Question

A pulley having radius r and moment of inertia $I$ about its axis is fixed at the top of an inclined plane of inclination $\theta$ as shown in the figure. A string is wrapped around the pulley and its free end supports a block of mass $m$ which can slide on the plane. Initially, the pulley is rotating at a speed ${\omega }_0$ in a direction such that the block slides up the plane. Calculate the distance moved by the block before stopping?

Answer 1

Given,

Initial angular velocity ${\omega }_o$

Radius of pulley is $r$

On inclination component of gravity acceleration is $g\sin \theta$ act on block.

initial velocity of block $u=r{\omega }_o$

Final velocity of block $v=0$

Apply kinematic equation of motion

$v^2-u^2=2as$

$0-{\left(r{\omega }_o\right)}^2=2(-g\sin \theta )s$

$s=\frac{r^2{\omega }_o^2}{2\sin \theta }$Distance moved by block before stop is$\frac{r^2{\omega }_o^2}{2\sin \theta }$