Acceleration of Gravity inside a planet

[JoeT21]

Homework Statement

Find the acceleration of gravity within the mantle (at some radius R) of a planet that has a core and a mantle, each with different constant density.

Rp= radius of planet
Rc= radius of core
Pm= density of the mantle
Pc= density of the core
P= density

I know what the answer is but I cannot get there!
ANSWER=
a= (-4/3)G(Pi)[((Pc-Pm)*Rc^3/R^2) + Pm*R]

Homework Equations

.
Gravitational Potential Equations

The Attempt at a Solution

I have made 3 potential equations:

(R>Rc):
V= (-4(pi)GP/3R)*(R^3-Rc^3)

(Rp>R>Rc):
V= [(-4(Pi)GP/3R)*(R^3-Rc^3) + (-2(Pi)GP)*(Rp^3-R^3)]

(Rp>R):
V= -2(Pi)GP*(Rp^3-R^3)

I cannot get to the correct answer any help would be great!

 

[borgwal]

You should use Gauss' law. Perhaps you know that law from electrostatics, where the law refers to electric fields and the total charge inside a volume. The gravitational equivalent refers to g and the total mass inside a volume.