hey guys. maybe not exactly a gre problem, but certainly something that should be understood for the GRE.

Im trying to find the gravitation potential per unit mass (U) inside a spherical shell. I thought this would be easy but cant reproduce the answer in Marion,Thornton 5ed,p188. I wanted to try a different approach than they did.

So i imagine bringing in a point mass from infinity into a sphere with outer radius (a) and inner (b) and uniform mass distribution; U(inf) = 0

= +

where i expressed M in terms of rho. Thats was easy. Now for the potential difference across the shell.

again subbing mass in terms of density and radius...

After integrating...

adding U_a

Some simplifying

whereas the answer in the book...

love a good latex workout, feeling the burn. anyway, its probably something stupid im messing up on, but please help. Im so frustrated at this.

and if there are other ways you would solve this, please suggest them