Question

A metaI rod of length '$L$' and mass '$m$' is pivoted at one end. A thin disk of mass '$M$' and radius ${}^{\prime }{R}^{\prime }(<L)$ is attached at its center to the free end of the rod. Consider two ways the disc is attached: (case A) The disc is not free to rotate about its center and (case B) the disc is free to rotate about its center. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is (are) true?

• A. Restoring torque $in$ case $A=$ Restoring torque in case $B$
• B. Restoring torque in case $A<$ Restoring torque in case $B$
• C. Angular frequency for case $A>$ Angular frequency for case B.
• D. Angular frequency for case $A<$ Angular frequency for case B.

Answer 1

Answer: (A), (D)

Restoring torque $in$ case $A=$ Restoring torque in case $B$
Angular frequency for case $A<$ Angular frequency for case B.

Restoring torque is same in both cases,

$\propto =\frac{T}{I}=-{\omega }^2\theta$

In case A the moment of Inertia is more as compared to B, So

${\omega }_B>{\omega }_A$

A & D are correct answer