Work (physics) - Wikipedia

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id="toc-Early_concepts_of_work" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Early_concepts_of_work"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Early concepts of work</span> </div> </a> <ul id="toc-Early_concepts_of_work-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Etymology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Etymology"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Etymology</span> </div> </a> <ul id="toc-Etymology-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Units" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Units"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Units</span> </div> </a> <ul id="toc-Units-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Work_and_energy" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Work_and_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Work and energy</span> </div> </a> <ul id="toc-Work_and_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Constraint_forces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Constraint_forces"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Constraint forces</span> </div> </a> <ul id="toc-Constraint_forces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_calculation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mathematical_calculation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Mathematical calculation</span> </div> </a> <button aria-controls="toc-Mathematical_calculation-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>Toggle Mathematical calculation subsection</span> </button> <ul id="toc-Mathematical_calculation-sublist" class="vector-toc-list"> <li id="toc-Work_done_by_a_variable_force" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Work_done_by_a_variable_force"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Work done by a variable force</span> </div> </a> <ul id="toc-Work_done_by_a_variable_force-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Torque_and_rotation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Torque_and_rotation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Torque and rotation</span> </div> </a> <ul id="toc-Torque_and_rotation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Work_and_potential_energy" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Work_and_potential_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Work and potential energy</span> </div> </a> <button aria-controls="toc-Work_and_potential_energy-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>Toggle Work and potential energy subsection</span> </button> <ul id="toc-Work_and_potential_energy-sublist" class="vector-toc-list"> <li id="toc-Path_dependence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Path_dependence"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Path dependence</span> </div> </a> <ul id="toc-Path_dependence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Path_independence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Path_independence"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Path independence</span> </div> </a> <ul id="toc-Path_independence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Work_by_gravity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Work_by_gravity"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Work by gravity</span> </div> </a> <ul id="toc-Work_by_gravity-sublist" class="vector-toc-list"> <li id="toc-In_space" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#In_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3.1</span> <span>In space</span> </div> </a> <ul id="toc-In_space-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Work_by_a_spring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Work_by_a_spring"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Work by a spring</span> </div> </a> <ul id="toc-Work_by_a_spring-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Work_by_a_gas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Work_by_a_gas"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Work by a gas</span> </div> </a> <ul id="toc-Work_by_a_gas-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Work–energy_principle" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Work–energy_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Work–energy principle</span> </div> </a> <button aria-controls="toc-Work–energy_principle-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>Toggle Work–energy principle subsection</span> </button> <ul id="toc-Work–energy_principle-sublist" class="vector-toc-list"> <li id="toc-Derivation_for_a_particle_moving_along_a_straight_line" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation_for_a_particle_moving_along_a_straight_line"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Derivation for a particle moving along a straight line</span> </div> </a> <ul id="toc-Derivation_for_a_particle_moving_along_a_straight_line-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_derivation_of_the_work–energy_principle_for_a_particle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_derivation_of_the_work–energy_principle_for_a_particle"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>General derivation of the work–energy principle for a particle</span> </div> </a> <ul id="toc-General_derivation_of_the_work–energy_principle_for_a_particle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_for_a_particle_in_constrained_movement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation_for_a_particle_in_constrained_movement"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Derivation for a particle in constrained movement</span> </div> </a> <ul id="toc-Derivation_for_a_particle_in_constrained_movement-sublist" class="vector-toc-list"> <li id="toc-Vector_formulation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Vector_formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3.1</span> <span>Vector formulation</span> </div> </a> <ul id="toc-Vector_formulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tangential_and_normal_components" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Tangential_and_normal_components"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3.2</span> <span>Tangential and normal components</span> </div> </a> <ul id="toc-Tangential_and_normal_components-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Moving_in_a_straight_line_(skid_to_a_stop)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Moving_in_a_straight_line_(skid_to_a_stop)"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Moving in a straight line (skid to a stop)</span> </div> </a> <ul id="toc-Moving_in_a_straight_line_(skid_to_a_stop)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coasting_down_an_inclined_surface_(gravity_racing)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Coasting_down_an_inclined_surface_(gravity_racing)"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5</span> <span>Coasting down an inclined surface (gravity racing)</span> </div> </a> <ul id="toc-Coasting_down_an_inclined_surface_(gravity_racing)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Work_of_forces_acting_on_a_rigid_body" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Work_of_forces_acting_on_a_rigid_body"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Work of forces acting on a rigid body</span> </div> </a> <ul id="toc-Work_of_forces_acting_on_a_rigid_body-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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="Contents" 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="Toggle the table of contents" > <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">Toggle the table of contents</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">Work (physics)</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="Go to an article in another language. Available in 102 languages" > <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 languages</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 – Afrikaans" lang="af" hreflang="af" data-title="Arbeid" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" 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) – Alemannic" lang="gsw" hreflang="gsw" data-title="Arbeit (Physik)" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" 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 – Amharic" lang="am" hreflang="am" data-title="ሥራ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</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) – Arabic" lang="ar" hreflang="ar" data-title="شغل (فيزياء)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Treballo_mecanico" title="Treballo mecanico – Aragonese" lang="an" hreflang="an" data-title="Treballo mecanico" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</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 – Assamese" lang="as" hreflang="as" data-title="কাৰ্য্য" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" 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) – Asturian" lang="ast" hreflang="ast" data-title="Trabayu (física)" data-language-autonym="Asturianu" data-language-local-name="Asturian" 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 – Azerbaijani" lang="az" hreflang="az" data-title="Mexaniki iş" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</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) – Bangla" lang="bn" hreflang="bn" data-title="কাজ (পদার্থবিজ্ঞান)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Kang_(bu%CC%8Dt-l%C3%AD-ha%CC%8Dk)" title="Kang (bu̍t-lí-ha̍k) – Minnan" lang="nan" hreflang="nan" data-title="Kang (bu̍t-lí-ha̍k)" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</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 – Belarusian" lang="be" hreflang="be" data-title="Механічная работа" data-language-autonym="Беларуская" data-language-local-name="Belarusian" 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 – Bulgarian" lang="bg" hreflang="bg" data-title="Механична работа" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Rad_(fizika)" title="Rad (fizika) – Bosnian" lang="bs" hreflang="bs" data-title="Rad (fizika)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</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 – Breton" lang="br" hreflang="br" data-title="Labour un nerzh" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</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) – Catalan" lang="ca" hreflang="ca" data-title="Treball (física)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</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 – Chuvash" lang="cv" hreflang="cv" data-title="Механикăлла ĕç" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</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) – Czech" lang="cs" hreflang="cs" data-title="Práce (fyzika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Basa_(fundoyetsimba)" title="Basa (fundoyetsimba) – Shona" lang="sn" hreflang="sn" data-title="Basa (fundoyetsimba)" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Arbejde_(fysik)" title="Arbejde (fysik) – Danish" lang="da" hreflang="da" data-title="Arbejde (fysik)" data-language-autonym="Dansk" data-language-local-name="Danish" 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) – German" lang="de" hreflang="de" data-title="Arbeit (Physik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</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 – Estonian" lang="et" hreflang="et" data-title="Mehaaniline töö" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</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) – Greek" lang="el" hreflang="el" data-title="Έργο (φυσική)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</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) – Spanish" lang="es" hreflang="es" data-title="Trabajo (física)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Laboro_(fiziko)" title="Laboro (fiziko) – Esperanto" lang="eo" hreflang="eo" data-title="Laboro (fiziko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Lan_(fisika)" title="Lan (fisika) – Basque" lang="eu" hreflang="eu" data-title="Lan (fisika)" data-language-autonym="Euskara" data-language-local-name="Basque" 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) – Persian" lang="fa" hreflang="fa" data-title="کار (فیزیک)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</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 – French" lang="fr" hreflang="fr" data-title="Travail d'une force" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Obair_(fisic)" title="Obair (fisic) – Irish" lang="ga" hreflang="ga" data-title="Obair (fisic)" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Obbyr_(fishig)" title="Obbyr (fishig) – Manx" lang="gv" hreflang="gv" data-title="Obbyr (fishig)" data-language-autonym="Gaelg" data-language-local-name="Manx" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Obair_(fiosaigs)" title="Obair (fiosaigs) – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Obair (fiosaigs)" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" 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) – Galician" lang="gl" hreflang="gl" data-title="Traballo (física)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</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) – Korean" lang="ko" hreflang="ko" data-title="일 (물리학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</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) – Armenian" lang="hy" hreflang="hy" data-title="Աշխատանք (ֆիզիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</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) – Hindi" lang="hi" hreflang="hi" data-title="कार्य (भौतिकी)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" 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) – Croatian" lang="hr" hreflang="hr" data-title="Rad (fizika)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Usaha_(fisika)" title="Usaha (fisika) – Indonesian" lang="id" hreflang="id" data-title="Usaha (fisika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" 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) – Icelandic" lang="is" hreflang="is" data-title="Vinna (eðlisfræði)" data-language-autonym="Íslenska" data-language-local-name="Icelandic" 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) – Italian" lang="it" hreflang="it" data-title="Lavoro (fisica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</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) – Hebrew" lang="he" hreflang="he" data-title="עבודה (פיזיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</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 – Kannada" lang="kn" hreflang="kn" data-title="ಕೆಲಸ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</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 – Georgian" lang="ka" hreflang="ka" data-title="მუშაობა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</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) – Kashubian" lang="csb" hreflang="csb" data-title="Robòta (fizyka)" data-language-autonym="Kaszëbsczi" data-language-local-name="Kashubian" class="interlanguage-link-target"><span>Kaszëbsczi</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 – Kazakh" lang="kk" hreflang="kk" data-title="Механикалық жұмыс" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Travay_(fizik)" title="Travay (fizik) – Haitian Creole" lang="ht" hreflang="ht" data-title="Travay (fizik)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</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) – Kurdish" lang="ku" hreflang="ku" data-title="Kar (fizîk)" data-language-autonym="Kurdî" data-language-local-name="Kurdish" 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 – Kyrgyz" lang="ky" hreflang="ky" data-title="Механикалык жумуш" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" 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) – Latin" lang="la" hreflang="la" data-title="Labor (physica)" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Darbs_(fizika)" title="Darbs (fizika) – Latvian" lang="lv" hreflang="lv" data-title="Darbs (fizika)" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Mechaninis_darbas" title="Mechaninis darbas – Lithuanian" lang="lt" hreflang="lt" data-title="Mechaninis darbas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Labora_(fisica)" title="Labora (fisica) – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Labora (fisica)" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" 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) – Lombard" lang="lmo" hreflang="lmo" data-title="Laô (fisica)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Mechanikai_munka" title="Mechanikai munka – Hungarian" lang="hu" hreflang="hu" data-title="Mechanikai munka" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</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) – Macedonian" lang="mk" hreflang="mk" data-title="Работа (физика)" data-language-autonym="Македонски" data-language-local-name="Macedonian" 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 – Malayalam" lang="ml" hreflang="ml" data-title="പ്രവൃത്തി" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</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) – Marathi" lang="mr" hreflang="mr" data-title="कार्य (भौतिकशास्त्र)" data-language-autonym="मराठी" data-language-local-name="Marathi" 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) – Malay" lang="ms" hreflang="ms" data-title="Kerja (fizik)" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</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) – Mongolian" lang="mn" hreflang="mn" data-title="Ажил (физик)" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</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) – Burmese" lang="my" hreflang="my" data-title="အလုပ် (ရူပဗေဒ)" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Arbeid_(natuurkunde)" title="Arbeid (natuurkunde) – Dutch" lang="nl" hreflang="nl" data-title="Arbeid (natuurkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</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) – Nepali" lang="ne" hreflang="ne" data-title="कार्य (भौतिकशास्त्र)" data-language-autonym="नेपाली" data-language-local-name="Nepali" 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) – Newari" lang="new" hreflang="new" data-title="वेलै (सन् १९९८या संकिपा)" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</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) – Japanese" lang="ja" hreflang="ja" data-title="仕事 (物理学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Werk" title="Werk – Northern Frisian" lang="frr" hreflang="frr" data-title="Werk" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Arbeid_(fysikk)" title="Arbeid (fysikk) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Arbeid (fysikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</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 – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Arbeid i fysikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Trabalh_(fisica)" title="Trabalh (fisica) – Occitan" lang="oc" hreflang="oc" data-title="Trabalh (fisica)" data-language-autonym="Occitan" data-language-local-name="Occitan" 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 – Oromo" lang="om" hreflang="om" data-title="Dalagaa" data-language-autonym="Oromoo" data-language-local-name="Oromo" 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) – Punjabi" lang="pa" hreflang="pa" data-title="ਕੰਮ (ਭੌਤਿਕ ਵਿਗਿਆਨ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</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) – Khmer" lang="km" hreflang="km" data-title="កម្មន្ត (រូបវិទ្យា)" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Travaj" title="Travaj – Piedmontese" lang="pms" hreflang="pms" data-title="Travaj" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Praca_(fizyka)" title="Praca (fizyka) – Polish" lang="pl" hreflang="pl" data-title="Praca (fizyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</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) – Portuguese" lang="pt" hreflang="pt" data-title="Trabalho (física)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Lucru_mecanic" title="Lucru mecanic – Romanian" lang="ro" hreflang="ro" data-title="Lucru mecanic" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Ruray" title="Ruray – Quechua" lang="qu" hreflang="qu" data-title="Ruray" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</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 – Russian" lang="ru" hreflang="ru" data-title="Механическая работа" data-language-autonym="Русский" data-language-local-name="Russian" 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 – Santali" lang="sat" hreflang="sat" data-title="ᱠᱟᱹᱢᱤ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="Santali" class="interlanguage-link-target"><span>ᱥᱟᱱᱛᱟᱲᱤ</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) – Albanian" lang="sq" hreflang="sq" data-title="Puna (fizikë)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</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) – Sinhala" lang="si" hreflang="si" data-title="කාර්යය (භෞතික විද්‍යාව)" data-language-autonym="සිංහල" data-language-local-name="Sinhala" 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-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) – Sindhi" lang="sd" hreflang="sd" data-title="ڪم (طبعيات)" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</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) – Slovak" lang="sk" hreflang="sk" data-title="Práca (fyzika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" 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) – Slovenian" lang="sl" hreflang="sl" data-title="Delo (fizika)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</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) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="کار (فیزیک)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</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 – Serbian" lang="sr" hreflang="sr" data-title="Механички рад" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Rad_(fizika)" title="Rad (fizika) – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Rad (fizika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</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 – Sundanese" lang="su" hreflang="su" data-title="Usaha mékanik" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</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) – Finnish" lang="fi" hreflang="fi" data-title="Työ (fysiikka)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Arbete_(fysik)" title="Arbete (fysik) – Swedish" lang="sv" hreflang="sv" data-title="Arbete (fysik)" data-language-autonym="Svenska" data-language-local-name="Swedish" 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) – Tamil" lang="ta" hreflang="ta" data-title="வேலை (இயற்பியல்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" 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 – Telugu" lang="te" hreflang="te" data-title="పని" data-language-autonym="తెలుగు" data-language-local-name="Telugu" 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) – Thai" lang="th" hreflang="th" data-title="งาน (ฟิสิกส์)" data-language-autonym="ไทย" data-language-local-name="Thai" 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) – Turkish" lang="tr" hreflang="tr" data-title="İş (fizik)" data-language-autonym="Türkçe" data-language-local-name="Turkish" 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) – Ukrainian" lang="uk" hreflang="uk" data-title="Робота (фізика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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) – Urdu" lang="ur" hreflang="ur" data-title="کام (طبیعیات)" data-language-autonym="اردو" data-language-local-name="Urdu" 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) – Vietnamese" lang="vi" hreflang="vi" data-title="Công (vật lý học)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</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 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="功" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</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) – Wolof" lang="wo" hreflang="wo" data-title="Liggéey (jëmm)" data-language-autonym="Wolof" data-language-local-name="Wolof" 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 – Wu" lang="wuu" hreflang="wuu" data-title="功" data-language-autonym="吴语" data-language-local-name="Wu" 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) – Yiddish" lang="yi" hreflang="yi" data-title="ארבעט (פיזיק)" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8A%9F" title="功 – Cantonese" lang="yue" hreflang="yue" data-title="功" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8A%9F" title="功 – Chinese" lang="zh" hreflang="zh" data-title="功" data-language-autonym="中文" data-language-local-name="Chinese" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Process of energy transfer to an object via force application through displacement</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses of "Work" in physics, see <a href="/wiki/Work_(electric_field)" title="Work (electric field)">Work (electric field)</a> and <a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work (thermodynamics)</a>.</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Work</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><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/250px-Baseball_pitching_motion_2004.jpg" decoding="async" width="250" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Baseball_pitching_motion_2004.jpg/375px-Baseball_pitching_motion_2004.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/25/Baseball_pitching_motion_2004.jpg/500px-Baseball_pitching_motion_2004.jpg 2x" data-file-width="749" data-file-height="262" /></a></span><div class="infobox-caption">A <a href="/wiki/Baseball" title="Baseball">baseball</a> <a href="/wiki/Pitcher" title="Pitcher">pitcher</a> does positive work on the ball by applying a force to it over the distance it moves while in his grip.</div></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data"><span class="texhtml"><i>W</i></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data"><a href="/wiki/Joule" title="Joule">joule</a> (J)</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Other units</div></th><td class="infobox-data"><a href="/wiki/Foot-pound_(energy)" title="Foot-pound (energy)">Foot-pound</a>, <a href="/wiki/Erg" title="Erg">Erg</a></td></tr><tr><th scope="row" class="infobox-label">In <a href="/wiki/SI_base_unit" title="SI base unit"><span class="wrap">SI base units</span></a></th><td class="infobox-data">1 <a href="/wiki/Kilogram" title="Kilogram">kg</a>⋅<a href="/wiki/Metre" title="Metre">m</a><sup>2</sup>⋅<a href="/wiki/Second" title="Second">s</a><sup>−2</sup></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Derivations from<br />other quantities</div></th><td class="infobox-data"><i>W</i> = <b><a href="/wiki/Force" title="Force">F</a></b> ⋅ <b><a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">s</a></b> <br /> <i>W</i> = <i><a href="/wiki/Torque" title="Torque">τ</a></i> <i><a href="/wiki/Angle" title="Angle">θ</a></i></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><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 {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30444cf24043ed5dec95ef56ddad687597057d34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.263ex; height:2.676ex;" alt="{\displaystyle {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-2}}"></span></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1129693374">.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 .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl 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style="padding-left:0.9em;padding-right:0.9em;"><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></th></tr><tr><td class="sidebar-image"><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 {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">F</mtext> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ad0a6d6780c3abc5247abd82bd8a2249d56ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.318ex; height:5.509ex;" alt="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}"></span><div class="sidebar-caption" style="font-size:90%;padding:0.6em 0;font-style:italic;"><a href="/wiki/Second_law_of_motion" class="mw-redirect" title="Second law of motion">Second law of motion</a></div></td></tr><tr><th class="sidebar-heading" style="font-weight: bold; display:block;margin-bottom:1.0em;"> <div class="hlist"> <ul><li><a href="/wiki/History_of_classical_mechanics" title="History of classical mechanics">History</a></li> <li><a href="/wiki/Timeline_of_classical_mechanics" title="Timeline of classical mechanics">Timeline</a></li> <li><a href="/wiki/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics" title="List of textbooks on classical mechanics and quantum mechanics">Textbooks</a></li></ul> </div></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Branches</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Applied_mechanics" title="Applied mechanics">Applied</a></li> <li><a href="/wiki/Celestial_mechanics" title="Celestial mechanics">Celestial</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum</a></li> <li><a href="/wiki/Analytical_dynamics" class="mw-redirect" title="Analytical dynamics">Dynamics</a></li> <li><a href="/wiki/Classical_field_theory" title="Classical field theory">Field theory</a></li> <li><a href="/wiki/Kinematics" title="Kinematics">Kinematics</a></li> <li><a href="/wiki/Kinetics_(physics)" title="Kinetics (physics)">Kinetics</a></li> <li><a href="/wiki/Statics" title="Statics">Statics</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Acceleration" title="Acceleration">Acceleration</a></li> <li><a href="/wiki/Angular_momentum" title="Angular momentum">Angular momentum</a></li> <li><a href="/wiki/Couple_(mechanics)" title="Couple (mechanics)">Couple</a></li> <li><a href="/wiki/D%27Alembert%27s_principle" title="D'Alembert's principle">D'Alembert's principle</a></li> <li><a href="/wiki/Energy" title="Energy">Energy</a> <ul><li><a href="/wiki/Kinetic_energy#Newtonian_kinetic_energy" title="Kinetic energy">kinetic</a></li> <li><a href="/wiki/Potential_energy" title="Potential energy">potential</a></li></ul></li> <li><a href="/wiki/Force" title="Force">Force</a></li> <li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li> <li><a href="/wiki/Impulse_(physics)" title="Impulse (physics)">Impulse</a></li> <li><span class="nowrap"><a href="/wiki/Inertia" title="Inertia">Inertia</a> / <a href="/wiki/Moment_of_inertia" title="Moment of inertia">Moment of inertia</a></span></li> <li><a href="/wiki/Mass" title="Mass">Mass</a></li> <li><br /><a href="/wiki/Mechanical_power_(physics)" class="mw-redirect" title="Mechanical power (physics)">Mechanical power</a></li> <li><a class="mw-selflink selflink">Mechanical work</a></li> <li><br /><a href="/wiki/Moment_(physics)" title="Moment (physics)">Moment</a></li> <li><a href="/wiki/Momentum" title="Momentum">Momentum</a></li> <li><a href="/wiki/Space" title="Space">Space</a></li> <li><a href="/wiki/Speed" title="Speed">Speed</a></li> <li><a href="/wiki/Time" title="Time">Time</a></li> <li><a href="/wiki/Torque" title="Torque">Torque</a></li> <li><a href="/wiki/Velocity" title="Velocity">Velocity</a></li> <li><a href="/wiki/Virtual_work" title="Virtual work">Virtual work</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"> <ul><li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></b></div></li> <li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a></b> <div class="plainlist"><ul><li><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li><li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li><li><a href="/wiki/Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a></li><li><a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li><li><a href="/wiki/Appell%27s_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</a></li><li><a href="/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann mechanics</a></li></ul></div></div></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Core topics</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Damping" title="Damping">Damping</a></li> <li><a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">Displacement</a></li> <li><a href="/wiki/Equations_of_motion" title="Equations of motion">Equations of motion</a></li> <li><a href="/wiki/Euler%27s_laws_of_motion" title="Euler's laws of motion"><span class="wrap">Euler's laws of motion</span></a></li> <li><a href="/wiki/Fictitious_force" title="Fictitious force">Fictitious force</a></li> <li><a href="/wiki/Friction" title="Friction">Friction</a></li> <li><a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li></ul> </div> <ul><li><span class="nowrap"><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial</a> / <a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">Non-inertial reference frame</a></span></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Motion" title="Motion">Motion</a> (<a href="/wiki/Linear_motion" title="Linear motion">linear</a>)</li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation"><span class="wrap">Newton's law of universal gravitation</span></a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li> <li><a href="/wiki/Relative_velocity" title="Relative velocity">Relative velocity</a></li> <li><a href="/wiki/Rigid_body" title="Rigid body">Rigid body</a> <ul><li><a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">dynamics</a></li> <li><a href="/wiki/Euler%27s_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's equations</a></li></ul></li> <li><a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</a></li> <li><a href="/wiki/Vibration" title="Vibration">Vibration</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)"><a href="/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">Rotation</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Circular_motion" title="Circular motion">Circular motion</a></li> <li><a href="/wiki/Rotating_reference_frame" title="Rotating reference frame">Rotating reference frame</a></li> <li><a href="/wiki/Centripetal_force" title="Centripetal force">Centripetal force</a></li> <li><a href="/wiki/Centrifugal_force" title="Centrifugal force">Centrifugal force</a> <ul><li><a href="/wiki/Reactive_centrifugal_force" title="Reactive centrifugal force">reactive</a></li></ul></li> <li><a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a></li> <li><a href="/wiki/Pendulum_(mechanics)" title="Pendulum (mechanics)">Pendulum</a></li> <li><a href="/wiki/Tangential_speed" title="Tangential speed">Tangential speed</a></li> <li><a href="/wiki/Rotational_frequency" title="Rotational frequency">Rotational frequency</a></li></ul> </div> <ul><li><a href="/wiki/Angular_acceleration" title="Angular acceleration">Angular acceleration</a> / <a href="/wiki/Angular_displacement" title="Angular displacement">displacement</a> / <a href="/wiki/Angular_frequency" title="Angular frequency">frequency</a> / <a href="/wiki/Angular_velocity" title="Angular velocity">velocity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible 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.navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Classical_mechanics" title="Template:Classical mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classical_mechanics" title="Template talk:Classical mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classical_mechanics" title="Special:EditPage/Template:Classical mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In science, <b>work</b> is the <a href="/wiki/Energy" title="Energy">energy</a> transferred to or from an <a href="/wiki/Physical_object" title="Physical object">object</a> via the application of <a href="/wiki/Force" title="Force">force</a> along a <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a>. In its simplest form, for a constant force aligned with the direction of motion, the work equals the <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a> of the force strength and the distance traveled. A force is said to do <i>positive work</i> if it has a component in the direction of the displacement of the <a href="/wiki/Point_of_application" class="mw-redirect" title="Point of application">point of application</a>. A force does <i>negative work</i> if it has a component opposite to the direction of the displacement at the point of application of the force.<sup id="cite_ref-:1_1-0" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). If the ball is thrown upwards, the work done by the gravitational force is negative, and is equal to the weight multiplied by the displacement in the upwards direction. </p><p>Both force and displacement are <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a>. The work done is given by the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of the two vectors, where the result is a <a href="/wiki/Scalar_(physics)" title="Scalar (physics)">scalar</a>. When the force <span class="texhtml mvar" style="font-style:italic;">F</span> is constant and the angle <span class="texhtml">θ</span> between the force and the displacement <span class="texhtml mvar" style="font-style:italic;">s</span> is also constant, then the work done is given by: <span class="mwe-math-element" data-qid="Q42213"><a href="/w/index.php?title=Special:MathWikibase&qid=Q42213" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=Fs\cos {\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>F</mi> <mi>s</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=Fs\cos {\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466757d5d25550923ab9611f48f8e9e3da4d8513" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.341ex; height:2.176ex;" alt="{\displaystyle W=Fs\cos {\theta }}"></a></span> </p><p>If the force is variable, then work is given by the <a href="/wiki/Line_integral" title="Line integral">line integral</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 W=\int {\vec {F}}\cdot d{\vec {s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <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> <mi>d</mi> <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 W=\int {\vec {F}}\cdot d{\vec {s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fba7934e22940e978b42e3327df0273c823a899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.003ex; height:5.676ex;" alt="{\displaystyle W=\int {\vec {F}}\cdot d{\vec {s}}}"></span> </p><p>where <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{\vec {s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <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 d{\vec {s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb1d8aba0a3e8c1f59efc33344f15261740b20b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.439ex; height:2.343ex;" alt="{\displaystyle d{\vec {s}}}"></span> is the tiny change in displacement vector. </p><p>Work is a <a href="/wiki/Scalar_(physics)" title="Scalar (physics)">scalar quantity</a>,<sup id="cite_ref-Young_2-0" class="reference"><a href="#cite_note-Young-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> so it has only magnitude and no direction. Work transfers energy from one place to another, or one form to another. The <a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a> of work is the <a href="/wiki/Joule" title="Joule">joule</a> (J), the same unit as for energy. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Ancient_Greek_technology" title="Ancient Greek technology">ancient Greek understanding of physics</a> was limited to the <a href="/wiki/Statics" title="Statics">statics</a> of simple machines (the balance of forces), and did not include <a href="/wiki/Dynamics_(mechanics)" class="mw-redirect" title="Dynamics (mechanics)">dynamics</a> or the concept of work. During the <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a> the dynamics of the <i>Mechanical Powers</i>, as the <a href="/wiki/Simple_machine" title="Simple machine">simple machines</a> were called, began to be studied from the standpoint of how far they could lift a load, in addition to the force they could apply, leading eventually to the new concept of mechanical work. The complete dynamic theory of simple machines was worked out by Italian scientist <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a> in 1600 in <i>Le Meccaniche</i> (<i>On Mechanics</i>), in which he showed the underlying mathematical similarity of the machines as force amplifiers.<sup id="cite_ref-Krebs_3-0" class="reference"><a href="#cite_note-Krebs-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Stephen_4-0" class="reference"><a href="#cite_note-Stephen-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> He was the first to explain that simple machines do not create energy, only transform it.<sup id="cite_ref-Krebs_3-1" class="reference"><a href="#cite_note-Krebs-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Early_concepts_of_work">Early concepts of work</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=2" title="Edit section: Early concepts of work"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although <i>work</i> was not formally used until 1826, similar concepts existed before then. Early names for the same concept included <i>moment of activity, quantity of action, latent live force, dynamic effect, efficiency</i>, and even <i>force</i>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In 1637, the French philosopher <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> wrote:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>Lifting 100 lb one foot twice over is the same as lifting 200 lb one foot, or 100 lb two feet.</p><div class="templatequotecite">— <cite>René Descartes, Letter to Huygens</cite></div></blockquote> <p>In 1686, the German philosopher <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Leibniz</a> wrote:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>The same force ["work" in modern terms] is necessary to raise body A of 1 pound (libra) to a height of 4 yards (ulnae), as is necessary to raise body B of 4 pounds to a height of 1 yard.</p><div class="templatequotecite">— <cite>Gottfried Leibniz, Brevis demonstratio</cite></div></blockquote> <p>In 1759, <a href="/wiki/John_Smeaton" title="John Smeaton">John Smeaton</a> described a quantity that he called "power" "to signify the exertion of strength, gravitation, impulse, or pressure, as to produce motion." Smeaton continues that this quantity can be calculated if "the weight raised is multiplied by the height to which it can be raised in a given time," making this definition remarkably similar to <a href="/wiki/Gaspard-Gustave_de_Coriolis" title="Gaspard-Gustave de Coriolis">Coriolis</a>'s.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Etymology">Etymology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=3" title="Edit section: Etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>According to the 1957 physics textbook by <a href="/wiki/Max_Jammer" title="Max Jammer">Max Jammer</a>,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> the term <i>work</i> was introduced in 1826 by the French mathematician <a href="/wiki/Gaspard-Gustave_Coriolis" class="mw-redirect" title="Gaspard-Gustave Coriolis">Gaspard-Gustave Coriolis</a><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> as "weight <i>lifted</i> through a height", which is based on the use of early <a href="/wiki/Steam_engine" title="Steam engine">steam engines</a> to lift buckets of water out of flooded ore mines. According to Rene Dugas, French engineer and historian, it is to <a href="/wiki/Salomon_de_Caus" title="Salomon de Caus">Solomon of Caux</a> "that we owe the term <i>work</i> in the sense that it is used in mechanics now".<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Units">Units</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=4" title="Edit section: Units"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/International_System_of_Units" title="International System of Units">SI</a> unit of work is the <a href="/wiki/Joule" title="Joule">joule</a> (J), named after English physicist <a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">James Prescott Joule</a> (1818-1889), which is defined as the work required to exert a force of one <a href="/wiki/Newton_(unit)" title="Newton (unit)">newton</a> through a displacement of one <a href="/wiki/Metre" title="Metre">metre</a>. </p><p>The dimensionally equivalent <a href="/wiki/Newton-metre" title="Newton-metre">newton-metre</a> (N⋅m) is sometimes used as the measuring unit for work, but this can be confused with the measurement unit of <a href="/wiki/Torque" title="Torque">torque</a>. Usage of N⋅m is discouraged by the <a href="/wiki/General_Conference_on_Weights_and_Measures" title="General Conference on Weights and Measures">SI authority</a>, since it can lead to confusion as to whether the quantity expressed in newton-metres is a torque measurement, or a measurement of work.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Another unit for work is the <a href="/wiki/Foot-pound_(energy)" title="Foot-pound (energy)">foot-pound</a>, which comes from the English system of measurement. As the unit name suggests, it is the product of pounds for the unit of force and feet for the unit of displacement. One joule is equivalent to 0.07376 ft-lbs.<sup id="cite_ref-:0_13-0" class="reference"><a href="#cite_note-:0-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Non-SI units of work include the newton-metre, <a href="/wiki/Erg" title="Erg">erg</a>, the foot-pound, the <a href="/wiki/Foot-poundal" title="Foot-poundal">foot-poundal</a>, the <a href="/wiki/Kilowatt_hour" class="mw-redirect" title="Kilowatt hour">kilowatt hour</a>, the <a href="/w/index.php?title=Litre-atmosphere&action=edit&redlink=1" class="new" title="Litre-atmosphere (page does not exist)">litre-atmosphere</a>, and the <a href="/wiki/Horsepower" title="Horsepower">horsepower-hour</a>. Due to work having the same <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">physical dimension</a> as <a href="/wiki/Heat" title="Heat">heat</a>, occasionally measurement units typically reserved for heat or energy content, such as <a href="/wiki/Therm" title="Therm">therm</a>, <a href="/wiki/BTU" class="mw-redirect" title="BTU">BTU</a> and <a href="/wiki/Calorie" title="Calorie">calorie</a>, are used as a measuring unit. </p> <div class="mw-heading mw-heading2"><h2 id="Work_and_energy">Work and energy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=5" title="Edit section: Work and energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The work <span class="texhtml mvar" style="font-style:italic;">W</span> done by a constant force of magnitude <span class="texhtml mvar" style="font-style:italic;">F</span> on a point that moves a displacement <span class="texhtml mvar" style="font-style:italic;">s</span> in a straight line in the direction of the force is the product <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W={\vec {F}}\cdot {\vec {s}}}"> <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>s</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W={\vec {F}}\cdot {\vec {s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54c85b27e7a84e7989196c693d9e29d16f8b201f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.207ex; height:2.843ex;" alt="{\displaystyle W={\vec {F}}\cdot {\vec {s}}}"></span> </p><p>For example, if a force of 10 newtons (<span class="texhtml"><i>F</i> = 10 N</span>) acts along a point that travels 2 metres (<span class="texhtml"><i>s</i> = 2 m</span>), then <span class="texhtml"><i>W</i> = <i>Fs</i> = (10 N) (2 m) = 20 J</span>. This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. </p><p>The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. </p><p>Work is closely related to <a href="/wiki/Energy" title="Energy">energy</a>. Energy shares the same unit of measurement with work (Joules) because the energy from the object doing work is transferred to the other objects it interacts with when work is being done.<sup id="cite_ref-:0_13-1" class="reference"><a href="#cite_note-:0-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The work–energy principle states that an increase in the kinetic energy of a <a href="/wiki/Rigid_body" title="Rigid body">rigid body</a> is caused by an equal amount of positive work done on the body by the resultant force acting on that body. Conversely, a decrease in kinetic energy is caused by an equal amount of negative work done by the resultant force. Thus, if the net work is positive, then the particle's kinetic energy increases by the amount of the work. If the net work done is negative, then the particle's kinetic energy decreases by the amount of work.<sup id="cite_ref-walker_14-0" class="reference"><a href="#cite_note-walker-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>From <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's second law</a>, it can be shown that work on a free (no fields), rigid (no internal degrees of freedom) body, is equal to the change in kinetic energy <span class="texhtml"><i>E</i><sub>k</sub></span> corresponding to the linear velocity and <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> of that body, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\Delta E_{\text{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"> <mtext>k</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\Delta E_{\text{k}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99fd42b68cf01252cc1cd66aba086b58a5a1533f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.932ex; height:2.509ex;" alt="{\displaystyle W=\Delta E_{\text{k}}.}"></span> The work of forces generated by a potential function is known as <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> and the forces are said to be <a href="/wiki/Conservative_force" title="Conservative force">conservative</a>. Therefore, work on an object that is merely displaced in a conservative force <a href="/wiki/Field_(physics)" title="Field (physics)">field</a>, without change in velocity or rotation, is equal to <i>minus</i> the change of potential energy <span class="texhtml"><i>E</i><sub>p</sub></span> of the object, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=-\Delta E_{\text{p}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=-\Delta E_{\text{p}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5a87dcaf2529acf7306203192aea7f78e689f51" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.786ex; height:2.843ex;" alt="{\displaystyle W=-\Delta E_{\text{p}}.}"></span> These formulas show that work is the energy associated with the action of a force, so work subsequently possesses the <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">physical dimensions</a>, and units, of energy. The work/energy principles discussed here are identical to electric work/energy principles. </p> <div class="mw-heading mw-heading2"><h2 id="Constraint_forces">Constraint forces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=6" title="Edit section: Constraint forces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Constraint forces determine the object's displacement in the system, limiting it within a range. For example, in the case of a <a href="/wiki/Slope" title="Slope">slope</a> plus gravity, the object is <i>stuck to</i> the slope and, when attached to a taut string, it cannot move in an outwards direction to make the string any 'tauter'. It eliminates all displacements in that direction, that is, the velocity in the direction of the constraint is limited to 0, so that the constraint forces do not perform work on the system. </p><p>For a <a href="/wiki/Mechanical_system" class="mw-redirect" title="Mechanical system">mechanical system</a>,<sup id="cite_ref-goldstein_15-0" class="reference"><a href="#cite_note-goldstein-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> constraint forces eliminate movement in directions that characterize the constraint. Thus the <a href="/wiki/Virtual_work" title="Virtual work">virtual work</a> done by the forces of constraint is zero, a result which is only true if friction forces are excluded.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Fixed, frictionless constraint forces do not perform work on the system,<sup id="cite_ref-Feynman_17-0" class="reference"><a href="#cite_note-Feynman-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> as the angle between the motion and the constraint forces is always <a href="/wiki/Right_angle" title="Right angle">90°</a>.<sup id="cite_ref-Feynman_17-1" class="reference"><a href="#cite_note-Feynman-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Examples of workless constraints are: rigid interconnections between particles, sliding motion on a frictionless surface, and rolling contact without slipping.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>For example, in a pulley system like the <a href="/wiki/Atwood_machine" title="Atwood machine">Atwood machine</a>, the internal forces on the rope and at the supporting pulley do no work on the system. Therefore, work need only be computed for the gravitational forces acting on the bodies. Another example is the <a href="/wiki/Centripetal_force" title="Centripetal force">centripetal force</a> exerted <i>inwards</i> by a string on a ball in uniform <a href="/wiki/Circular_motion" title="Circular motion">circular motion</a> <i>sideways</i> constrains the ball to circular motion restricting its movement away from the centre of the circle. This force does zero work because it is perpendicular to the velocity of the ball. </p><p>The <a href="/wiki/Magnetic_force" class="mw-redirect" title="Magnetic force">magnetic force</a> on a charged particle is <span class="texhtml"><b>F</b> = <i>q</i><b>v</b> × <b>B</b></span>, where <span class="texhtml mvar" style="font-style:italic;">q</span> is the charge, <span class="texhtml"><b>v</b></span> is the velocity of the particle, and <span class="texhtml"><b>B</b></span> is the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>. The result of a <a href="/wiki/Cross_product" title="Cross product">cross product</a> is always perpendicular to both of the original vectors, so <span class="texhtml"><b>F</b> ⊥ <b>v</b></span>. The <a href="/wiki/Dot_product" title="Dot product">dot product</a> of two perpendicular vectors is always zero, so the work <span class="texhtml"><i>W</i> = <b>F</b> ⋅ <b>v</b> = 0</span>, and the magnetic force does not do work. It can change the direction of motion but never change the speed. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_calculation">Mathematical calculation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=7" title="Edit section: Mathematical calculation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For moving objects, the quantity of work/time (power) is integrated along the trajectory of the point of application of the force. Thus, at any instant, the rate of the work done by a force (measured in joules/second, or <a href="/wiki/Watt" title="Watt">watts</a>) is the <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a> of the force (a vector), and the velocity vector of the point of application. This scalar product of force and velocity is known as instantaneous <a href="/wiki/Power_(physics)" title="Power (physics)">power</a>. Just as velocities may be integrated over time to obtain a total distance, by the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a>, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.<sup id="cite_ref-Resnick_19-0" class="reference"><a href="#cite_note-Resnick-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>Work is the result of a force on a point that follows a curve <span class="texhtml"><b>X</b></span>, with a velocity <span class="texhtml"><b>v</b></span>, at each instant. The small amount of work <span class="texhtml"><i>δW</i></span> that occurs over an instant of time <span class="texhtml"><i>dt</i></span> is calculated as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\mathbf {F} \cdot d\mathbf {s} =\mathbf {F} \cdot \mathbf {v} dt}"> <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> <mi>d</mi> <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> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\mathbf {F} \cdot d\mathbf {s} =\mathbf {F} \cdot \mathbf {v} dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2277d4841d3e704a83a98c202f65edf42acf962f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.142ex; height:2.343ex;" alt="{\displaystyle \delta W=\mathbf {F} \cdot d\mathbf {s} =\mathbf {F} \cdot \mathbf {v} dt}"></span> where the <span class="texhtml"><b>F</b> ⋅ <b>v</b></span> is the power over the instant <span class="texhtml"><i>dt</i></span>. The sum of these small amounts of work over the trajectory of the point yields the work, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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}}\mathbf {F} \cdot {\tfrac {d\mathbf {s} }{dt}}\,dt=\int _{C}\mathbf {F} \cdot 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" /> <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">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <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">F</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo>,</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}}\mathbf {F} \cdot {\tfrac {d\mathbf {s} }{dt}}\,dt=\int _{C}\mathbf {F} \cdot d\mathbf {s} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62bae8e044a7815446bf8c42b1a29c91e732e866" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.651ex; height:6.509ex;" alt="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} \,dt=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot {\tfrac {d\mathbf {s} }{dt}}\,dt=\int _{C}\mathbf {F} \cdot d\mathbf {s} ,}"></span> where <i>C</i> is the trajectory from <b>x</b>(<i>t</i><sub>1</sub>) to <b>x</b>(<i>t</i><sub>2</sub>). This integral is computed along the trajectory of the particle, and is therefore said to be <i>path dependent</i>. </p><p>If the force is always directed along this line, and the magnitude of the force is <span class="texhtml"><i>F</i></span>, then this integral simplifies to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{C}F\,ds}"> <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" /> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{C}F\,ds}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db7d02ae3321f49301f20f492c5b7784881f4055" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.129ex; height:5.676ex;" alt="{\displaystyle W=\int _{C}F\,ds}"></span> where <span class="texhtml mvar" style="font-style:italic;">s</span> is displacement along the line. If <span class="texhtml"><b>F</b></span> is constant, in addition to being directed along the line, then the integral simplifies further to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{C}F\,ds=F\int _{C}ds=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" /> <mi>d</mi> <mi>s</mi> <mo>=</mo> <mi>F</mi> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mi>d</mi> <mi>s</mi> <mo>=</mo> <mi>F</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{C}F\,ds=F\int _{C}ds=Fs}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb498cc293d618a52c688cec3f0760af9fc1ad0f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.752ex; height:5.676ex;" alt="{\displaystyle W=\int _{C}F\,ds=F\int _{C}ds=Fs}"></span> where <i>s</i> is the displacement of the point along the line. </p><p>This calculation can be generalized for a constant force that is not directed along the line, followed by the particle. In this case the <a href="/wiki/Dot_product" title="Dot product">dot product</a> <span class="texhtml"><b>F</b> ⋅ <i>d</i><b>s</b> = <i>F</i> cos <i>θ</i> <i>ds</i></span>, where <span class="texhtml mvar" style="font-style:italic;">θ</span> is the angle between the force vector and the direction of movement,<sup id="cite_ref-Resnick_19-1" class="reference"><a href="#cite_note-Resnick-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> that is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{C}\mathbf {F} \cdot 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> <mi>d</mi> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {s} =Fs\cos \theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f14d27e0f49e195993daf06d71b65dfff36b10b5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.88ex; height:5.676ex;" alt="{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {s} =Fs\cos \theta .}"></span> </p><p>When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a <a href="/wiki/Central_force" title="Central force">central force</a>), no work is done, since the cosine of 90° is zero.<sup id="cite_ref-walker_14-1" class="reference"><a href="#cite_note-walker-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Thus, no work can be performed by gravity on a planet with a circular orbit (this is ideal, as all orbits are slightly elliptical). Also, no work is done on a body moving circularly at a constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge. </p> <div class="mw-heading mw-heading3"><h3 id="Work_done_by_a_variable_force">Work done by a variable force</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=8" title="Edit section: Work done by a variable force"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Calculating the work as "force times straight path segment" would only apply in the most simple of circumstances, as noted above. If force is changing, or if the body is moving along a curved path, possibly rotating and not necessarily rigid, then only the path of the application point of the force is relevant for the work done, and only the component of the force parallel to the application point <a href="/wiki/Velocity" title="Velocity">velocity</a> is doing work (positive work when in the same direction, and negative when in the opposite direction of the velocity). This component of force can be described by the scalar quantity called <i>scalar tangential component</i> (<span class="texhtml"><i>F</i> cos(<i>θ</i>)</span>, where <span class="texhtml mvar" style="font-style:italic;">θ</span> is the angle between the force and the velocity). And then the most general definition of work can be formulated as follows: </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Force-distance-diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Force-distance-diagram.svg/220px-Force-distance-diagram.svg.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Force-distance-diagram.svg/330px-Force-distance-diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Force-distance-diagram.svg/440px-Force-distance-diagram.svg.png 2x" data-file-width="899" data-file-height="889" /></a><figcaption><a href="/wiki/Area_under_the_curve" class="mw-redirect" title="Area under the curve">Area under the curve</a> gives work done by F(x).</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.5em;"><i>Work done by a variable force is the line integral of its scalar tangential component along the path of its application point.</i><div class="paragraphbreak" style="margin-top:0.5em"></div> <p>If the force varies (e.g. compressing a spring) we need to use calculus to find the work done. If the force as a variable of <span class="texhtml mvar" style="font-style:italic;"><b>x</b></span> is given by <span class="texhtml"><b>F</b>(<b>x</b>)</span>, then the work done by the force along the x-axis from <span class="texhtml mvar" style="font-style:italic;"><b>x</b><sub>1</sub></span> to <span class="texhtml mvar" style="font-style:italic;"><b>x</b><sub>2</sub></span> is: </p> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\lim _{\Delta \mathbf {x} \to 0}\sum _{x_{1}}^{x_{2}}\mathbf {F(x)} \Delta \mathbf {x} =\int _{x_{1}}^{x_{2}}\mathbf {F(x)} d\mathbf {x} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold">x</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold">x</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\lim _{\Delta \mathbf {x} \to 0}\sum _{x_{1}}^{x_{2}}\mathbf {F(x)} \Delta \mathbf {x} =\int _{x_{1}}^{x_{2}}\mathbf {F(x)} d\mathbf {x} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad796c2a1ea2936ac1395da70f846d3abd8b9d18" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.414ex; height:7.176ex;" alt="{\displaystyle W=\lim _{\Delta \mathbf {x} \to 0}\sum _{x_{1}}^{x_{2}}\mathbf {F(x)} \Delta \mathbf {x} =\int _{x_{1}}^{x_{2}}\mathbf {F(x)} d\mathbf {x} .}"></span></div> <p>Thus, the work done for a variable force can be expressed as a <a href="/wiki/Definite_integral" class="mw-redirect" title="Definite integral">definite integral</a> of force over displacement.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>If the displacement as a variable of time is given by <span class="texhtml mvar" style="font-style:italic;">∆<b>x</b>(t)</span>, then work done by the variable force from <span class="texhtml mvar" style="font-style:italic;">t<sub>1</sub></span> to <span class="texhtml mvar" style="font-style:italic;">t<sub>2</sub></span> is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} (t)\cdot \mathbf {v} (t)dt=\int _{t_{1}}^{t_{2}}P(t)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">F</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <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>P</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} (t)\cdot \mathbf {v} (t)dt=\int _{t_{1}}^{t_{2}}P(t)dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce9a9935838284020baaf5ab2ba96f24453238ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.872ex; height:6.509ex;" alt="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} (t)\cdot \mathbf {v} (t)dt=\int _{t_{1}}^{t_{2}}P(t)dt.}"></span></dd></dl> <p>Thus, the work done for a variable force can be expressed as a definite integral of <a href="/wiki/Power_(physics)" title="Power (physics)">power</a> over time. </p> <div class="mw-heading mw-heading3"><h3 id="Torque_and_rotation">Torque and rotation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=9" title="Edit section: Torque and rotation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Couple_(mechanics)" title="Couple (mechanics)">force couple</a> results from equal and opposite forces, acting on two different points of a rigid body. The sum (resultant) of these forces may cancel, but their effect on the body is the couple or torque <b>T</b>. The work of the torque is calculated as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\mathbf {T} \cdot {\boldsymbol {\omega }}\,dt,}"> <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"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\mathbf {T} \cdot {\boldsymbol {\omega }}\,dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afc0fa6c868b74b713e7ca73faff73ee52aa89da" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.879ex; height:2.676ex;" alt="{\displaystyle \delta W=\mathbf {T} \cdot {\boldsymbol {\omega }}\,dt,}"></span> where the <span class="texhtml"><b>T</b> ⋅ <i><b>ω</b></i></span> is the power over the instant <span class="texhtml"><i>dt</i></span>. The sum of these small amounts of work over the trajectory of the rigid body yields the work, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot {\boldsymbol {\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"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot {\boldsymbol {\omega }}\,dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8241dbf14cabd8e06e55fcf01722b13e273d1239" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.339ex; height:6.509ex;" alt="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot {\boldsymbol {\omega }}\,dt.}"></span> This integral is computed along the trajectory of the rigid body with an angular velocity <span class="texhtml"><i><b>ω</b></i></span> that varies with time, and is therefore said to be <i>path dependent</i>. </p><p>If the angular velocity vector maintains a constant direction, then it takes the form, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}={\dot {\phi }}\mathbf {S} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}={\dot {\phi }}\mathbf {S} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/630f101188b8dcfd75a20673987a40ca0326a3b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.367ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {\omega }}={\dot {\phi }}\mathbf {S} ,}"></span> where <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> is the angle of rotation about the constant unit vector <span class="texhtml"><b>S</b></span>. In this case, the work of the torque becomes, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot {\boldsymbol {\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"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mspace width="thinmathspace" /> <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> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ϕ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot {\boldsymbol {\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/08c92bf2fef89f86d461368a7b138a3d669dcddc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.734ex; height:6.509ex;" alt="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {T} \cdot {\boldsymbol {\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> where <span class="texhtml"><i>C</i></span> is the trajectory from <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 \phi (t_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <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 \phi (t_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6dcdb8c90f4fde4032fe0be7502b86b993328f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.089ex; height:2.843ex;" alt="{\displaystyle \phi (t_{1})}"></span> to <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 \phi (t_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <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 \phi (t_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea998c4e24018529cfd2134466189545dbeea34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.089ex; height:2.843ex;" alt="{\displaystyle \phi (t_{2})}"></span>. This integral depends on the rotational trajectory <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 \phi (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23781b983d21d78467b65e7e32b9e7bc05d625f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.034ex; height:2.843ex;" alt="{\displaystyle \phi (t)}"></span>, and is therefore path-dependent. </p><p>If the torque <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> is aligned with the angular velocity vector so that, <span class="mwe-math-element" data-qid="Q48103"><a href="/w/index.php?title=Special:MathWikibase&qid=Q48103" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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> <mo>,</mo> </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/30b72ae28f4a7b461e35030e877fefe7eda7489f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.292ex; height:2.509ex;" alt="{\displaystyle \mathbf {T} =\tau \mathbf {S} ,}"></a></span> and both the torque and angular velocity are constant, then the work takes the form,<sup id="cite_ref-Young_2-1" class="reference"><a href="#cite_note-Young-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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> <mspace width="thinmathspace" /> <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> <mo>.</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/21f14349459a1f7d9356fed47582e80ed141bdc2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.631ex; height:6.509ex;" alt="{\displaystyle W=\int _{t_{1}}^{t_{2}}\tau {\dot {\phi }}\,dt=\tau (\phi _{2}-\phi _{1}).}"></span> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Work_on_lever_arm.png" class="mw-file-description"><img alt="Work on lever arm" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Work_on_lever_arm.png/250px-Work_on_lever_arm.png" decoding="async" width="250" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Work_on_lever_arm.png/375px-Work_on_lever_arm.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Work_on_lever_arm.png/500px-Work_on_lever_arm.png 2x" data-file-width="800" data-file-height="698" /></a><figcaption>A force of constant magnitude and perpendicular to the lever arm</figcaption></figure> <p>This result can be understood more simply by considering the torque as arising from a force of constant magnitude <span class="texhtml"><i>F</i></span>, being applied perpendicularly to a lever arm at a distance <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, as shown in the figure. This force will act through the distance along the circular arc <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 l=s=r\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mi>s</mi> <mo>=</mo> <mi>r</mi> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=s=r\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d3a8ccdf3723ca3bc3fadfda54fbef7a25c023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.415ex; height:2.509ex;" alt="{\displaystyle l=s=r\phi }"></span>, so the work done is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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> <mo>.</mo> </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/355e2ef762e06b29d51d93290865421f53ac1798" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.285ex; height:2.509ex;" alt="{\displaystyle W=Fs=Fr\phi .}"></span> Introduce the torque <span class="texhtml"><i>τ</i> = <i>Fr</i></span>, to obtain <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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> <mo>,</mo> </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/c8369efe7b4f97ad2afb02e0fb60e305d7457804" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.041ex; height:2.509ex;" alt="{\displaystyle W=Fr\phi =\tau \phi ,}"></span> as presented above. </p><p>Notice that only the component of torque in the direction of the angular velocity vector contributes to the work. </p> <div class="mw-heading mw-heading2"><h2 id="Work_and_potential_energy">Work and potential energy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=10" title="Edit section: Work and potential energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The scalar product of a force <span class="texhtml"><b>F</b></span> and the velocity <span class="texhtml"><b>v</b></span> of its point of application defines the <a href="/wiki/Power_(physics)" title="Power (physics)">power</a> input to a system at an instant of time. Integration of this power over the trajectory of the point of application, <span class="texhtml"><i>C</i> = <b>x</b>(<i>t</i>)</span>, defines the work input to the system by the force. </p> <div class="mw-heading mw-heading3"><h3 id="Path_dependence">Path dependence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=11" title="Edit section: Path dependence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Therefore, the <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">work</a> done by a force <span class="texhtml"><b>F</b></span> on an object that travels along a curve <span class="texhtml"><i>C</i></span> is given by the <a href="/wiki/Line_integral" title="Line integral">line integral</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} dt,}"> <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> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad41ea906a31472351d54b4d3a6c1644fef82757" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.766ex; height:6.509ex;" alt="{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} dt,}"></span> where <span class="texhtml"><i>dx</i>(<i>t</i>)</span> defines the trajectory <span class="texhtml"><i>C</i></span> and <span class="texhtml"><b>v</b></span> is the velocity along this trajectory. In general this integral requires that the path along which the velocity is defined, so the evaluation of work is said to be path dependent. </p><p>The time derivative of the integral for work yields the instantaneous power, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dW}{dt}}=P(t)=\mathbf {F} \cdot \mathbf {v} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>W</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dW}{dt}}=P(t)=\mathbf {F} \cdot \mathbf {v} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f66dfa886e6b38066ef4a832dd1d3cdaf51d758f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.498ex; height:5.509ex;" alt="{\displaystyle {\frac {dW}{dt}}=P(t)=\mathbf {F} \cdot \mathbf {v} .}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Path_independence">Path independence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=12" title="Edit section: Path independence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the work for an applied force is independent of the path, then the work done by the force, by the <a href="/wiki/Gradient_theorem" title="Gradient theorem">gradient theorem</a>, defines a potential function which is evaluated at the start and end of the trajectory of the point of application. This means that there is a potential function <span class="texhtml"><i>U</i>(<b>x</b>)</span>, that can be evaluated at the two points <span class="texhtml"><b>x</b>(<i>t</i><sub>1</sub>)</span> and <span class="texhtml"><b>x</b>(<i>t</i><sub>2</sub>)</span> to obtain the work over any trajectory between these two points. It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, that is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{\mathbf {x} (t_{1})}^{\mathbf {x} (t_{2})}\mathbf {F} \cdot d\mathbf {x} =U(\mathbf {x} (t_{1}))-U(\mathbf {x} (t_{2})).}"> <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> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <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> </mrow> <mrow class="MJX-TeXAtom-ORD"> <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> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mi>U</mi> <mo stretchy="false">(</mo> <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> <mo stretchy="false">)</mo> <mo>−</mo> <mi>U</mi> <mo stretchy="false">(</mo> <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> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{\mathbf {x} (t_{1})}^{\mathbf {x} (t_{2})}\mathbf {F} \cdot d\mathbf {x} =U(\mathbf {x} (t_{1}))-U(\mathbf {x} (t_{2})).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d905f560f0b933af6c093e432c493d1b0318daf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:54.554ex; height:6.676ex;" alt="{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{\mathbf {x} (t_{1})}^{\mathbf {x} (t_{2})}\mathbf {F} \cdot d\mathbf {x} =U(\mathbf {x} (t_{1}))-U(\mathbf {x} (t_{2})).}"></span> </p><p>The function <span class="texhtml"><i>U</i>(<b>x</b>)</span> is called the <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> associated with the applied force. The force derived from such a potential function is said to be <a href="/wiki/Conservative_force" title="Conservative force">conservative</a>. Examples of forces that have potential energies are gravity and spring forces. </p><p>In this case, the <a href="/wiki/Gradient" title="Gradient">gradient</a> of work yields <span class="mwe-math-element" data-qid="Q11402"><a href="/w/index.php?title=Special:MathWikibase&qid=Q11402" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla W=-\nabla U=-\left({\frac {\partial U}{\partial x}},{\frac {\partial U}{\partial y}},{\frac {\partial U}{\partial z}}\right)=\mathbf {F} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>U</mi> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla W=-\nabla U=-\left({\frac {\partial U}{\partial x}},{\frac {\partial U}{\partial y}},{\frac {\partial U}{\partial z}}\right)=\mathbf {F} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992dd2e43776610739af8c23ec67b3856c2e8d4c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.017ex; height:6.176ex;" alt="{\displaystyle \nabla W=-\nabla U=-\left({\frac {\partial U}{\partial x}},{\frac {\partial U}{\partial y}},{\frac {\partial U}{\partial z}}\right)=\mathbf {F} ,}"></a></span> and the force <b>F</b> is said to be "derivable from a potential."<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>Because the potential <span class="texhtml mvar" style="font-style:italic;">U</span> defines a force <span class="texhtml"><b>F</b></span> at every point <span class="texhtml"><b>x</b></span> in space, the set of forces is called a <a href="/wiki/Force_field_(physics)" title="Force field (physics)">force field</a>. The power applied to a body by a force field is obtained from the gradient of the work, or potential, in the direction of the velocity <span class="texhtml"><b>V</b></span> of the body, that is <span class="mwe-math-element" data-qid="Q25342"><a href="/w/index.php?title=Special:MathWikibase&qid=Q25342" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(t)=-\nabla U\cdot \mathbf {v} =\mathbf {F} \cdot \mathbf {v} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>U</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(t)=-\nabla U\cdot \mathbf {v} =\mathbf {F} \cdot \mathbf {v} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39cdfc586392f80d5ef3b7de7d2989289b475e9e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.628ex; height:2.843ex;" alt="{\displaystyle P(t)=-\nabla U\cdot \mathbf {v} =\mathbf {F} \cdot \mathbf {v} .}"></a></span> </p> <div class="mw-heading mw-heading3"><h3 id="Work_by_gravity">Work by gravity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=13" title="Edit section: Work by gravity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Work_of_gravity_F_dot_d_equals_mgh.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6c/Work_of_gravity_F_dot_d_equals_mgh.JPG" decoding="async" width="168" height="138" class="mw-file-element" data-file-width="168" data-file-height="138" /></a><figcaption>Gravity <span class="texhtml"><i>F</i> = <i>mg</i></span> does work <span class="texhtml"><i>W</i> = <i>mgh</i></span> along any descending path</figcaption></figure> <p>In the absence of other forces, gravity results in a constant downward acceleration of every freely moving object. Near Earth's surface the acceleration due to gravity is <span class="texhtml"><i>g</i> = 9.8 m⋅s<sup>−2</sup></span> and the gravitational force on an object of mass <i>m</i> is <span class="texhtml"><b>F</b><sub>g</sub> = <i>mg</i></span>. It is convenient to imagine this gravitational force concentrated at the <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> of the object. </p><p>If an object with weight <span class="texhtml"><i>mg</i></span> is displaced upwards or downwards a vertical distance <span class="texhtml"><i>y</i><sub>2</sub> − <i>y</i><sub>1</sub></span>, the work <span class="texhtml"><i>W</i></span> done on the object is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=F_{g}(y_{2}-y_{1})=F_{g}\Delta y=mg\Delta y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <mo>=</mo> <mi>m</mi> <mi>g</mi> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=F_{g}(y_{2}-y_{1})=F_{g}\Delta y=mg\Delta y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7e5b98f03cbe3578c21760d80edc9e4b277dfdc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.138ex; height:3.009ex;" alt="{\displaystyle W=F_{g}(y_{2}-y_{1})=F_{g}\Delta y=mg\Delta y}"></span> where <i>F<sub>g</sub></i> is weight (pounds in imperial units, and newtons in SI units), and Δ<i>y</i> is the change in height <i>y</i>. Notice that the work done by gravity depends only on the vertical movement of the object. The presence of friction does not affect the work done on the object by its weight. </p> <div class="mw-heading mw-heading4"><h4 id="In_space">In space</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=14" title="Edit section: In space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The force of gravity exerted by a mass <span class="texhtml mvar" style="font-style:italic;">M</span> on another mass <span class="texhtml mvar" style="font-style:italic;">m</span> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =-{\frac {GMm}{r^{2}}}{\hat {\mathbf {r} }}=-{\frac {GMm}{r^{3}}}\mathbf {r} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =-{\frac {GMm}{r^{2}}}{\hat {\mathbf {r} }}=-{\frac {GMm}{r^{3}}}\mathbf {r} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4737d4b91184725b7e94b1b1aa49276a396eb6d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.698ex; height:5.676ex;" alt="{\displaystyle \mathbf {F} =-{\frac {GMm}{r^{2}}}{\hat {\mathbf {r} }}=-{\frac {GMm}{r^{3}}}\mathbf {r} ,}"></span> where <span class="texhtml"><b>r</b></span> is the position vector from <span class="texhtml mvar" style="font-style:italic;">M</span> to <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml"><b>r̂</b></span> is the unit vector in the direction of <span class="texhtml"><b>r</b></span>. </p><p>Let the mass <span class="texhtml mvar" style="font-style:italic;">m</span> move at the velocity <span class="texhtml"><b>v</b></span>; then the work of gravity on this mass as it moves from position <span class="texhtml"><b>r</b>(<i>t</i><sub>1</sub>)</span> to <span class="texhtml"><b>r</b>(<i>t</i><sub>2</sub>)</span> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=-\int _{\mathbf {r} (t_{1})}^{\mathbf {r} (t_{2})}{\frac {GMm}{r^{3}}}\mathbf {r} \cdot d\mathbf {r} =-\int _{t_{1}}^{t_{2}}{\frac {GMm}{r^{3}}}\mathbf {r} \cdot \mathbf {v} \,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <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"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=-\int _{\mathbf {r} (t_{1})}^{\mathbf {r} (t_{2})}{\frac {GMm}{r^{3}}}\mathbf {r} \cdot d\mathbf {r} =-\int _{t_{1}}^{t_{2}}{\frac {GMm}{r^{3}}}\mathbf {r} \cdot \mathbf {v} \,dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3be5bae571f2ae04285631a40816e371c185f7d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.77ex; height:6.676ex;" alt="{\displaystyle W=-\int _{\mathbf {r} (t_{1})}^{\mathbf {r} (t_{2})}{\frac {GMm}{r^{3}}}\mathbf {r} \cdot d\mathbf {r} =-\int _{t_{1}}^{t_{2}}{\frac {GMm}{r^{3}}}\mathbf {r} \cdot \mathbf {v} \,dt.}"></span> Notice that the position and velocity of the mass <span class="texhtml mvar" style="font-style:italic;">m</span> are given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} =r\mathbf {e} _{r},\qquad \mathbf {v} ={\frac {d\mathbf {r} }{dt}}={\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mi>r</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =r\mathbf {e} _{r},\qquad \mathbf {v} ={\frac {d\mathbf {r} }{dt}}={\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f73b6c9cb5d76c660236e3c1c86c65ffe57697d9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.323ex; height:5.509ex;" alt="{\displaystyle \mathbf {r} =r\mathbf {e} _{r},\qquad \mathbf {v} ={\frac {d\mathbf {r} }{dt}}={\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t},}"></span> where <span class="texhtml"><b>e</b><sub><i>r</i></sub></span> and <span class="texhtml"><b>e</b><sub><i>t</i></sub></span> are the radial and tangential unit vectors directed relative to the vector from <span class="texhtml mvar" style="font-style:italic;">M</span> to <span class="texhtml mvar" style="font-style:italic;">m</span>, and we use the fact that <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\mathbf {e} _{r}/dt={\dot {\theta }}\mathbf {e} _{t}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {e} _{r}/dt={\dot {\theta }}\mathbf {e} _{t}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/475a7bc4105ad6c62ca2d880af6936e73186067a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.785ex; height:3.343ex;" alt="{\displaystyle d\mathbf {e} _{r}/dt={\dot {\theta }}\mathbf {e} _{t}.}"></span> Use this to simplify the formula for work of gravity to, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=-\int _{t_{1}}^{t_{2}}{\frac {GmM}{r^{3}}}(r\mathbf {e} _{r})\cdot \left({\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t}\right)dt=-\int _{t_{1}}^{t_{2}}{\frac {GmM}{r^{3}}}r{\dot {r}}dt={\frac {GMm}{r(t_{2})}}-{\frac {GMm}{r(t_{1})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <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"> <mfrac> <mrow> <mi>G</mi> <mi>m</mi> <mi>M</mi> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <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"> <mfrac> <mrow> <mi>G</mi> <mi>m</mi> <mi>M</mi> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=-\int _{t_{1}}^{t_{2}}{\frac {GmM}{r^{3}}}(r\mathbf {e} _{r})\cdot \left({\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t}\right)dt=-\int _{t_{1}}^{t_{2}}{\frac {GmM}{r^{3}}}r{\dot {r}}dt={\frac {GMm}{r(t_{2})}}-{\frac {GMm}{r(t_{1})}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac439499279b8e642d41ded4740159dbe1e6928" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:84.344ex; height:6.509ex;" alt="{\displaystyle W=-\int _{t_{1}}^{t_{2}}{\frac {GmM}{r^{3}}}(r\mathbf {e} _{r})\cdot \left({\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t}\right)dt=-\int _{t_{1}}^{t_{2}}{\frac {GmM}{r^{3}}}r{\dot {r}}dt={\frac {GMm}{r(t_{2})}}-{\frac {GMm}{r(t_{1})}}.}"></span> This calculation uses the fact that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}r^{-1}=-r^{-2}{\dot {r}}=-{\frac {\dot {r}}{r^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}r^{-1}=-r^{-2}{\dot {r}}=-{\frac {\dot {r}}{r^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68f74984cb177d2d1be704437e3b93467b37cf36" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.345ex; height:5.676ex;" alt="{\displaystyle {\frac {d}{dt}}r^{-1}=-r^{-2}{\dot {r}}=-{\frac {\dot {r}}{r^{2}}}.}"></span> The function <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=-{\frac {GMm}{r}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=-{\frac {GMm}{r}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e4ba4c6266dadc7bda4935bf42846bd225e4ed1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.481ex; height:5.343ex;" alt="{\displaystyle U=-{\frac {GMm}{r}},}"></span> is the gravitational potential function, also known as <a href="/wiki/Gravitational_potential_energy" class="mw-redirect" title="Gravitational potential energy">gravitational potential energy</a>. The negative sign follows the convention that work is gained from a loss of potential energy. </p> <div class="mw-heading mw-heading3"><h3 id="Work_by_a_spring">Work by a spring</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=15" title="Edit section: Work by a spring"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Analogie_ressorts_contrainte.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Analogie_ressorts_contrainte.svg/170px-Analogie_ressorts_contrainte.svg.png" decoding="async" width="170" height="262" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Analogie_ressorts_contrainte.svg/255px-Analogie_ressorts_contrainte.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Analogie_ressorts_contrainte.svg/340px-Analogie_ressorts_contrainte.svg.png 2x" data-file-width="278" data-file-height="429" /></a><figcaption>Forces in springs assembled in parallel</figcaption></figure> <p>Consider a spring that exerts a horizontal force <span class="texhtml"><b>F</b> = (−<i>kx</i>, 0, 0)</span> that is proportional to its deflection in the <i>x</i> direction independent of how a body moves. The work of this spring on a body moving along the space with the curve <span class="texhtml"><b>X</b>(<i>t</i>) = (<i>x</i>(<i>t</i>), <i>y</i>(<i>t</i>), <i>z</i>(<i>t</i>))</span>, is calculated using its velocity, <span class="texhtml"><b>v</b> = (<i>v</i><sub>x</sub>, <i>v</i><sub>y</sub>, <i>v</i><sub>z</sub>)</span>, to obtain <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{0}^{t}\mathbf {F} \cdot \mathbf {v} dt=-\int _{0}^{t}kxv_{x}dt=-{\frac {1}{2}}kx^{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"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </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> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>k</mi> <mi>x</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>k</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{0}^{t}\mathbf {F} \cdot \mathbf {v} dt=-\int _{0}^{t}kxv_{x}dt=-{\frac {1}{2}}kx^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae7e7a0e814645c626066f3f2ce845cb0697bc81" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.054ex; height:6.176ex;" alt="{\displaystyle W=\int _{0}^{t}\mathbf {F} \cdot \mathbf {v} dt=-\int _{0}^{t}kxv_{x}dt=-{\frac {1}{2}}kx^{2}.}"></span> For convenience, consider contact with the spring occurs at <span class="texhtml"><i>t</i> = 0</span>, then the integral of the product of the distance <span class="texhtml mvar" style="font-style:italic;">x</span> and the x-velocity, <span class="texhtml"><i>xv</i><sub>x</sub><i>dt</i></span>, over time <span class="texhtml mvar" style="font-style:italic;">t</span> is <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>x</i><sup>2</sup></span>. The work is the product of the distance times the spring force, which is also dependent on distance; hence the <span class="texhtml"><i>x</i><sup>2</sup></span> result. </p> <div class="mw-heading mw-heading3"><h3 id="Work_by_a_gas">Work by a gas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=16" title="Edit section: Work by a gas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The work <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> done by a body of gas on its surroundings is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{a}^{b}P\,dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>P</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{a}^{b}P\,dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c346817bc06ef7caec483ab7455606d05009860" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.458ex; height:6.343ex;" alt="{\displaystyle W=\int _{a}^{b}P\,dV}"></span> where <span class="texhtml mvar" style="font-style:italic;">P</span> is pressure, <span class="texhtml mvar" style="font-style:italic;">V</span> is volume, and <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are initial and final volumes. </p> <div class="mw-heading mw-heading2"><h2 id="Work–energy_principle"><span id="Work.E2.80.93energy_principle"></span>Work–energy principle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=17" title="Edit section: Work–energy principle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The principle of work and <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> (also known as the <b>work–energy principle</b>) states that <i>the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle.</i><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> That is, the work <i>W</i> done by the <a href="/wiki/Resultant_force" title="Resultant force">resultant force</a> on a <a href="/wiki/Particle" title="Particle">particle</a> equals the change in the particle's kinetic energy <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_{\text{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce71f608d367b0bb170515bcdb0cf74f7c6dcf73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.815ex; height:2.509ex;" alt="{\displaystyle E_{\text{k}}}"></span>,<sup id="cite_ref-Young_2-2" class="reference"><a href="#cite_note-Young-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\Delta E_{\text{k}}={\frac {1}{2}}mv_{2}^{2}-{\frac {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"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\Delta E_{\text{k}}={\frac {1}{2}}mv_{2}^{2}-{\frac {1}{2}}mv_{1}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd15b6b12dd6c8b8e37a4a03caaa40702523e6c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.666ex; height:5.176ex;" alt="{\displaystyle W=\Delta E_{\text{k}}={\frac {1}{2}}mv_{2}^{2}-{\frac {1}{2}}mv_{1}^{2}}"></span> where <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> and <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> are the <a href="/wiki/Speed" title="Speed">speeds</a> of the particle before and after the work is done, and <span class="texhtml mvar" style="font-style:italic;">m</span> is its <a href="/wiki/Mass" title="Mass">mass</a>. </p><p>The derivation of the <i>work–energy principle</i> begins with <a href="/wiki/Newton%27s_second_law_of_motion" class="mw-redirect" title="Newton's second law of motion">Newton's second law of motion</a> and the resultant force on a particle. Computation of the scalar product of the force with the velocity of the particle evaluates the instantaneous power added to the system.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> (Constraints define the direction of movement of the particle by ensuring there is no component of velocity in the direction of the constraint force. This also means the constraint forces do not add to the instantaneous power.) The time integral of this scalar equation yields work from the instantaneous power, and kinetic energy from the scalar product of acceleration with velocity. The fact that the work–energy principle eliminates the constraint forces underlies <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>This section focuses on the work–energy principle as it applies to particle dynamics. In more general systems work can change the <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> of a mechanical device, the thermal energy in a thermal system, or the <a href="/wiki/Electrical_energy" title="Electrical energy">electrical energy</a> in an electrical device. Work transfers energy from one place to another or one form to another. </p> <div class="mw-heading mw-heading3"><h3 id="Derivation_for_a_particle_moving_along_a_straight_line">Derivation for a particle moving along a straight line</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=18" title="Edit section: Derivation for a particle moving along a straight line"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the case the <a href="/wiki/Resultant_force" title="Resultant force">resultant force</a> <span class="texhtml"><b>F</b></span> is constant in both magnitude and direction, and parallel to the velocity of the particle, the particle is moving with constant acceleration <i>a</i> along a straight line.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> The relation between the net force and the acceleration is given by the equation <span class="texhtml"><i>F</i> = <i>ma</i></span> (<a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a>), and the particle <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a> <span class="texhtml mvar" style="font-style:italic;">s</span> can be expressed by the equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\frac {v_{2}^{2}-v_{1}^{2}}{2a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</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 s={\frac {v_{2}^{2}-v_{1}^{2}}{2a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b53737c9a970c27853de621935e7a4635018fd4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.229ex; height:5.843ex;" alt="{\displaystyle s={\frac {v_{2}^{2}-v_{1}^{2}}{2a}}}"></span> which follows from <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}+2as}"> <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>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{2}^{2}=v_{1}^{2}+2as}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6564ed829a2fb73bb9e9cd3a2b978665d23801f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.785ex; height:3.176ex;" alt="{\displaystyle v_{2}^{2}=v_{1}^{2}+2as}"></span> (see <a href="/wiki/Equations_of_motion" title="Equations of motion">Equations of motion</a>). </p><p>The work of the net force is calculated as the product of its magnitude and the particle displacement. Substituting the above equations, one obtains: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=Fs=mas=ma{\frac {v_{2}^{2}-v_{1}^{2}}{2a}}={\frac {1}{2}}mv_{2}^{2}-{\frac {1}{2}}mv_{1}^{2}=\Delta E_{\text{k}}}"> <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>m</mi> <mi>a</mi> <mi>s</mi> <mo>=</mo> <mi>m</mi> <mi>a</mi> <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 class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </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> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=Fs=mas=ma{\frac {v_{2}^{2}-v_{1}^{2}}{2a}}={\frac {1}{2}}mv_{2}^{2}-{\frac {1}{2}}mv_{1}^{2}=\Delta E_{\text{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39336e38bdb0dd6e3b3e434a9cf3244ada453ade" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:56.463ex; height:5.843ex;" alt="{\displaystyle W=Fs=mas=ma{\frac {v_{2}^{2}-v_{1}^{2}}{2a}}={\frac {1}{2}}mv_{2}^{2}-{\frac {1}{2}}mv_{1}^{2}=\Delta E_{\text{k}}}"></span> </p><p>Other derivation: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=Fs=mas=m{\frac {v_{2}^{2}-v_{1}^{2}}{2s}}s={\frac {1}{2}}mv_{2}^{2}-{\frac {1}{2}}mv_{1}^{2}=\Delta E_{\text{k}}}"> <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>m</mi> <mi>a</mi> <mi>s</mi> <mo>=</mo> <mi>m</mi> <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>s</mi> </mrow> </mfrac> </mrow> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </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> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=Fs=mas=m{\frac {v_{2}^{2}-v_{1}^{2}}{2s}}s={\frac {1}{2}}mv_{2}^{2}-{\frac {1}{2}}mv_{1}^{2}=\Delta E_{\text{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730095b0ac730ccb71926128d853c1be3a51c285" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:56.324ex; height:5.843ex;" alt="{\displaystyle W=Fs=mas=m{\frac {v_{2}^{2}-v_{1}^{2}}{2s}}s={\frac {1}{2}}mv_{2}^{2}-{\frac {1}{2}}mv_{1}^{2}=\Delta E_{\text{k}}}"></span> </p><p>In the general case of rectilinear motion, when the net force <span class="texhtml"><b>F</b></span> is not constant in magnitude, but is constant in direction, and parallel to the velocity of the particle, the work must be integrated along the path of the particle: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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\,v\,dt=\int _{t_{1}}^{t_{2}}ma\,v\,dt=m\int _{t_{1}}^{t_{2}}v\,{\frac {dv}{dt}}\,dt=m\int _{v_{1}}^{v_{2}}v\,dv={\tfrac {1}{2}}m\left(v_{2}^{2}-v_{1}^{2}\right).}"> <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> <mspace width="thinmathspace" /> <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> <mspace width="thinmathspace" /> <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> <mrow> <mo>(</mo> <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> <mo>)</mo> </mrow> <mo>.</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\,v\,dt=\int _{t_{1}}^{t_{2}}ma\,v\,dt=m\int _{t_{1}}^{t_{2}}v\,{\frac {dv}{dt}}\,dt=m\int _{v_{1}}^{v_{2}}v\,dv={\tfrac {1}{2}}m\left(v_{2}^{2}-v_{1}^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06e2910100f46f4d2774c3cfca4710c9b057a309" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:93.831ex; height:6.509ex;" alt="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} dt=\int _{t_{1}}^{t_{2}}F\,v\,dt=\int _{t_{1}}^{t_{2}}ma\,v\,dt=m\int _{t_{1}}^{t_{2}}v\,{\frac {dv}{dt}}\,dt=m\int _{v_{1}}^{v_{2}}v\,dv={\tfrac {1}{2}}m\left(v_{2}^{2}-v_{1}^{2}\right).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="General_derivation_of_the_work–energy_principle_for_a_particle"><span id="General_derivation_of_the_work.E2.80.93energy_principle_for_a_particle"></span>General derivation of the work–energy principle for a particle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=19" title="Edit section: General derivation of the work–energy principle for a particle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any net force acting on a particle moving along any curvilinear path, it can be demonstrated that its work equals the change in the kinetic energy of the particle by a simple derivation analogous to the equation above. It is known as <b>the work–energy principle</b>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} dt=m\int _{t_{1}}^{t_{2}}\mathbf {a} \cdot \mathbf {v} dt={\frac {m}{2}}\int _{t_{1}}^{t_{2}}{\frac {dv^{2}}{dt}}\,dt={\frac {m}{2}}\int _{v_{1}^{2}}^{v_{2}^{2}}dv^{2}={\frac {mv_{2}^{2}}{2}}-{\frac {mv_{1}^{2}}{2}}=\Delta E_{\text{k}}}"> <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> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <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"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <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 class="MJX-TeXAtom-ORD"> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </msubsup> <mi>d</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <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> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} dt=m\int _{t_{1}}^{t_{2}}\mathbf {a} \cdot \mathbf {v} dt={\frac {m}{2}}\int _{t_{1}}^{t_{2}}{\frac {dv^{2}}{dt}}\,dt={\frac {m}{2}}\int _{v_{1}^{2}}^{v_{2}^{2}}dv^{2}={\frac {mv_{2}^{2}}{2}}-{\frac {mv_{1}^{2}}{2}}=\Delta E_{\text{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e957d837642873e498ca1ce161aff619de2212e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:89.276ex; height:7.176ex;" alt="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot \mathbf {v} dt=m\int _{t_{1}}^{t_{2}}\mathbf {a} \cdot \mathbf {v} dt={\frac {m}{2}}\int _{t_{1}}^{t_{2}}{\frac {dv^{2}}{dt}}\,dt={\frac {m}{2}}\int _{v_{1}^{2}}^{v_{2}^{2}}dv^{2}={\frac {mv_{2}^{2}}{2}}-{\frac {mv_{1}^{2}}{2}}=\Delta E_{\text{k}}}"></span> </p><p>The identity <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="{\textstyle \mathbf {a} \cdot \mathbf {v} ={\frac {1}{2}}{\frac {dv^{2}}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {a} \cdot \mathbf {v} ={\frac {1}{2}}{\frac {dv^{2}}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2750c20f340add6f838880b7c49e51470a9fcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.471ex; height:4.176ex;" alt="{\textstyle \mathbf {a} \cdot \mathbf {v} ={\frac {1}{2}}{\frac {dv^{2}}{dt}}}"></span> requires some algebra. From the identity <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="{\textstyle v^{2}=\mathbf {v} \cdot \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle v^{2}=\mathbf {v} \cdot \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd8a3af513a73d4e14a652906b9a465bd2dc78d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.781ex; height:2.509ex;" alt="{\textstyle v^{2}=\mathbf {v} \cdot \mathbf {v} }"></span> and definition <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="{\textstyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97a6ba6b4a6a710467141d55b931377d91a6fb4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.092ex; height:3.843ex;" alt="{\textstyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}}"></span> it follows <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dv^{2}}{dt}}={\frac {d(\mathbf {v} \cdot \mathbf {v} )}{dt}}={\frac {d\mathbf {v} }{dt}}\cdot \mathbf {v} +\mathbf {v} \cdot {\frac {d\mathbf {v} }{dt}}=2{\frac {d\mathbf {v} }{dt}}\cdot \mathbf {v} =2\mathbf {a} \cdot \mathbf {v} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dv^{2}}{dt}}={\frac {d(\mathbf {v} \cdot \mathbf {v} )}{dt}}={\frac {d\mathbf {v} }{dt}}\cdot \mathbf {v} +\mathbf {v} \cdot {\frac {d\mathbf {v} }{dt}}=2{\frac {d\mathbf {v} }{dt}}\cdot \mathbf {v} =2\mathbf {a} \cdot \mathbf {v} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0239e9088c061a5877cd5b110332083d2f552c26" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:54.851ex; height:5.843ex;" alt="{\displaystyle {\frac {dv^{2}}{dt}}={\frac {d(\mathbf {v} \cdot \mathbf {v} )}{dt}}={\frac {d\mathbf {v} }{dt}}\cdot \mathbf {v} +\mathbf {v} \cdot {\frac {d\mathbf {v} }{dt}}=2{\frac {d\mathbf {v} }{dt}}\cdot \mathbf {v} =2\mathbf {a} \cdot \mathbf {v} .}"></span> </p><p>The remaining part of the above derivation is just simple calculus, same as in the preceding rectilinear case. </p> <div class="mw-heading mw-heading3"><h3 id="Derivation_for_a_particle_in_constrained_movement">Derivation for a particle in constrained movement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=20" title="Edit section: Derivation for a particle in constrained movement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In particle dynamics, a formula equating work applied to a system to its change in kinetic energy is obtained as a first integral of <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's second law of motion</a>. It is useful to notice that the resultant force used in Newton's laws can be separated into forces that are applied to the particle and forces imposed by constraints on the movement of the particle. Remarkably, the work of a constraint force is zero, therefore only the work of the applied forces need be considered in the work–energy principle. </p><p>To see this, consider a particle P that follows the trajectory <span class="texhtml"><b>X</b>(<i>t</i>)</span> with a force <span class="texhtml"><b>F</b></span> acting on it. Isolate the particle from its environment to expose constraint forces <span class="texhtml"><b>R</b></span>, then Newton's Law takes the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} +\mathbf {R} =m{\ddot {\mathbf {X} }},}"> <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">R</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} +\mathbf {R} =m{\ddot {\mathbf {X} }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c62f7a4f5084fa2fc19729d1067f80bf0bc2957" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.332ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} +\mathbf {R} =m{\ddot {\mathbf {X} }},}"></span> where <span class="texhtml mvar" style="font-style:italic;">m</span> is the mass of the particle. </p> <div class="mw-heading mw-heading4"><h4 id="Vector_formulation">Vector formulation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=21" title="Edit section: Vector formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Note that n dots above a vector indicates its nth <a href="/wiki/Time_derivative" title="Time derivative">time derivative</a>. The <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a> of each side of Newton's law with the velocity vector yields <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} \cdot {\dot {\mathbf {X} }}=m{\ddot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }},}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} \cdot {\dot {\mathbf {X} }}=m{\ddot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28373471328c2dcf276b694462b4a7959a329c08" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.885ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} \cdot {\dot {\mathbf {X} }}=m{\ddot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }},}"></span> because the constraint forces are perpendicular to the particle velocity. Integrate this equation along its trajectory from the point <span class="texhtml"><b>X</b>(<i>t</i><sub>1</sub>)</span> to the point <span class="texhtml"><b>X</b>(<i>t</i><sub>2</sub>)</span> to obtain <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{t_{1}}^{t_{2}}\mathbf {F} \cdot {\dot {\mathbf {X} }}dt=m\int _{t_{1}}^{t_{2}}{\ddot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{t_{1}}^{t_{2}}\mathbf {F} \cdot {\dot {\mathbf {X} }}dt=m\int _{t_{1}}^{t_{2}}{\ddot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a8547be94819c5b9ee8d4a7bea1260cca0129e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.401ex; height:6.509ex;" alt="{\displaystyle \int _{t_{1}}^{t_{2}}\mathbf {F} \cdot {\dot {\mathbf {X} }}dt=m\int _{t_{1}}^{t_{2}}{\ddot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}dt.}"></span> </p><p>The left side of this equation is the work of the applied force as it acts on the particle along the trajectory from time <span class="texhtml"><i>t</i><sub>1</sub></span> to time <span class="texhtml"><i>t</i><sub>2</sub></span>. This can also be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot {\dot {\mathbf {X} }}dt=\int _{\mathbf {X} (t_{1})}^{\mathbf {X} (t_{2})}\mathbf {F} \cdot d\mathbf {X} .}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <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> </mrow> <mrow class="MJX-TeXAtom-ORD"> <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> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot {\dot {\mathbf {X} }}dt=\int _{\mathbf {X} (t_{1})}^{\mathbf {X} (t_{2})}\mathbf {F} \cdot d\mathbf {X} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3907b3bc1acc361fd355c76e388ae36ea31dc08b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.038ex; height:6.676ex;" alt="{\displaystyle W=\int _{t_{1}}^{t_{2}}\mathbf {F} \cdot {\dot {\mathbf {X} }}dt=\int _{\mathbf {X} (t_{1})}^{\mathbf {X} (t_{2})}\mathbf {F} \cdot d\mathbf {X} .}"></span> This integral is computed along the trajectory <span class="texhtml"><b>X</b>(<i>t</i>)</span> of the particle and is therefore path dependent. </p><p>The right side of the first integral of Newton's equations can be simplified using the following identity <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}{\frac {d}{dt}}({\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }})={\ddot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}{\frac {d}{dt}}({\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }})={\ddot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33b9e72f9617e1c6fec1094924889c60dd514c28" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.881ex; height:5.509ex;" alt="{\displaystyle {\frac {1}{2}}{\frac {d}{dt}}({\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }})={\ddot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }},}"></span> (see <a href="/wiki/Product_rule" title="Product rule">product rule</a> for derivation). Now it is integrated explicitly to obtain the change in kinetic energy, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta K=m\int _{t_{1}}^{t_{2}}{\ddot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}dt={\frac {m}{2}}\int _{t_{1}}^{t_{2}}{\frac {d}{dt}}({\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }})dt={\frac {m}{2}}{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}(t_{2})-{\frac {m}{2}}{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}(t_{1})={\frac {1}{2}}m\Delta \mathbf {v} ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>K</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <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"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mi mathvariant="normal">Δ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta K=m\int _{t_{1}}^{t_{2}}{\ddot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}dt={\frac {m}{2}}\int _{t_{1}}^{t_{2}}{\frac {d}{dt}}({\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }})dt={\frac {m}{2}}{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}(t_{2})-{\frac {m}{2}}{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}(t_{1})={\frac {1}{2}}m\Delta \mathbf {v} ^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66808063c4a31017415a51f1ee8a6c8276e2973" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:87.875ex; height:6.509ex;" alt="{\displaystyle \Delta K=m\int _{t_{1}}^{t_{2}}{\ddot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}dt={\frac {m}{2}}\int _{t_{1}}^{t_{2}}{\frac {d}{dt}}({\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }})dt={\frac {m}{2}}{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}(t_{2})-{\frac {m}{2}}{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}(t_{1})={\frac {1}{2}}m\Delta \mathbf {v} ^{2},}"></span> where the kinetic energy of the particle is defined by the scalar quantity, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\frac {m}{2}}{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}={\frac {1}{2}}m{\mathbf {v} ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {m}{2}}{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}={\frac {1}{2}}m{\mathbf {v} ^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d48e5c6ec2251d5f80fea040511daeb6fdcc265" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.362ex; height:5.176ex;" alt="{\displaystyle K={\frac {m}{2}}{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}={\frac {1}{2}}m{\mathbf {v} ^{2}}}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Tangential_and_normal_components">Tangential and normal components</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=22" title="Edit section: Tangential and normal components"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is useful to resolve the velocity and acceleration vectors into tangential and normal components along the trajectory <span class="texhtml"><b>X</b>(<i>t</i>)</span>, such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\mathbf {X} }}=v\mathbf {T} \quad {\text{and}}\quad {\ddot {\mathbf {X} }}={\dot {v}}\mathbf {T} +v^{2}\kappa \mathbf {N} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>+</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {X} }}=v\mathbf {T} \quad {\text{and}}\quad {\ddot {\mathbf {X} }}={\dot {v}}\mathbf {T} +v^{2}\kappa \mathbf {N} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6fe9936631c57e939c852de6a48043771b7bf94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.801ex; height:3.009ex;" alt="{\displaystyle {\dot {\mathbf {X} }}=v\mathbf {T} \quad {\text{and}}\quad {\ddot {\mathbf {X} }}={\dot {v}}\mathbf {T} +v^{2}\kappa \mathbf {N} ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=|{\dot {\mathbf {X} }}|={\sqrt {{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=|{\dot {\mathbf {X} }}|={\sqrt {{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/497eeaf3496febf1f7563692fc070d89bbf6442c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.326ex; height:3.676ex;" alt="{\displaystyle v=|{\dot {\mathbf {X} }}|={\sqrt {{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}}}.}"></span> Then, the <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a> of velocity with acceleration in Newton's second law takes the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta K=m\int _{t_{1}}^{t_{2}}{\dot {v}}v\,dt={\frac {m}{2}}\int _{t_{1}}^{t_{2}}{\frac {d}{dt}}v^{2}\,dt={\frac {m}{2}}v^{2}(t_{2})-{\frac {m}{2}}v^{2}(t_{1}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>K</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <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"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta K=m\int _{t_{1}}^{t_{2}}{\dot {v}}v\,dt={\frac {m}{2}}\int _{t_{1}}^{t_{2}}{\frac {d}{dt}}v^{2}\,dt={\frac {m}{2}}v^{2}(t_{2})-{\frac {m}{2}}v^{2}(t_{1}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee00b1666c6648b3c39cf478c0f01ec0c9714c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:61.329ex; height:6.509ex;" alt="{\displaystyle \Delta K=m\int _{t_{1}}^{t_{2}}{\dot {v}}v\,dt={\frac {m}{2}}\int _{t_{1}}^{t_{2}}{\frac {d}{dt}}v^{2}\,dt={\frac {m}{2}}v^{2}(t_{2})-{\frac {m}{2}}v^{2}(t_{1}),}"></span> where the kinetic energy of the particle is defined by the scalar quantity, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\frac {m}{2}}v^{2}={\frac {m}{2}}{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {m}{2}}v^{2}={\frac {m}{2}}{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/501492d3610f9075d69db4fce889438d6aa673cc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.563ex; height:4.676ex;" alt="{\displaystyle K={\frac {m}{2}}v^{2}={\frac {m}{2}}{\dot {\mathbf {X} }}\cdot {\dot {\mathbf {X} }}.}"></span> </p><p>The result is the work–energy principle for particle dynamics, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\Delta K.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>K</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\Delta K.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e84ac5fe80d334fa1c83987c6552d659459c2c0e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.182ex; height:2.176ex;" alt="{\displaystyle W=\Delta K.}"></span> This derivation can be generalized to arbitrary rigid body systems. </p> <div class="mw-heading mw-heading3"><h3 id="Moving_in_a_straight_line_(skid_to_a_stop)"><span id="Moving_in_a_straight_line_.28skid_to_a_stop.29"></span>Moving in a straight line (skid to a stop)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=23" title="Edit section: Moving in a straight line (skid to a stop)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the case of a vehicle moving along a straight horizontal trajectory under the action of a driving force and gravity that sum to <span class="texhtml"><b>F</b></span>. The constraint forces between the vehicle and the road define <span class="texhtml"><b>R</b></span>, and we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} +\mathbf {R} =m{\ddot {\mathbf {X} }}.}"> <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">R</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} +\mathbf {R} =m{\ddot {\mathbf {X} }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e89fbfe7c74133802580327c00a2a78bc96945" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.332ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} +\mathbf {R} =m{\ddot {\mathbf {X} }}.}"></span> For convenience let the trajectory be along the X-axis, so <span class="texhtml"><b>X</b> = (<i>d</i>, 0)</span> and the velocity is <span class="texhtml"><b>V</b> = (<i>v</i>, 0)</span>, then <span class="texhtml"><b>R</b> ⋅ <b>V</b> = 0</span>, and <span class="texhtml"><b>F</b> ⋅ <b>V</b> = <i>F</i><sub>x</sub><i>v</i></span>, where <i>F</i><sub>x</sub> is the component of <b>F</b> along the X-axis, so <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{x}v=m{\dot {v}}v.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>v</mi> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{x}v=m{\dot {v}}v.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d67ac1d0acb5f16e2cd5761b74780ded8ea771" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.935ex; height:2.509ex;" alt="{\displaystyle F_{x}v=m{\dot {v}}v.}"></span> Integration of both sides yields <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{t_{1}}^{t_{2}}F_{x}vdt={\frac {m}{2}}v^{2}(t_{2})-{\frac {m}{2}}v^{2}(t_{1}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>v</mi> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{t_{1}}^{t_{2}}F_{x}vdt={\frac {m}{2}}v^{2}(t_{2})-{\frac {m}{2}}v^{2}(t_{1}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63dc83b0c381be0abe86e19df8acb04c67b6b0c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.468ex; height:6.509ex;" alt="{\displaystyle \int _{t_{1}}^{t_{2}}F_{x}vdt={\frac {m}{2}}v^{2}(t_{2})-{\frac {m}{2}}v^{2}(t_{1}).}"></span> If <span class="texhtml"><i>F</i><sub>x</sub></span> is constant along the trajectory, then the integral of velocity is distance, so <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{x}(d(t_{2})-d(t_{1}))={\frac {m}{2}}v^{2}(t_{2})-{\frac {m}{2}}v^{2}(t_{1}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{x}(d(t_{2})-d(t_{1}))={\frac {m}{2}}v^{2}(t_{2})-{\frac {m}{2}}v^{2}(t_{1}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4276b02ad41bb585b8e05cad7b9caf9c3891265c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.264ex; height:4.676ex;" alt="{\displaystyle F_{x}(d(t_{2})-d(t_{1}))={\frac {m}{2}}v^{2}(t_{2})-{\frac {m}{2}}v^{2}(t_{1}).}"></span> </p><p>As an example consider a car skidding to a stop, where <i>k</i> is the coefficient of friction and <i>W</i> is the weight of the car. Then the force along the trajectory is <span class="texhtml"><i>F</i><sub>x</sub> = −<i>kW</i></span>. The velocity <i>v</i> of the car can be determined from the length <span class="texhtml mvar" style="font-style:italic;">s</span> of the skid using the work–energy principle, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle kWs={\frac {W}{2g}}v^{2},\quad {\text{or}}\quad v={\sqrt {2ksg}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>W</mi> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <mrow> <mn>2</mn> <mi>g</mi> </mrow> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> <mspace width="1em" /> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>k</mi> <mi>s</mi> <mi>g</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle kWs={\frac {W}{2g}}v^{2},\quad {\text{or}}\quad v={\sqrt {2ksg}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050ecceea06ece5cabc389cd2d0b92a4c2af520" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.819ex; height:5.676ex;" alt="{\displaystyle kWs={\frac {W}{2g}}v^{2},\quad {\text{or}}\quad v={\sqrt {2ksg}}.}"></span> This formula uses the fact that the mass of the vehicle is <span class="texhtml"><i>m</i> = <i>W</i>/<i>g</i></span>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:LotusType119B.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/LotusType119B.jpg/220px-LotusType119B.jpg" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/LotusType119B.jpg/330px-LotusType119B.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/LotusType119B.jpg/440px-LotusType119B.jpg 2x" data-file-width="3888" data-file-height="2592" /></a><figcaption>Lotus type 119B gravity racer at Lotus 60th celebration</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Campeonato_de_carrinhos_de_rolim%C3%A3_em_Campos_Novos_SC.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Campeonato_de_carrinhos_de_rolim%C3%A3_em_Campos_Novos_SC.jpg/220px-Campeonato_de_carrinhos_de_rolim%C3%A3_em_Campos_Novos_SC.jpg" decoding="async" width="220" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Campeonato_de_carrinhos_de_rolim%C3%A3_em_Campos_Novos_SC.jpg/330px-Campeonato_de_carrinhos_de_rolim%C3%A3_em_Campos_Novos_SC.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Campeonato_de_carrinhos_de_rolim%C3%A3_em_Campos_Novos_SC.jpg/440px-Campeonato_de_carrinhos_de_rolim%C3%A3_em_Campos_Novos_SC.jpg 2x" data-file-width="800" data-file-height="509" /></a><figcaption>Gravity racing championship in Campos Novos, Santa Catarina, Brazil, 8 September 2010</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Coasting_down_an_inclined_surface_(gravity_racing)"><span id="Coasting_down_an_inclined_surface_.28gravity_racing.29"></span>Coasting down an inclined surface (gravity racing)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=24" title="Edit section: Coasting down an inclined surface (gravity racing)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the case of a vehicle that starts at rest and coasts down an inclined surface (such as mountain road), the work–energy principle helps compute the minimum distance that the vehicle travels to reach a velocity <span class="texhtml"><i>V</i></span>, of say 60 mph (88 fps). Rolling resistance and air drag will slow the vehicle down so the actual distance will be greater than if these forces are neglected. </p><p>Let the trajectory of the vehicle following the road be <span class="texhtml"><b>X</b>(<i>t</i>)</span> which is a curve in three-dimensional space. The force acting on the vehicle that pushes it down the road is the constant force of gravity <span class="texhtml"><b>F</b> = (0, 0, <i>W</i>)</span>, while the force of the road on the vehicle is the constraint force <span class="texhtml"><b>R</b></span>. Newton's second law yields, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} +\mathbf {R} =m{\ddot {\mathbf {X} }}.}"> <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">R</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} +\mathbf {R} =m{\ddot {\mathbf {X} }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e89fbfe7c74133802580327c00a2a78bc96945" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.332ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} +\mathbf {R} =m{\ddot {\mathbf {X} }}.}"></span> The <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a> of this equation with the velocity, <span class="texhtml"><b>V</b> = (<i>v</i><sub>x</sub>, <i>v</i><sub>y</sub>, <i>v</i><sub>z</sub>)</span>, yields <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Wv_{z}=m{\dot {V}}V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Wv_{z}=m{\dot {V}}V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/913d4966a7b228ed3f0d6c83cc6bf8c97543f572" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.925ex; height:3.009ex;" alt="{\displaystyle Wv_{z}=m{\dot {V}}V,}"></span> where <span class="texhtml"><i>V</i></span> is the magnitude of <span class="texhtml"><b>V</b></span>. The constraint forces between the vehicle and the road cancel from this equation because <span class="texhtml"><b>R</b> ⋅ <b>V</b> = 0</span>, which means they do no work. Integrate both sides to obtain <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{t_{1}}^{t_{2}}Wv_{z}dt={\frac {m}{2}}V^{2}(t_{2})-{\frac {m}{2}}V^{2}(t_{1}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>W</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{t_{1}}^{t_{2}}Wv_{z}dt={\frac {m}{2}}V^{2}(t_{2})-{\frac {m}{2}}V^{2}(t_{1}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c097ed1b0e1b8375cbc4f49c292bbec2a7d91d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.816ex; height:6.509ex;" alt="{\displaystyle \int _{t_{1}}^{t_{2}}Wv_{z}dt={\frac {m}{2}}V^{2}(t_{2})-{\frac {m}{2}}V^{2}(t_{1}).}"></span> The weight force <i>W</i> is constant along the trajectory and the integral of the vertical velocity is the vertical distance, therefore, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\Delta z={\frac {m}{2}}V^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mi mathvariant="normal">Δ</mi> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W\Delta z={\frac {m}{2}}V^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/564ac47e1f80ccbff880054fe66b9c2a8a1639f9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.052ex; height:4.676ex;" alt="{\displaystyle W\Delta z={\frac {m}{2}}V^{2}.}"></span> Recall that V(<i>t</i><sub>1</sub>)=0. Notice that this result does not depend on the shape of the road followed by the vehicle. </p><p>In order to determine the distance along the road assume the downgrade is 6%, which is a steep road. This means the altitude decreases 6 feet for every 100 feet traveled—for angles this small the sin and tan functions are approximately equal. Therefore, the distance <span class="texhtml mvar" style="font-style:italic;">s</span> in feet down a 6% grade to reach the velocity <span class="texhtml mvar" style="font-style:italic;">V</span> is at least <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\frac {\Delta z}{0.06}}=8.3{\frac {V^{2}}{g}},\quad {\text{or}}\quad s=8.3{\frac {88^{2}}{32.2}}\approx 2000\mathrm {ft} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>z</mi> </mrow> <mn>0.06</mn> </mfrac> </mrow> <mo>=</mo> <mn>8.3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>g</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> <mspace width="1em" /> <mi>s</mi> <mo>=</mo> <mn>8.3</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>88</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>32.2</mn> </mfrac> </mrow> <mo>≈</mo> <mn>2000</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">t</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\frac {\Delta z}{0.06}}=8.3{\frac {V^{2}}{g}},\quad {\text{or}}\quad s=8.3{\frac {88^{2}}{32.2}}\approx 2000\mathrm {ft} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7101676a2f2d45510e904de9caafa1342ace377b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.086ex; height:6.176ex;" alt="{\displaystyle s={\frac {\Delta z}{0.06}}=8.3{\frac {V^{2}}{g}},\quad {\text{or}}\quad s=8.3{\frac {88^{2}}{32.2}}\approx 2000\mathrm {ft} .}"></span> This formula uses the fact that the weight of the vehicle is <span class="texhtml"><i>W</i> = <i>mg</i></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Work_of_forces_acting_on_a_rigid_body">Work of forces acting on a rigid body</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=25" title="Edit section: Work of forces acting on a rigid body"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The work of forces acting at various points on a single rigid body can be calculated from the work of a <a href="/wiki/Resultant_force" title="Resultant force">resultant force and torque</a>. To see this, let the forces <b>F</b><sub>1</sub>, <b>F</b><sub>2</sub>, ..., <b>F</b><sub>n</sub> act on the points <b>X</b><sub>1</sub>, <b>X</b><sub>2</sub>, ..., <b>X</b><sub><i>n</i></sub> in a rigid body. </p><p>The trajectories of <b>X</b><sub><i>i</i></sub>, <i>i</i> = 1, ..., <i>n</i> are defined by the movement of the rigid body. This movement is given by the set of rotations [<i>A</i>(<i>t</i>)] and the trajectory <b>d</b>(<i>t</i>) of a reference point in the body. Let the coordinates <b>x</b><sub><i>i</i></sub> <i>i</i> = 1, ..., <i>n</i> define these points in the moving rigid body's <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">reference frame</a> <i>M</i>, so that the trajectories traced in the fixed frame <i>F</i> are given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} _{i}(t)=[A(t)]\mathbf {x} _{i}+\mathbf {d} (t)\quad i=1,\ldots ,n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} _{i}(t)=[A(t)]\mathbf {x} _{i}+\mathbf {d} (t)\quad i=1,\ldots ,n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/007f113cb9f2c64f292276ef03331e590deddaca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.043ex; height:2.843ex;" alt="{\displaystyle \mathbf {X} _{i}(t)=[A(t)]\mathbf {x} _{i}+\mathbf {d} (t)\quad i=1,\ldots ,n.}"></span> </p><p>The velocity of the points <span class="texhtml"><b>X</b><sub><i>i</i></sub></span> along their trajectories are <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {V} _{i}={\boldsymbol {\omega }}\times (\mathbf {X} _{i}-\mathbf {d} )+{\dot {\mathbf {d} }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V} _{i}={\boldsymbol {\omega }}\times (\mathbf {X} _{i}-\mathbf {d} )+{\dot {\mathbf {d} }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/237ce1ddf7f8eaf77781f48d2fc278366a80e77e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.354ex; height:3.176ex;" alt="{\displaystyle \mathbf {V} _{i}={\boldsymbol {\omega }}\times (\mathbf {X} _{i}-\mathbf {d} )+{\dot {\mathbf {d} }},}"></span> where <span class="texhtml"><i><b>ω</b></i></span> is the angular velocity vector obtained from the skew symmetric matrix <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\Omega ]={\dot {A}}A^{\mathsf {T}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>˙</mo> </mover> </mrow> </mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\Omega ]={\dot {A}}A^{\mathsf {T}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f944ca8607581db78ecdfff09726268f85986c90" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.587ex; height:3.343ex;" alt="{\displaystyle [\Omega ]={\dot {A}}A^{\mathsf {T}},}"></span> known as the angular velocity matrix. </p><p>The small amount of work by the forces over the small displacements <span class="texhtml"><i>δ</i><b>r</b><sub><i>i</i></sub></span> can be determined by approximating the displacement by <span class="texhtml"><i>δ</i><b>r</b> = <b>v</b><i>δt</i></span> so <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\mathbf {F} _{1}\cdot \mathbf {V} _{1}\delta t+\mathbf {F} _{2}\cdot \mathbf {V} _{2}\delta t+\ldots +\mathbf {F} _{n}\cdot \mathbf {V} _{n}\delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>δ</mi> <mi>t</mi> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>δ</mi> <mi>t</mi> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\mathbf {F} _{1}\cdot \mathbf {V} _{1}\delta t+\mathbf {F} _{2}\cdot \mathbf {V} _{2}\delta t+\ldots +\mathbf {F} _{n}\cdot \mathbf {V} _{n}\delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74f789481eaff3c7ba6de27d86fec0c190fa146a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:46.289ex; height:2.676ex;" alt="{\displaystyle \delta W=\mathbf {F} _{1}\cdot \mathbf {V} _{1}\delta t+\mathbf {F} _{2}\cdot \mathbf {V} _{2}\delta t+\ldots +\mathbf {F} _{n}\cdot \mathbf {V} _{n}\delta t}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot ({\boldsymbol {\omega }}\times (\mathbf {X} _{i}-\mathbf {d} )+{\dot {\mathbf {d} }})\delta t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mi>δ</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot ({\boldsymbol {\omega }}\times (\mathbf {X} _{i}-\mathbf {d} )+{\dot {\mathbf {d} }})\delta t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55eac4883556f9fd7848e722a51d73fd5bc3a21" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.619ex; height:6.843ex;" alt="{\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot ({\boldsymbol {\omega }}\times (\mathbf {X} _{i}-\mathbf {d} )+{\dot {\mathbf {d} }})\delta t.}"></span> </p><p>This formula can be rewritten to obtain <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\left(\sum _{i=1}^{n}\mathbf {F} _{i}\right)\cdot {\dot {\mathbf {d} }}\delta t+\left(\sum _{i=1}^{n}\left(\mathbf {X} _{i}-\mathbf {d} \right)\times \mathbf {F} _{i}\right)\cdot {\boldsymbol {\omega }}\delta t=\left(\mathbf {F} \cdot {\dot {\mathbf {d} }}+\mathbf {T} \cdot {\boldsymbol {\omega }}\right)\delta t,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mi>δ</mi> <mi>t</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mi>δ</mi> <mi>t</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mi>δ</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\left(\sum _{i=1}^{n}\mathbf {F} _{i}\right)\cdot {\dot {\mathbf {d} }}\delta t+\left(\sum _{i=1}^{n}\left(\mathbf {X} _{i}-\mathbf {d} \right)\times \mathbf {F} _{i}\right)\cdot {\boldsymbol {\omega }}\delta t=\left(\mathbf {F} \cdot {\dot {\mathbf {d} }}+\mathbf {T} \cdot {\boldsymbol {\omega }}\right)\delta t,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b61985a21a75980cc0bd026fee7539a018812e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:73.008ex; height:7.509ex;" alt="{\displaystyle \delta W=\left(\sum _{i=1}^{n}\mathbf {F} _{i}\right)\cdot {\dot {\mathbf {d} }}\delta t+\left(\sum _{i=1}^{n}\left(\mathbf {X} _{i}-\mathbf {d} \right)\times \mathbf {F} _{i}\right)\cdot {\boldsymbol {\omega }}\delta t=\left(\mathbf {F} \cdot {\dot {\mathbf {d} }}+\mathbf {T} \cdot {\boldsymbol {\omega }}\right)\delta t,}"></span> where <b>F</b> and <b>T</b> are the <a href="/wiki/Resultant_force" title="Resultant force">resultant force and torque</a> applied at the reference point <b>d</b> of the moving frame <i>M</i> in the rigid body. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=26" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-:1-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-:1_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFNCERT2020" class="citation web cs1"><a href="/wiki/National_Council_of_Educational_Research_and_Training" title="National Council of Educational Research and Training">NCERT</a> (2020). <a rel="nofollow" class="external text" href="https://www.ncert.nic.in/ncerts/l/keph106.pdf">"Physics Book"</a> <span class="cs1-format">(PDF)</span>. <i>ncert.nic.in</i><span class="reference-accessdate">. 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(1971). <a rel="nofollow" class="external text" href="https://nature.berkeley.edu/departments/espm/env-hist/articles/2.pdf">"Leibniz and the vis viva controversy"</a> <span class="cs1-format">(PDF)</span>. <i>Isis</i>. <b>62</b> (1): 21–35 (specifically p. 24). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F350705">10.1086/350705</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:143652761">143652761</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Isis&rft.atitle=Leibniz+and+the+vis+viva+controversy&rft.volume=62&rft.issue=1&rft.pages=21-35+%28specifically+p.+24%29&rft.date=1971&rft_id=info%3Adoi%2F10.1086%2F350705&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A143652761%23id-name%3DS2CID&rft.aulast=Iltis&rft.aufirst=C.&rft_id=https%3A%2F%2Fnature.berkeley.edu%2Fdepartments%2Fespm%2Fenv-hist%2Farticles%2F2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmeaton1759" class="citation journal cs1">Smeaton, John (1759). <a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1759.0019">"Experimental Enquiry Concerning the Natural Powers of Water and Wind to Turn Mills and Other Machines Depending on a Circular Motion"</a>. <i>Philosophical Transactions of the Royal Society</i>. <b>51</b>: 105. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1759.0019">10.1098/rstl.1759.0019</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:186213498">186213498</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society&rft.atitle=Experimental+Enquiry+Concerning+the+Natural+Powers+of+Water+and+Wind+to+Turn+Mills+and+Other+Machines+Depending+on+a+Circular+Motion&rft.volume=51&rft.pages=105&rft.date=1759&rft_id=info%3Adoi%2F10.1098%2Frstl.1759.0019&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186213498%23id-name%3DS2CID&rft.aulast=Smeaton&rft.aufirst=John&rft_id=https%3A%2F%2Fdoi.org%2F10.1098%252Frstl.1759.0019&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJammer1957" class="citation book cs1">Jammer, Max (1957). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CZtEBcmOe6gC"><i>Concepts of Force</i></a>. Dover Publications, Inc. p. 167; footnote 14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-40689-X" title="Special:BookSources/0-486-40689-X"><bdi>0-486-40689-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Concepts+of+Force&rft.pages=167%3B+footnote+14&rft.pub=Dover+Publications%2C+Inc.&rft.date=1957&rft.isbn=0-486-40689-X&rft.aulast=Jammer&rft.aufirst=Max&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCZtEBcmOe6gC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoriolis1829" class="citation book cs1">Coriolis, Gustave (1829). <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k1068268/f2"><i>Calculation of the Effect of Machines, or Considerations on the Use of Engines and their Evaluation</i></a>. Carilian-Goeury, Libraire (Paris).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculation+of+the+Effect+of+Machines%2C+or+Considerations+on+the+Use+of+Engines+and+their+Evaluation&rft.pub=Carilian-Goeury%2C+Libraire+%28Paris%29&rft.date=1829&rft.aulast=Coriolis&rft.aufirst=Gustave&rft_id=https%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k1068268%2Ff2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDugas1955" class="citation book cs1">Dugas, R. (1955). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmachani000518mbp"><i>A History of Mechanics</i></a>. Switzerland: Éditions du Griffon. p. 128.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mechanics&rft.place=Switzerland&rft.pages=128&rft.pub=%C3%89ditions+du+Griffon&rft.date=1955&rft.aulast=Dugas&rft.aufirst=R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmachani000518mbp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130420184231/http://www.bipm.org/en/si/si_brochure/chapter2/2-2/2-2-2.html">"Units with special names and symbols; units that incorporate special names and symbols"</a>. <i>The International System of Units (SI)</i> (8th ed.). <a href="/wiki/International_Bureau_of_Weights_and_Measures" title="International Bureau of Weights and Measures">International Bureau of Weights and Measures</a>. 2006. Archived from <a rel="nofollow" class="external text" href="http://www.bipm.org/en/si/si_brochure/chapter2/2-2/2-2-2.html">the original</a> on 2013-04-20<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-10-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Units+with+special+names+and+symbols%3B+units+that+incorporate+special+names+and+symbols&rft.btitle=The+International+System+of+Units+%28SI%29&rft.edition=8th&rft.pub=International+Bureau+of+Weights+and+Measures&rft.date=2006&rft_id=http%3A%2F%2Fwww.bipm.org%2Fen%2Fsi%2Fsi_brochure%2Fchapter2%2F2-2%2F2-2-2.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-:0-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcGrath2010" class="citation book cs1">McGrath, Kimberley A., ed. 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Work and potential energy. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7876-3651-7" title="Special:BookSources/978-0-7876-3651-7"><bdi>978-0-7876-3651-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=World+of+physics&rft.place=Detroit&rft.pages=Work+and+potential+energy&rft.edition=1st&rft.pub=Gale&rft.date=2010-05-05&rft.isbn=978-0-7876-3651-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-walker-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-walker_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-walker_14-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWalkerHallidayResnick2011" class="citation book cs1">Walker, Jearl; Halliday, David; Resnick, Robert (2011). <i>Fundamentals of physics</i> (9th ed.). 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Retrieved <span class="nowrap">2023-10-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ng.cengage.com&rft.atitle=MindTap+-+Cengage+Learning&rft_id=https%3A%2F%2Fng.cengage.com%2Fstatic%2Fnb%2Fui%2Fevo%2Findex.html%3FsnapshotId%3D1529049%26id%3D677758950%26eISBN%3D9780357049105&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylor2005" class="citation book cs1">Taylor, John R. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=P1kCtNr-pJsC&pg=PA117"><i>Classical Mechanics</i></a>. University Science Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-891389-22-1" title="Special:BookSources/978-1-891389-22-1"><bdi>978-1-891389-22-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Mechanics&rft.pub=University+Science+Books&rft.date=2005&rft.isbn=978-1-891389-22-1&rft.aulast=Taylor&rft.aufirst=John+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DP1kCtNr-pJsC%26pg%3DPA117&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndrew_PytelJaan_Kiusalaas2010" class="citation book cs1">Andrew Pytel; Jaan Kiusalaas (2010). <i>Engineering Mechanics: Dynamics – SI Version, Volume 2</i> (3rd ed.). Cengage Learning. p. 654. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780495295631" title="Special:BookSources/9780495295631"><bdi>9780495295631</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics%3A+Dynamics+%E2%80%93+SI+Version%2C+Volume+2&rft.pages=654&rft.edition=3rd&rft.pub=Cengage+Learning&rft.date=2010&rft.isbn=9780495295631&rft.au=Andrew+Pytel&rft.au=Jaan+Kiusalaas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaul1979" class="citation book cs1">Paul, Burton (1979). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3UdSAAAAMAAJ&q=instantaneous+power"><i>Kinematics and Dynamics of Planar Machinery</i></a>. Prentice-Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-516062-6" title="Special:BookSources/978-0-13-516062-6"><bdi>978-0-13-516062-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kinematics+and+Dynamics+of+Planar+Machinery&rft.pub=Prentice-Hall&rft.date=1979&rft.isbn=978-0-13-516062-6&rft.aulast=Paul&rft.aufirst=Burton&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3UdSAAAAMAAJ%26q%3Dinstantaneous%2Bpower&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhittaker1904" class="citation book cs1"><a href="/wiki/E._T._Whittaker" title="E. T. Whittaker">Whittaker, E. T.</a> (1904). <a rel="nofollow" class="external text" href="https://archive.org/details/atreatiseonanal00whitgoog"><i>A treatise on the analytical dynamics of particles and rigid bodies</i></a>. Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+treatise+on+the+analytical+dynamics+of+particles+and+rigid+bodies&rft.pub=Cambridge+University+Press&rft.date=1904&rft.aulast=Whittaker&rft.aufirst=E.+T.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fatreatiseonanal00whitgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120530075449/http://faculty.wwu.edu/vawter/PhysicsNet/Topics/Work/WorkEngergyTheorem.html">"Work–energy principle"</a>. <i>www.wwu.edu</i>. Archived from <a rel="nofollow" class="external text" href="http://faculty.wwu.edu/vawter/PhysicsNet/Topics/Work/WorkEngergyTheorem.html">the original</a> on 2012-05-30<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-08-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.wwu.edu&rft.atitle=Work%E2%80%93energy+principle&rft_id=http%3A%2F%2Ffaculty.wwu.edu%2Fvawter%2FPhysicsNet%2FTopics%2FWork%2FWorkEngergyTheorem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=27" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerway,_Raymond_A.Jewett,_John_W.2004" class="citation book cs1">Serway, Raymond A.; Jewett, John W. (2004). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/physicssciengv2p00serw"><i>Physics for Scientists and Engineers</i></a></span> (6th ed.). Brooks/Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-40842-7" title="Special:BookSources/0-534-40842-7"><bdi>0-534-40842-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers&rft.edition=6th&rft.pub=Brooks%2FCole&rft.date=2004&rft.isbn=0-534-40842-7&rft.au=Serway%2C+Raymond+A.&rft.au=Jewett%2C+John+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fphysicssciengv2p00serw&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTipler,_Paul1991" class="citation book cs1">Tipler, Paul (1991). <i>Physics for Scientists and Engineers: Mechanics</i> (3rd ed., extended version ed.). W. H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-87901-432-6" title="Special:BookSources/0-87901-432-6"><bdi>0-87901-432-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers%3A+Mechanics&rft.edition=3rd+ed.%2C+extended+version&rft.pub=W.+H.+Freeman&rft.date=1991&rft.isbn=0-87901-432-6&rft.au=Tipler%2C+Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWork+%28physics%29" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Work_(physics)&action=edit&section=28" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120530075449/http://faculty.wwu.edu/vawter/PhysicsNet/Topics/Work/WorkEngergyTheorem.html">Work–energy principle</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output 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dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Classical_mechanics_SI_units" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Classical_mechanics_SI_units" title="Template:Classical mechanics SI units"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classical_mechanics_SI_units" title="Template talk:Classical mechanics SI units"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classical_mechanics_SI_units" title="Special:EditPage/Template:Classical mechanics SI units"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Classical_mechanics_SI_units" style="font-size:114%;margin:0 4em"><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a> <a href="/wiki/International_System_of_Units" title="International System of Units">SI units</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><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> <td colspan="4" style="border:none;backgound:none; font-weight:bold;">Linear/translational quantities</td> <td rowspan="12" style="border:none;backgound:none;"></td> <td colspan="4" style="border:none;backgound:none; font-weight:bold;">Angular/rotational quantities</td> </tr> <tr> <th style="font-weight:normal;font-size:80%;">Dimensions</th> <th style="font-weight:normal;">1</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%;">Dimensions</th> <th style="font-weight:normal;">1</th> <th style="font-weight:normal;"><span class="texhtml"><i>θ</i></span></th> <th style="font-weight:normal;"><span class="texhtml"><i>θ</i></span><sup>2</sup></th> </tr> <tr> <th style="font-weight:normal;">T</th> <td><a href="/wiki/Time" title="Time">time</a>: <span class="texhtml"><i>t</i></span><br /><a href="/wiki/Second" title="Second">s</a></td> <td><a href="/wiki/Absement" title="Absement">absement</a>: <span class="texhtml"><b>A</b></span><br /><a href="/wiki/Meter_second" class="mw-redirect" title="Meter second">m s</a></td> <td></td> <th style="font-weight:normal;">T</th> <td><a href="/wiki/Time" title="Time">time</a>: <span class="texhtml"><i>t</i></span><br /><a href="/wiki/Second" title="Second">s</a></td> <td></td> <td></td> </tr> <tr> <th style="font-weight:normal;">1</th> <td></td> <td><a href="/wiki/Distance" title="Distance">distance</a>: <span class="texhtml"><i>d</i></span>, <span class="nowrap"><a href="/wiki/Position_(vector)" class="mw-redirect" title="Position (vector)">position</a>: <span class="texhtml"><b>r</b></span>, <span class="texhtml"><b>s</b></span>, <span class="texhtml"><b>x</b></span></span>, <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a><br /><a href="/wiki/Metre" title="Metre">m</a></td> <td><a href="/wiki/Area" title="Area">area</a>: <span class="texhtml"><i>A</i></span><br /><a href="/wiki/Square_metre" title="Square metre">m<sup>2</sup></a></td> <th style="font-weight:normal;">1</th> <td></td> <td><a href="/wiki/Angle" title="Angle">angle</a>: <span class="texhtml"><i>θ</i></span>, <a href="/wiki/Angular_displacement" title="Angular displacement">angular displacement</a>: <span class="texhtml"><i><b>θ</b></i></span><br /><a href="/wiki/Radian" title="Radian">rad</a></td> <td><span class="nowrap"><a href="/wiki/Solid_angle" title="Solid angle">solid angle</a>: <span class="texhtml">Ω</span><br /><a href="/wiki/Steradian" title="Steradian">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/Frequency" title="Frequency">frequency</a>: <span class="texhtml"><i>f</i></span></span><br /><a href="/wiki/Inverse_second" title="Inverse second">s<sup>−1</sup></a>, <a href="/wiki/Hertz" title="Hertz">Hz</a></td> <td><a href="/wiki/Speed" title="Speed">speed</a>: <span class="texhtml"><i>v</i></span>, <a href="/wiki/Velocity" title="Velocity">velocity</a>: <span class="texhtml"><b>v</b></span><br /><a href="/wiki/Metre_per_second" title="Metre per second">m s<sup>−1</sup></a></td> <td><a href="/wiki/Kinematic_viscosity" class="mw-redirect" title="Kinematic viscosity">kinematic viscosity</a>: <span class="texhtml"><i>ν</i></span>,<br /><a href="/wiki/Specific_angular_momentum" title="Specific angular momentum">specific angular momentum</a>: <span class="texhtml"><b>h</b></span><br />m<sup>2</sup> s<sup>−1</sup></td> <th style="font-weight:normal;">T<sup>−1</sup></th> <td><span class="nowrap"><a href="/wiki/Frequency" title="Frequency">frequency</a>: <span class="texhtml"><i>f</i></span></span>, <span class="nowrap"><a href="/wiki/Rotational_speed" class="mw-redirect" title="Rotational speed">rotational speed</a>: <span class="texhtml"><i>n</i></span></span>, <span class="nowrap"><a href="/wiki/Rotational_velocity" class="mw-redirect" title="Rotational velocity">rotational velocity</a>: <span class="texhtml"><i><b>n</b></i></span></span><br /><a href="/wiki/Inverse_second" title="Inverse second">s<sup>−1</sup></a>, <a href="/wiki/Hertz" title="Hertz">Hz</a></td> <td><a href="/wiki/Angular_speed" class="mw-redirect" title="Angular speed">angular speed</a>: <span class="texhtml"><i>ω</i></span>, <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a>: <span class="texhtml"><i><b>ω</b></i></span><br /><a href="/wiki/Radian_per_second" title="Radian per second">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/Acceleration" title="Acceleration">acceleration</a>: <span class="texhtml"><b>a</b></span><br /><a href="/wiki/Metre_per_second_squared" title="Metre per second squared">m s<sup>−2</sup></a></td> <td></td> <th style="font-weight:normal;">T<sup>−2</sup></th> <td><span class="nowrap"><a href="/wiki/Rotational_acceleration" class="mw-redirect" title="Rotational acceleration">rotational acceleration</a></span><br /><a href="/wiki/Inverse_square_second" class="mw-redirect" title="Inverse square second">s<sup>−2</sup></a></td> <td><a href="/wiki/Angular_acceleration" title="Angular acceleration">angular acceleration</a>: <span class="texhtml"><i><b>α</b></i></span><br /><a href="/wiki/Radian_per_second_squared" class="mw-redirect" title="Radian per second squared">rad<span style="letter-spacing:0.1em"> </span>s<sup>−2</sup></a></td> <td></td> </tr> <tr> <th style="font-weight:normal;">T<sup>−3</sup></th> <td></td> <td><a href="/wiki/Jerk_(physics)" title="Jerk (physics)">jerk</a>: <span class="texhtml"><b>j</b></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/Jerk_(physics)#Jerk_in_rotation" title="Jerk (physics)">angular jerk</a>: <span class="texhtml"><i><b>ζ</b></i></span><br />rad<span style="letter-spacing:0.1em"> </span>s<sup>−3</sup></td> <td></td> </tr> <tr style="border-top: 3px double #a2a9b1;"> <th style="font-weight:normal;">M</th> <td><a href="/wiki/Mass" title="Mass">mass</a>: <span class="texhtml"><i>m</i></span><br /><a href="/wiki/Kilogram" title="Kilogram">kg</a></td> <td>weighted position: <span class="texhtml"><i>M</i> ⟨<i>x</i>⟩ = ∑ <i>m</i> <i>x</i></span> </td> <td></td> <th style="font-weight:normal;">ML<sup>2</sup></th> <td><a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>: <span class="texhtml"><i>I</i></span><br /><a href="/wiki/Kilogram_square_metre" class="mw-redirect" title="Kilogram square metre">kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup></a></td> <td></td> <td></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−1</sup></th> <td><a href="/wiki/Mass_flow_rate" title="Mass flow rate">Mass flow rate</a>: <span class="texhtml"><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 {\dot {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad59b9876301e8fb75b9ddbf08de594b87251d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:2.176ex;" alt="{\displaystyle {\dot {m}}}"></span></span><br /><a href="/wiki/Kilogram_per_second" class="mw-redirect" title="Kilogram per second">kg<span style="letter-spacing:0.1em"> </span>s<sup>−1</sup></a></td> <td><a href="/wiki/Momentum" title="Momentum">momentum</a>: <span class="texhtml"><b>p</b></span>, <a href="/wiki/Impulse_(physics)" title="Impulse (physics)">impulse</a>: <span class="texhtml"><b>J</b></span><br /><a href="/wiki/Kilogram_metre_per_second" class="mw-redirect" title="Kilogram metre per second">kg<span style="letter-spacing:0.1em"> </span>m s<sup>−1</sup></a>, <a href="/wiki/Newton_second" class="mw-redirect" title="Newton second">N s</a></td> <td><a href="/wiki/Action_(physics)" title="Action (physics)">action</a>: <span class="texhtml">𝒮</span>, <a href="/wiki/Absement#Applications" title="Absement">actergy</a>: <span class="texhtml">ℵ</span><br /><a href="/wiki/Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup> s<sup>−1</sup></a>, <a href="/wiki/Joule-second" title="Joule-second">J s</a></td> <th style="font-weight:normal;">ML<sup>2</sup>T<sup>−1</sup></th> <td></td> <td><a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>: <span class="texhtml"><b>L</b></span>, <a href="/wiki/List_of_equations_in_classical_mechanics#Derived_dynamic_quantities" title="List of equations in classical mechanics">angular impulse</a>: <span class="texhtml">Δ<b>L</b></span><br /><a href="/wiki/Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup> s<sup>−1</sup></a></td> <td><a href="/wiki/Action_(physics)" title="Action (physics)">action</a>: <span class="texhtml">𝒮</span>, <a href="/wiki/Absement#Applications" title="Absement">actergy</a>: <span class="texhtml">ℵ</span><br /><a href="/wiki/Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup> s<sup>−1</sup></a>, <a href="/wiki/Joule-second" title="Joule-second">J s</a></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−2</sup></th> <td></td> <td><a href="/wiki/Force" title="Force">force</a>: <span class="texhtml"><b>F</b></span>, <a href="/wiki/Weight" title="Weight">weight</a>: <span class="texhtml"><b>F</b><sub>g</sub></span><br /><span style="margin-right:0.1em;">kg </span> m s<sup>−2</sup>, <a href="/wiki/Newton_(unit)" title="Newton (unit)">N</a></td> <td><a href="/wiki/Energy" title="Energy">energy</a>: <span class="texhtml"><i>E</i></span>, <a class="mw-selflink selflink">work</a>: <span class="texhtml"><i>W</i></span>, <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a>: <span class="texhtml"><i>L</i></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−2</sup>, <a href="/wiki/Joule" title="Joule">J</a></td> <th style="font-weight:normal;">ML<sup>2</sup>T<sup>−2</sup></th> <td></td> <td><a href="/wiki/Torque" title="Torque">torque</a>: <span class="texhtml"><i><b>τ</b></i></span>, <a href="/wiki/Torque#Terminology" title="Torque">moment</a>: <span class="texhtml"><b>M</b></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−2</sup>, <a href="/wiki/Newton-metre" title="Newton-metre">N m</a></td> <td><a href="/wiki/Energy" title="Energy">energy</a>: <span class="texhtml"><i>E</i></span>, <a class="mw-selflink selflink">work</a>: <span class="texhtml"><i>W</i></span>, <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a>: <span class="texhtml"><i>L</i></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−2</sup>, <a href="/wiki/Joule" title="Joule">J</a></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−3</sup></th> <td></td> <td><a href="/wiki/Yank_(physics)" class="mw-redirect" title="Yank (physics)">yank</a>: <span class="texhtml"><b>Y</b></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/Power_(physics)" title="Power (physics)">power</a>: <span class="texhtml"><i>P</i></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−3</sup>, <a href="/wiki/Watt" title="Watt">W</a></td> <th style="font-weight:normal;">ML<sup>2</sup>T<sup>−3</sup></th> <td></td> <td><a href="/wiki/Rotatum" class="mw-redirect" title="Rotatum">rotatum</a>: <span class="texhtml"><b>P</b></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/Power_(physics)" title="Power (physics)">power</a>: <span class="texhtml"><i>P</i></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2 </sup>s<sup>−3</sup>, <a href="/wiki/Watt" title="Watt">W</a></td> </tr> </tbody></table></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Energy" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Energy_footer" title="Template:Energy footer"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Energy_footer" title="Template talk:Energy footer"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Energy_footer" title="Special:EditPage/Template:Energy footer"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Energy" style="font-size:114%;margin:0 4em"><a href="/wiki/Energy" title="Energy">Energy</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_energy" title="History of energy">History</a></li> <li><a href="/wiki/Index_of_energy_articles" title="Index of energy articles">Index</a></li> <li><a href="/wiki/Outline_of_energy" title="Outline of energy">Outline</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamental <br />concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Conservation_of_energy" title="Conservation of energy">Conservation of energy</a></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Energetics</a></li> <li><a href="/wiki/Energy" title="Energy">Energy</a> <ul><li><a href="/wiki/Units_of_energy" title="Units of energy">Units</a></li></ul></li> <li><a href="/wiki/Energy_condition" title="Energy condition">Energy condition</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a></li> <li><a href="/wiki/Energy_system" title="Energy system">Energy system</a></li> <li><a href="/wiki/Energy_transformation" title="Energy transformation">Energy transformation</a></li> <li><a href="/wiki/Energy_transition" title="Energy transition">Energy transition</a></li> <li><a href="/wiki/Mass" title="Mass">Mass</a> <ul><li><a href="/wiki/Negative_mass" title="Negative mass">Negative mass</a></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence</a></li></ul></li> <li><a href="/wiki/Power_(physics)" title="Power (physics)">Power</a></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a> <ul><li><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a></li> <li><a href="/wiki/Entropic_force" title="Entropic force">Entropic force</a></li> <li><a href="/wiki/Entropy" title="Entropy">Entropy</a></li> <li><a href="/wiki/Exergy" title="Exergy">Exergy</a></li> <li><a href="/wiki/Free_entropy" title="Free entropy">Free entropy</a></li> <li><a href="/wiki/Heat_capacity" title="Heat capacity">Heat capacity</a></li> <li><a href="/wiki/Heat_transfer" title="Heat transfer">Heat transfer</a></li> <li><a href="/wiki/Irreversible_process" title="Irreversible process">Irreversible process</a></li> <li><a href="/wiki/Isolated_system" title="Isolated system">Isolated system</a></li> <li><a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">Laws of thermodynamics</a></li> <li><a href="/wiki/Negentropy" title="Negentropy">Negentropy</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li> <li><a href="/wiki/Thermal_equilibrium" title="Thermal equilibrium">Thermal equilibrium</a></li> <li><a href="/wiki/Thermal_reservoir" title="Thermal reservoir">Thermal reservoir</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Thermodynamic equilibrium</a></li> <li><a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">Thermodynamic free energy</a></li> <li><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Thermodynamic potential</a></li> <li><a href="/wiki/Thermodynamic_state" title="Thermodynamic state">Thermodynamic state</a></li> <li><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Thermodynamic system</a></li> <li><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Thermodynamic temperature</a></li> <li><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume (thermodynamics)</a></li> <li><a class="mw-selflink selflink">Work</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binding_energy" title="Binding energy">Binding</a> <ul><li><a href="/wiki/Nuclear_binding_energy" title="Nuclear binding energy">Nuclear</a></li></ul></li> <li><a href="/wiki/Chemical_energy" title="Chemical energy">Chemical</a></li> <li><a href="/wiki/Dark_energy" title="Dark energy">Dark</a></li> <li><a href="/wiki/Elastic_energy" title="Elastic energy">Elastic</a></li> <li><a href="/wiki/Electric_potential_energy" title="Electric potential energy">Electric potential energy</a></li> <li><a href="/wiki/Electrical_energy" title="Electrical energy">Electrical</a></li> <li><a href="/wiki/Gravitational_energy" title="Gravitational energy">Gravitational</a> <ul><li><a href="/wiki/Gravitational_binding_energy" title="Gravitational binding energy">Binding</a></li></ul></li> <li><a href="/wiki/Interatomic_potential" title="Interatomic potential">Interatomic potential</a></li> <li><a href="/wiki/Internal_energy" title="Internal energy">Internal</a></li> <li><a href="/wiki/Ionization_energy" title="Ionization energy">Ionization</a></li> <li><a href="/wiki/Kinetic_energy" title="Kinetic energy">Kinetic</a></li> <li><a href="/wiki/Magnetic_energy" title="Magnetic energy">Magnetic</a></li> <li><a href="/wiki/Mechanical_energy" title="Mechanical energy">Mechanical</a></li> <li><a href="/wiki/Negative_energy" title="Negative energy">Negative</a></li> <li><a href="/wiki/Phantom_energy" title="Phantom energy">Phantom</a></li> <li><a href="/wiki/Potential_energy" title="Potential energy">Potential</a></li> <li><a href="/wiki/Quantum_chromodynamics_binding_energy" title="Quantum chromodynamics binding energy">Quantum chromodynamics binding energy</a></li> <li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Quantum_potential" title="Quantum potential">Quantum potential</a></li> <li><a href="/wiki/Quintessence_(physics)" title="Quintessence (physics)">Quintessence</a></li> <li><a href="/wiki/Radiant_energy" title="Radiant energy">Radiant</a></li> <li><a href="/wiki/Rest_energy" class="mw-redirect" title="Rest energy">Rest</a></li> <li><a href="/wiki/Sound_energy" title="Sound energy">Sound</a></li> <li><a href="/wiki/Surface_energy" title="Surface energy">Surface</a></li> <li><a href="/wiki/Thermal_energy" title="Thermal energy">Thermal</a></li> <li><a href="/wiki/Vacuum_energy" title="Vacuum energy">Vacuum</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Energy_carrier" title="Energy carrier">Energy carriers</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Electric_battery" title="Electric battery">Battery</a></li> <li><a href="/wiki/Capacitor" title="Capacitor">Capacitor</a></li> <li><a href="/wiki/Electricity" title="Electricity">Electricity</a></li> <li><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a></li> <li><a href="/wiki/Fuel" title="Fuel">Fuel</a> <ul><li><a href="/wiki/Fossil_fuel" title="Fossil fuel">Fossil</a></li> <li><a href="/wiki/Fuel_oil" title="Fuel oil">Oil</a></li></ul></li> <li><a href="/wiki/Heat" title="Heat">Heat</a> <ul><li><a href="/wiki/Latent_heat" title="Latent heat">Latent heat</a></li></ul></li> <li><a href="/wiki/Hydrogen" title="Hydrogen">Hydrogen</a> <ul><li><a href="/wiki/Hydrogen_fuel" class="mw-redirect" title="Hydrogen fuel">Hydrogen fuel</a></li></ul></li> <li><a href="/wiki/Mechanical_wave" title="Mechanical wave">Mechanical wave</a></li> <li><a href="/wiki/Radiation" title="Radiation">Radiation</a></li> <li><a href="/wiki/Sound_wave" class="mw-redirect" title="Sound wave">Sound wave</a></li> <li><a class="mw-selflink selflink">Work</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Primary_energy" title="Primary energy">Primary energy</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bioenergy" title="Bioenergy">Bioenergy</a></li> <li><a href="/wiki/Fossil_fuel" title="Fossil fuel">Fossil fuel</a> <ul><li><a href="/wiki/Coal" title="Coal">Coal</a></li> <li><a href="/wiki/Natural_gas" title="Natural gas">Natural gas</a></li> <li><a href="/wiki/Petroleum" title="Petroleum">Petroleum</a></li></ul></li> <li><a href="/wiki/Geothermal_energy" title="Geothermal energy">Geothermal</a></li> <li><a href="/wiki/Gravitational_energy" title="Gravitational energy">Gravitational</a></li> <li><a href="/wiki/Hydropower" title="Hydropower">Hydropower</a></li> <li><a href="/wiki/Marine_energy" title="Marine energy">Marine</a></li> <li><a href="/wiki/Nuclear_fuel" title="Nuclear fuel">Nuclear fuel</a> <ul><li><a href="/wiki/Natural_uranium" title="Natural uranium">Natural uranium</a></li></ul></li> <li><a href="/wiki/Radiant_energy" title="Radiant energy">Radiant</a></li> <li><a href="/wiki/Solar_energy" title="Solar energy">Solar</a></li> <li><a href="/wiki/Wind_power" title="Wind power">Wind</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Energy_system" title="Energy system">Energy system</a><br />components</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Biomass" title="Biomass">Biomass</a></li> <li><a href="/wiki/Electric_power" title="Electric power">Electric power</a></li> <li><a href="/wiki/Electricity_delivery" title="Electricity delivery">Electricity delivery</a></li> <li><a href="/wiki/Energy_engineering" title="Energy engineering">Energy engineering</a></li> <li><a href="/wiki/Fossil_fuel_power_station" title="Fossil fuel power station">Fossil fuel power station</a> <ul><li><a href="/wiki/Cogeneration" title="Cogeneration">Cogeneration</a></li> <li><a href="/wiki/Integrated_gasification_combined_cycle" title="Integrated gasification combined cycle">Integrated gasification combined cycle</a></li></ul></li> <li><a href="/wiki/Geothermal_power" title="Geothermal power">Geothermal power</a></li> <li><a href="/wiki/Hydropower" title="Hydropower">Hydropower</a> <ul><li><a href="/wiki/Hydroelectricity" title="Hydroelectricity">Hydroelectricity</a></li> <li><a href="/wiki/Tidal_power" title="Tidal power">Tidal power</a></li> <li><a href="/wiki/Wave_farm" class="mw-redirect" title="Wave farm">Wave farm</a></li></ul></li> <li><a href="/wiki/Nuclear_power" title="Nuclear power">Nuclear power</a> <ul><li><a href="/wiki/Nuclear_power_plant" title="Nuclear power plant">Nuclear power plant</a></li> <li><a href="/wiki/Radioisotope_thermoelectric_generator" title="Radioisotope thermoelectric generator">Radioisotope thermoelectric generator</a></li></ul></li> <li><a href="/wiki/Oil_refinery" title="Oil refinery">Oil refinery</a></li> <li><a href="/wiki/Solar_power" title="Solar power">Solar power</a> <ul><li><a href="/wiki/Concentrated_solar_power" title="Concentrated solar power">Concentrated solar power</a></li> <li><a href="/wiki/Photovoltaic_system" title="Photovoltaic system">Photovoltaic system</a></li></ul></li> <li><a href="/wiki/Solar_thermal_energy" title="Solar thermal energy">Solar thermal energy</a> <ul><li><a href="/wiki/Solar_furnace" title="Solar furnace">Solar furnace</a></li> <li><a href="/wiki/Solar_power_tower" title="Solar power tower">Solar power tower</a></li></ul></li> <li><a href="/wiki/Wind_power" title="Wind power">Wind power</a> <ul><li><a href="/wiki/Airborne_wind_energy" title="Airborne wind energy">Airborne wind energy</a></li> <li><a href="/wiki/Wind_farm" title="Wind farm">Wind farm</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Use and<br /><a href="/wiki/Energy_supply" title="Energy supply">supply</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Efficient_energy_use" title="Efficient energy use">Efficient energy use</a> <ul><li><a href="/wiki/Energy_efficiency_in_agriculture" title="Energy efficiency in agriculture">Agriculture</a></li> <li><a href="/wiki/Power_usage_effectiveness" title="Power usage effectiveness">Computing</a></li> <li><a href="/wiki/Energy_efficiency_in_transport" title="Energy efficiency in transport">Transport</a></li></ul></li> <li><a href="/wiki/Energy_conservation" title="Energy conservation">Energy conservation</a></li> <li><a href="/wiki/Energy_consumption" title="Energy consumption">Energy consumption</a></li> <li><a href="/wiki/Energy_policy" title="Energy policy">Energy policy</a> <ul><li><a href="/wiki/Energy_development" title="Energy development">Energy development</a></li></ul></li> <li><a href="/wiki/Energy_security" title="Energy security">Energy security</a></li> <li><a href="/wiki/Energy_storage" title="Energy storage">Energy storage</a></li> <li><a href="/wiki/Renewable_energy" title="Renewable energy">Renewable energy</a></li> <li><a href="/wiki/Sustainable_energy" title="Sustainable energy">Sustainable energy</a></li> <li><a href="/wiki/World_energy_supply_and_consumption" title="World energy supply and consumption">World energy supply and consumption</a></li> <li><a href="/wiki/Energy_in_Africa" title="Energy in Africa">Africa</a></li> <li><a href="/wiki/Energy_in_Asia" class="mw-redirect" title="Energy in Asia">Asia</a></li> <li><a href="/wiki/Energy_in_Australia" title="Energy in Australia">Australia</a></li> <li><a href="/wiki/Energy_policy_of_Canada" title="Energy policy of Canada">Canada</a></li> <li><a href="/wiki/Energy_in_Europe" title="Energy in Europe">Europe</a></li> <li><a href="/wiki/Energy_in_Mexico" title="Energy in Mexico">Mexico</a></li> <li><a href="/wiki/Energy_in_South_America" title="Energy in South America">South America</a></li> <li><a href="/wiki/Energy_in_the_United_States" title="Energy in the United States">United States</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Misc.</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carbon_footprint" title="Carbon footprint">Carbon footprint</a></li> <li><a href="/wiki/Energy_democracy" title="Energy democracy">Energy democracy</a></li> <li><a href="/wiki/Energy_recovery" title="Energy recovery">Energy recovery</a></li> <li><a href="/wiki/Energy_recycling" title="Energy recycling">Energy recycling</a></li> <li><a href="/wiki/Jevons_paradox" title="Jevons paradox">Jevons paradox</a></li> <li><a href="/wiki/Waste-to-energy" title="Waste-to-energy">Waste-to-energy</a> 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