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? $Image$
$(A)$ 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 $B$.
IIT 2011, Advanced
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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
$\omega_{ B } > \omega_{ A }$
$A$ and $D$ are correct answer
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