System Of Particles And Rotational Motion Ques 1
1. 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 of mass $m$ be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity:
[2009, 1998]
(a) $\frac{\omega M}{M+2 m}$
(b) $\frac{\omega(M+2 m)}{M}$
(c) $\frac{\omega M}{M+m}$
(d) $\frac{\omega(M-2 m)}{M+2 m}$
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Answer:
Correct Answer: 1.(a)
Solution: (a) In absence of external torque, $\mathrm{L}=\mathrm{I} \omega$ $=$ constant
$ \mathrm{I}_1 \omega_1=\mathrm{I}_2 \omega_2, \mathrm{I}_1=\mathrm{MR}^2, \mathrm{I}_2=\mathrm{MR}^2+2 \mathrm{mR}^2 $
(Moment of inertia of a thin circular ring about an axis vertical to its plane $=\mathrm{MR}^2$ )
$ \therefore \quad \omega_2=\frac{\mathrm{I}_1}{\mathrm{I}_2} \omega=\frac{\mathrm{M}}{\mathrm{M}+2 \mathrm{m}} \omega . $
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