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 one of the spheres due to the other two.

Answer 1

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

$F1=\frac{GMM}{{4a}^2}$

similarly

$F_2=\frac{GMM}{4a^2}$

As mass and radius of all sphere is same.

So on one sphere two forces of equal magnitude are acting at angle of 60 degrees.

$F=\sqrt{{f_1}^2+{f_2}^2+2f_1{f}_{2}cos60}$

$F=\sqrt{3}\frac{GMM}{4a^2}$