A disc of mass $M$ and radius $R$ is free to rotate about its vertical axis as shown in the figure. A…
A disc of mass $M$ and radius $R$ is free to rotate about its vertical axis as shown in the figure. A battery operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass $M$ and radius $R / 2$ is fixed to the motor's thin shaft. Initially, both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed $\omega$. If the angular speed at which the large disc rotates is $\omega / n$, then the value of $n$ is ______ .
Solution
On applying conservation of angular momentum about axis of larger disc.
$\begin{aligned} & \frac{1}{2} \cdot \mathrm{M}\left(\frac{\mathrm{R}}{2}\right)^2 \cdot \omega-\mathrm{M}\left(\omega^{\prime} \mathrm{R}\right) \cdot \mathrm{R}-\frac{\mathrm{MR}^2}{2} \cdot \omega^{\prime}=0 \\ & \Rightarrow \frac{\omega}{8}=\frac{3 \omega^{\prime}}{2} \\ & \Rightarrow \omega^{\prime}=\frac{\omega}{12} \quad \text { Hence, } \mathrm{n}=12\end{aligned}$
On applying conservation of angular momentum about axis of larger disc.
$\begin{aligned} & \frac{1}{2} \cdot \mathrm{M}\left(\frac{\mathrm{R}}{2}\right)^2 \cdot \omega-\mathrm{M}\left(\omega^{\prime} \mathrm{R}\right) \cdot \mathrm{R}-\frac{\mathrm{MR}^2}{2} \cdot \omega^{\prime}=0 \\ & \Rightarrow \frac{\omega}{8}=\frac{3 \omega^{\prime}}{2} \\ & \Rightarrow \omega^{\prime}=\frac{\omega}{12} \quad \text { Hence, } \mathrm{n}=12\end{aligned}$