Find the potential energy of a system of four particles each of mass $m$ placed at the vertices of a

Question

Find the potential energy of a system of four particles each of mass $m$ placed at the vertices of a square of side $l$.

Professor PC Thomas Classes · Answer

A square has 4 sides of length $l$ and 2 diagonals of length $\sqrt{2}l$. The total gravitational potential energy for the 6 pairs of masses is $U = -4\frac{Gm^2}{l} - 2\frac{Gm^2}{\sqrt{2}l} = -\frac{Gm^2}{l}(4 + \sqrt{2})$.