14.3 : Gravitation Between Spherically Symmetric Masses

The gravitational potential energy between two spherically symmetric bodies can be calculated from the masses and the distance between the bodies, assuming that the center of mass is concentrated at the respective centers of the bodies.

Consider that a spherically symmetric mass distribution comprises multiple concentric spherical shells. A point mass is placed at a distance 'r' from the center of mass of the spherical shell. All the particles in a given spherical ring on the surface of the shell are at equal distances from the point mass.

The potential between the point mass and the ring can be obtained from the ring's mass. Integrating the expression for the potential energy between a point mass and a ring over the limits of distance gives the gravitational potential energy, which is the same as the potential energy between two point masses.

If the point mass is inside the shell, then the limits of integration change. This shows that the potential energy does not depend on the distance and is the same everywhere for all points inside the shell. However, no work is done on the point mass if it is inside the shell.