Question

A solid cylinder is projected on a rough surface having coefficient of friction $\mu$, with velocity $v_0$ and angular velocity ${\omega }_0=\frac{v_0}{2r}$, (whre $r$ is radius). What is the time $t$ at which rolling without slipping occurs is

• A. $\frac{v_0}{3\mu g}$
• B. $\frac{v_0}{4\mu g}$
• C. $\frac{v_0}{2\mu g}$
• D. $\frac{v_0}{\mu g}$

Answer 1

The correct option is B $\frac{v_0}{4\mu g}$

Rolling without slipping will occur at $v_o=r\omega$ or at $\omega =\frac{v_o}{r}$.
Also using torque equilibrium we have $\mu mgr=I\alpha$
or
$\mu mgr=\frac{1}{2}m{r}^2\alpha$
or
$\alpha =\frac{2\mu g}{r}$
Now using the equation ${\omega }_f={\omega }_i+\alpha t$ we have
$\frac{v_o}{r}=\frac{v_o}{2r}+\frac{2\mu g}{r}t$
or
$t=\frac{v_o}{4\mu g}$