A metal rod of length 'L' and mass 'm' is pivoted at one end. A thin disc of mass 'M' and radius 'R' (ltL) is attached at its center to the free end of the rod. Consider two ways the disc is attached : (case A). The dise is not free to rotate about its centre and (case B) the disc is free to rotate about its centre. The rod disc system perfoms (SHM) in vertical plane after being released from the same displacement position. Which of the following statement (s) is (are) true ?
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Options

A. Restoring torque in case `A` = Resorting torque in case `B`
B. Restoring torque in case `A` lt Restoring torque in case `B`
C. Angular frequency for case `A` gt Angular frequency for case `B`
D. Angular frequency for case `A` lt Angular frequency for case `B`.

Answer

A, D

Explanation

Torque is same for both the cases.
`T=2pi sqrt((I)/((M+m)gd))`
`I_(A) =(mL^(2))/(3) +((MR^(2))/(2)+ ML^(2))`
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 by
`I_(B)=(mL^(2))/(3)+ ML^(2)`
`I_(A) gt I_(B) rArr T_(A) gt T_(B) rArr omega_(A) lt omega_(B)`.