## Gravitational potential

Glossary Definition for 16-19

#### Description

The gravitational potential at a point in a gravitational field is the work done per unit mass that would have to be done by some externally applied force to bring a massive object to that point from some defined position of zero potential, usually infinity

. It is the gravitational potential difference between the chosen point and the position of zero potential.

Gravitational potential is often represented by the symbol *V*.

If the field is due to an isolated massive point object (or any object of finite size), then it is conventional to define the potential to be zero at an infinite distance from the object; the potential is negative everywhere else because the gravitational force is always attractive.

Gravitational potential is also defined as the gravitational potential energy per unit mass relative to a defined position of zero potential energy. The two definitions are equivalent.

#### Discussion

There is a strong similarity between gravitational potential and electrostatic potential. In both cases, the underlying forces depend on the separation, *r*, of interacting objects as 1*r*^{ 2} and, in both cases, the change in the potential is defined via the work done in changing the separation between the interacting objects. The difference lies in the nature of the force: charges may be positive or negative, so the electrostatic interaction may be attractive or repulsive. The force of gravity is always attractive.

#### SI unit

J kg^{-1}

#### Expressed in SI base units

m^{2} s^{-2}

#### Other commonly used unit(s)

none

#### Mathematical expressions

- Raising an object through a height Δ
*h*at the surface of the Earth leads to a change of gravitational potential

Δ*V*=*g*Δ*h*
where g is the gravitational field at the surface of the Earth and Δ - More generally,

Δ*V*=*G**M*R -*G**M*R+Δ*h*=*G**M*R Δ*h**R*+Δ*h*

where*M*and*R*are, respectively, the mass and radius of the Earth, and*G*is the universal gravitational constant.

*h*is much less than the radius of the Earth.

#### Related entries

- Gravitational field
- Potential energy

#### In context

The difference in gravitational potential between sea level and the summit of Mount Everest is about 8.7 × 10^{4} J kg^{–1}. If a mountaineer of mass 100 kg travels from sea level to the Everest summit, the gravitational potential energy of the Earth-mountaineer system increases by about 8.7 × 10^{6} J.