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<span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Work_and_potential_energy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Work_and_potential_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</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-Derivable_from_a_potential" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivable_from_a_potential"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Derivable from a potential</span> </div> </a> <ul id="toc-Derivable_from_a_potential-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computing_potential_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computing_potential_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Computing potential energy</span> </div> </a> <ul id="toc-Computing_potential_energy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Potential_energy_for_near-Earth_gravity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Potential_energy_for_near-Earth_gravity"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Potential energy for near-Earth gravity</span> </div> </a> <ul id="toc-Potential_energy_for_near-Earth_gravity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Potential_energy_for_a_linear_spring" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Potential_energy_for_a_linear_spring"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Potential energy for a linear spring</span> </div> </a> <ul id="toc-Potential_energy_for_a_linear_spring-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Potential_energy_for_gravitational_forces_between_two_bodies" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Potential_energy_for_gravitational_forces_between_two_bodies"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Potential energy for gravitational forces between two bodies</span> </div> </a> <button aria-controls="toc-Potential_energy_for_gravitational_forces_between_two_bodies-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 Potential energy for gravitational forces between two bodies subsection</span> </button> <ul id="toc-Potential_energy_for_gravitational_forces_between_two_bodies-sublist" class="vector-toc-list"> <li id="toc-Derivation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Derivation</span> </div> </a> <ul id="toc-Derivation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Potential_energy_for_electrostatic_forces_between_two_bodies" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Potential_energy_for_electrostatic_forces_between_two_bodies"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Potential energy for electrostatic forces between two bodies</span> </div> </a> <ul id="toc-Potential_energy_for_electrostatic_forces_between_two_bodies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reference_level" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reference_level"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Reference level</span> </div> </a> <ul id="toc-Reference_level-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gravitational_potential_energy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Gravitational_potential_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Gravitational potential energy</span> </div> </a> <button aria-controls="toc-Gravitational_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 Gravitational potential energy subsection</span> </button> <ul id="toc-Gravitational_potential_energy-sublist" class="vector-toc-list"> <li id="toc-Local_approximation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Local_approximation"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Local approximation</span> </div> </a> <ul id="toc-Local_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>General formula</span> </div> </a> <ul id="toc-General_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Negative_gravitational_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Negative_gravitational_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3</span> <span>Negative gravitational energy</span> </div> </a> <ul id="toc-Negative_gravitational_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uses" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uses"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.4</span> <span>Uses</span> </div> </a> <ul id="toc-Uses-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Chemical_potential_energy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Chemical_potential_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Chemical potential energy</span> </div> </a> <ul id="toc-Chemical_potential_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electric_potential_energy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Electric_potential_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Electric potential energy</span> </div> </a> <button aria-controls="toc-Electric_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 Electric potential energy subsection</span> </button> <ul id="toc-Electric_potential_energy-sublist" class="vector-toc-list"> <li id="toc-Electrostatic_potential_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electrostatic_potential_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Electrostatic potential energy</span> </div> </a> <ul id="toc-Electrostatic_potential_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetic_potential_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Magnetic_potential_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.2</span> <span>Magnetic potential energy</span> </div> </a> <ul id="toc-Magnetic_potential_energy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Nuclear_potential_energy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Nuclear_potential_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Nuclear potential energy</span> </div> </a> <ul id="toc-Nuclear_potential_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Forces_and_potential_energy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Forces_and_potential_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Forces and potential energy</span> </div> </a> <ul id="toc-Forces_and_potential_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">16</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">Potential energy</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 87 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-87" 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">87 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7%D8%A7%D9%82%D8%A9_%D9%88%D8%B6%D8%B9" title="طاقة وضع – Arabic" lang="ar" hreflang="ar" data-title="طاقة وضع" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BF%E0%A6%A4%E0%A6%BF_%E0%A6%B6%E0%A6%95%E0%A7%8D%E0%A6%A4%E0%A6%BF" title="স্থিতি শক্তি – Assamese" lang="as" hreflang="as" data-title="স্থিতি শক্তি" 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/Enerx%C3%ADa_potencial" title="Enerxía potencial – Asturian" lang="ast" hreflang="ast" data-title="Enerxía potencial" 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/Potensial_enerji" title="Potensial enerji – Azerbaijani" lang="az" hreflang="az" data-title="Potensial enerji" 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%AC%E0%A6%BF%E0%A6%AD%E0%A6%AC_%E0%A6%B6%E0%A6%95%E0%A7%8D%E0%A6%A4%E0%A6%BF" title="বিভব শক্তি – Bangla" lang="bn" hreflang="bn" data-title="বিভব শক্তি" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B0%D1%82%D1%8D%D0%BD%D1%86%D1%8B%D1%8F%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D1%96%D1%8F" title="Патэнцыяльная энергія – Belarusian" lang="be" hreflang="be" data-title="Патэнцыяльная энергія" 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%9F%D0%B0%D1%82%D1%8D%D0%BD%D1%86%D1%8B%D0%B9%D0%BD%D0%B0%D1%8F_%D1%8D%D0%BD%D1%8D%D1%80%D0%B3%D1%96%D1%8F" title="Патэнцыйная энэргія – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Патэнцыйная энэргія" 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-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Enerhiyang_potensyal" title="Enerhiyang potensyal – Central Bikol" lang="bcl" hreflang="bcl" data-title="Enerhiyang potensyal" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Потенциална енергия – Bulgarian" lang="bg" hreflang="bg" data-title="Потенциална енергия" 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/Potencijalna_energija" title="Potencijalna energija – Bosnian" lang="bs" hreflang="bs" data-title="Potencijalna energija" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Energia_potencial" title="Energia potencial – Catalan" lang="ca" hreflang="ca" data-title="Energia potencial" 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%9F%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%BB%D0%B0_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8" title="Потенциалла энерги – Chuvash" lang="cv" hreflang="cv" data-title="Потенциалла энерги" 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/Potenci%C3%A1ln%C3%AD_energie" title="Potenciální energie – Czech" lang="cs" hreflang="cs" data-title="Potenciální energie" 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/Simba_reChakunga" title="Simba reChakunga – Shona" lang="sn" hreflang="sn" data-title="Simba reChakunga" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Egni_potensial" title="Egni potensial – Welsh" lang="cy" hreflang="cy" data-title="Egni potensial" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Potentiel_energi" title="Potentiel energi – Danish" lang="da" hreflang="da" data-title="Potentiel energi" 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/Potentielle_Energie" title="Potentielle Energie – German" lang="de" hreflang="de" data-title="Potentielle Energie" 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/Potentsiaalne_energia" title="Potentsiaalne energia – Estonian" lang="et" hreflang="et" data-title="Potentsiaalne energia" 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%94%CF%85%CE%BD%CE%B1%CE%BC%CE%B9%CE%BA%CE%AE_%CE%B5%CE%BD%CE%AD%CF%81%CE%B3%CE%B5%CE%B9%CE%B1" title="Δυναμική ενέργεια – Greek" lang="el" hreflang="el" data-title="Δυναμική ενέργεια" 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/Energ%C3%ADa_potencial" title="Energía potencial – Spanish" lang="es" hreflang="es" data-title="Energía potencial" 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/Potenciala_energio" title="Potenciala energio – Esperanto" lang="eo" hreflang="eo" data-title="Potenciala energio" 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/Energia_potentzial" title="Energia potentzial – Basque" lang="eu" hreflang="eu" data-title="Energia potentzial" 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/%D8%A7%D9%86%D8%B1%DA%98%DB%8C_%D9%BE%D8%AA%D8%A7%D9%86%D8%B3%DB%8C%D9%84" title="انرژی پتانسیل – Persian" lang="fa" hreflang="fa" data-title="انرژی پتانسیل" 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/%C3%89nergie_potentielle" title="Énergie potentielle – French" lang="fr" hreflang="fr" data-title="Énergie potentielle" 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/Fuinneamh_poit%C3%A9insi%C3%BAil" title="Fuinneamh poitéinsiúil – Irish" lang="ga" hreflang="ga" data-title="Fuinneamh poitéinsiúil" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Enerx%C3%ADa_potencial" title="Enerxía potencial – Galician" lang="gl" hreflang="gl" data-title="Enerxía potencial" 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%9C%84%EC%B9%98_%EC%97%90%EB%84%88%EC%A7%80" title="위치 에너지 – Korean" lang="ko" hreflang="ko" data-title="위치 에너지" 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/%D5%8A%D5%B8%D5%BF%D5%A5%D5%B6%D6%81%D5%AB%D5%A1%D5%AC_%D5%A7%D5%B6%D5%A5%D6%80%D5%A3%D5%AB%D5%A1" title="Պոտենցիալ էներգիա – Armenian" lang="hy" hreflang="hy" data-title="Պոտենցիալ էներգիա" 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%B8%E0%A5%8D%E0%A4%A5%E0%A4%BF%E0%A4%A4%E0%A4%BF%E0%A4%9C_%E0%A4%8A%E0%A4%B0%E0%A5%8D%E0%A4%9C%E0%A4%BE" title="स्थितिज ऊर्जा – Hindi" lang="hi" hreflang="hi" data-title="स्थितिज ऊर्जा" 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/Potencijalna_energija" title="Potencijalna energija – Croatian" lang="hr" hreflang="hr" data-title="Potencijalna energija" 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/Energi_potensial" title="Energi potensial – Indonesian" lang="id" hreflang="id" data-title="Energi potensial" 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-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Isidlakathi" title="Isidlakathi – Zulu" lang="zu" hreflang="zu" data-title="Isidlakathi" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/St%C3%B6%C3%B0uorka" title="Stöðuorka – Icelandic" lang="is" hreflang="is" data-title="Stöðuorka" 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/Energia_potenziale" title="Energia potenziale – Italian" lang="it" hreflang="it" data-title="Energia potenziale" 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%90%D7%A0%D7%A8%D7%92%D7%99%D7%94_%D7%A4%D7%95%D7%98%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%AA" title="אנרגיה פוטנציאלית – Hebrew" lang="he" hreflang="he" data-title="אנרגיה פוטנציאלית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Energi_Potensial" title="Energi Potensial – Javanese" lang="jv" hreflang="jv" data-title="Energi Potensial" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%85%E0%B2%82%E0%B2%A4%E0%B2%B8%E0%B3%8D%E0%B2%A5_%E0%B2%B6%E0%B2%95%E0%B3%8D%E0%B2%A4%E0%B2%BF" title="ಅಂತಸ್ಥ ಶಕ್ತಿ – Kannada" lang="kn" hreflang="kn" data-title="ಅಂತಸ್ಥ ಶಕ್ತಿ" 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%9E%E1%83%9D%E1%83%A2%E1%83%94%E1%83%9C%E1%83%AA%E1%83%98%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%94%E1%83%9C%E1%83%94%E1%83%A0%E1%83%92%E1%83%98%E1%83%90" title="პოტენციური ენერგია – Georgian" lang="ka" hreflang="ka" data-title="პოტენციური ენერგია" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9F%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%B4%D1%8B%D2%9B_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Потенциалдық энергия – Kazakh" lang="kk" hreflang="kk" data-title="Потенциалдық энергия" 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/En%C3%A8ji_potansy%C3%A8l_chimik" title="Enèji potansyèl chimik – Haitian Creole" lang="ht" hreflang="ht" data-title="Enèji potansyèl chimik" 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-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/%C3%89nerji_potansy%C3%A8l" title="Énerji potansyèl – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Énerji potansyèl" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Energia_potentialis" title="Energia potentialis – Latin" lang="la" hreflang="la" data-title="Energia potentialis" 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/Potenci%C4%81l%C4%81_ener%C4%A3ija" title="Potenciālā enerģija – Latvian" lang="lv" hreflang="lv" data-title="Potenciālā enerģija" 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/Potencin%C4%97_energija" title="Potencinė energija – Lithuanian" lang="lt" hreflang="lt" data-title="Potencinė energija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Potenci%C3%A1lis_energia" title="Potenciális energia – Hungarian" lang="hu" hreflang="hu" data-title="Potenciális energia" 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%9F%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%98%D0%B0" title="Потенцијална енергија – Macedonian" lang="mk" hreflang="mk" data-title="Потенцијална енергија" 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%B8%E0%B5%8D%E0%B4%A5%E0%B4%BF%E0%B4%A4%E0%B4%BF%E0%B4%95%E0%B5%8B%E0%B5%BC%E0%B4%9C%E0%B5%8D%E0%B4%9C%E0%B4%82" title="സ്ഥിതികോർജ്ജം – Malayalam" lang="ml" hreflang="ml" data-title="സ്ഥിതികോർജ്ജം" 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%B8%E0%A5%8D%E0%A4%A5%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%9C_%E0%A4%8A%E0%A4%B0%E0%A5%8D%E0%A4%9C%E0%A4%BE" title="स्थितीज ऊर्जा – Marathi" lang="mr" hreflang="mr" data-title="स्थितीज ऊर्जा" 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/Tenaga_keupayaan" title="Tenaga keupayaan – Malay" lang="ms" hreflang="ms" data-title="Tenaga keupayaan" 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%9F%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8" title="Потенциал энерги – Mongolian" lang="mn" hreflang="mn" data-title="Потенциал энерги" 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%90%E1%80%8A%E1%80%BA%E1%80%85%E1%80%BD%E1%80%99%E1%80%BA%E1%80%B8%E1%80%A1%E1%80%84%E1%80%BA" title="အတည်စွမ်းအင် – Burmese" lang="my" hreflang="my" data-title="အတည်စွမ်းအင်" 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/Potenti%C3%ABle_energie" title="Potentiële energie – Dutch" lang="nl" hreflang="nl" data-title="Potentiële energie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BD%8D%E7%BD%AE%E3%82%A8%E3%83%8D%E3%83%AB%E3%82%AE%E3%83%BC" title="位置エネルギー – Japanese" lang="ja" hreflang="ja" data-title="位置エネルギー" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Potensiell_energi" title="Potensiell energi – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Potensiell energi" 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/Potensiell_energi" title="Potensiell energi – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Potensiell energi" 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/Energia_potenciala" title="Energia potenciala – Occitan" lang="oc" hreflang="oc" data-title="Energia potenciala" 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/Anniisaa_Kuufamaa" title="Anniisaa Kuufamaa – Oromo" lang="om" hreflang="om" data-title="Anniisaa Kuufamaa" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Potensial_energiya" title="Potensial energiya – Uzbek" lang="uz" hreflang="uz" data-title="Potensial energiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A9%81%E0%A8%9F%E0%A9%88%E0%A8%82%E0%A8%B8%E0%A8%BC%E0%A8%B2_%E0%A8%8A%E0%A8%B0%E0%A8%9C%E0%A8%BE" title="ਪੁਟੈਂਸ਼ਲ ਊਰਜਾ – Punjabi" lang="pa" hreflang="pa" data-title="ਪੁਟੈਂਸ਼ਲ ਊਰਜਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Potenshal_enaji" title="Potenshal enaji – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Potenshal enaji" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Energ%C3%ACa_potensial" title="Energìa potensial – Piedmontese" lang="pms" hreflang="pms" data-title="Energìa potensial" 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/Energia_potencjalna" title="Energia potencjalna – Polish" lang="pl" hreflang="pl" data-title="Energia potencjalna" 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/Energia_potencial" title="Energia potencial – Portuguese" lang="pt" hreflang="pt" data-title="Energia potencial" 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/Energie_poten%C8%9Bial%C4%83" title="Energie potențială – Romanian" lang="ro" hreflang="ro" data-title="Energie potențială" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Потенциальная энергия – Russian" lang="ru" hreflang="ru" data-title="Потенциальная энергия" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Energjia_potenciale" title="Energjia potenciale – Albanian" lang="sq" hreflang="sq" data-title="Energjia potenciale" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Potential_energy" title="Potential energy – Simple English" lang="en-simple" hreflang="en-simple" data-title="Potential energy" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Potenci%C3%A1lna_energia" title="Potenciálna energia – Slovak" lang="sk" hreflang="sk" data-title="Potenciálna energia" 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/Potencialna_energija" title="Potencialna energija – Slovenian" lang="sl" hreflang="sl" data-title="Potencialna energija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Awood_Kaydsan" title="Awood Kaydsan – Somali" lang="so" hreflang="so" data-title="Awood Kaydsan" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%98%D0%B0" title="Потенцијална енергија – Serbian" lang="sr" hreflang="sr" data-title="Потенцијална енергија" 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/Potencijalna_energija" title="Potencijalna energija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Potencijalna energija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" 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.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 the Flash episode, see <a href="/wiki/Potential_Energy_(The_Flash)" class="mw-redirect" title="Potential Energy (The Flash)">Potential Energy (The Flash)</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">Potential energy</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Mediaeval_archery_reenactment.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Mediaeval_archery_reenactment.jpg/220px-Mediaeval_archery_reenactment.jpg" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Mediaeval_archery_reenactment.jpg/330px-Mediaeval_archery_reenactment.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Mediaeval_archery_reenactment.jpg/440px-Mediaeval_archery_reenactment.jpg 2x" data-file-width="480" data-file-height="640" /></a></span><div class="infobox-caption">In the case of a <a href="/wiki/Bow_and_arrow" title="Bow and arrow">bow and arrow</a>, when the archer does <a href="/wiki/Work_(physics)" title="Work (physics)">work</a> on the bow, drawing the string back, some of the chemical energy of the archer's body is transformed into <a href="/wiki/Elastic_potential_energy" class="mw-redirect" title="Elastic potential energy">elastic potential energy</a> in the bent limb of the bow. When the string is released, the force between the string and the arrow does work on the arrow. The potential energy in the bow limbs is transformed into the <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> of the arrow as it takes flight.</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"><i>PE</i>, <i>U</i>, or <i>V</i></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;">Derivations from<br />other quantities</div></th><td class="infobox-data"><i>U</i> = <i><a href="/wiki/Mass" title="Mass">m</a></i> ⋅ <i><a href="/wiki/Gravity_of_Earth" title="Gravity of Earth">g</a></i> ⋅ <i><a href="/wiki/Height" title="Height">h</a></i> (<a href="/wiki/Gravitational_potential_energy" class="mw-redirect" title="Gravitational potential energy">gravitational</a>)<br /> <p><i>U</i> = <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.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="frac"><span class="num">1</span>⁄<span class="den">2</span></span> ⋅ <i><a href="/wiki/Hooke%27s_law" title="Hooke's law">k</a></i> ⋅ <i><a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">x</a></i><sup>2</sup> (<a href="/wiki/Elastic_energy" title="Elastic energy">elastic</a>)<br /> <i>U</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> ⋅ <i><a href="/wiki/Capacitance" title="Capacitance">C</a></i> ⋅ <i><a href="/wiki/Electric_potential" title="Electric potential">V</a></i><sup>2</sup> (<a href="/wiki/Electric_potential_energy" title="Electric potential energy">electric</a>)<br /> <i>U</i> = −<i><a href="/wiki/Magnetic_moment" title="Magnetic moment">m</a></i> ⋅ <i><a href="/wiki/Magnetic_field" title="Magnetic field">B</a></i> (<a href="/wiki/Magnetic_potential_energy" class="mw-redirect" title="Magnetic potential energy">magnetic</a>) </p> <i>U</i> = <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 \int F(r)\,dr}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int F(r)\,dr}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24064daff0ad88bc1c3c5cd0dedc2012ca4f5609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.055ex; height:3.176ex;" alt="{\textstyle \int F(r)\,dr}"></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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href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle" 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 mw-collapsed"><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 class="mw-selflink selflink">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 href="/wiki/Work_(physics)" title="Work (physics)">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 mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler</a></li> <li><a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Jeremiah_Horrocks" title="Jeremiah Horrocks">Horrocks</a></li> <li><a href="/wiki/Edmond_Halley" title="Edmond Halley">Halley</a></li> <li><a href="/wiki/Pierre_Louis_Maupertuis" title="Pierre Louis Maupertuis">Maupertuis</a></li> <li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a></li> <li><a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a></li> 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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 <a href="/wiki/Physics" title="Physics">physics</a>, <b>potential energy</b> is the <a href="/wiki/Energy" title="Energy">energy</a> held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The term <i>potential energy</i> was introduced by the 19th-century Scottish engineer and physicist <a href="/wiki/William_Rankine" title="William Rankine">William Rankine</a>,<sup id="cite_ref-Rankin1853_3-0" class="reference"><a href="#cite_note-Rankin1853-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Roche2003_4-0" class="reference"><a href="#cite_note-Roche2003-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-SmithEnergy_5-0" class="reference"><a href="#cite_note-SmithEnergy-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> although it has links to the ancient Greek philosopher <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>'s concept of <a href="/wiki/Potentiality_and_Actuality" class="mw-redirect" title="Potentiality and Actuality"><i>potentiality</i></a>. </p><p>Common types of potential energy include the <a href="/wiki/Gravitational_potential_energy" class="mw-redirect" title="Gravitational potential energy">gravitational potential energy</a> of an object, the <a href="/wiki/Elastic_potential_energy" class="mw-redirect" title="Elastic potential energy">elastic potential energy</a> of a deformed spring, and the <a href="/wiki/Electric_potential_energy" title="Electric potential energy">electric potential energy</a> of an <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> in an <a href="/wiki/Electric_field" title="Electric field">electric field</a>. The unit for energy in the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a> (SI) is the <a href="/wiki/Joule" title="Joule">joule</a> (symbol J). </p><p>Potential energy is associated with forces that act on a body in a way that the total work done by these forces on the body depends only on the initial and final positions of the body in space. These forces, whose total work is path independent, are called <a href="/wiki/Conservative_force" title="Conservative force"><i>conservative forces</i></a>. If the force acting on a body varies over space, then one has a <a href="/wiki/Force_field_(physics)" title="Force field (physics)"><i>force field</i></a>; such a field is described by vectors at every point in space, which is in-turn called a <i><a href="/wiki/Vector_field" title="Vector field">vector field</a></i>. A conservative vector field can be simply expressed as the gradient of a certain scalar function, called a <i><a href="/wiki/Scalar_potential" title="Scalar potential">scalar potential</a></i>. The potential energy is related to, and can be obtained from, this potential function. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div> <p>There are various types of potential energy, each associated with a particular type of force. For example, the work of an <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> force is called elastic potential energy; work of the gravitational force is called gravitational potential energy; work of the <a href="/wiki/Coulomb_force" class="mw-redirect" title="Coulomb force">Coulomb force</a> is called <a href="/wiki/Electric_potential_energy" title="Electric potential energy">electric potential energy</a>; work of the <a href="/wiki/Strong_nuclear_force" class="mw-redirect" title="Strong nuclear force">strong nuclear force</a> or <a href="/wiki/Weak_nuclear_force" class="mw-redirect" title="Weak nuclear force">weak nuclear force</a> acting on the <a href="/wiki/Baryon" title="Baryon">baryon</a> <a href="/wiki/Charge_(physics)" title="Charge (physics)">charge</a> is called nuclear potential energy; work of <a href="/wiki/Intermolecular_forces" class="mw-redirect" title="Intermolecular forces">intermolecular forces</a> is called intermolecular potential energy. Chemical potential energy, such as the energy stored in <a href="/wiki/Fossil_fuels" class="mw-redirect" title="Fossil fuels">fossil fuels</a>, is the work of the Coulomb force during rearrangement of configurations of electrons and nuclei in atoms and molecules. Thermal energy usually has two components: the kinetic energy of random motions of particles and the potential energy of their configuration. </p><p>Forces derivable from a potential are also called <a href="/wiki/Conservative_force" title="Conservative force">conservative forces</a>. The work done by a conservative force 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=-\Delta U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=-\Delta U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/817c77fc4aa36cac8441f88e2646c8a82f7dbad8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.06ex; height:2.343ex;" alt="{\displaystyle W=-\Delta U}"></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 \Delta U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4716a2c49bbbe155e8b399117ca78342e802cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.718ex; height:2.176ex;" alt="{\displaystyle \Delta U}"></span> is the change in the potential energy associated with the force. The negative sign provides the convention that work done against a force field increases potential energy, while work done by the force field decreases potential energy. Common notations for potential energy are <i>PE</i>, <i>U</i>, <i>V</i>, and <i>E<sub>p</sub></i>. </p><p>Potential energy is the energy by virtue of an object's position relative to other objects.<sup id="cite_ref-:0_6-0" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Potential energy is often associated with restoring <a href="/wiki/Force" title="Force">forces</a> such as a <a href="/wiki/Spring_(device)" title="Spring (device)">spring</a> or the force of <a href="/wiki/Gravity" title="Gravity">gravity</a>. The action of stretching a spring or lifting a mass is performed by an external force that works against the force field of the potential. This work is stored in the force field, which is said to be stored as potential energy. If the external force is removed the force field acts on the body to perform the work as it moves the body back to the initial position, reducing the stretch of the spring or causing a body to fall. </p><p>Consider a ball whose mass is <span class="texhtml"><i>m</i></span> dropped from height <span class="texhtml"><i>h</i></span>. The acceleration <span class="texhtml"><i>g</i></span> of free fall is approximately constant, so the weight force of the ball <span class="texhtml"><i>mg</i></span> is constant. The product of force and displacement gives the work done, which is equal to the gravitational potential energy, thus <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_{g}=mgh.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mi>g</mi> <mi>h</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{g}=mgh.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9f9c213ee46740f1643b2552703e5714c10db4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.85ex; height:2.843ex;" alt="{\displaystyle U_{g}=mgh.}"></span> </p><p>The more formal definition is that potential energy is the energy difference between the energy of an object in a given position and its energy at a reference position. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></div> <p>From around 1840 scientists sought to define and understand energy and <a href="/wiki/Work_(physics)" title="Work (physics)"> work</a>.<sup id="cite_ref-SmithEnergy_5-1" class="reference"><a href="#cite_note-SmithEnergy-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The term "potential energy" was coined by <a href="/wiki/William_Rankine" title="William Rankine">William Rankine</a> a Scottish engineer and physicist in 1853 as part of a specific effort to develop terminology.<sup id="cite_ref-Rankin1853_3-1" class="reference"><a href="#cite_note-Rankin1853-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> He chose the term as part of the pair "actual" vs "potential" going back to work by <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>. In his 1867 discussion of the same topic Rankine describes potential energy as ‘energy of configuration’ in contrast to actual energy as 'energy of activity'. Also in 1867, <a href="/wiki/Lord_Kelvin" title="Lord Kelvin"> William Thomson</a> introduced "kinetic energy" as the opposite of "potential energy", asserting that all actual energy took the form of <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>mv<sup>2</sup>. Once this hypothesis became widely accepted, the term "actual energy" gradually faded.<sup id="cite_ref-Roche2003_4-1" class="reference"><a href="#cite_note-Roche2003-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Work_and_potential_energy">Work and potential energy</h2></div> <p>Potential energy is closely linked with <a href="/wiki/Force_(physics)" class="mw-redirect" title="Force (physics)">forces</a>. If the work done by a force on a body that moves from <i>A</i> to <i>B</i> does not depend on the path between these points (if the work is done by a conservative force), then the work of this force measured from <i>A</i> assigns a scalar value to every other point in space and defines a <a href="/wiki/Scalar_potential" title="Scalar potential">scalar potential</a> field. In this case, the force can be defined as the negative of the <a href="/wiki/Gradient" title="Gradient">vector gradient</a> of the potential field. </p><p>If the work for an applied force is independent of the path, then the work done by the force is evaluated from the start to the end of the trajectory of the point of application. This means that there is a function <i>U</i>(<b>x</b>), called a "potential", that can be evaluated at the two points <b>x</b><sub>A</sub> and <b>x</b><sub>B</sub> 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} =U(\mathbf {x} _{\text{A}})-U(\mathbf {x} _{\text{B}})}"> <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> <mi>U</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>U</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =U(\mathbf {x} _{\text{A}})-U(\mathbf {x} _{\text{B}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799ebee8649d879b556394a3ea57e2d0138519c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.488ex; height:5.676ex;" alt="{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =U(\mathbf {x} _{\text{A}})-U(\mathbf {x} _{\text{B}})}"></span> where <i>C</i> is the trajectory taken from A to B. Because the work done is independent of the path taken, then this expression is true for any trajectory, <i>C</i>, from A to B. </p><p>The function <i>U</i>(<b>x</b>) is called the potential energy associated with the applied force. Examples of forces that have potential energies are gravity and spring forces. </p> <div class="mw-heading mw-heading3"><h3 id="Derivable_from_a_potential">Derivable from a potential</h3></div> <p>In this section the relationship between work and potential energy is presented in more detail. The <a href="/wiki/Line_integral" title="Line integral">line integral</a> that defines work along curve <i>C</i> takes a special form if the force <b>F</b> is related to a scalar field <i>U</i>′(<b>x</b>) so 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 \mathbf {F} ={\nabla U'}=\left({\frac {\partial U'}{\partial x}},{\frac {\partial U'}{\partial y}},{\frac {\partial U'}{\partial z}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> </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> <msup> <mi>U</mi> <mo>′</mo> </msup> </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> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} ={\nabla U'}=\left({\frac {\partial U'}{\partial x}},{\frac {\partial U'}{\partial y}},{\frac {\partial U'}{\partial z}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5479c40f234d752098ffc916f11676a2a489cf5c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.905ex; height:6.176ex;" alt="{\displaystyle \mathbf {F} ={\nabla U'}=\left({\frac {\partial U'}{\partial x}},{\frac {\partial U'}{\partial y}},{\frac {\partial U'}{\partial z}}\right).}"></span> This means that the units of <i>U</i>′ must be this case, work along the curve 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 _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{C}\nabla U'\cdot d\mathbf {x} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> <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 _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{C}\nabla U'\cdot d\mathbf {x} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c2c1bf7334946f5846ab2853b6fdea47dac25b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.357ex; height:5.676ex;" alt="{\displaystyle W=\int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{C}\nabla U'\cdot d\mathbf {x} ,}"></span> which can be evaluated using the <a href="/wiki/Gradient_theorem" title="Gradient theorem">gradient theorem</a> 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=U'(\mathbf {x} _{\text{B}})-U'(\mathbf {x} _{\text{A}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msup> <mi>U</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>U</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=U'(\mathbf {x} _{\text{B}})-U'(\mathbf {x} _{\text{A}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/865d575b658050fe2275c038b7cb3da2b2f699f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.374ex; height:3.009ex;" alt="{\displaystyle W=U'(\mathbf {x} _{\text{B}})-U'(\mathbf {x} _{\text{A}}).}"></span> This shows that when forces are derivable from a scalar field, the work of those forces along a curve <i>C</i> is computed by evaluating the scalar field at the start point A and the end point B of the curve. This means the work integral does not depend on the path between A and B and is said to be independent of the path. </p><p>Potential energy <span class="texhtml"><i>U</i> = −<i>U</i>′(<b>x</b>)</span> is traditionally defined as the negative of this scalar field so that work by the force field decreases potential energy, 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=U(\mathbf {x} _{\text{A}})-U(\mathbf {x} _{\text{B}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>U</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>U</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=U(\mathbf {x} _{\text{A}})-U(\mathbf {x} _{\text{B}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa74c1b22fc5c8b86c09970bf4394c3d58009ca4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.887ex; height:2.843ex;" alt="{\displaystyle W=U(\mathbf {x} _{\text{A}})-U(\mathbf {x} _{\text{B}}).}"></span> </p><p>In this case, the application of the <a href="/wiki/Del_operator" class="mw-redirect" title="Del operator">del operator</a> to the work function 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 {\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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> <mi>W</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> <mi>U</mi> </mrow> <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/6eadee4df78e6e0a7163cc581adbe3e8fb31c433" 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} ,}"></span> and the force <b>F</b> is said to be "derivable from a potential".<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> This also necessarily implies that <b>F</b> must be a <a href="/wiki/Conservative_vector_field" title="Conservative vector field">conservative vector field</a>. The potential <i>U</i> defines a force <b>F</b> at every point <b>x</b> in space, so the set of forces is called a <a href="/wiki/Force_field_(physics)" title="Force field (physics)">force field</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Computing_potential_energy">Computing potential energy</h3></div> <p>Given a force field <b>F</b>(<b>x</b>), evaluation of the work integral using the <a href="/wiki/Gradient_theorem" title="Gradient theorem">gradient theorem</a> can be used to find the scalar function associated with potential energy. This is done by introducing a parameterized curve <span class="texhtml"><i>γ</i>(<i>t</i>) = <b>r</b>(<i>t</i>)</span> from <span class="texhtml"><i>γ</i>(<i>a</i>) = <i>A</i></span> to <span class="texhtml"><i>γ</i>(<i>b</i>) = <i>B</i></span>, and computing, <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 {\begin{aligned}\int _{\gamma }\nabla \Phi (\mathbf {r} )\cdot d\mathbf {r} &=\int _{a}^{b}\nabla \Phi (\mathbf {r} (t))\cdot \mathbf {r} '(t)dt,\\&=\int _{a}^{b}{\frac {d}{dt}}\Phi (\mathbf {r} (t))dt=\Phi (\mathbf {r} (b))-\Phi (\mathbf {r} (a))=\Phi \left(\mathbf {x} _{B}\right)-\Phi \left(\mathbf {x} _{A}\right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mtd> <mtd> <mi></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 mathvariant="normal">∇</mi> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>−</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{\gamma }\nabla \Phi (\mathbf {r} )\cdot d\mathbf {r} &=\int _{a}^{b}\nabla \Phi (\mathbf {r} (t))\cdot \mathbf {r} '(t)dt,\\&=\int _{a}^{b}{\frac {d}{dt}}\Phi (\mathbf {r} (t))dt=\Phi (\mathbf {r} (b))-\Phi (\mathbf {r} (a))=\Phi \left(\mathbf {x} _{B}\right)-\Phi \left(\mathbf {x} _{A}\right).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b0f2885c5c2505f4f884f71059ffbbbc7579f9d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:74.603ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\int _{\gamma }\nabla \Phi (\mathbf {r} )\cdot d\mathbf {r} &=\int _{a}^{b}\nabla \Phi (\mathbf {r} (t))\cdot \mathbf {r} '(t)dt,\\&=\int _{a}^{b}{\frac {d}{dt}}\Phi (\mathbf {r} (t))dt=\Phi (\mathbf {r} (b))-\Phi (\mathbf {r} (a))=\Phi \left(\mathbf {x} _{B}\right)-\Phi \left(\mathbf {x} _{A}\right).\end{aligned}}}"></span> </p><p>For the force field <b>F</b>, let <span class="texhtml"><b>v</b> = <i>d</i><b>r</b>/<i>dt</i></span>, then the <a href="/wiki/Gradient_theorem" title="Gradient theorem">gradient theorem</a> 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 {\begin{aligned}\int _{\gamma }\mathbf {F} \cdot d\mathbf {r} &=\int _{a}^{b}\mathbf {F} \cdot \mathbf {v} \,dt,\\&=-\int _{a}^{b}{\frac {d}{dt}}U(\mathbf {r} (t))\,dt=U(\mathbf {x} _{A})-U(\mathbf {x} _{B}).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</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">r</mi> </mrow> </mtd> <mtd> <mi></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> <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> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>U</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>U</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{\gamma }\mathbf {F} \cdot d\mathbf {r} &=\int _{a}^{b}\mathbf {F} \cdot \mathbf {v} \,dt,\\&=-\int _{a}^{b}{\frac {d}{dt}}U(\mathbf {r} (t))\,dt=U(\mathbf {x} _{A})-U(\mathbf {x} _{B}).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/995577cd127c003fdf63e5d48e2e1994c208570e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:50.531ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\int _{\gamma }\mathbf {F} \cdot d\mathbf {r} &=\int _{a}^{b}\mathbf {F} \cdot \mathbf {v} \,dt,\\&=-\int _{a}^{b}{\frac {d}{dt}}U(\mathbf {r} (t))\,dt=U(\mathbf {x} _{A})-U(\mathbf {x} _{B}).\end{aligned}}}"></span> </p><p>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 <b>v</b> of the point of application, 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 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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> <mi>U</mi> </mrow> <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/c83c632670f20b5c035568a8d32c5eafb4d24cbf" 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} .}"></span> </p><p>Examples of work that can be computed from potential functions are gravity and spring forces.<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-heading2"><h2 id="Potential_energy_for_near-Earth_gravity">Potential energy for near-Earth gravity</h2></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Trebuchet.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Trebuchet.jpg/220px-Trebuchet.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Trebuchet.jpg/330px-Trebuchet.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Trebuchet.jpg/440px-Trebuchet.jpg 2x" data-file-width="600" data-file-height="450" /></a><figcaption>A <a href="/wiki/Trebuchet" title="Trebuchet">trebuchet</a> uses the gravitational potential energy of the <a href="/wiki/Counterweight" title="Counterweight">counterweight</a> to throw projectiles over two hundred meters</figcaption></figure> <p>For small height changes, gravitational potential energy can be computed using <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_{g}=mgh,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mi>g</mi> <mi>h</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{g}=mgh,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87be837da2ee5ccfbff2299c4825d2c25e1a0513" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.85ex; height:2.843ex;" alt="{\displaystyle U_{g}=mgh,}"></span> where <i>m</i> is the mass in kilograms, <i>g</i> is the local gravitational field (9.8 metres per second squared on Earth), <i>h</i> is the height above a reference level in metres, and <i>U</i> is the energy in joules. </p><p>In classical physics, gravity exerts a constant downward force <span class="texhtml"><i><b>F</b></i> = (0, 0, <i>F<sub>z</sub></i>)</span> on the center of mass of a body moving near the surface of the Earth. The work of gravity on a body moving along a trajectory <span class="texhtml"><i><b>r</b></i>(<i>t</i>) = (<i>x</i>(<i>t</i>), <i>y</i>(<i>t</i>), <i>z</i>(<i>t</i>))</span>, such as the track of a roller coaster is calculated using its velocity, <span class="texhtml"><i><b>v</b></i> = (<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 _{t_{1}}^{t_{2}}{\boldsymbol {F}}\cdot {\boldsymbol {v}}\,dt=\int _{t_{1}}^{t_{2}}F_{z}v_{z}\,dt=F_{z}\Delta z.}"> <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-italic">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">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> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{t_{1}}^{t_{2}}{\boldsymbol {F}}\cdot {\boldsymbol {v}}\,dt=\int _{t_{1}}^{t_{2}}F_{z}v_{z}\,dt=F_{z}\Delta z.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/605a8a5a496b4b1086efc9d30a12c4436f637300" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.303ex; height:6.509ex;" alt="{\displaystyle W=\int _{t_{1}}^{t_{2}}{\boldsymbol {F}}\cdot {\boldsymbol {v}}\,dt=\int _{t_{1}}^{t_{2}}F_{z}v_{z}\,dt=F_{z}\Delta z.}"></span> where the integral of the vertical component of velocity is the vertical distance. The work of gravity depends only on the vertical movement of the curve <span class="texhtml"><i><b>r</b></i>(<i>t</i>)</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Potential_energy_for_a_linear_spring">Potential energy for a linear spring</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Elastic_potential_energy" class="mw-redirect" title="Elastic potential energy">Elastic potential energy</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Springs_009.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Springs_009.jpg/220px-Springs_009.jpg" decoding="async" width="220" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Springs_009.jpg/330px-Springs_009.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Springs_009.jpg/440px-Springs_009.jpg 2x" data-file-width="1836" data-file-height="1252" /></a><figcaption><a href="/wiki/Spring_(device)" title="Spring (device)">Springs</a> are used for storing <a href="/wiki/Elastic_potential_energy" class="mw-redirect" title="Elastic potential energy">elastic potential energy</a></figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Longbowmen.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Longbowmen.jpg/220px-Longbowmen.jpg" decoding="async" width="220" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Longbowmen.jpg/330px-Longbowmen.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Longbowmen.jpg/440px-Longbowmen.jpg 2x" data-file-width="1123" data-file-height="763" /></a><figcaption><a href="/wiki/Archery" title="Archery">Archery</a> is one of humankind's oldest applications of elastic potential energy</figcaption></figure> <p>A horizontal spring exerts a force <span class="texhtml"><b>F</b> = (−<i>kx</i>, 0, 0)</span> that is proportional to its deformation in the axial or <i>x</i> direction. The work of this spring on a body moving along the space curve <span class="texhtml"><b>s</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=-\int _{0}^{t}kx{\frac {dx}{dt}}dt=\int _{x(t_{0})}^{x(t)}kx\,dx={\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> <mspace width="thinmathspace" /> <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> <mspace width="thinmathspace" /> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>k</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{0}^{t}\mathbf {F} \cdot \mathbf {v} \,dt=-\int _{0}^{t}kxv_{x}\,dt=-\int _{0}^{t}kx{\frac {dx}{dt}}dt=\int _{x(t_{0})}^{x(t)}kx\,dx={\frac {1}{2}}kx^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf131cd5e4a9704538ac9f89ef6da4bcbbfa3c0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:72.791ex; height:6.676ex;" alt="{\displaystyle W=\int _{0}^{t}\mathbf {F} \cdot \mathbf {v} \,dt=-\int _{0}^{t}kxv_{x}\,dt=-\int _{0}^{t}kx{\frac {dx}{dt}}dt=\int _{x(t_{0})}^{x(t)}kx\,dx={\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 <i>x</i> and the <i>x</i>-velocity, <i>xv<sub>x</sub></i>, is <i>x</i><sup>2</sup>/2. </p><p>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(x)={\frac {1}{2}}kx^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</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 U(x)={\frac {1}{2}}kx^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c7e243bdec3b25a115573033e700cdc4cb01505" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.261ex; height:5.176ex;" alt="{\displaystyle U(x)={\frac {1}{2}}kx^{2},}"></span> is called the potential energy of a linear spring. </p><p>Elastic potential energy is the potential energy of an <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> object (for example a <a href="/wiki/Bow_(weapon)" class="mw-redirect" title="Bow (weapon)">bow</a> or a catapult) that is deformed under tension or compression (or <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stressed</a> in formal terminology). It arises as a consequence of a force that tries to restore the object to its original shape, which is most often the <a href="/wiki/Electromagnetic_force" class="mw-redirect" title="Electromagnetic force">electromagnetic force</a> between the atoms and molecules that constitute the object. If the stretch is released, the energy is transformed into <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Potential_energy_for_gravitational_forces_between_two_bodies">Potential energy for gravitational forces between two bodies</h2></div> <p>The gravitational potential function, also known as <a href="/wiki/Gravitational_potential_energy" class="mw-redirect" title="Gravitational potential energy">gravitational potential energy</a>, 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 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> </p><p>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="Derivation">Derivation</h3></div> <p>The gravitational force between two bodies of mass <i>M</i> and <i>m</i> separated by a distance <i>r</i> is given by <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</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 \mathbf {F} =-{\frac {GMm}{r^{2}}}\mathbf {\hat {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> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =-{\frac {GMm}{r^{2}}}\mathbf {\hat {r}} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/563afb133555aa37bf9874e83826101157b49eb4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.718ex; height:5.676ex;" alt="{\displaystyle \mathbf {F} =-{\frac {GMm}{r^{2}}}\mathbf {\hat {r}} ,}"></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 \mathbf {\hat {r}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {r}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe52dfe80c9a6604b3a46b24d65eb02c92c59e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {r}} }"></span> is <a href="/wiki/Unit_vector" title="Unit vector">a vector of length 1</a> pointing from <i>M</i> to <i>m</i> and <i>G</i> is the <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a>. </p><p>Let the mass <i>m</i> 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> The position and velocity of the mass <i>m</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 {r} =r\mathbf {e} _{r},\qquad \mathbf {v} ={\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"> <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} ={\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/5da8f86b161aec39cba82f8a57d47043a1f3b44e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.07ex; height:3.176ex;" alt="{\displaystyle \mathbf {r} =r\mathbf {e} _{r},\qquad \mathbf {v} ={\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t},}"></span> where <b>e</b><sub><i>r</i></sub> and <b>e</b><sub><i>t</i></sub> are the radial and tangential unit vectors directed relative to the vector from <i>M</i> to <i>m</i>. 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 ({\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t})\,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> <mo stretchy="false">(</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 stretchy="false">)</mo> <mspace width="thinmathspace" /> <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 ({\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t})\,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/9fd4ea9ea12925ef1d06407b6a326addf2cfe17e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:83.378ex; height:6.509ex;" alt="{\displaystyle W=-\int _{t_{1}}^{t_{2}}{\frac {GmM}{r^{3}}}(r\mathbf {e} _{r})\cdot ({\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t})\,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> </p><p>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> </p> <div class="mw-heading mw-heading2"><h2 id="Potential_energy_for_electrostatic_forces_between_two_bodies">Potential energy for electrostatic forces between two bodies</h2></div> <p>The electrostatic force exerted by a charge <i>Q</i> on another charge <i>q</i> separated by a distance <i>r</i> is given by <a href="/wiki/Coulomb%27s_Law" class="mw-redirect" title="Coulomb's Law">Coulomb's Law</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 \mathbf {F} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\hat {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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Q</mi> <mi>q</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> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\hat {r}} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/515c327baabebc2e5258e21e544a5e0f53075c84" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.977ex; height:5.843ex;" alt="{\displaystyle \mathbf {F} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\hat {r}} ,}"></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 \mathbf {\hat {r}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {r}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe52dfe80c9a6604b3a46b24d65eb02c92c59e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {r}} }"></span> is a vector of length 1 pointing from <i>Q</i> to <i>q</i> and <i>ε</i><sub>0</sub> is the <a href="/wiki/Vacuum_permittivity" title="Vacuum permittivity">vacuum permittivity</a>. </p><p>The work <i>W</i> required to move <i>q</i> from <i>A</i> to any point <i>B</i> in the electrostatic force field is given by the potential 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(r)={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Q</mi> <mi>q</mi> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(r)={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b43fe12913f8378546fff1563569d615edf263c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.598ex; height:5.843ex;" alt="{\displaystyle U(r)={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r}}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Reference_level">Reference level</h2></div> <p>The potential energy is a function of the state a system is in, and is defined relative to that for a particular state. This reference state is not always a real state; it may also be a limit, such as with the distances between all bodies tending to infinity, provided that the energy involved in tending to that limit is finite, such as in the case of <a href="/wiki/Inverse-square_law" title="Inverse-square law">inverse-square law</a> forces. Any arbitrary reference state could be used; therefore it can be chosen based on convenience. </p><p>Typically the potential energy of a system depends on the <i>relative</i> positions of its components only, so the reference state can also be expressed in terms of relative positions. </p> <div class="mw-heading mw-heading2"><h2 id="Gravitational_potential_energy">Gravitational potential energy</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Gravitational_potential" title="Gravitational potential">Gravitational potential</a>, <a href="/wiki/Gravitational_energy" title="Gravitational energy">Gravitational energy</a>, and <a href="/wiki/Gravity_field" class="mw-redirect" title="Gravity field">Gravity field</a></div> <p>Gravitational energy is the potential energy associated with <a href="/wiki/Gravitational_force" class="mw-redirect" title="Gravitational force">gravitational force</a>, as work is required to elevate objects against Earth's gravity. The potential energy due to elevated positions is called gravitational potential energy, and is evidenced by water in an elevated reservoir or kept behind a dam. If an object falls from one point to another point inside a gravitational field, the force of gravity will do positive work on the object, and the gravitational potential energy will decrease by the same amount. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Solar_sys.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Solar_sys.jpg/220px-Solar_sys.jpg" decoding="async" width="220" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Solar_sys.jpg/330px-Solar_sys.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Solar_sys.jpg/440px-Solar_sys.jpg 2x" data-file-width="1440" data-file-height="904" /></a><figcaption>Gravitational force keeps the planets in orbit around the <a href="/wiki/Sun" title="Sun">Sun</a></figcaption></figure> <p>Consider a book placed on top of a table. As the book is raised from the floor to the table, some external force works against the gravitational force. If the book falls back to the floor, the "falling" energy the book receives is provided by the gravitational force. Thus, if the book falls off the table, this potential energy goes to accelerate the mass of the book and is converted into <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a>. When the book hits the floor this kinetic energy is converted into heat, deformation, and sound by the impact. </p><p>The factors that affect an object's gravitational potential energy are its height relative to some reference point, its mass, and the strength of the gravitational field it is in. Thus, a book lying on a table has less gravitational potential energy than the same book on top of a taller cupboard and less gravitational potential energy than a heavier book lying on the same table. An object at a certain height above the Moon's surface has less gravitational potential energy than at the same height above the Earth's surface because the Moon's gravity is weaker. "Height" in the common sense of the term cannot be used for gravitational potential energy calculations when gravity is not assumed to be a constant. The following sections provide more detail. </p> <div class="mw-heading mw-heading3"><h3 id="Local_approximation">Local approximation</h3></div> <p>The strength of a gravitational field varies with location. However, when the change of distance is small in relation to the distances from the center of the source of the gravitational field, this variation in field strength is negligible and we can assume that the force of gravity on a particular object is constant. Near the surface of the Earth, for example, we assume that the acceleration due to gravity is a constant <span class="texhtml"><i>g</i> = 9.8 m/s<sup>2</sup></span> (<a href="/wiki/Standard_gravity" title="Standard gravity">standard gravity</a>). In this case, a simple expression for gravitational potential energy can be derived using the <span class="texhtml"><i>W</i> = <i>Fd</i></span> equation for <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">work</a>, and 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 W_{F}=-\Delta U_{F}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{F}=-\Delta U_{F}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d61a32277976422a7ef51af2a7501d52743bdc27" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.197ex; height:2.509ex;" alt="{\displaystyle W_{F}=-\Delta U_{F}.}"></span> </p><p>The amount of gravitational potential energy held by an elevated object is equal to the work done against gravity in lifting it. The work done equals the force required to move it upward multiplied with the vertical distance it is moved (remember <span class="texhtml"><i>W</i> = <i>Fd</i></span>). The upward force required while moving at a constant velocity is equal to the weight, <span class="texhtml"><i>mg</i></span>, of an object, so the work done in lifting it through a height <span class="texhtml mvar" style="font-style:italic;">h</span> is the product <span class="texhtml"><i>mgh</i></span>. Thus, when accounting only for <a href="/wiki/Mass" title="Mass">mass</a>, <a href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravity</a>, and <a href="/wiki/Altitude" title="Altitude">altitude</a>, the equation is:<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> <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=mgh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>m</mi> <mi>g</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=mgh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2630448f947077ab71ede5b40839f03d3d906272" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.376ex; height:2.509ex;" alt="{\displaystyle U=mgh}"></span> where <span class="texhtml"><i>U</i></span> is the potential energy of the object relative to its being on the Earth's surface, <span class="texhtml"><i>m</i></span> is the mass of the object, <span class="texhtml"><i>g</i></span> is the acceleration due to gravity, and <i>h</i> is the altitude of the object.<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> </p><p>Hence, the potential difference 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 \Delta U=mg\Delta h.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo>=</mo> <mi>m</mi> <mi>g</mi> <mi mathvariant="normal">Δ</mi> <mi>h</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U=mg\Delta h.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f7c0c78b3e299137246d874be98dac6f2fc572" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.895ex; height:2.509ex;" alt="{\displaystyle \Delta U=mg\Delta h.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="General_formula">General formula</h3></div> <p>However, over large variations in distance, the approximation that <span class="texhtml"><i>g</i></span> is constant is no longer valid, and we have to use <a href="/wiki/Calculus" title="Calculus">calculus</a> and the general mathematical definition of work to determine gravitational potential energy. For the computation of the potential energy, we can <a href="/wiki/Integral" title="Integral">integrate</a> the gravitational force, whose magnitude is given by <a href="/wiki/Law_of_universal_gravitation" class="mw-redirect" title="Law of universal gravitation">Newton's law of gravitation</a>, with respect to the distance <span class="texhtml"><i>r</i></span> between the two bodies. Using that definition, the gravitational potential energy of a system of masses <span class="texhtml"><i>m</i><sub>1</sub></span> and <span class="texhtml"><i>M</i><sub>2</sub></span> at a distance <span class="texhtml"><i>r</i></span> using the <a href="/wiki/Newtonian_constant_of_gravitation" class="mw-redirect" title="Newtonian constant of gravitation">Newtonian constant of gravitation</a> <span class="texhtml"><i>G</i></span> is </p><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 U=-G{\frac {m_{1}M_{2}}{r}}+K,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>−</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>+</mo> <mi>K</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=-G{\frac {m_{1}M_{2}}{r}}+K,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0141c6cbb8bdcbd17613e5e8b370aeb77a0dd0da" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.308ex; height:5.176ex;" alt="{\displaystyle U=-G{\frac {m_{1}M_{2}}{r}}+K,}"></span> </p><p>where <span class="texhtml"><i>K</i></span> is an arbitrary constant dependent on the choice of datum from which potential is measured. Choosing the convention that <span class="texhtml"><i>K</i> = 0</span> (i.e. in relation to a point at infinity) makes calculations simpler, albeit at the cost of making <span class="texhtml"><i>U</i></span> negative; for why this is physically reasonable, see below. </p><p>Given this formula for <span class="texhtml"><i>U</i></span>, the total potential energy of a system of <span class="texhtml mvar" style="font-style:italic;">n</span> bodies is found by summing, for all <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 {\frac {n(n-1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {n(n-1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daf76f7f60595b709cbca7bc7e45ca2d7d3e8785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.188ex; height:4.176ex;" alt="{\textstyle {\frac {n(n-1)}{2}}}"></span> pairs of two bodies, the potential energy of the system of those two bodies. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gravitational_potential_summation_2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Gravitational_potential_summation_2.png/220px-Gravitational_potential_summation_2.png" decoding="async" width="220" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Gravitational_potential_summation_2.png/330px-Gravitational_potential_summation_2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Gravitational_potential_summation_2.png/440px-Gravitational_potential_summation_2.png 2x" data-file-width="592" data-file-height="448" /></a><figcaption>Gravitational potential summation <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 U=-m\left(G{\frac {M_{1}}{r_{1}}}+G{\frac {M_{2}}{r_{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>−</mo> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=-m\left(G{\frac {M_{1}}{r_{1}}}+G{\frac {M_{2}}{r_{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8425aefd5785206e19b574034ece0eee18084f0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.321ex; height:6.176ex;" alt="{\displaystyle U=-m\left(G{\frac {M_{1}}{r_{1}}}+G{\frac {M_{2}}{r_{2}}}\right)}"></span></figcaption></figure><p>Considering the system of bodies as the combined set of small particles the bodies consist of, and applying the previous on the particle level we get the negative <a href="/wiki/Gravitational_binding_energy" title="Gravitational binding energy">gravitational binding energy</a>. This potential energy is more strongly negative than the total potential energy of the system of bodies as such since it also includes the negative gravitational binding energy of each body. The potential energy of the system of bodies as such is the negative of the energy needed to separate the bodies from each other to infinity, while the gravitational binding energy is the energy needed to separate all particles from each other to infinity. </p><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 U=-m\left(G{\frac {M_{1}}{r_{1}}}+G{\frac {M_{2}}{r_{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>−</mo> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=-m\left(G{\frac {M_{1}}{r_{1}}}+G{\frac {M_{2}}{r_{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8425aefd5785206e19b574034ece0eee18084f0f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.321ex; height:6.176ex;" alt="{\displaystyle U=-m\left(G{\frac {M_{1}}{r_{1}}}+G{\frac {M_{2}}{r_{2}}}\right)}"></span> 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 U=-m\sum G{\frac {M}{r}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>−</mo> <mi>m</mi> <mo>∑</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <mi>r</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=-m\sum G{\frac {M}{r}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29eafaf6d8c11ea69b1472c0f3a94d7dcff2cc50" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.611ex; height:5.176ex;" alt="{\displaystyle U=-m\sum G{\frac {M}{r}},}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Negative_gravitational_energy">Negative gravitational energy</h3></div> <p>As with all potential energies, only differences in gravitational potential energy matter for most physical purposes, and the choice of zero point is arbitrary. Given that there is no reasonable criterion for preferring one particular finite <i>r</i> over another, there seem to be only two reasonable choices for the distance at which <span class="texhtml"><i>U</i></span> becomes zero: <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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894a83e863728b4ee2e12f3a999a09f5f2bf1c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=0}"></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 r=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d38c8164ddb69351cdab28da290255fde3b846d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.471ex; height:1.676ex;" alt="{\displaystyle r=\infty }"></span>. The choice of <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 U=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d868ee7902023d29169252eb059f8faff9f08fc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.044ex; height:2.176ex;" alt="{\displaystyle U=0}"></span> at infinity may seem peculiar, and the consequence that gravitational energy is always negative may seem counterintuitive, but this choice allows gravitational potential energy values to be finite, albeit negative. </p><p>The <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularity</a> at <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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894a83e863728b4ee2e12f3a999a09f5f2bf1c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=0}"></span> in the formula for gravitational potential energy means that the only other apparently reasonable alternative choice of convention, with <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 U=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d868ee7902023d29169252eb059f8faff9f08fc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.044ex; height:2.176ex;" alt="{\displaystyle U=0}"></span> for <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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894a83e863728b4ee2e12f3a999a09f5f2bf1c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=0}"></span>, would result in potential energy being positive, but infinitely large for all nonzero values of <span class="texhtml"><i>r</i></span>, and would make calculations involving sums or differences of potential energies beyond what is possible with the <a href="/wiki/Real_number" title="Real number">real number</a> system. Since physicists abhor infinities in their calculations, and <span class="texhtml"><i>r</i></span> is always non-zero in practice, the choice of <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 U=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d868ee7902023d29169252eb059f8faff9f08fc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.044ex; height:2.176ex;" alt="{\displaystyle U=0}"></span> at infinity is by far the more preferable choice, even if the idea of negative energy in a <a href="/wiki/Gravity_well" class="mw-redirect" title="Gravity well">gravity well</a> appears to be peculiar at first. </p><p>The negative value for gravitational energy also has deeper implications that make it seem more reasonable in cosmological calculations where the total energy of the universe can meaningfully be considered; see <a href="/wiki/Inflation_theory" class="mw-redirect" title="Inflation theory">inflation theory</a> for more on this.<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-heading3"><h3 id="Uses">Uses</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Gravitational_potential_energy_storage" class="mw-redirect" title="Gravitational potential energy storage">Gravitational potential energy storage</a></div> <p>Gravitational potential energy has a number of practical uses, notably the generation of <a href="/wiki/Pumped-storage_hydroelectricity" title="Pumped-storage hydroelectricity">pumped-storage hydroelectricity</a>. For example, in <a href="/wiki/Dinorwig_Power_Station" title="Dinorwig Power Station">Dinorwig</a>, Wales, there are two lakes, one at a higher elevation than the other. At times when surplus electricity is not required (and so is comparatively cheap), water is pumped up to the higher lake, thus converting the electrical energy (running the pump) to gravitational potential energy. At times of peak demand for electricity, the water flows back down through electrical generator turbines, converting the potential energy into kinetic energy and then back into electricity. The process is not completely efficient and some of the original energy from the surplus electricity is in fact lost to friction.<sup id="cite_ref-EconomistPSH_12-0" class="reference"><a href="#cite_note-EconomistPSH-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-thier_13-0" class="reference"><a href="#cite_note-thier-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Levine_14-0" class="reference"><a href="#cite_note-Levine-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-yang_15-0" class="reference"><a href="#cite_note-yang-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-heco_16-0" class="reference"><a href="#cite_note-heco-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p> Gravitational potential energy is also used to power clocks in which falling weights operate the mechanism.</p><div style="clear:right;" class=""></div><p> It is also used by <a href="/wiki/Counterweight" title="Counterweight">counterweights</a> for lifting up an <a href="/wiki/Elevator" title="Elevator">elevator</a>, crane, or <a href="/wiki/Sash_window" title="Sash window">sash window</a>. </p><p><a href="/wiki/Rollercoasters" class="mw-redirect" title="Rollercoasters">Roller coasters</a> are an entertaining way to utilize potential energy – chains are used to move a car up an incline (building up gravitational potential energy), to then have that energy converted into kinetic energy as it falls. </p><p>Another practical use is utilizing gravitational potential energy to descend (perhaps coast) downhill in transportation such as the descent of an automobile, truck, railroad train, bicycle, airplane, or fluid in a pipeline. In some cases the <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> obtained from the potential energy of descent may be used to start ascending the next grade such as what happens when a road is undulating and has frequent dips. The commercialization of stored energy (in the form of rail cars raised to higher elevations) that is then converted to electrical energy when needed by an electrical grid, is being undertaken in the United States in a system called <a href="/wiki/List_of_energy_storage_projects#Gravitational_potential_energy_storage" class="mw-redirect" title="List of energy storage projects">Advanced Rail Energy Storage</a> (ARES).<sup id="cite_ref-Economist-2012.03.03_17-0" class="reference"><a href="#cite_note-Economist-2012.03.03-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Bloomberg-2012.09.06_18-0" class="reference"><a href="#cite_note-Bloomberg-2012.09.06-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Chemical_potential_energy">Chemical potential energy</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Chemical_energy" title="Chemical energy">Chemical energy</a></div> <p>Chemical potential energy is a form of potential energy related to the structural arrangement of atoms or molecules. This arrangement may be the result of <a href="/wiki/Chemical_bond" title="Chemical bond">chemical bonds</a> within a molecule or otherwise. Chemical energy of a chemical substance can be transformed to other forms of energy by a <a href="/wiki/Chemical_reaction" title="Chemical reaction">chemical reaction</a>. As an example, when a fuel is burned the chemical energy is converted to heat, same is the case with digestion of food metabolized in a biological organism. Green plants transform <a href="/wiki/Solar_energy" title="Solar energy">solar energy</a> to chemical energy through the process known as <a href="/wiki/Photosynthesis" title="Photosynthesis">photosynthesis</a>, and electrical energy can be converted to chemical energy through <a href="/wiki/Electrochemical" class="mw-redirect" title="Electrochemical">electrochemical</a> reactions. </p><p>The similar term <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a> is used to indicate the potential of a substance to undergo a change of configuration, be it in the form of a chemical reaction, spatial transport, particle exchange with a reservoir, etc. </p> <div class="mw-heading mw-heading2"><h2 id="Electric_potential_energy">Electric potential energy</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Electric_potential_energy" title="Electric potential energy">Electric potential energy</a></div> <p>An object can have potential energy by virtue of its <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> and several forces related to their presence. There are two main types of this kind of potential energy: electrostatic potential energy, electrodynamic potential energy (also sometimes called magnetic potential energy). </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Plasma-lamp_2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Plasma-lamp_2.jpg/220px-Plasma-lamp_2.jpg" decoding="async" width="220" height="223" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Plasma-lamp_2.jpg/330px-Plasma-lamp_2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Plasma-lamp_2.jpg/440px-Plasma-lamp_2.jpg 2x" data-file-width="1589" data-file-height="1609" /></a><figcaption><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a> formed inside a gas filled sphere</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Electrostatic_potential_energy">Electrostatic potential energy</h3></div> <p>Electrostatic potential energy between two bodies in space is obtained from the force exerted by a charge <i>Q</i> on another charge <i>q</i> which 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} _{e}=-{\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\hat {r}} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Q</mi> <mi>q</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> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{e}=-{\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\hat {r}} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76f6140a681ec789469a10c10819183ae3fd8c3c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.784ex; height:5.843ex;" alt="{\displaystyle \mathbf {F} _{e}=-{\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\hat {r}} ,}"></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 \mathbf {\hat {r}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {r}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe52dfe80c9a6604b3a46b24d65eb02c92c59e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {r}} }"></span> is a vector of length 1 pointing from <i>Q</i> to <i>q</i> and <i>ε</i><sub>0</sub> is the <a href="/wiki/Vacuum_permittivity" title="Vacuum permittivity">vacuum permittivity</a>. </p><p>If the electric charge of an object can be assumed to be at rest, then it has potential energy due to its position relative to other charged objects. The <a href="/wiki/Electric_potential_energy" title="Electric potential energy">electrostatic potential energy</a> is the energy of an electrically charged particle (at rest) in an electric field. It is defined as the <a href="/wiki/Work_(physics)" title="Work (physics)">work</a> that must be done to move it from an infinite distance away to its present location, adjusted for non-electrical forces on the object. This energy will generally be non-zero if there is another electrically charged object nearby. </p><p>The work <i>W</i> required to move <i>q</i> from <i>A</i> to any point <i>B</i> in the electrostatic force field 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 \Delta U_{AB}({\mathbf {r} })=-\int _{A}^{B}\mathbf {F_{e}} \cdot d\mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> </msub> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta U_{AB}({\mathbf {r} })=-\int _{A}^{B}\mathbf {F_{e}} \cdot d\mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee90d4592943c2af5e32481b0a226946672ffc27" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.55ex; height:6.176ex;" alt="{\displaystyle \Delta U_{AB}({\mathbf {r} })=-\int _{A}^{B}\mathbf {F_{e}} \cdot d\mathbf {r} }"></span> typically given in J for Joules. A related quantity called <i><a href="/wiki/Electric_potential" title="Electric potential">electric potential</a></i> (commonly denoted with a <i>V</i> for voltage) is equal to the electric potential energy per unit charge. </p> <div class="mw-heading mw-heading3"><h3 id="Magnetic_potential_energy">Magnetic potential energy</h3></div> <p>The energy of a <a href="/wiki/Magnetic_moment" title="Magnetic moment">magnetic moment</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\mu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\mu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1aee7d7b4a36d96dfb35bfee9c7751bba1fdfbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.646ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\mu }}}"></span> in an externally produced <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic B-field</a> <span class="texhtml"><b>B</b></span> has potential energy<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> <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=-{\boldsymbol {\mu }}\cdot \mathbf {B} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=-{\boldsymbol {\mu }}\cdot \mathbf {B} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f207b3923b54bab1f029d006f1d8ace1d52f3f1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.562ex; height:2.509ex;" alt="{\displaystyle U=-{\boldsymbol {\mu }}\cdot \mathbf {B} .}"></span> </p><p>The <a href="/wiki/Magnetization" title="Magnetization">magnetization</a> <span class="texhtml"><b>M</b></span> in a field 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 U=-{\frac {1}{2}}\int \mathbf {M} \cdot \mathbf {B} \,dV,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=-{\frac {1}{2}}\int \mathbf {M} \cdot \mathbf {B} \,dV,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb25c09400910ac33b1b363bd2d767b9d62b5ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.81ex; height:5.676ex;" alt="{\displaystyle U=-{\frac {1}{2}}\int \mathbf {M} \cdot \mathbf {B} \,dV,}"></span> where the integral can be over all space or, equivalently, where <span class="texhtml"><b>M</b></span> is nonzero.<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> Magnetic potential energy is the form of energy related not only to the distance between magnetic materials, but also to the orientation, or alignment, of those materials within the field. For example, the needle of a compass has the lowest magnetic potential energy when it is aligned with the north and south poles of the Earth's magnetic field. If the needle is moved by an outside force, torque is exerted on the magnetic dipole of the needle by the Earth's magnetic field, causing it to move back into alignment. The magnetic potential energy of the needle is highest when its field is in the same direction as the Earth's magnetic field. Two magnets will have potential energy in relation to each other and the distance between them, but this also depends on their orientation. If the opposite poles are held apart, the potential energy will be higher the further they are apart and lower the closer they are. Conversely, like poles will have the highest potential energy when forced together, and the lowest when they spring apart.<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><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> </p> <div class="mw-heading mw-heading2"><h2 id="Nuclear_potential_energy">Nuclear potential energy</h2></div> <p>Nuclear potential energy is the potential energy of the <a href="/wiki/Subatomic_particle" title="Subatomic particle">particles</a> inside an <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">atomic nucleus</a>. The nuclear particles are bound together by the <a href="/wiki/Strong_nuclear_force" class="mw-redirect" title="Strong nuclear force">strong nuclear force</a>. <a href="/wiki/Weak_nuclear_force" class="mw-redirect" title="Weak nuclear force">Weak nuclear forces</a> provide the potential energy for certain kinds of radioactive decay, such as <a href="/wiki/Beta_decay" title="Beta decay">beta decay</a>. </p><p>Nuclear particles like protons and neutrons are not destroyed in fission and fusion processes, but collections of them can have less mass than if they were individually free, in which case this mass difference can be liberated as heat and radiation in nuclear reactions (the heat and radiation have the missing mass, but it often escapes from the system, where it is not measured). The energy from the <a href="/wiki/Sun" title="Sun">Sun</a> is an example of this form of energy conversion. In the Sun, the process of hydrogen fusion converts about 4 million tonnes of solar matter per second into <a href="/wiki/Electromagnetic_energy" class="mw-redirect" title="Electromagnetic energy">electromagnetic energy</a>, which is radiated into space. </p> <div class="mw-heading mw-heading2"><h2 id="Forces_and_potential_energy">Forces and potential energy</h2></div> <p>Potential energy is closely linked with <a href="/wiki/Force_(physics)" class="mw-redirect" title="Force (physics)">forces</a>. If the work done by a force on a body that moves from <i>A</i> to <i>B</i> does not depend on the path between these points, then the work of this force measured from <i>A</i> assigns a scalar value to every other point in space and defines a <a href="/wiki/Scalar_potential" title="Scalar potential">scalar potential</a> field. In this case, the force can be defined as the negative of the <a href="/wiki/Gradient" title="Gradient">vector gradient</a> of the potential field. </p><p>For example, gravity is a <a href="/wiki/Conservative_force" title="Conservative force">conservative force</a>. The associated potential is the <a href="/wiki/Gravitational_potential" title="Gravitational potential">gravitational potential</a>, often denoted by <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> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, corresponding to the energy per unit mass as a function of position. The gravitational potential energy of two particles of mass <i>M</i> and <i>m</i> separated by a distance <i>r</i> 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 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/139f5ca37da97d23736e4844ab2857f10bb44c17" 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> The gravitational potential (<a href="/wiki/Specific_orbital_energy" title="Specific orbital energy">specific energy</a>) of the two bodies 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 \phi =-\left({\frac {GM}{r}}+{\frac {Gm}{r}}\right)=-{\frac {G(M+m)}{r}}=-{\frac {GMm}{\mu r}}={\frac {U}{\mu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>m</mi> </mrow> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo>+</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> <mi>m</mi> </mrow> <mrow> <mi>μ</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>U</mi> <mi>μ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =-\left({\frac {GM}{r}}+{\frac {Gm}{r}}\right)=-{\frac {G(M+m)}{r}}=-{\frac {GMm}{\mu r}}={\frac {U}{\mu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b13fa222d176ada345d04dd8887e185876b9756e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.22ex; height:6.343ex;" alt="{\displaystyle \phi =-\left({\frac {GM}{r}}+{\frac {Gm}{r}}\right)=-{\frac {G(M+m)}{r}}=-{\frac {GMm}{\mu r}}={\frac {U}{\mu }}}"></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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is the <a href="/wiki/Reduced_mass" title="Reduced mass">reduced mass</a>. </p><p>The work done against gravity by moving an <a href="/wiki/Test_particle" title="Test particle">infinitesimal mass</a> from point A with <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 U=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25f7c64dfadca83cc9f4b12a39401157d08c6676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.111ex; height:2.176ex;" alt="{\displaystyle U=a}"></span> to point B with <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 U=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90ee1a6dbef47df856564b9621590ec4865e3eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.879ex; height:2.176ex;" alt="{\displaystyle U=b}"></span> is <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 (b-a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b-a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/438f310cfd162b48bf8476ee318726770a92bce9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.877ex; height:2.843ex;" alt="{\displaystyle (b-a)}"></span> and the work done going back the other way is <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 (a-b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a-b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de142beb9731d08ae0cf1d37fbb1e7bc6ccd4dc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.877ex; height:2.843ex;" alt="{\displaystyle (a-b)}"></span> so that the total work done in moving from A to B and returning to A 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 U_{A\to B\to A}=(b-a)+(a-b)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{A\to B\to A}=(b-a)+(a-b)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fed6784345f5da7649dfc4173f0bfc55255e13b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.419ex; height:2.843ex;" alt="{\displaystyle U_{A\to B\to A}=(b-a)+(a-b)=0.}"></span> If the potential is redefined at A to be <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 a+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c25f3213d18c8ab37a1c4125462d7568047567d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.077ex; height:2.176ex;" alt="{\displaystyle a+c}"></span> and the potential at B to be <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 b+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7db36971debb247704a03ec6c3832d8fecdb05b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.845ex; height:2.343ex;" alt="{\displaystyle b+c}"></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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> is a constant (i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> can be any number, positive or negative, but it must be the same at A as it is at B) then the work done going from A to B 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 U_{A\to B}=(b+c)-(a+c)=b-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{A\to B}=(b+c)-(a+c)=b-a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41fc6dc68501bb759f43d6762ab08906bfddf81b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.588ex; height:2.843ex;" alt="{\displaystyle U_{A\to B}=(b+c)-(a+c)=b-a}"></span> as before. </p><p>In practical terms, this means that one can set the zero of <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></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 \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> anywhere one likes. One may set it to be zero at the surface of the <a href="/wiki/Earth" title="Earth">Earth</a>, or may find it more convenient to set zero at infinity (as in the expressions given earlier in this section). </p><p>A conservative force can be expressed in the language of <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> as a <a href="/wiki/Closed_differential_form" class="mw-redirect" title="Closed differential form">closed form</a>. As <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> is <a href="/wiki/Contractible_space" title="Contractible space">contractible</a>, its <a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">de Rham cohomology</a> vanishes, so every closed form is also an <a href="/wiki/Exact_differential_form" class="mw-redirect" title="Exact differential form">exact form</a>, and can be expressed as the gradient of a scalar field. This gives a mathematical justification of the fact that all conservative forces are gradients of a potential field. </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></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"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</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="CITEREFJain,_Mahesh_C.2009" class="citation book cs1">Jain, Mahesh C. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DqZlU3RJTywC&pg=PA10">"Fundamental forces and laws: a brief review"</a>. <i>Textbook of Engineering Physics, Part 1</i>. PHI Learning Pvt. Ltd. p. 10. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-203-3862-3" title="Special:BookSources/978-81-203-3862-3"><bdi>978-81-203-3862-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Fundamental+forces+and+laws%3A+a+brief+review&rft.btitle=Textbook+of+Engineering+Physics%2C+Part+1&rft.pages=10&rft.pub=PHI+Learning+Pvt.+Ltd.&rft.date=2009&rft.isbn=978-81-203-3862-3&rft.au=Jain%2C+Mahesh+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDqZlU3RJTywC%26pg%3DPA10&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcCall,_Robert_P.2010" class="citation book cs1">McCall, Robert P. (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LSyC41h6CG8C&pg=PA74">"Energy, Work and Metabolism"</a>. <a rel="nofollow" class="external text" href="https://archive.org/details/physicsofhumanbo0000mcca/page/74"><i>Physics of the Human Body</i></a>. JHU Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/physicsofhumanbo0000mcca/page/74">74</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8018-9455-8" title="Special:BookSources/978-0-8018-9455-8"><bdi>978-0-8018-9455-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Energy%2C+Work+and+Metabolism&rft.btitle=Physics+of+the+Human+Body&rft.pages=74&rft.pub=JHU+Press&rft.date=2010&rft.isbn=978-0-8018-9455-8&rft.au=McCall%2C+Robert+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLSyC41h6CG8C%26pg%3DPA74&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" class="Z3988"></span></span> </li> <li id="cite_note-Rankin1853-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rankin1853_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rankin1853_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">William John Macquorn Rankine (1853) "On the general law of the transformation of energy", <i>Proceedings of the Philosophical Society of Glasgow</i>, vol. 3, no. 5, pages 276–280; reprinted in: <b>(1)</b> <i>Philosophical Magazine</i>, series 4, vol. 5, no. 30, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3Ov22-gFMnEC&pg=PA106">pp. 106–117</a> (February 1853); and <b>(2)</b> W. J. Millar, ed., <i>Miscellaneous Scientific Papers: by W. J. Macquorn Rankine</i>, ... (London, England: Charles Griffin and Co., 1881), part II, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-kRB9v6KRvsC&pg=PA203">pp. 203–208</a>.</span> </li> <li id="cite_note-Roche2003-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Roche2003_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Roche2003_4-1"><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="CITEREFRoche2003" class="citation journal cs1">Roche, John (1 March 2003). <a rel="nofollow" class="external text" href="https://iopscience.iop.org/article/10.1088/0143-0807/24/2/359">"What is potential energy?"</a>. <i>European Journal of Physics</i>. <b>24</b> (2): 185–196. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0143-0807%2F24%2F2%2F359">10.1088/0143-0807/24/2/359</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:250895349">250895349</a><span class="reference-accessdate">. Retrieved <span class="nowrap">15 February</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=European+Journal+of+Physics&rft.atitle=What+is+potential+energy%3F&rft.volume=24&rft.issue=2&rft.pages=185-196&rft.date=2003-03-01&rft_id=info%3Adoi%2F10.1088%2F0143-0807%2F24%2F2%2F359&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250895349%23id-name%3DS2CID&rft.aulast=Roche&rft.aufirst=John&rft_id=https%3A%2F%2Fiopscience.iop.org%2Farticle%2F10.1088%2F0143-0807%2F24%2F2%2F359&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" class="Z3988"></span></span> </li> <li id="cite_note-SmithEnergy-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-SmithEnergy_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-SmithEnergy_5-1"><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="CITEREFSmith1998" class="citation book cs1">Smith, Crosbie (1998). <i>The Science of Energy – a Cultural History of Energy Physics in Victorian Britain</i>. The University of Chicago Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-226-76420-6" title="Special:BookSources/0-226-76420-6"><bdi>0-226-76420-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Science+of+Energy+%E2%80%93+a+Cultural+History+of+Energy+Physics+in+Victorian+Britain&rft.pub=The+University+of+Chicago+Press&rft.date=1998&rft.isbn=0-226-76420-6&rft.aulast=Smith&rft.aufirst=Crosbie&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" class="Z3988"></span></span> </li> <li id="cite_note-:0-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown2006" class="citation book cs1">Brown, Theodore L. (2006). <a rel="nofollow" class="external text" href="https://archive.org/details/chemistry00theo_0/page/168"><i>Chemistry The Central Science</i></a>. Upper Saddle River, New Jersey: Pearson Education, Inc. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/chemistry00theo_0/page/168">168</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-109686-9" title="Special:BookSources/0-13-109686-9"><bdi>0-13-109686-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chemistry+The+Central+Science&rft.place=Upper+Saddle+River%2C+New+Jersey&rft.pages=168&rft.pub=Pearson+Education%2C+Inc.&rft.date=2006&rft.isbn=0-13-109686-9&rft.aulast=Brown&rft.aufirst=Theodore+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fchemistry00theo_0%2Fpage%2F168&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Robert_Taylor2005" class="citation book cs1">John Robert Taylor (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. p. 117. <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.pages=117&rft.pub=University+Science+Books&rft.date=2005&rft.isbn=978-1-891389-22-1&rft.au=John+Robert+Taylor&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DP1kCtNr-pJsC%26pg%3DPA117&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" 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="CITEREFBurton_Paul1979" class="citation book cs1">Burton Paul (1979). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3UdSAAAAMAAJ"><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.au=Burton+Paul&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3UdSAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" 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"><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_13.html">The Feynman Lectures on Physics Vol. I Ch. 13: Work and Potential Energy (A)</a></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 class="citation web cs1"><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/Hbase/gpot.html">"Hyperphysics – Gravitational Potential Energy"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Hyperphysics+%E2%80%93+Gravitational+Potential+Energy&rft_id=http%3A%2F%2Fhyperphysics.phy-astr.gsu.edu%2FHbase%2Fgpot.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" 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="CITEREFGuth,_Alan1997" class="citation book cs1">Guth, Alan (1997). "Appendix A, Gravitational Energy". <i>The Inflationary Universe</i>. Perseus Books. pp. 289–293. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-14942-7" title="Special:BookSources/0-201-14942-7"><bdi>0-201-14942-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Appendix+A%2C+Gravitational+Energy&rft.btitle=The+Inflationary+Universe&rft.pages=289-293&rft.pub=Perseus+Books&rft.date=1997&rft.isbn=0-201-14942-7&rft.au=Guth%2C+Alan&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" class="Z3988"></span></span> </li> <li id="cite_note-EconomistPSH-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-EconomistPSH_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="http://www.economist.com/node/21548495?frsc=dg%7Ca">"Energy storage – Packing some power"</a>. <i><a href="/wiki/The_Economist" title="The Economist">The Economist</a></i>. 3 March 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Economist&rft.atitle=Energy+storage+%E2%80%93+Packing+some+power&rft.date=2011-03-03&rft_id=http%3A%2F%2Fwww.economist.com%2Fnode%2F21548495%3Ffrsc%3Ddg%257Ca&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" class="Z3988"></span></span> </li> <li id="cite_note-thier-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-thier_13-0">^</a></b></span> <span class="reference-text">Jacob, Thierry.<a rel="nofollow" class="external text" href="http://www.stucky.ch/en/contenu/pdf/Pumped_storage_in_Switzerland_Dr_Jacob.pdf">Pumped storage in Switzerland – an outlook beyond 2000</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120317091142/http://www.stucky.ch/en/contenu/pdf/Pumped_storage_in_Switzerland_Dr_Jacob.pdf">Archived</a> 17 March 2012 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> <i>Stucky</i>. Accessed: 13 February 2012.</span> </li> <li id="cite_note-Levine-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Levine_14-0">^</a></b></span> <span class="reference-text">Levine, Jonah G. <a rel="nofollow" class="external text" href="http://www.colorado.edu/engineering/energystorage/files/MSThesis_JGLevine_final.pdf">Pumped Hydroelectric Energy Storage and Spatial Diversity of Wind Resources as Methods of Improving Utilization of Renewable Energy Sources</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140801113053/http://www.colorado.edu/engineering/energystorage/files/MSThesis_JGLevine_final.pdf">Archived</a> 1 August 2014 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> page 6, <i><a href="/wiki/University_of_Colorado" title="University of Colorado">University of Colorado</a></i>, December 2007. Accessed: 12 February 2012.</span> </li> <li id="cite_note-yang-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-yang_15-0">^</a></b></span> <span class="reference-text">Yang, Chi-Jen. <a rel="nofollow" class="external text" href="http://www.duke.edu/~cy42/PHS.pdf">Pumped Hydroelectric Storage</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120905193845/http://www.duke.edu/~cy42/PHS.pdf">Archived</a> 5 September 2012 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> <i><a href="/wiki/Duke_University" title="Duke University">Duke University</a></i>. Accessed: 12 February 2012.</span> </li> <li id="cite_note-heco-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-heco_16-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.heco.com/portal/site/heco/menuitem.508576f78baa14340b4c0610c510b1ca/?vgnextoid=94600420af0db110VgnVCM1000005c011bacRCRD&vgnextchannel=ab020420af0db110VgnVCM1000005c011bacRCRD&vgnextfmt=default&vgnextrefresh=1&level=0&ct=article">Energy Storage</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140407064054/http://www.heco.com/portal/site/heco/menuitem.508576f78baa14340b4c0610c510b1ca/?vgnextoid=94600420af0db110VgnVCM1000005c011bacRCRD&vgnextchannel=ab020420af0db110VgnVCM1000005c011bacRCRD&vgnextfmt=default&vgnextrefresh=1&level=0&ct=article">Archived</a> 7 April 2014 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> <i><a href="/wiki/Hawaiian_Electric_Company" class="mw-redirect" title="Hawaiian Electric Company">Hawaiian Electric Company</a></i>. Accessed: 13 February 2012.</span> </li> <li id="cite_note-Economist-2012.03.03-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-Economist-2012.03.03_17-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.economist.com/node/21548495">Packing Some Power: Energy Technology: Better ways of storing energy are needed if electricity systems are to become cleaner and more efficient</a>, <i><a href="/wiki/The_Economist" title="The Economist">The Economist</a></i>, 3 March 2012</span> </li> <li id="cite_note-Bloomberg-2012.09.06-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bloomberg-2012.09.06_18-0">^</a></b></span> <span class="reference-text">Downing, Louise. <a rel="nofollow" class="external text" href="https://www.bloomberg.com/news/print/2012-08-27/ski-lifts-help-open-25-billion-market-for-storing-power-energy.html">Ski Lifts Help Open $25 Billion Market for Storing Power</a>, <a href="/wiki/Bloomberg_News" title="Bloomberg News">Bloomberg News</a> online, 6 September 2012</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Kernan, Aedan. <a rel="nofollow" class="external text" href="http://www.leonardo-energy.org/storing-energy-rail-tracks">Storing Energy on Rail Tracks</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140412182442/http://www.leonardo-energy.org/storing-energy-rail-tracks">Archived</a> 12 April 2014 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Leonardo-Energy.org website, 30 October 2013</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAharoni1996" class="citation book cs1">Aharoni, Amikam (1996). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth00ahar"><i>Introduction to the theory of ferromagnetism</i></a> (Repr. ed.). Oxford: Clarendon Pr. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-851791-2" title="Special:BookSources/0-19-851791-2"><bdi>0-19-851791-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+theory+of+ferromagnetism&rft.place=Oxford&rft.edition=Repr.&rft.pub=Clarendon+Pr.&rft.date=1996&rft.isbn=0-19-851791-2&rft.aulast=Aharoni&rft.aufirst=Amikam&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoth00ahar&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" 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="CITEREFJackson1975" class="citation book cs1"><a href="/wiki/John_David_Jackson_(physicist)" title="John David Jackson (physicist)">Jackson, John David</a> (1975). <a rel="nofollow" class="external text" href="https://archive.org/details/classicalelectro00jack_0"><i>Classical electrodynamics</i></a> (2d ed.). New York: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-43132-X" title="Special:BookSources/0-471-43132-X"><bdi>0-471-43132-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=Classical+electrodynamics&rft.place=New+York&rft.edition=2d&rft.pub=Wiley&rft.date=1975&rft.isbn=0-471-43132-X&rft.aulast=Jackson&rft.aufirst=John+David&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fclassicalelectro00jack_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" 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="CITEREFLivingston2011" class="citation book cs1">Livingston, James D. (2011). <i>Rising Force: The Magic of Magnetic Levitation</i>. <a href="/wiki/President_and_Fellows_of_Harvard_College" title="President and Fellows of Harvard College">President and Fellows of Harvard College</a>. p. 152.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Rising+Force%3A+The+Magic+of+Magnetic+Levitation&rft.pages=152&rft.pub=President+and+Fellows+of+Harvard+College&rft.date=2011&rft.aulast=Livingston&rft.aufirst=James+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" 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="CITEREFKumar2004" class="citation book cs1">Kumar, Narinder (2004). <i>Comprehensive Physics XII</i>. Laxmi Publications. p. 713.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Comprehensive+Physics+XII&rft.pages=713&rft.pub=Laxmi+Publications&rft.date=2004&rft.aulast=Kumar&rft.aufirst=Narinder&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerway,_Raymond_A.Jewett,_John_W.2010" class="citation book cs1">Serway, Raymond A.; Jewett, John W. (2010). <i>Physics for Scientists and Engineers</i> (8th ed.). Brooks/Cole cengage. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4390-4844-3" title="Special:BookSources/978-1-4390-4844-3"><bdi>978-1-4390-4844-3</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=8th&rft.pub=Brooks%2FCole+cengage&rft.date=2010&rft.isbn=978-1-4390-4844-3&rft.au=Serway%2C+Raymond+A.&rft.au=Jewett%2C+John+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTipler,_Paul2004" class="citation book cs1">Tipler, Paul (2004). <i>Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics</i> (5th ed.). W. H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-0809-4" title="Special:BookSources/0-7167-0809-4"><bdi>0-7167-0809-4</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%2C+Oscillations+and+Waves%2C+Thermodynamics&rft.edition=5th&rft.pub=W.+H.+Freeman&rft.date=2004&rft.isbn=0-7167-0809-4&rft.au=Tipler%2C+Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3APotential+energy" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130111094907/http://www.kineticenergys.com/potential-energy/">What is potential energy?</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" 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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 href="/wiki/Work_(physics)" title="Work (physics)">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 class="mw-selflink selflink">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 href="/wiki/Work_(physics)" title="Work (physics)">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 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