Question

A thin circular ring of mass $M$ and radius $r$ is rotating about its axis with a constant angular velocity $\omega$. Four objects each of mass $m$, are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be-

• A. $\frac{M\omega }{4m}$
• B. $\frac{M\omega }{M+2m}$
• C. $\frac{(M+4m)\omega }{M}$
• D. $\frac{(M+4m)\omega }{M+4m}$

Answer 1

Answer: (B)

$\frac{M\omega }{M+2m}$

By using conservation of angular momentum

Initial angular $p$ $=$ Final angular $P$

$\begin{array}{c}I\omega =I^{\prime }{\omega }^{\prime }\\ M{R}^2\omega =\left(M{R}^2+m{R}^2+m{R}^2\right){\omega }^{\prime }\\ M{R}^2\omega =\left(M{R}^2+2m{R}^2\right){\omega }^{\prime }\\ M\omega =\left(M+2m\right){\omega }^{\prime }\\ {\omega }^{\prime }=\frac{M\omega }{M+2m}\end{array}$

Hence, Option $B$ is the correct answer.