Question

Two identical small balls each of mass $m$ are rigidly affixed at the ends of light rigid rod of length $1.5\, m$ and the assembly is placed symmetrically on an elevated protrusion of width $l$ as shown in the figure. The rod-ball assembly is tilted by a small angle $\theta$ in the vertical plane as shown in the figure and released. Estimate time-period (in seconds) of the oscillation of the rod-ball assembly assuming collisions of the rod with corners of the protrusion to be perfectly elastic and the rod does not slide on the corners. Assume $\theta=g l$ (numerically in radians), where $g$ is the acceleration of free fall.

Answer 1

$\tau=I \alpha$
$M g(a +l) \cos \theta-M g a \cos a=2 M a^{2} \alpha$
$M g l \cos \theta=2 m a^{2} \alpha$
$\alpha=\frac{g l}{2 a^{2}} \cos \theta \simeq \frac{g l}{2 a^{2}}$
$\theta=\frac{1}{2} \alpha t^{2}$
$\theta=\frac{1}{2} \frac{g l}{2 a^{2}} t^{2}$
$t=2 a$
$T=4 t=8 a=8 \times\left(\frac{1.5}{2}\right)$
$T=6\, s$