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.

 

[ogb p]

I would first start by defining all the variables and assumptions in the problem. From the given information, we know that there are two spheres, one with mass m and one with mass M and radius R. The distance between the two spheres is d, which includes the radius R of the larger sphere. We are also told that the larger sphere has a spherical hollow inside which touches the surface of the sphere and the center of the sphere.

Next, I would use the equation for gravitational force, F = GMm/r^2, and try to find an expression for the distance r between the two spheres. Since the larger sphere has a hollow inside, the distance between the centers of the two spheres would be (d - R). However, we also need to take into account the radius of the smaller sphere, so the actual distance between the center of the smaller sphere and the surface of the larger sphere would be (d - R - R) = (d - 2R). Therefore, the distance r would be (d - 2R + R) = (d - R).

Now, we can plug this expression for r into the equation for gravitational force and we get:

F = GMm/(d-R)^2

However, this is not the final answer, as we still need to take into account the spherical hollow inside the larger sphere. To do this, we can use the volume of the hollow, which is 7πR^3/6, and divide it by the total volume of the larger sphere, which is 4/3 πR^3. This gives us a fraction of (7/6)/(4/3) = 7/8.

Finally, we can plug this fraction into our expression for gravitational force and we get the final answer:

F = (GMm/d^2)( 1 - 1/(8(1-R/2d)^2))

This expression takes into account the effects of the spherical hollow inside the larger sphere and gives us the correct gravitational force between the two spheres.