Question

A uniform circular disc of radius $r$ is placed on a rough horizontal surface and given a linear velocity $v_0$ and angular velocity $\omega$ as shown. The disc comes to rest after moving some distance to the right. It follows that :

• A. $3v_0=2{\omega }_{0}r$
• B. $2v={\omega }_{0}r$
• C. $v_0={\omega }_{0}r$
• D. $2v_0=3{\omega }_{0}r$

Answer 1

Answer: (B)

$2v={\omega }_{0}r$

Since the disc comes to rest, it stops rotating and translating simultaneously $v=0$ and $\omega =0$. That means, the angular momentum about the instantaneous point of contact just after the time of stopping is zero. We know that the angular momentum of the disc about P remains constant because frictional force f, N and mg pass through point P and thus produce no torque about this point.
$\Rightarrow L_{initial}=L_{final}\Rightarrow mvr=l_0{\omega }_0=0$
$\Rightarrow mvr=\frac{1}{2}m{r}^2{\omega }_0\Rightarrow 2v={\omega }_{0}r$