A thin circular ring of mass $\mathrm{M}$ and radius $\mathrm{R}$ is rotating in a horizontal plane about an…

A thin circular ring of mass $\mathrm{M}$ and radius $\mathrm{R}$ is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity $\omega$. If two objects each mass $m$ be attached gently to the opposite ends of a diameter of the ring, the ring, will then rotate with an angular velocity:
  1. $\frac{\omega \mathrm{M}}{\mathrm{M}+\mathrm{m}}$
  2. $\frac{\omega(M-2 m)}{M+2 m}$
  3. $\frac{\omega \mathrm{M}}{\mathrm{M}+2 \mathrm{~m}}$
  4. $\frac{\omega(M+2 m)}{M}$

Solution

Apply conservation of angular momentum. $\begin{aligned} \mathrm{L}_{\mathrm{i}} & =\mathrm{L}_{\mathrm{f}} \\ \mathrm{MR}^2 \omega & =(\mathrm{M}+2 \mathrm{~m}) \mathrm{R}^2 \omega^{\prime} \\ \omega^{\prime} & =\frac{\mathrm{M} \omega}{\mathrm{M}+2 \mathrm{~m}} \end{aligned}$ ~

Asked in: NEET 2009 (Mains)

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