A uniform metal rod of length $L$ and mass $M$ is rotating about an axis passing through one of the ends and…
A uniform metal rod of length $L$ and mass $M$ is rotating about an axis passing through one of the ends and perpendicular to the rod with angular speed $\omega$. If the temperature increases by $t^{\circ} \mathrm{C}$, then the change in its angular velocity is proportional to which of the following?
(Coefficient of linear expansion of rod $=\alpha$ )
$\sqrt{\omega}$
$\omega$
$\omega^2$
$\frac{1}{\omega}$
Solution
From law of conservation of angular momentum
$L=$ constant
$\begin{aligned} I_1 \omega_1 & =I_2 \omega_2=L \\ \omega_1 & =\frac{L}{I_1}, \omega_2=\frac{L}{I_2}\end{aligned}$
Change in angular velocity
$\begin{aligned} \Delta \omega & =\omega_2-\omega_1=\frac{L}{I_2}-\frac{L}{I_1}=L\left(\frac{1}{I_2}-\frac{1}{I_1}\right) \\ & =L\left(\frac{I_1-I_2}{I_1 I_2}\right)=\frac{L}{I_1}\left(\frac{I_1}{I_2}-1\right) \\ & =\omega_1\left(\frac{I_1}{I_2}-1\right) \\ \Delta \omega & \propto \omega_1 \\ \Delta \omega & \propto \omega\end{aligned}$
,