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 of the spheres due to the other two.

Explanation

As per question, three point masses, each of mass `M` are lying at the vertices of an equilaternal triangle of side `2 R`, Fig.

Force of attraction on sphere `C` due to sphere `A`
is `F_(1) = (Gm xxm)/((2R)^(2)) = (Gm^(2))/(4 R^(2))` along `CA`
Force of attraction on sphere `C` due to sphere `B`
is `F_(2) = (Gm xxm)/((2R)^(2)) = (Gm^(2))/(4 R^(2))` along `CB`
These force `vec(F_(1))` and `vec(F_(2))` are inclined at an angule `60^(@)`
So the resultant force on sphere `C` is
`F = sqrt(F_(1)^(2) + F_(2)^(2) + 2 F_(1) F_(2) cos 60^(@))`
`= sqrt(F_(1)^(2) + F_(1)^(2) + 2 F_(1) F_(1) xx 1//2)`
`= sqrt(3) F_(1) = (sqrt(3))/(4) (Gm^(2))/(R^(2))`