Английская Википедия:Gravitational energy
Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (converted into kinetic energy) when the objects fall towards each other. Gravitational potential energy increases when two objects are brought further apart.
Formulation
For two pairwise interacting point particles, the gravitational potential energy <math>U</math> is given by <math display="block">U = -\frac{GMm}{R},</math> where <math>M</math> and <math>m</math> are the masses of the two particles, <math>R</math> is the distance between them, and <math>G</math> is the gravitational constant.[1]
Close to the Earth's surface, the gravitational field is approximately constant, and the gravitational potential energy of an object reduces to <math display="block">U = mgh</math> where <math>m</math> is the object's mass, <math display="inline">g = {GM_{\oplus}} / {R_{\oplus}^2}</math> is the gravity of Earth, and <math>h</math> is the height of the object's center of mass above a chosen reference level.[1]
Newtonian mechanics
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, <math>M</math>, and another point mass, <math>m</math>, is given by Newton's law of gravitation:[3] <math display="block">F = \frac {GMm}{r^2}</math>
To get the total work done by an external force to bring point mass <math>m</math> from infinity to the final distance <math>R</math> (for example the radius of Earth) of the two mass points, the force is integrated with respect to displacement: <math display="block">W = \int_\infty^R \frac {GMm}{r^2}dr = -\left . \frac{G M m}{r} \right|_{\infty}^{R}</math>
Because <math display="inline">\lim_{r\to \infty} \frac{1}{r} = 0</math>, the total work done on the object can be written as:[4] Шаблон:Equation box 1
In the common situation where a much smaller mass <math>m</math> is moving near the surface of a much larger object with mass <math>M</math>, 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 <math>R</math> from the center) to a height <math>h</math> above the surface is <math display="block">\begin{align} \Delta U &= \frac{GMm}{R}-\frac{GMm}{R+h} \\ &= \frac{GMm}{R}\left(1-\frac{1}{1+h/R}\right). \end{align}</math> If <math>h/R</math> is small, as it must be close to the surface where <math>g</math> is constant, then this expression can be simplified using the binomial approximation <math display="block">\frac{1}{1+h/R} \approx 1-\frac{h}{R}</math> to <math display="block">\begin{align} \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{align}</math> As the gravitational field is <math>g = GM / R^2</math>, this reduces to <math display="block">\Delta U \approx mgh.</math> Taking <math>U = 0</math> at the surface (instead of at infinity), the familiar expression for gravitational potential energy emerges:[5] <math display="block">U = mgh.</math>
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.Шаблон:Citation needed
See also
- Gravitational binding energy
- Gravitational potential
- Gravitational potential energy storage
- Positive energy theorem
References
Шаблон:Footer energy Шаблон:Authority control
- ↑ 1,0 1,1 Шаблон:Cite web
- ↑ 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), Шаблон:ISBN, Appendix A—Gravitational Energy.
- ↑ Шаблон:Cite book Extract of page 10
- ↑ Шаблон:Cite book Extract of page 143
- ↑ Шаблон:Cite web
- ↑ Lev Davidovich Landau & Evgeny Mikhailovich Lifshitz, The Classical Theory of Fields, (1951), Pergamon Press, Шаблон:ISBN
