Gravitation of non-uniform density

[Mindscrape]

Homework Statement

A gaseous planet, with radius R, has a radially dependent density function
\rho (r') = \rho_0 [\frac{r'}{R}]^2
where r' is the distance from the center planet. Find the magnitude of force for a mass m inside and outside of the planet.

Homework Equations

F = -\frac{GmM}{r^2}
F = -G m \int_V \frac{\rho(r') \hat{e_r}}{r^2}dv'

The Attempt at a Solution

I'm pretty sure I did it right, but would like some confirmation. The mass inside the planet, taking a sample shell, would be

M = \int_0^r 4\pi r'^2 dr' \rho(r')

which would give

M = \int_0^r 4\pi r'^2 dr' \rho_0 (\frac{r'}{R})^2

and evaluates to

M = \frac{4}{5} \pi r^5 \frac{\rho_0}{R^2}

such that the force inside will be

F = \frac{-Gm r^3}{r^2}\rho_0 \hat{e_r}

Outside

the same idea applies

M = \int_0^R 4 \pi r'^2 dr' \rho(r')

and gives

M = \frac{4}{5} \pi R^5 \frac{\rho_0}{R^2}

so that the force is

F = \frac{-GmR^2}{r^2}\rho_0 \hat{e_r}

Both answers, more or less, make sense. Inside, as you move further away that the force will increase. Outside, the force will decrease as you move further away.

 

[StatMechGuy]

It's the same problem as a Gauss' law problem for a ball of non-uniform charge, and that's how the solution goes, so that all looks right.