# Linear Gravity Inside Solid Sphere Derivation

I am currently taking a dynamics module at university and am revising for a midterm exam.

One of the problems on a past paper was to consider a mass falling through a hole in the Earth (pole to pole). We are to consider the Earth to be a uniform sphere with constant density. The problem which I am stuck on is to show that the force due to gravity inside the Earth is linear with radial distance r.

I did something very similar in an electromagnetism module where I modelled the sphere as being made up of a series of concentric shells. Obviously my current problem is analogous to the electromagnetism one just with gravity. However I can't quite remember how to go about solving the problem.

I've been searching online and all I can find are derivations for the force being zero inside a hollow shell, not a solid sphere. Any help would be much appreciated!

Thanks, Sean.

Gauss's Law for Gravity: $\iint\limits_{\partial V} \overrightarrow{E_G} \cdot d S = - 4 \pi G M$

Do you know Gauss's Theorem for $\vec{E}$ in electrostatics? All the same here for gravity. Apply it, and you'll just work it out.
A very simple derivation which does not require even memorizing the constant factor in Gauss' law is as follows: take a sphere of radius $r < R$, concentric to the gravitating ball and apply Gauss' law. Using symmetry we argue that the field can only be parallel (or anti-parallel) to $\vec r$. Plugging $m(r) \propto r^3$ and $A(r) \propto r^2$ in, we find that $g \propto r$. Using the boundary $g(R) = GM/R^2$, we conclude that $g(r) = GMr/R^3$ inside the ball.