Question

Two solid spheres of same size of a certain metal are placed in contact with each other. Prove that the gravitiational force acting between them is directly proportional to the fourth power of their radius.

Answer 1

The weights of the spheres may be assumed to be concentrated at their centres.
so, $F=\frac{G\left[\frac{4}{3}\pi R^3\rho \right]\times \left[\frac{4}{3}\pi R^3\rho \right]}{(2R{)}^2}=\frac{4}{9}\left(G{\pi }^2{\rho }^2\right)R^4$
$\therefore F\propto R^4$