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<title>Specific potential energy</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Specific potential energy</span></span>

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</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Specific potential energy</th></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">pe, or <i>e</i><sub>u</sub></td></tr><tr><th scope="row" class="infobox-label"><a href="SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data">J/kg</td></tr><tr><th scope="row" class="infobox-label">In <a href="SI_base_unit" title="SI base unit"><span class="wrap">SI base units</span></a></th><td class="infobox-data">m<sup>2</sup>/s<sup>2</sup></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Derivations from<br>other quantities</div></th><td class="infobox-data"><i>e</i><sub>u</sub> = <i><a href="Gravitational_acceleration" title="Gravitational acceleration">g</a></i> <a href="Height" title="Height">h</a></td></tr></tbody></table>

<p><b>Specific potential energy</b> is <a href="Potential_energy" title="Potential energy">potential energy</a> of an object per unit of mass of that object. In a gravitational field it is the acceleration of gravity times height, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{u}=gh}">

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<div class="mw-heading mw-heading2"><h2 id="Mathematical_form">Mathematical form</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="Gravitational_potential#Mathematical_form" title="Gravitational potential">Gravitational potential § Mathematical form</a>.<span class="mw-editsection-like "><span class="mw-editsection-bracket">[</span><a class="external text external" href="https://en.wikipedia.org/w/index.php?title=Gravitational_potential&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt">

<p>The gravitational <a href="Scalar_potential" title="Scalar potential">potential</a> <i>V</i> at a distance <i>x</i> from a <a href="Point_particle" title="Point particle">point mass</a> of mass <i>M</i> can be defined as the work <i>W</i> that needs to be done by an external agent to bring a unit mass in from infinity to that point:<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><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\mathbf {x} )={\frac {W}{m}}={\frac {1}{m}}\int _{\infty }^{x}\mathbf {F} \left(\mathbf {x} '\right)\cdot d\mathbf {x} '={\frac {1}{m}}\int _{\infty }^{x}{\frac {GmM}{x'^{2}}}dx'=-{\frac {GM}{x}},}">

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<annotation encoding="application/x-tex">{\displaystyle V(\mathbf {x} )={\frac {W}{m}}={\frac {1}{m}}\int _{\infty }^{x}\mathbf {F} \left(\mathbf {x} '\right)\cdot d\mathbf {x} '={\frac {1}{m}}\int _{\infty }^{x}{\frac {GmM}{x'^{2}}}dx'=-{\frac {GM}{x}},}</annotation>

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where <i>G</i> is the <a href="Gravitational_constant" title="Gravitational constant">gravitational constant</a>, and F is the gravitational force. The product <i>GM</i> is the <a href="Standard_gravitational_parameter" title="Standard gravitational parameter">standard gravitational parameter</a> and is often known to higher precision than <i>G</i> or <i>M</i> separately. The potential has <a href="Units_of_energy" title="Units of energy">units of energy</a> per mass, e.g., J/kg in the <a href="MKS_system_of_units" class="mw-redirect" title="MKS system of units">MKS</a> system. By convention, it is always negative where it is defined, and as <i>x</i> tends to infinity, it approaches zero.

</p><p>The <a href="Gravitational_field" title="Gravitational field">gravitational field</a>, and thus the acceleration of a small body in the space around the massive object, is the negative <a href="Gradient" title="Gradient">gradient</a> of the gravitational potential. Thus the negative of a negative gradient yields positive acceleration toward a massive object. Because the potential has no angular components, its gradient is

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =-{\frac {GM}{x^{3}}}\mathbf {x} =-{\frac {GM}{x^{2}}}{\hat {\mathbf {x} }},}</annotation>

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where x is a vector of length <i>x</i> pointing from the point mass toward the small body and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {x} }}}">

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<annotation encoding="application/x-tex">{\displaystyle \|\mathbf {a} \|={\frac {GM}{x^{2}}}.}</annotation>

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</p><p>The potential associated with a <a href="Mass_distribution" title="Mass distribution">mass distribution</a> is the superposition of the potentials of point masses. If the mass distribution is a finite collection of point masses, and if the point masses are located at the points x<sub>1</sub>, ..., x<sub><i>n</i></sub> and have masses <i>m</i><sub>1</sub>, ..., <i>m</i><sub><i>n</i></sub>, then the potential of the distribution at the point x is

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<annotation encoding="application/x-tex">{\displaystyle V(\mathbf {x} )=\sum _{i=1}^{n}-{\frac {Gm_{i}}{\|\mathbf {x} -\mathbf {x} _{i}\|}}.}</annotation>

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<p>If the mass distribution is given as a mass <a href="Borel_measure" title="Borel measure">measure</a> <i>dm</i> on three-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> R<sup>3</sup>, then the potential is the <a href="Convolution" title="Convolution">convolution</a> of <span class="texhtml">−<i>G</i>/|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">r</span>|</span> with <i>dm</i>. In good cases this equals the integral

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<annotation encoding="application/x-tex">{\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{\|\mathbf {x} -\mathbf {r} \|}}\,dm(\mathbf {r} ),}</annotation>

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where <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">x − r</span>|</span> is the <a href="Euclidean_distance" title="Euclidean distance">distance</a> between the points x and r. If there is a function <i>ρ</i>(r) representing the density of the distribution at r, so that <span class="texhtml"><i>dm</i>(r) = <i>ρ</i>(r) <i>dv</i>(r)</span>, where <i>dv</i>(r) is the Euclidean <a href="Volume_element" title="Volume element">volume element</a>, then the gravitational potential is the <a href="Volume_integral" title="Volume integral">volume integral</a>

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<annotation encoding="application/x-tex">{\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{\|\mathbf {x} -\mathbf {r} \|}}\,\rho (\mathbf {r} )dv(\mathbf {r} ).}</annotation>

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</p><p>If <i>V</i> is a potential function coming from a continuous mass distribution <i>ρ</i>(r), then <i>ρ</i> can be recovered using the <a href="Laplace_operator" title="Laplace operator">Laplace operator</a>, <span class="texhtml">Δ</span>:

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<annotation encoding="application/x-tex">{\displaystyle \rho (\mathbf {x} )={\frac {1}{4\pi G}}\Delta V(\mathbf {x} ).}</annotation>

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This holds pointwise whenever <i>ρ</i> is continuous and is zero outside of a bounded set. In general, the mass measure <i>dm</i> can be recovered in the same way if the Laplace operator is taken in the sense of <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>. As a consequence, the gravitational potential satisfies <a href="Poisson's_equation" title="Poisson's equation">Poisson's equation</a>. See also <a href="Green's_function_for_the_three-variable_Laplace_equation" title="Green's function for the three-variable Laplace equation">Green's function for the three-variable Laplace equation</a> and <a href="Newtonian_potential" title="Newtonian potential">Newtonian potential</a>.

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The integral may be expressed in terms of known transcendental functions for all ellipsoidal shapes, including the symmetrical and degenerate ones.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> These include the sphere, where the three semi axes are equal; the oblate (see <a href="Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">reference ellipsoid</a>) and prolate spheroids, where two semi axes are equal; the degenerate ones where one semi axes is infinite (the elliptical and circular cylinder) and the unbounded sheet where two semi axes are infinite. All these shapes are widely used in the applications of the gravitational potential integral (apart from the constant <i>G</i>, with 𝜌 being a constant <a href="Charge_density" title="Charge density">charge density</a>) to electromagnetism.</div></div>

<ul><li><a href="Specific_mechanical_energy" title="Specific mechanical energy">Specific mechanical energy</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFMarionThornton1995" class="citation book cs1">Marion, J.B.; Thornton, S.T. (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/classicaldynamic00mari_0/page/192"><i>Classical Dynamics of particles and systems</i></a></span> (4th ed.). Harcourt Brace & Company. p. <a rel="nofollow" class="external text" href="https://archive.org/details/classicaldynamic00mari_0/page/192">192</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-03-097302-3</bdi>.</cite></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"><cite id="CITEREFArfkenWeber2005" class="citation book cs1">Arfken, George B.; Weber, Hans J. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tNtijk2iBSMC&pg=PA72"><i>Mathematical Methods For Physicists International Student Edition</i></a> (6th ed.). <a href="Academic_Press" title="Academic Press">Academic Press</a>. p. 72. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-08-047069-6</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFSangJonesChadhaWoodside2014" class="citation book cs1">Sang, David; Jones, Graham; Chadha, Gurinder; Woodside, Richard; Stark, Will; Gill, Aidan (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SjsDBAAAQBAJ&pg=PA276"><i>Cambridge International AS and A Level Physics Coursebook</i></a> (illustrated ed.). <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 276. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-107-69769-0</bdi>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFMuncaster1993" class="citation book cs1">Muncaster, Roger (1993). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Knov8XAyf2cC&pg=PA106"><i>A-level Physics</i></a> (illustrated ed.). <a href="Nelson_Thornes" title="Nelson Thornes">Nelson Thornes</a>. p. 106. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7487-1584-8</bdi>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFMacMillan1958" class="citation book cs1">MacMillan, W.D. (1958). <i>The Theory of the Potential</i>. Dover Press.</cite></span>

</li>

</ol></div></div>

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