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
B
3GMm4a2
C
3GMm8a2
D
3GMm7a2

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$$\displaystyle =\dfrac{\sqrt{3}GMm}{8a^{2}}$$ Physics

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