Question

A heavy plank is moving with horizontal acceleration $a=$ $2m/s^2$ as shown in figure. A hollow cylinder of mass $m=3$ kg and radius = $25$ cm is on the upper surface of the plank and moves without slipping. Find the angular acceleration (about the axis in $rad/s^2$) of the hollow cylinder?

Answer 1

As friction is the only force supporting the motion of hollow cylinder :
$f=m{a}_{C.M.}$....(i)
Taking torque of frictional about cylinder axis
$fR=I\alpha$...(ii)
$f=M{R}^2\alpha$....(iii)
As cylinder is rolling without slipping the acceleration of point of contact should be 2 $m/s^2$
$a_{C.M.}+R\alpha =2m/s^2$
from (i) and (iii) $a_C.M.=R\alpha$
$2R\alpha =2m/s^2$
$\alpha =4rad/s^2$