Question

A string of negligible thickness is wrapped several times around a cylinder kept on a rough horizontal surface. A man standing at a distance l from the cylinder holds one end of the string and pulls the cylinder towards him (figure 10-Q4). There is no slipping anywhere. The length of the string passed through the hand of the man while the cylinder reaches his hands is
(a) l
(b) 2l
(c) 3l
(d) 4l.
Figure

Answer 1

(b) 2l

For pure rolling, $\omega r=v_0$

Figure:

As shown in the figure, the velocity of the string will be resultant of v₀ and ωr.
$$\begin{gathered} v_{net}=v_0+\left(\omega r\right) \\ {v}_{net}=2v_0 \end{gathered}$$
Let:
Linear distance travelled by the cylinder in time (t), $v_{0}t=l$
∴ Linear distance travelled by the string in same time, $2v_{0}t=2l$