A metal rod of length ‘L’ and mass ‘m’ is pivoted at one end. A thin disc of mass ‘M’ and radius ‘R’ (

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 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 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? [2011]

Smart Achievers · Accepted Answer

(1, 4)

Applying, torque $τ = I α$

For case A: $m g (\frac{ℓ}{2} \sin θ) + M g (ℓ \sin θ) = I_{A} α_{A} ...(i)$

For case B: $m g (\frac{ℓ}{2} \sin θ) + M g (ℓ \sin θ) = I_{B} α_{B} ...(ii)$

From eq. (i) & (ii)

$I_{A} α_{A} = I_{B} α_{B}$

Since $I_{A} > I_{B}$ $∴$ $α_{A} < α_{B}$

Hence, $ω_{A} < ω_{B}$

Solution

1 Restoring torque in case A = Restoring torque in case B

2 Restoring torque in case A < Restoring torque in case B

3 Angular frequency for case A > angular frequency for case B.

4 Angular frequency for case A < angular frequency for case B.

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