Question

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 $r\sqrt{3}$ as shown in the figure. The gravitational force exerted by the sphere on the ring will be

• A. $\frac{GMm}{8r^{2}}$
• B. $\frac{GMm}{4r^{2}}$
• C. $\sqrt{3}\frac{GMm}{8r^{2}}$
• D. $\frac{GMm}{8r^{3}\sqrt{3}}$

Answer 1

Answer: (C)

$\sqrt{3}\frac{GMm}{8r^{2}}$

Answer 2

$dF = G \frac{Mdm}{4r^{2}}$ $F = \Sigma dF\,cos\,\theta$ $= \Sigma \frac{GMdm}{4r^{2}} cos\,\theta$ $= \frac{GM}{4r^{2}} \times \frac{\sqrt{3}r}{2r} \Sigma dm$ $= \frac{\sqrt{3}GMm}{8r^{2}}$