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