Question

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

In case A
$mg(\ell /2)\sin \theta +Mg\ell \sin \theta$
$=\left(\frac{m{\ell }^2}{3}+\frac{M{R}^2}{2}+M{\ell }^2\right){\alpha }_A$
In case B
$mg(\ell /2)\sin \theta +Mg\ell \sin \theta$
$=\left(\frac{m{\ell }^2}{3}+M{\ell }^2\right){\alpha }_B$
${\tau }_A={\tau }_B,{\omega }_A<{\omega }_B$