Question

A uniform rod AB of length $l$ is rotating with an angular velocity $\omega$ while its centre moves with a linear velocity $v=\frac{\omega l}{6}$. If the end A of the rod is suddenly fixed, the angular velocity of the rod will be:

• A. $\frac{3}{4}\omega$
• B. $\frac{\omega }{3}$
• C. $\frac{\omega }{2}$
• D. $\frac{2}{3}\omega$

Answer 1

The correct option is C $\frac{\omega }{2}$

Applying conservation of angular momentum about end A,
$I\omega +mv\frac{l}{2}=\frac{m{l}^2}{3}{\omega }^{\prime }$
$\frac{m{l}^2}{12}\omega +m\left(\frac{\omega l}{6}\right)\frac{l}{2}=\frac{m{l}^2}{3}{\omega }^{\prime }$
$\Rightarrow \frac{\omega }{12}+\frac{\omega }{12}=\frac{{\omega }^{\prime }}{3}$
$\Rightarrow \frac{{\omega }^{\prime }}{3}=\frac{2\omega }{12}$
$\Rightarrow {\omega }^{\prime }=\frac{\omega }{2}$.