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

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

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

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

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

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Solution

The correct option is A $\sqrt{3} \frac{G M m}{8 r^{2}}$
$d F = G \frac{m d m}{4 r^{2}}$
$F = \sum d F c o s θ$
or $F = \sum \frac{G M d m}{4 r^{2}} c o s θ$
or $F = \frac{G M}{4 r^{2}} \times \frac{\sqrt{3} r}{2 r} \sum d m$
or $F = \frac{\sqrt{3} G M m}{8 r^{2}}$

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or $F = \frac{G M}{4 r^{2}} \times \frac{\sqrt{3} r}{2 r} \sum d m$
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