question_answer

If spheres of same material and same radius r are touching each other, then show that the gravitational force between them is directly proportional to \[{{r}^{\mathbf{4}}}\].

Answer:

                If \[\rho \]is the density of the material, then \[F=G\frac{{{m}_{1}}{{m}_{2}}}{{{(2r)}^{2}}}=G\frac{\frac{4}{3}\pi {{r}^{3}}\rho \times \frac{4}{3}\pi {{r}^{3}}\rho }{(2{{r}^{2}})}=\frac{4}{9}{{\pi }^{2}}{{\rho }^{2}}G{{r}^{4}}\] Clearly, \[F\propto {{r}^{4}}\]