Question

A thin perfectly rigid weightless rod with a point-like ball fixed at one end is deflected through a small angle $\alpha$ from its equilibrium position and then released. At the moment when the rod forms an angle $\beta <\alpha$ with the vertical, the ball undergoes a perfectly elastic collision with an inclined wall.
Determine the ration $T_1/T$ of the period of oscillations of this pendulum to the period of oscillations of a simple pendulum having the same length.

Answer 1

In the absence of the wall, the angle of deflection of the simple pendulum varies harmonically with a period $T$ and an angular amplitude $\alpha$. the projecttion of the point rotating in a circle of radius $\alpha$ at an angular velocity $\omega =2\pi /T$ perform the same motion. The perfectly elastic collision of the rigid rod with the wall at angle of deflection $\beta$ corresponds to an instantaneous jump reduced by
$\Delta t=2\gamma /\omega$,
where
$\gamma =arc\cos \left(\beta /\alpha \right)$.
$T_1=\frac{2\pi }{\omega }-\frac{2\gamma }{\omega },$and the sought solution is$\frac{T_1}{T}=1-\frac{1}{\pi }arc\cos \frac{\beta }{\alpha }$.