Question

A uniform rod $AB$ of length $2l$ and mass $m$ is rotating in a horizontal plane about a vertical axis through $A$ with angular velocity $\omega$ when the mid-point of the rod strikes a fixed nail and is brought immediately to rest. If the impulse exerted by the nail is $\frac{4}{x}ml\omega$, find the value of '$x$'.

Answer 1

Angular impulse $=$ change in angular momentum
$\therefore (J\times r_{\perp })=0-I\omega =-I\omega$
where, $J=$ linear impulse $=\frac{I\omega }{r_{\perp }}$
where $r_{\perp }=\frac{2l}{2}=l$ as rod strikes nail at its mid point
Moment of inertia of rod rotating about an axis passing through it's end point and perpendicular to its length is given by
$I=\frac{M{L}^2}{3}=\frac{m(2l{)}^2}{3}$
$J=\frac{\left(m\right)\frac{{\left(2l\right)}^2}{3}\cdot \omega }{l}$
$=\frac{4}{3}ml\omega$