Question

A metal rod of length $L$ and mass m is pivoted at one end. A thin disc 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. 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)

$ {\tau} _A = {\tau} _B = \bigg( mg \frac{L}{2} sin \theta + M g sin \theta \bigg) $
Restoring torque about point O .
In case A, moment of inertia will be more.
Hence, angular acceleration (a = x //) will
be less. Therefore, angular frequency will be
less.
$ \therefore $ Correct options are (a) and (d).