Three uniform spheres each of mass m and radius R are kept in such a way that each touches the orher two. Find the magnitude of the gravitational force on any one of the spheres due to the other two.

Explanation

Mass m and radius R are same for all spheres.
Force between 1,3 spheres `=F_1(G*m^2)/((2R)^2)`
Force between 1,2 spheres `=F_2(G*m^2)/((2R)^2)`

Now `F_1" and "F_2` will act with an angle `theta=60^(@)` between them so from Parallelogram law
Reaultant force F= `sqrt(F_1^2+F_2^2+2F_1F_2 cos theta)`
`=sqrt(F^2+F^2+2F^2""(1)/(2))= sqrt(3) F`
Force on 1st sphere due to other two spheres `=(sqrt(3)Gm^2)/(4R^2)`.