A thin circular ring of mass $M$ and radius $r$ is rotating about its axis with constant angular velocity…

A thin circular ring of mass $M$ and radius $r$ is rotating about its axis with constant angular velocity $\omega$. Two objects each of mass $m$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with angular velocity given by
  1. $\frac{(M+2 m) \omega}{2 m}$
  2. $\frac{2 \mathrm{M} \omega}{\mathrm{M}+2 \mathrm{~m}}$
  3. $\frac{(M+2 m) \omega}{M}$
  4. $\frac{M \omega}{M+2 m}$

Solution

In the absence of external torque, angular momentum remain constant $\begin{aligned} \mathrm{L} & =\mathrm{I} \omega=\mathrm{I}^{\prime} \omega^{\prime} \\ \therefore \quad \mathrm{MR}^2 \omega & =(\mathrm{M}+2 \mathrm{~m}) \mathrm{R}^2 \omega^{\prime} \\ \omega^{\prime} & =\frac{\mathrm{M} \omega}{(\mathrm{M}+2 \mathrm{~m})} \end{aligned}$

Asked in: NEET 2010 (Mains)

Practice more Center of Mass Momentum and Collision questions on Aicharya