Can the Gauss Law for Gravitation be used to prove the Shell Theorem?

[SHISHKABOB]

Homework Statement

Prove that an object within a spherically symmetric shell with uniform density will feel no gravitational force due to the mass of the shell. Let the density of the shell be ρ, the mass of the object be m, the radii to the inner and outer surfaces be r₁ and r₂ respectively, and let the mass of the shell be M.

Homework Equations

F = GmM/r^2

The Attempt at a Solution

so far I've drawn my setup. I tried to emulate the wikipedia article's drawing so that now I have two shells basically. I'm not exactly sure where I should go from here.

I have:

dF_R = GmdM/($s^{2}_{1}$)cos($\varphi_{1}$) + GmdM/($s^{2}_{2}$)cos($\varphi_{2}$)

but... can't I just take the distance and angle right in the middle of those two? I'll call them x and $\alpha$

so that I have

dF_R = GmdM/($x^{2}$)cos($\alpha$)

where x = (s₁ + s₂)/2

and $\alpha$ = ($\varphi_{1}$ + $\varphi_{2}$)/2

?

if so that would make things a lot easier

I think that it would be okay to do that because it says that the shell has uniform density

 

[vivekrai]

I think you would like to use the ' Gauss Law for Gravitation '.