Question

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 _____

Answer 1

Correct Answer: 12

Conservation of Angular Momentum 

By the conservation of angular momentum about the center of the large disc:

\( L_i = L_f = 0 \)

Initially, the system is at rest, so the initial angular momentum is zero. When the motor is switched on, the clockwise angular momentum of the smaller disc is balanced by the counterclockwise angular momentum of the large disc.

Angular Momentum Calculation

The angular momentum of the smaller disc is:

\[ L_1 = I_1 \cdot \omega_1 = \frac{1}{2} M \left(\frac{R}{2}\right)^2 \cdot \omega = \frac{1}{8} MR^2 \cdot \omega \]

The angular momentum of the large disc is:

\[ L_2 = -MvR = -M(\omega_1 R)R = -MR^2 \cdot \omega_1 \]

Equating \( L_1 \) and \( L_2 \):

\[ \frac{1}{8} MR^2 \cdot \omega = MR^2 \cdot \omega_1 \]

Simplifying to find \( \omega_1 \):

\[ \omega_1 = \frac{\omega}{8} \]

Finding the Value of \( n \)

Since the angular speed of the large disc is given by \( \omega/n \):

\[ \frac{\omega}{n} = \frac{\omega}{12} \]

Thus, the value of \( n \) is:

n = 12

Final Answer:

n = 12