A uniform disc of radius R and mass M is free to oscillate about the axis A as shown in the figure

Question

A uniform disc of radius R and mass M is free to oscillate about the axis A as shown in the figure. For small oscillations the time period is ______. (g is acceleration due to gravity) [2026]

Smart Achievers · Accepted Answer

(1)

$I_{disc} = \frac{m R^{2}}{4} + m R^{2} = \frac{5 m R^{2}}{4}$

$∴$ $For small displacement$

$τ = - m g R θ$

$∴$ $α = - \frac{m g R}{\frac{5 m R^{2}}{4}} θ$

$∴$ $ω = \sqrt{\frac{4 g}{5 R}}$

$∴$ $T = 2 π \sqrt{\frac{5 R}{4 g}}$

Solution

Q.
A uniform disc of radius R and mass M is free to oscillate about the axis A as shown in the figure. For small oscillations the time period is ______. (g is acceleration due to gravity) [2026]

1 $2 π \sqrt{\frac{5 R}{4 g}}$

2 $2 π \sqrt{\frac{2 R}{3 g}}$

3 $2 π \sqrt{\frac{3 R}{2 g}}$

4 $2 π \sqrt{\frac{3 R}{g}}$

Ans.
(1)

$I_{disc} = \frac{m R^{2}}{4} + m R^{2} = \frac{5 m R^{2}}{4}$

$∴$ $For small displacement$

$τ = - m g R θ$

$∴$ $α = - \frac{m g R}{\frac{5 m R^{2}}{4}} θ$

$∴$ $ω = \sqrt{\frac{4 g}{5 R}}$

$∴$ $T = 2 π \sqrt{\frac{5 R}{4 g}}$

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Solution

Q. A uniform disc of radius R and mass M is free to oscillate about the axis A as shown in the figure. For small oscillations the time period is ______. (g is acceleration due to gravity) [2026]

1 2π5R4g

2 2π2R3g

3 2π3R2g

4 2π3Rg

Ans. (1) Idisc=mR24+mR2=5mR24 ∴ For small displacement τ=-mgRθ ∴ α=-mgR5mR24θ ∴ ω=4g5R ∴ T=2π5R4g