Hard gravitational force problem

[alexwaylo2008]

Hard gravitational force problem!

Homework Statement

[/b]
Derive an expression for the gravitational force between a sphere with mass m and another sphere with mass M and radius R, IF the sphere of mass M has a spherical hollow inside which touches the surface of the sphere and the center of the sphere. The distance between M and m is d (including the radius R and ignoring the radius of m).

Homework Equations

F = GMm/r^2

The Attempt at a Solution

I tried to calculate the volume of the sphere with the hollow and I got = 7piR^3/6, but then I don't know what to do. The answer is supposed to be:
F = (GMm/d^2)( 1 - 1/(8(1-R/2d)^2))

 

[Astronuc]

Find the gravitational force between mass M and m without the hole.

Then find the gravitational force of the mass that would be in the hole, and subtract it from the other force found above. Think superposition of vectors.

If d is the distance between m and M (center), what is the distance between m and the hole in M? That is the key.