Q: A uniform ring of mass *m* and radius *r* is placed directly above a uniform sphere of mass M and of equal radius. The centre of the ring is directly above the centre of the sphere at a distance as shown in the figure. The gravitational force exerted by the sphere on the ring will be

(a) $ \displaystyle \frac{G M m}{8 r^2}$

(b) $ \displaystyle \frac{G M m}{4 r^2}$

(c) $ \displaystyle \sqrt3 \frac{G M m}{8 r^2}$

(d) $ \displaystyle \frac{G M m}{8 r^3 \sqrt3}$

Ans: (c)

Sol:

Gravitational field due to ring at the centre of sphere is

$ \displaystyle E = \frac{G m r \sqrt 3 }{((r\sqrt 3)^2 + r^2)^{3/2}} $

Force on sphere is

F = M E