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

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$\frac{G M m}{8 r^{2}}$

$\frac{G M m}{8 r^{3} \sqrt{3}}$

$\frac{G M m}{4 r^{2}}$

$\sqrt{3} \frac{G M m}{8 r^{2}}$

answer is C.

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Detailed Solution

Consider a small length element of the ring with mass $d m$ . Gravitational force between element $d m$ and sphere,

$d F = G \frac{M d m}{4 r^{2}} T o t a l f o r c e, F = Σ d F \cos θ = Σ \frac{G M d m}{4 r^{2}} \cos θ = \frac{G M}{4 r^{2}} \times \frac{\sqrt{3} r}{2 r} Σ d m = \frac{\sqrt{3} G M m}{8 r^{2}}$

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