Gravitational potential energy between two objects

In classical mechanics, two or more masses always have a gravitational potential. Conservation of energy requires that this gravitational field energy is always negative, so that it is zero when the objects are infinitely far apart.[2] The gravitational potential energy is the potential energy an object has because it is within a gravitational field.

The force between a point mass, M {\displaystyle M}  , and another point mass, m {\displaystyle m}  , is given by Newton's law of gravitation:[3]

F = G M m r 2 {\displaystyle F={\frac {GMm}{r^{2}}}}

To get the total work done by an external force to bring point mass m {\displaystyle m}   from infinity to the final distance R {\displaystyle R}   (for example the radius of Earth) of the two mass points, the force is integrated with respect to displacement:

W = ∫ ∞ R G M m r 2 d r = − G M m r | ∞ R {\displaystyle W=\int _{\infty }^{R}{\frac {GMm}{r^{2}}}dr=-\left.{\frac {GMm}{r}}\right|_{\infty }^{R}}

Because lim r → ∞ 1 r = 0 {\textstyle \lim _{r\to \infty }{\frac {1}{r}}=0}  , the total work done on the object can be written as:[4]

Gravitational Potential Energy

U = − G M m R {\displaystyle U=-{\frac {GMm}{R}}}  

In the common situation where a much smaller mass m {\displaystyle m}   is moving near the surface of a much larger object with mass M {\displaystyle M}  , the gravitational field is nearly constant and so the expression for gravitational energy can be considerably simplified. The change in potential energy moving from the surface (a distance R {\displaystyle R}   from the center) to a height h {\displaystyle h}   above the surface is

Δ U = G M m R − G M m R + h = G M m R ( 1 − 1 1 + h / R ) . {\displaystyle {\begin{aligned}\Delta U&={\frac {GMm}{R}}-{\frac {GMm}{R+h}}\\&={\frac {GMm}{R}}\left(1-{\frac {1}{1+h/R}}\right).\end{aligned}}}

1 1 + h / r ≈ 1 − h R {\displaystyle {\frac {1}{1+h/r}}\approx 1-{\frac {h}{R}}}

Δ U ≈ G M m R [ 1 − ( 1 − h R ) ] Δ U ≈ G M m h R 2 Δ U ≈ m ( G M R 2 ) h . {\displaystyle {\begin{aligned}\Delta U&\approx {\frac {GMm}{R}}\left[1-\left(1-{\frac {h}{R}}\right)\right]\\\Delta U&\approx {\frac {GMmh}{R^{2}}}\\\Delta U&\approx m\left({\frac {GM}{R^{2}}}\right)h.\end{aligned}}}

Δ U ≈ m g h . {\displaystyle \Delta U\approx mgh.}

U = m g h . {\displaystyle U=mgh.}

Main article: Mass in general relativity

In general relativity gravitational energy is extremely complex, and there is no single agreed upon definition of the concept. It is sometimes modelled via the Landau–Lifshitz pseudotensor[6] that allows retention for the energy–momentum conservation laws of classical mechanics. Addition of the matter stress–energy tensor to the Landau–Lifshitz pseudotensor results in a combined matter plus gravitational energy pseudotensor that has a vanishing 4-divergence in all frames—ensuring the conservation law. Some people object to this derivation on the grounds that pseudotensors are inappropriate in general relativity, but the divergence of the combined matter plus gravitational energy pseudotensor is a tensor.

• Gravitational binding energy
• Gravitational potential
• Gravitational potential energy storage

1. ^ a b "Gravitational Potential Energy". hyperphysics.phy-astr.gsu.edu. Retrieved 10 January 2017.
2. ^ For a demonstration of the negativity of gravitational energy, see Alan Guth, The Inflationary Universe: The Quest for a New Theory of Cosmic Origins (Random House, 1997), ISBN 0-224-04448-6, Appendix A—Gravitational Energy.
3. ^ MacDougal, Douglas W. (2012). Newton's Gravity: An Introductory Guide to the Mechanics of the Universe (illustrated ed.). Springer Science & Business Media. p. 10. ISBN 978-1-4614-5444-1. Extract of page 10
4. ^ Tsokos, K. A. (2010). Physics for the IB Diploma Full Colour (revised ed.). Cambridge University Press. p. 143. ISBN 978-0-521-13821-5. Extract of page 143
5. ^ Fitzpatrick, Richard (2006-02-02). "Gravitational potential energy". farside.ph.utexas.edu. The University of Texas at Austin.
6. ^ Lev Davidovich Landau & Evgeny Mikhailovich Lifshitz, The Classical Theory of Fields, (1951), Pergamon Press, ISBN 7-5062-4256-7