Three uniform spheres each having a mass $M$ and radius $a$ are kept in such a way that each touches

Question

Three uniform spheres each having a mass $M$ and radius $a$ are kept in such a way that each touches the other two. Find the magnitude of the gravitational force on any of the spheres due to the other two.

Professor PC Thomas Classes · Answer

Distance between centers is $2a$. Force by one sphere is $F = \frac{GM^2}{(2a)^2} = \frac{GM^2}{4a^2}$. The two forces are at $60^\circ$, so resultant $F_{net} = \sqrt{F^2 + F^2 + 2F^2\cos(60^\circ)} = \sqrt{3}F = \frac{\sqrt{3}GM^2}{4a^2}$.