Question

A uniform rod $A B$ of length $l$ 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

Answer: (C)

Applying conservation of angular momentum,
$I \omega+m v \frac{l}{2}=\frac{m l^{2}}{3} \omega^{\prime} $
$\frac{m l^{2}}{12} \omega+m\left(\frac{\omega l}{6}\right) \cdot \frac{l}{2}=\frac{m l^{2}}{3} \omega^{\prime}$
$\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}$