Question

Consider a physical pendulum consisting of an inverted "$T$" shaped rigid body made out of two identical, thin, uniform rods of mass $M$ and length $L$ each joined together and hinged at the top point as shown, free to rotate in a vertical plane about an axis perpendicular to the plane of the figure, the time period for 'small' angular oscillations is $T=2\pi \sqrt{\frac{\alpha L}{\beta g}}$. Find $|\alpha -\beta |$.

Answer 1

Answer: 1

Use direct formula $T=2\pi \sqrt{\frac{I}{Mg{y}_{CM}}}$
where $I$ is the moment of inertia about the axis of rotation at the hinge,
$M$ the total mass and $y_{CM}$ the position of the center of mass w.r.t the axis of rotation.
Here, $I=\left[\frac{M{L}^2}{3}\right]+\left[\frac{M{L}^2}{12}+M{L}^2\right]$
$=\frac{17M{L}^2}{12}$ and total mass $=2M$
and $y_{CM}=3L/4$