Eight particles of mass '$m$' each are placed at the vertices of a cube of side '$a$'. Find gravitational potential energy of this system.

Total particles are $n=8$

$\therefore$ Total number of pairs are $\frac{n(n-1)}{2}$ or $28$ .

In $12$ pairs, distance between the particles is ${}^{*}{a}^{*}.$

In $12$ pairs, distance is $\sqrt{2}a$ and in remaining $4$ pairs distance is $\sqrt{3}a$.

$\therefore U=12\left(-\frac{Gmm}{a}\right)+12\left(\frac{-Gmm}{\sqrt{2}a}\right)+4\left(\frac{-Gmm}{\sqrt{3}a}\right)$                                        $=-\frac{G{m}^2}{a}\left(12+6\sqrt{2}+\frac{4}{\sqrt{3}}\right)$