Question

A uniform rod $AB$ of mass $3m$and length $4l$ which is free to tum in a vertical plane about a smooth horizontal axis through $,A$ is released from rest when horizontal. When the rod first becomes vertical, a point $C$ of the rod, where $AC=3l$ strikes a fixed peg. Find the linear impulse exerted by the peg on the rod if the rod is brought to rest by the peg.

Answer 1

From conservation of mechanical energy.
$\left(3m\right)\left(g\right)\left(2l\right)=\frac{1}{2}I{\omega }^2$
$=\frac{1}{2}\left[\frac{\left(3m\right)(4l{)}^2}{3}\right]{\omega }^2=8m{l}^2{\omega }^2$
$\therefore \omega =\frac{1}{2}\sqrt{\frac{3g}{l}}$
Applying, angular impulse $=$ change in angular momentum
$J\left(3l\right)=I\omega$ or $3Jl=\left(16m{l}^2\right)\left(\frac{1}{2}\sqrt{\frac{3g}{l}}\right)$
or $J=\frac{8}{3}ml\sqrt{\frac{3g}{l}}$ or
$J=\frac{8}{3}m\sqrt{3gl}$