Question

Three uniform spheres of mass M and radius R each 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{\sqrt{3}}{4}$$\frac{{GM}^2}{R^2}$
• B. $\frac{3}{2}$$\frac{{GM}^2}{R^2}$
• C. $\sqrt{3}$$\frac{{GM}^2}{R^2}$
• D. $\frac{\sqrt{3}}{2}$$\frac{{GM}^2}{R^2}$

Answer 1

The correct option is A

$\frac{\sqrt{3}}{4}$$\frac{{GM}^2}{R^2}$

Force between any two spheres will be :

F = $\frac{G\left(M\right)\left(M\right)}{2R^2}$ = $\frac{GM}{{4R}^2}$

The two forces of equal magnitude F are acting at angle ${60}^{\circ }$ on any of the sphere

$\therefore$ $F_{net}$ = $\sqrt{F^2+F^2+2\left(F\right)\left(F\right)cos{60}^{\circ }}$ = $\sqrt{3}$F

or $F_{net}$ = $\frac{\sqrt{3}}{4}$ $\frac{{GM}^2}{R^2}$