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Gravitational energy or gravitational potential energy is the
potential energy a
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
ive object has in relation to another massive object due to
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
. It is the potential energy associated with the
gravitational field, which is released (converted into
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
) when the objects
fall
Autumn, also known as fall in American English and Canadian English, is one of the four temperate seasons on Earth. Outside the tropics, autumn marks the transition from summer to winter, in September (Northern Hemisphere) or March ( Southe ...
towards each other. Gravitational potential energy increases when two objects are brought further apart.
For two pairwise interacting point particles, the gravitational potential energy
is given by
where
and
are the masses of the two particles,
is the distance between them, and
is the
gravitational constant.
Close to the Earth's surface, the gravitational field is approximately constant, and the gravitational potential energy of an object reduces to
where
is the object's mass,
is the
gravity of Earth, and
is the height of the object's
center of mass above a chosen reference level.
[
]
Newtonian mechanics
In classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, two or more mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
es always have a gravitational potential
In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
. Conservation of energy requires that this gravitational field energy is always negative, so that it is zero when the objects are infinitely far apart. The gravitational potential energy is the potential energy an object has because it is within a gravitational field.
The force between a point mass, , and another point mass, , is given by Newton's law of gravitation
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
:
To get the total work done by an external force to bring point mass from infinity to the final distance (for example the radius of Earth) of the two mass points, the force is integrated with respect to displacement:
Because , the total work done on the object can be written as:
In the common situation where a much smaller mass is moving near the surface of a much larger object with mass , 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 from the center) to a height above the surface is
If is small, as it must be close to the surface where is constant, then this expression can be simplified using the binomial approximation
The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number ''x''. It states that
: (1 + x)^\alpha \approx 1 + \alpha x.
It is valid when , x, -1 and \alpha \geq 1.
Derivations Using linear ...
to
As the gravitational field is , this reduces to
Taking at the surface (instead of at infinity), the familiar expression for gravitational potential energy emerges:
General relativity
In general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
gravitational energy is extremely complex, and there is no single agreed upon definition of the concept. It is sometimes modelled via the Landau–Lifshitz pseudotensorLev Davidovich Landau
Lev Davidovich Landau (russian: Лев Дави́дович Ланда́у; 22 January 1908 – 1 April 1968) was a Soviet-Azerbaijani physicist of Jewish descent who made fundamental contributions to many areas of theoretical physics.
His ac ...
& Evgeny Mikhailovich Lifshitz
Evgeny Mikhailovich Lifshitz (russian: Евге́ний Миха́йлович Ли́фшиц; February 21, 1915, Kharkiv, Russian Empire – October 29, 1985, Moscow, Russian SFSR) was a leading Soviet physicist and brother of the physicist ...
, ''The Classical Theory of Fields'', (1951), Pergamon Press, that allows retention for the energy–momentum conservation laws of classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
. 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 vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
in all frames—ensuring the conservation law. Some people object to this derivation on the grounds that pseudotensor
In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g. a proper rotation) but additionally changes sign under an orientation-reversing coordin ...
s are inappropriate in general relativity, but the divergence of the combined matter plus gravitational energy pseudotensor is a tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
.
See also
* Gravitational binding energy
The gravitational binding energy of a system is the minimum energy which must be added to it in order for the system to cease being in a gravitationally bound state. A gravitationally bound system has a lower (''i.e.'', more negative) gravitati ...
* Gravitational potential
In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
* Gravitational potential energy storage
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
References
{{Footer energy
Forms of energy
Gravity
Conservation laws
Tensors in general relativity
Potentials