Question

A cylindrical drum, pushed along by a board rolls forward on the ground. There is no slipping at any contact. Find the distance moved by the man who is pushing the board, when axis of the cylinder covers a distance $L$

Answer 1

Let $v_0$ be the linear speed of the axis of the cylinder and $\omega$ be its angular speed about the axis. As it does not slip on the ground hence $\omega =\frac{v_0}{R}$, where $R$ is the radius of the cylinder.
Speed of the topmost point is $v=v_0+\omega R=2v_0$
Time taken by the axis to move a distance $L$ is equal to $t=L/v_0$
In the same interval of time distance moved by the topmost point is:
$S=2v_0\times =\frac{L}{v_0}=2L$
As there is no slipping between any point of contact
Hence distance moved by the man is $2L$