>−1</sup>, <span class="ilh-all" data-orig-title="J s" data-lang-code="en" data-lang-name="英语" data-foreign-title="joule-second"><span class="ilh-page"><a href="/w/index.php?title=J_s&action=edit&redlink=1" class="new" title="J s（页面不存在）">J s</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/joule-second" class="extiw" title="en:joule-second"><span lang="en" dir="auto">joule-second</span></a></span>）</span></span></td> <th style="font-weight:normal;">ML<sup>2</sup>T<sup>−1</sup></th> <td></td> <td><a href="/wiki/%E8%A7%92%E5%8A%A8%E9%87%8F" title="角动量">角动量</a>: <span class="serif"><span class="texhtml"><b>L</b></span></span>, <a href="/wiki/%E7%B6%93%E5%85%B8%E5%8A%9B%E5%AD%B8%E6%96%B9%E7%A8%8B%E5%88%97%E8%A1%A8#導出的動力學物理量" title="經典力學方程列表">角衝量</a>: <span class="serif"><span class="texhtml"><b>ι</b></span></span><br />kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup> s<sup>−1</sup></td> <td><a href="/wiki/%E4%BD%9C%E7%94%A8%E9%87%8F" title="作用量">作用量</a>: <span class="serif"><span class="texhtml">𝒮</span></span>, <a href="/wiki/Actergy" class="mw-redirect" title="Actergy">actergy</a>: <span class="serif"><span class="texhtml">ℵ</span></span><br />kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup> s<sup>−1</sup>, <span class="ilh-all" data-orig-title="J s" data-lang-code="en" data-lang-name="英语" data-foreign-title="joule-second"><span class="ilh-page"><a href="/w/index.php?title=J_s&action=edit&redlink=1" class="new" title="J s（页面不存在）">J s</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/joule-second" class="extiw" title="en:joule-second"><span lang="en" dir="auto">joule-second</span></a></span>）</span></span></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−2</sup></th> <td></td> <td><a href="/wiki/%E5%8A%9B" title="力">力</a>: <span class="serif"><span class="texhtml"><b>F</b></span></span>, <a href="/wiki/%E9%87%8D%E9%87%8F" title="重量">重量</a>: <span class="serif"><span class="texhtml"><b>F</b><sub>g</sub></span></span><br /><span style="margin-right:0.1em;">kg </span> m s<sup>−2</sup>, <a href="/wiki/%E7%89%9B%E9%A0%93_(%E5%96%AE%E4%BD%8D)" title="牛頓 (單位)">N</a></td> <td><a href="/wiki/%E8%83%BD%E9%87%8F" title="能量">能量</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">E</span></span>, <a class="mw-selflink selflink">功</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">W</span></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−2</sup>, <a href="/wiki/%E7%84%A6%E8%80%B3" title="焦耳">J</a></td> <th style="font-weight:normal;">ML<sup>2</sup>T<sup>−2</sup></th> <td></td> <td><a href="/wiki/%E5%8A%9B%E7%9F%A9" title="力矩">力矩</a>: <span class="serif"><span class="texhtml"><b>τ</b></span></span>, <span class="ilh-all" data-orig-title="Moment (physics)" data-lang-code="en" data-lang-name="英语" data-foreign-title="Moment (physics)"><span class="ilh-page"><a href="/w/index.php?title=Moment_(physics)&action=edit&redlink=1" class="new" title="Moment (physics)（页面不存在）">moment</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Moment_(physics)" class="extiw" title="en:Moment (physics)"><span lang="en" dir="auto">Moment (physics)</span></a></span>）</span></span>: <span class="serif"><span class="texhtml"><b>M</b></span></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−2</sup>, <a href="/wiki/%E7%89%9B%E9%A0%93%E7%B1%B3" title="牛頓米">N m</a></td> <td><a href="/wiki/%E8%83%BD%E9%87%8F" title="能量">能量</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">E</span></span>, <a class="mw-selflink selflink">功</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">W</span></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−2</sup>, <a href="/wiki/%E7%84%A6%E8%80%B3" title="焦耳">J</a></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−3</sup></th> <td></td> <td><a href="/wiki/%E5%8A%A0%E5%8A%9B" title="加力">加力</a>: <span class="serif"><span class="texhtml"><b>Y</b></span></span><br /><span style="margin-right:0.1em;">kg</span> m s<sup>−3</sup>, N s<sup>−1</sup></td> <td><a href="/wiki/%E5%8A%9F%E7%8E%87" title="功率">功率</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">P</span></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−3</sup>, <a href="/wiki/%E7%93%A6%E7%89%B9" title="瓦特">W</a></td> <th style="font-weight:normal;">ML<sup>2</sup>T<sup>−3</sup></th> <td></td> <td><span class="ilh-all" data-orig-title="rotatum" data-lang-code="en" data-lang-name="英语" data-foreign-title="rotatum"><span class="ilh-page"><a href="/w/index.php?title=Rotatum&action=edit&redlink=1" class="new" title="Rotatum（页面不存在）">rotatum</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/rotatum" class="extiw" title="en:rotatum"><span lang="en" dir="auto">rotatum</span></a></span>）</span></span>: <span class="serif"><span class="texhtml"><b>P</b></span></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−3</sup>, N m s<sup>−1</sup></td> <td><a href="/wiki/%E5%8A%9F%E7%8E%87" title="功率">功率</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">P</span></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−3</sup>, <a href="/wiki/%E7%93%A6%E7%89%B9" title="瓦特">W</a></td> </tr> </tbody></table></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> 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