Question

A uniform ring of mass ‘m’ and radius ‘a’ is placed directly above a uniform sphere of mass ‘M’ and of equal radius. The centre of the ring is at a distance $\sqrt{3}a$ from the centre of the sphere. The gravitational force exerted by the sphere on the ring is

• A. $\frac{GMm}{8a^2}$
• B. $\frac{8GMm}{\sqrt{3}{a}^2}$
• C. $\frac{\sqrt{3}GMm}{8a^2}$
• D. $\frac{GMm}{\sqrt{3}{a}^2}$

Answer 1

The correct option is C $\frac{\sqrt{3}GMm}{8a^2}$

The gravitational field due to the ring at a distance \(d = \sqrt {3}a\) on its axis is
$E=\frac{Gmd}{(a^2+d^2{)}^{\frac{3}{2}}}=\frac{\sqrt{3}Gm}{8a^2}$
The force on a particle of mass ‘M’ placed here is F = ME
$\Rightarrow F=\frac{\sqrt{3}GMm}{8a^2}$
This is also the force due to the sphere on the ring.