3 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.

Dear Student

The system can be considered at three particles located at the vertices of equilateral triangle having side 2a.
Gravitational force between two sphere is given as
F₁ = GMM/(2a)² 

As mass and radius of all sphere is same.
So on one sphere two forces of equal magnitude are acting at angle of 60^o.

So resultant gravitational force on one sphere will be

$$\begin{gathered} F =\sqrt{{F_1}^2+{F_1}^2+2F_1{F}_1\cos 60}=\sqrt{3F_1} \\ F =\sqrt{3}\frac{G{M}^2}{4a^2} \end{gathered}$$