Question

A block of mass $m$ is attached to a pulley disc of equal mass $m$, radius $r$ by means of a slack string as shown. The pulley is hinged about its centre on a horizontal table and the block is projected with an initial velocity of $5m/s$. The velocity when the string becomes taut is given as $\frac{10}{x}m/s$. Find $x$.

Answer 1

$\because$ there is no external torque on 'block plus disc pulley' system,
$\therefore$ there will be no change in angular momentum of system about the axis of disc

Initial angular momentum of system: $\left(L_1\right)=0\left(disc\right)+mvr\left(block\right)$
Final angular momentum of system $\left(L_2\right)=I\omega +m{v}^{\prime }r$;

$v^{\prime }=$ velocity of block when string becomes taut

So, $mvr=Iw+m{v}^{\prime }r$, where moment of inertia: $I=\frac{1}{2}m{r}^2$
$\Rightarrow$ $mvr=\frac{1}{2}m{r}^{2}w+m{v}^{\prime }r$ ; now $v^{\prime }=rw$
$\Rightarrow$ $mvr=\frac{1}{2}m{v}^{\prime }r+m{v}^{\prime }r$
or, $\frac{3}{2}{v}^{\prime }=v$
or, $v^{\prime }=\frac{2}{3}v$ ; given, $v=5m/s$
$\Rightarrow$ $v^{\prime }=\frac{10}{3}m/s$

So, $x=3$