Question

A thin uniform circular disco f mass $M$ and radius $R$ is rotating in a horizontal plane about an axis perpendicular to the plane at an angular velocity $\omega$. Another disc of mass $\frac{M}{3}$ but same radius is placed gently on the first disc coaxially. The angular velocity of the system now is

• A. $\frac{4\omega}{3}$
• B. $\omega$
• C. $\frac{3\omega}{4}$
• D. $\frac{3\omega}{8}$

Answer 1

Answer: (C)

$\frac{3\omega}{4}$

Answer 2

The principle of conservation of angular momentum is applied. Initial angular momentum of the first disc: $L_i = I_1 \omega = \frac{1}{2}MR^2 \omega$. The second disc is initially at rest. Moment of inertia of the first disc: $I_1 = \frac{1}{2}MR^2$. Moment of inertia of the second disc: $I_2 = \frac{1}{2}(\frac{M}{3})R^2 = \frac{1}{6}MR^2$. The total moment of inertia of the combined system is $I_f = I_1 + I_2 = \frac{1}{2}MR^2 + \frac{1}{6}MR^2 = \frac{2}{3}MR^2$. By conservation of angular momentum, $L_i = I_f \omega'$, where $\omega'$ is the final angular velocity. $\frac{1}{2}MR^2 \omega = \frac{2}{3}MR^2 \omega'$. Solving for $\omega'$ gives $\omega' = \frac{3}{4}\omega$.