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.

Options

A. `(Gm^(2))/(r^(2))`
B. `(Gm^(2))/(4r^(2))`
C. `sqrt2(Gm^(2))/(4r^(2))`
D. `sqrt(3)(Gm^(2))/(4r^(2))`

Answer

D

Explanation

The given system may be regarded as a system of three particles located at the three vertidcals of an equilateral triangle of side `2r`.
Now, `F_(A)=F_(B)=(Gm^(2))/((2r)^(2))=(Gm^(2))/(4r^(2))`
`F_(A)` and `F_(b)` are inclined to each other at the angle of `60^@`. If `F` is the resultant of `F_(A)` and `F_(b)` then
`F=sqrt(3)xx(GM^(2))/(4r^(2))`