Gravitational potential for various matter configurations

[Afonso Campos]

Homework Statement

Consider the Earth as

1. with a constant density of matter,
2. as a thin shell empty sphere and
3. with a constant linear density of matter $\rho(r) = \rho_{0}r$.

In all cases, calculate the gravitational potential and the gravitational field everywhere and make a sketch.

Homework Equations

The Attempt at a Solution

1. The gravitational potential outside the Earth is equal to the gravitational potential of a point particle of the mass of the Earth, that is,

$$V = - \frac{GM}{r}.$$

Therefore, the gravitational field is simply

$$g = - \nabla V = - \frac{GM}{r^{2}}.$$

To compute the gravitational potential within the Earth, do I have to slice up the Earth into thin shells of radius $dr$ and integrate over shells which contribute to the potential?

 

[mfb]

To compute the gravitational potential within the Earth, do I have to slice up the Earth into thin shells of radius $dr$ and integrate over shells which contribute to the potential?

Right.