Two identical spheres each of radius $r$ are placed in contact with each other. Show that the gravitational force between them is proportional to $r^4$.

(N/A) The mass $m$ of each sphere can be expressed in terms of density $\rho$ and radius $r$ as $m = \text{Volume} \times \text{Density} = \frac{4}{3} \pi r^3 \rho$.
The distance between the centers of the two spheres in contact is $d = r + r = 2r$.
According to Newton's Law of Gravitation,the force $F$ is given by $F = \frac{G m_1 m_2}{d^2}$.
Substituting the values,$F = \frac{G (\frac{4}{3} \pi r^3 \rho) (\frac{4}{3} \pi r^3 \rho)}{(2r)^2}$.
$F = \frac{G \cdot \frac{16}{9} \pi^2 r^6 \rho^2}{4r^2}$.
$F = (\frac{4}{9} \pi^2 \rho^2 G) r^4$.
Since $\frac{4}{9} \pi^2 \rho^2 G$ is a constant,we have $F \propto r^4$.