Potential inside homogenous sphere

[llandau]

How to calculate gravitational potential inside a homogenous sphere of mass m? I am curious because I had to solve the classic problem of the tunnel through Earth and wanted to generalize when the tunnel is not a diameter.

 

[HallsofIvy]

The gravitational force inside a homogeneous sphere depends only on the mass closer to the center- the force vectors due to mass farther out cancel each other.

If the radius of the sphere is R, then its volume is $(4/3)\pi R^3$ and the density is $(3/4)m/(\pi R^3)$. That means that the mass of the sphere below radius r is $(3/4)m/(\pi R^3)(4/3)\pi r^3= m(r/R)^3$. The force on an object of mass M then is $-Gm(r/R)^3M/r^2= -Gmr/R^3$.

So the force is proportional to r and the potential energy is $-Gmr^2/(2R^3)$.

 

[llandau]

You have been helpful, thanks (I believe you lost an M, but it really doesn't matter).