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Question

A uniform ring of mass $$m$$ and radius a is placed directly above a uniform sphere of mass $$M$$ and of equal radius. The centre of the ring is at a distance $$\sqrt{3}a $$ from the centre of the sphere. Find the gravitational force exerted by the sphere on the ring.


A
2GMm8a2
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B
3GMm4a2
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C
3GMm8a2
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D
3GMm7a2
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Solution

The correct option is C $$\displaystyle \frac{\sqrt{3}GMm}{8a^{2}}$$
$$dF =$$ force on a small mass $$'dm'$$ of the ring by the sphere.
Net force on ring$$=\sum \left ( dF\sin \theta  \right )$$ or $$\int dF\sin \theta $$

   $$\displaystyle =\sum \dfrac{GM\left ( dm \right )}{\left ( 2a \right )^{2}}\times \frac{\sqrt{3}}{2}$$

   $$=\displaystyle \dfrac{\sqrt{3}GM}{8a^{2}}\sum \left ( dm \right )$$

But $$\sum \left ( dm \right )=m$$, the mass of whole ring.

$$\therefore $$   Net force$$\displaystyle =\dfrac{\sqrt{3}GMm}{8a^{2}}$$ 

517169_219946_ans.png

Physics

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