Question

A uniform ring of mass M and radius R is placed directly above a uniform sphere of mass 8 M and of same radius R. The centre of the ring is at a distance of $d=\sqrt{3}R$ from the centre of the sphere. The gravitational attraction between the sphere and the ring is :

• A. $\frac{8G{M}^2}{R^2}$
• B. $\frac{2G{M}^2}{\sqrt{3}{R}^2}$
• C. $\frac{3G{M}^2}{2R^2}$
• D. $\frac{\sqrt{3}G{M}^2}{R^2}$

Answer 1

The correct option is D $\frac{\sqrt{3}G{M}^2}{R^2}$

Gravitational field due to ring at a distance $d=\sqrt{3}R$ on its axis is

$E=\frac{GMd}{(R^2+d^2{)}^{3/2}}=\frac{GM\sqrt{3}R}{(R^2+3R^2{)}^{3/2}}=\sqrt{3}\frac{GMR}{8R^3}=\frac{\sqrt{3}GM}{8R^2}$

Therefore, Force on the sphere = Mass of the sphere $\times E$

Force on the sphere $=8ME=\frac{\sqrt{3}G{M}^2}{R^2}$