5.4.4 Gravitational potential and energy

Gravitational potential
The gravitational potential (Vg) at a point in a gravitational field is defined as the work done per unit mass to move an object to that point from infinity. It has the units J kg^-1 and infinity refers to a distance so far from the object producing the gravitational field that the gravitational field strength is zero. Gravitational potential is a scalar quantity. At infinity gravitational potential is at a maximum 0J kg^-1. Since all masses attract each other it takes energy (external work must be done) to move objects apart.

The gravitational potential at any point in a radial field around a point mass depends on:

• the distance 'r' from the point mass producing the gravitational field to that point
• the mass M of the point mass.

We can use the ideas of work done on a force and Newton's law of gravitation to form the equation:

Vg = -GM/r

A graph of Vg against 1/r will produce a straight line through the origin with a gradient equal to -GM. If a graph is drawn of the Vg around the Earth against distance r from the centre of mass of the Earth, the smallest value of r on the graph will be the Earths radius.

Gravitational potential energy
The gravitational potential energy 'E' of any object with the mass m within a gravitational field is defined as the work done to move the mass from infinity to a point in a gravitational field:

E = m Vg

In a uniform gravitational field, in order to change the gravitational potential energy of an object its height above the surface must be changed as this results in a change in gravitational potential.

In a radial field (remember Vg = -GM/r) the gravitational potential energy can be written as:

E = m Vg = -GMm/r

Force-distance graphs
We know, from Newton's law of gravitation, that the gravitational force between objects decreases with distance in an inverse-square relationship. The area under a force-distance graph is equal to the work done (remember this can be negative if the object falls as the object loses gravitational potential energy).

Escape velocity
In order to escape the gravitational field of a mass an object must be supplied with energy equal to the gain in gravitational potential energy needed to lift it out of the field:

1.5mv² = GMm/r

The minimum velocity for this condition to be met is known as the escape velocity. It is the same for all objects regardless of their mass:

v = √(2GM/r)