A metal rod of length L and mass m is pivoted at one end. A thin...

A metal rod of length

L

and mass

m

is pivoted at one end. A thin disc of mass

M

and radius

R( < L)

is attached at its centre 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 centre and case

B

-the disc is free to rotate about its centre. 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 metal rod of length L and mass m is pivoted at one end. A thin disc of mass M and radius R(

A metal rod of length L and mass m is pivoted at one end. A thin disc of mass M and radius R(

Options

ARestoring torque in case $A=$ Restoring torque in case $B$
BRestoring torque in case $A <$ Restoring torque in case $B$
CAngular frequency for case $A>$ Angular frequency for case $B$
DAngular frequency for case $A <$ Angular frequency for case $B$

Correct answer

A. Restoring torque in case $A=$ Restoring torque in case $B$

Step-by-step solution

\tau_A=\tau_B=m g \frac{L}{2} \sin \theta+\operatorname{MgL} \sin \theta

= Restoring torque about point

O

. In case

A

, moment of inertia will be more. Hence, angular acceleration

(\alpha=\tau / I)

will be less. Therefore, angular frequency will be less. A metal rod of length L and mass m is pivoted at one end. A thin disc of mass M and radius R(

\therefore

Correct options are (a) and (d). Analysis of Question Question is difficult from my point of view. Because this type of SHM is rarely taught in the class and questions of this type are not given in standard books.

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