Question

Two identical small balls each of mass $m$ are rigidly affixed at the ends of light rigid rod of length $1.5m$ 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 $\theta =gl$ (numerically in radians), where $g$ is the acceleration of free fall.

Answer 1

Answer: 6

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