A square plate of mass M and side length ‘a’ is hinged at one of its vertex (A) and is free to rotate about it the time period of small oscillations (in seconds ) is equal to 2πN2a3g ifThe plate performs oscillations in the vertical plane of the figure. (Axis is perpendicular to figure.) Find the value of N.

Moment of inertia about the axis of oscillation is IA=23 Ma2 Distance of COM from point of suspension is l=a2∴ T=2πIAmgl ∴T=2π23Ma2Mga2 =2πNa23g

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A square plate of mass M and side length ‘a’ is hinged at one of its vertex (A) and is free to rotate about it the time period of small oscillations (in seconds ) is equal to
$2 π \sqrt{\frac{N \sqrt{2} a}{3 g}} if$
The plate performs oscillations in the vertical plane of the figure. (Axis is perpendicular to figure.) Find the value of N.

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Detailed Solution

Moment of inertia about the axis of oscillation is $I_{A} = \frac{2}{3} M a^{2}$
Distance of COM from point of suspension is $l = \frac{a}{\sqrt{2}}$
$∴ T = 2 π \sqrt{\frac{I_{A}}{m g l}} ∴ T = 2 π \sqrt{\frac{\frac{2}{3} M a^{2}}{M g \frac{a}{\sqrt{2}}}} = 2 π \sqrt{\frac{N a \sqrt{2}}{3 g}}$