Question

A uniform circular cylinder of mass $m$ and radius $r$ is given an initial angular velocity ${\omega }_0$ and no initial translational velocity. It is placed in contact with a plane inclined at an angle a to the horizontal. If there is a coefficient of friction $\mu$ for sliding between the cylinder and plane. Find the distance the cylinder moves up before sliding stops. Also, calculate the maximum distance it travels up the plane. Assume $\mu >\tan \alpha$

Answer 1

Given $\mu >\tan \alpha \Rightarrow \mu mg\cos \alpha >mgsin\alpha$

$a=(\mu g\cos \alpha -g\sin \alpha )$

$\alpha =\frac{\left(\mu mg\cos \alpha \right)r}{\frac{1}{2}m{r}^2}=\frac{2\mu g\cos \alpha }{r}$

Slipping will stop when,

$v=r\omega$ or $at=r({\omega }_0-\alpha t)$

$\therefore t=\frac{r{\omega }_0}{a+r\alpha }=\left(\frac{r{\omega }_0}{3\mu g\cos \alpha -g\sin \alpha }\right)$

$d_1=\frac{1}{2}a{t}^2$

$=\frac{1}{2}(\mu g\cos \alpha -g\sin \alpha ){\left(\frac{r{\omega }_0}{3\mu g\cos \alpha -g\sin \alpha }\right)}^2$

$=\frac{r^2{\omega }_0^2(\mu \cos \alpha -\sin \alpha )}{2g(3\mu \cos \alpha -\sin \alpha {)}^2}$

$v=at=(\mu g\cos \alpha -g\sin \alpha )\left(\frac{r{\omega }_0}{3\mu g\cos \alpha -g\sin \alpha }\right)$

$=\frac{r{\omega }_0(\mu \cos \alpha -\sin \alpha )}{(3\mu \cos \alpha -\sin \alpha )}$

Once slipping is stopped, retardation in cylinder,

$a^{\prime }=\frac{g\sin \alpha }{1+\frac{I}{m{r}^2}}=\frac{g\sin \alpha }{1+\frac{1}{2}}=\frac{2}{3}g\sin \alpha$
$d_2=\frac{v^2}{2a^{\prime }}=\frac{3r^2{\omega }_0^2(\mu \cos \alpha -\sin \alpha {)}^2}{(3\mu \cos \alpha -\sin \alpha {)}^2(4g\sin \alpha )}$

$\therefore d_{max}=d_1+d_2$

$=\frac{r^2{\omega }_0^2(\mu \cos \alpha -\sin \alpha )}{2g(3\mu \cos \alpha -\sin \alpha {)}^2}$ $[1+\frac{3(\mu \cos \alpha -\sin \alpha )}{2\sin \alpha }]$

$=\frac{r^2{\omega }_0^2(\mu \cos \alpha -\sin \alpha )}{4g\sin \alpha (3\mu \cos \alpha -\sin \alpha )}$

Note : Once slipping was stopped, pure rolling continues if

$\mu >\frac{\tan \alpha }{1+\frac{m{r}^2}{I}}$ or $\mu >\frac{\tan \alpha }{1+2}$ or

$\mu >\frac{\tan \alpha }{3}$
and
already in the question it is given that $\mu >\tan \alpha$ That's
why we bave taken $a^{\prime }=\frac{2}{3}g\sin \alpha$