A ring of mass ' $m$ ' and radius $3 R$ is rotating with constant angular speed $ω (ω = \sqrt{\frac{G M}{9 R^{3}}})$ around a planet of mass $M$ and radius $R$ . Center of ring and planet coincide with each other. Tension in the ring is given as $T = \frac{G M m}{3 n π R^{2}} .$ Find value of ' $n$ '. (Assume elements of the ring don’t gravitationally interact with each other)

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Detailed Solution

$2 T \sin \frac{θ}{2} + \frac{G M (d m)}{{(3 R)}^{2}} = (d m) ω^{2} (3 R)$

For small $θ \Rightarrow T θ + \frac{G M (\frac{m}{2 π} . θ)}{9 R^{2}} = (\frac{m θ}{2 π}) ω^{2} (3 R) \Rightarrow T = \frac{G M m}{9 π R^{2}}$

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