Question

what is the time period of oscillation for rigid hinged body

Answer 1

solution:-

consider a rigid body of mass (M) hinged at point R as shown in figure.there is no friction between hinge and body.

if body is displaced from equilibrium by an angle $\theta$ . then restoring torque can be written as

$\tau=-Mgl\sin\theta$ .....eq.1

where $l$ is the distance from hinged point R to center of gravity.

for small oscillation

$\sin\theta≈\theta$

and torque for a rigid body can be given

$\tau=I\alpha$ ............eq.2

where $I$ is moment of inertia of body about hinged point and $\alpha$ is angular acceleration of body.

by the equation 1 and 2 we can written as

$-Mgl\theta=I\alpha$

$\frac{\theta}{\alpha}=-\frac{I}{Mgl}$

there are $-\theta$ is proportional to $\alpha$ so its simple harmonic motion. and time period can be given as

$T=2\pi\sqrt\vert\frac{\theta}{\alpha}\vert$

$\fcolorbox{green}{yellow}{$T=2\pi\sqrt\frac{I}{Mgl}$}$

this is the time period of oscillation for rigid hinged body.