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A uniform ring of mass $m$ is lying at a distance $\sqrt{3}$ a from the centre of a sphere of mass $M$ just over the sphere (where $a$ ) is the radius of the ring as well as that of the sphere). Find the magnitude of gravitational force between them.

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

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

$\frac{\sqrt{3} G M m}{4 a^{2}}$

None of the above

answer is A.

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Detailed Solution

Net force on ring $= \int_{whole ring} d F \sin 60 °$

$= \int_{wole ring} \frac{G M (d m)}{(2 a)^{2}} \frac{\sqrt{3}}{2}$

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

as $\int_{whole ring} d m = m$

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