Question

A metal rod of length 'L' and mass ' $m$ ' is pivoted at one end. A thin disk of mass ' $M$ ' and radius ' $R'(< L)$ 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)

In case A
$mg (\ell / 2) \sin \theta +Mg \ell \sin \theta$
$=\left(\frac{ m \ell^{2}}{3}+\frac{ MR ^{2}}{2}+ M \ell^{2}\right) \alpha_{ A }$
In case B
$mg (\ell / 2) \sin \theta+ Mg \ell \sin \theta$
$=\left(\frac{ m \ell^{2}}{3}+ M \ell^{2}\right) \alpha_{ B }$
$\tau_{ A }=\tau_{ B }, \omega_{ A }<\omega_{ B }$