Question

Disc of mass $M=10\text{kg}$ and radius $R=50\text{cm}$ is revolving about fixed vertical axis passing through its centre and perpendicular to its plane, on smooth horizontal surface with angular velocity ${\omega }_0=10\text{rad/sec}$. A block of mass $m=5\text{kg}$ is connected through disc through slack string as shown in the diagram. Find the angular velocity (in $\text{rad/s}$) when string becomes taut.
Assume that rope doesnot slip over the disc

Answer 1

Let, angular velocity $\left({\omega }_f\right)=\omega$ after string becomes taut.
As there is no external torque exerted on system, angular momentum will be conserved.
Initial angular momentum = final angular momentum
$\left(\frac{1}{2}M{R}^2\right){\omega }_0=\frac{1}{2}M{R}^2\times \omega +mvr$ {here $r=R$}
$\frac{1}{2}\times M\times R^2{\omega }_0=\frac{1}{2}M{R}^2\omega +mvR$
As $v=R\omega$
$\Rightarrow \frac{1}{2}M{\omega }_0=(\frac{1}{2}M+m)\omega$
$\Rightarrow \omega =\frac{M{\omega }_0}{M+2m}=\frac{10\times 10}{10+10}=5\text{rad/s}$