We can define a quantity that gives a measure of the amount of energy a body of unit mass has when in a gravitational field. This quantity is called the gravitational potential and is defined as
where
is the work that must be done to remove a body to a distant point where is has no gravitational potential energy.is negative since work must be done to remove the body of mass
to this distant point.
The SI units of gravitational potential are J/Kg. Gravitational potential labels a point in space, so is a scalar quantity.
An alternative and more useful expression for the gravitational potential in the presence of a spherically symmetric body of mass
can be derived from Newton's Law of Gravity. It is
The graph of
against
is shown below for the Earth. It may be thought that at
the gravitational potential energy is infinity, but this is not so, since any mass has a finite size. The equation above does not apply inside in Earth. Inside the Earth, we consider only that mass inside the sphere defined by the distance from the centre.
The expression
is more general however.
Once you have the potential at one point, say
and at some other point, say
the work per unit mass needed to take a mass from one point to the other is given by