Four massive particles, each of mass m, are kept at the vertices of square of side $\ell$. With what speed should each particle be projected so as to stabilise the system ?

$\sqrt{\frac{Gm}{\ell}\left(1+\frac{1}{2 \sqrt{2}}\right)}$

In order to stabilise the system, each mass should move in a circular path of the same radius about the centre of the square. How is it possible? It is possible when each particle moves perpendicular to the corresponding diagonal with a speed, let v. As a result the net gravitational force on the particle due to the others will provide necessary centripetal acceleration v²/r, where r = $\ell / \sqrt{2}$.

$\Rightarrow F_g=m \frac{v^2}{r}=\frac{m v^2}{\left(\frac{\ell}{\sqrt{2}}\right)}$          .....(1)

where F_g = Net gravitational force on each particle. 

Resolving the individual gravitational force along the diameter, we obtain,

$F_g=\frac{F}{\sqrt{2}}+\frac{F}{\sqrt{2}}+F'=F'+\sqrt{2} F$

Putting $F = \frac{G m^2}{\ell^2}$  and  $F'=\frac{G m^2}{2 \ell^2}$

$\Rightarrow F_g=\frac{G m^2}{\ell^2}\left(\sqrt{2}+\frac{1}{2}\right)$        ......(2)

Using (1) and (2), we obtain,

$v=\sqrt{\frac{G m}{\ell}\left(1+\frac{1}{2 \sqrt{2}}\right)}$