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 the object from a fixed reference location to the location of the object. It is analogous to the electric potential with mass playing the role of charge. The reference location, where the potential is zero, is by convention infinitely far away from any mass, resulting in a negative potential at any distance.

In mathematics, the gravitational potential is also known as the Newtonian potential and is fundamental in the study of potential theory.

The gravitational potential (V) is the gravitational potential energy (U) per unit mass:

where m is the mass of the object. Potential energy is equal (in magnitude, but negative) to the work done by the gravitational field moving a body to its given position in space from infinity. If the body has a mass of 1 unit, then the potential energy to be assigned to that body is equal to the gravitational potential. So the potential can be interpreted as the negative of the work done by the gravitational field moving a unit mass in from infinity.

In some situations, the equations can be simplified by assuming a field that is nearly independent of position. For instance, in a region close to the surface of the Earth, the gravitational acceleration, g, can be considered constant. In that case, the difference in potential energy from one height to another is, to a good approximation, linearly related to the difference in height:

The potential V of a unit mass m at a distance x from a point mass of mass M can be defined as the work W that needs to be done by an external agent to bring the unit mass in from infinity to that point:

where G is the gravitational constant, and F is the gravitational force. The potential has units of energy per unit mass, e.g., J/kg in the MKS system. By convention, it is always negative where it is defined, and as x tends to infinity, it approaches zero.

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