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

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