Three uniform spheres of mass M and radius R earth are kept... | Physics | SolveeIt | Solveeit

Today, August 23

Question

Three uniform spheres of mass M and radius R earth are kept in such a way that each touches the other two. The magnitude of the gravitational force on any of the spheres due to the other two is

A

$\frac{\sqrt3}{4}\frac{GM^2}{R^2}$

B

$\frac{3}{2}\frac{GM^2}{R^2}$

C

$\frac{\sqrt3GM^2}{R^2}$

D

$\frac{\sqrt3}{2}\frac{GM^2}{R^2}$

Answer

$\frac{\sqrt3}{4}\frac{GM^2}{R^2}$

Explanation

Solution

Force between any two spheres will be $F =\frac{G(M) (M) }{(2R)^2} = \frac{GM^2}{4R^2}$ The two forces of equal magnitude F are acting at angle $60^\circ$ on any o f the sphere $F_{net} = \sqrt{F^2 + F^2 + 2(F) (F) \, cos \, 60^\circ} = \sqrt{3}F$