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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:
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1703 Upvotes
Verified Answer
The correct answer is:
$\frac{\omega \mathrm{M}}{\mathrm{M}+2 \mathrm{~m}}$
Apply conservation of angular momentum.
$$
\begin{aligned}
\mathrm{L}_{\mathrm{i}} & =\mathrm{L}_{\mathrm{f}} \\
\mathrm{MR}^2 \omega & =(\mathrm{M}+2 \mathrm{~m}) \mathrm{R}^2 \omega^{\prime} \\
\omega^{\prime} & =\frac{\mathrm{M} \omega}{\mathrm{M}+2 \mathrm{~m}}
\end{aligned}
$$
$$
\begin{aligned}
\mathrm{L}_{\mathrm{i}} & =\mathrm{L}_{\mathrm{f}} \\
\mathrm{MR}^2 \omega & =(\mathrm{M}+2 \mathrm{~m}) \mathrm{R}^2 \omega^{\prime} \\
\omega^{\prime} & =\frac{\mathrm{M} \omega}{\mathrm{M}+2 \mathrm{~m}}
\end{aligned}
$$
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