# Find gravitational potential inside uniform ball

• MasterNewbie

## Homework Statement

Find the potential and force on a mass m outside and inside the Earth in terms of g, the acceleration due to gravity, assuming Earth has uniform density and radius R.

## Homework Equations

For a mass m, the potential energy of it in the gravitational field of a spherical shell of radius r' and uniform mass distribution m' is given by
γ=const. if m is inside the shell
γ=-Cm'/r if m is outside the shell, where r is the distance from the center of the sphere to the mass and C is a constant

Outside:
F=mgR^2/r^2, γ=-mgR^2/r
Inside:
F=mgr/R, γ=mg/2R*(r^2-3R^2).

As a hint, to find the constants, when r=R, F=mg.

## The Attempt at a Solution

I managed to get the first part outside the earth, though whether or not the method I used is legitimate I don't know.

I integrated from 0 to R $\int-Cm'/rdr'$ to find the total potential β=-CMR/r, where M is the mass of the Earth (integrating from 0 to R, the sum of the masses of the shells will total M). Taking the negative gradient to find the force, I find F=-CMR/r^2er. At r=R, F=mg, so C=mgR/M. β=-mgR^2/r and F=-mgR^2/r^2er

Getting the inside potential is a little more difficult. I know that it involves integrating from 0 to r $\int-Cm'/rdr'$ and maybe r to R $\int Adr'$. I've tried two separate methods, where the second term was the integral as above and the second as just a constant in itself. I do know that at r=R, both inside/outside forces and inside/outside potentials should be equal, but neither attempt yielded anything remotely similar to the answer provided in the problem.

Since the method I used in part 1 didn't work for this, I believe it is at least partly a fluke.

What I just tried doing was 0 to r $\int-Cm'/rdr'$ = -CM(r^3/R^3) In this case I am not integrating over the whole ball so I can't put down M as the mass, r^3/R^3 is the ratio of volumes, so times M would give the mass of the portion I integrated over. This part of the potential has an extra r in it. I am not certain how to approach integrating over the constant potentials provided by the shells from r to R.