A metal rod of length `'L'` and mass `'m'` is piovted at one end. A thin disk of mass `'M'` and radius `'R' (lt 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 center. The rod-disc sustem perform `SHM` in vertical plane after being released from the same displaced potion. Which of the following statement(s) is (are) true?

Options

A. Restoring torque in case A = Restoring 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

AD

Explanation

Torque is same both the cases.
`T=2pisqrt((I)/((M+m)gd))` (Where d is the distance of C.M from pivot)
`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_(E)=(mL^(2))/(3)+ML^(2)` because `I_(A)gtI_(B) implies T_(A)gtT_(B)implies omega_(A)ltomega_(B)`