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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 centre to the free end. A thin disk 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
When rod-disc system is lowest position total kinetic energy for case A = total kinetic energy for case B
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B
Restoring torque in case A = Restoring torque in case B
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C
Angular frequency for case A > Angular frequency for case B
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D
Angular frequency for case A < Angular frequency for case B
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Solution

## The correct options are B Restoring torque in case A = Restoring torque in case B C Angular frequency for case A > Angular frequency for case BGiven:- A metal rod of length L and mass mSolution:- Torque is same for both the cases.τA=τB=2π√I(M+m)gdHere,IA=mL33+(MR22+ML2)When the disc is free to rotate about its centre, it will not rotate w.r.t. ground. It will act as a point mass, so moment of inertia is given byIB=mL23+ML2So,IA>IB⇒TA>TB⇒ωA<ωBHence the correct option is B and C

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