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

Answer 1

The correct option is A l

At instant in between of a motion $FBD$ will be as :

For pure rolling

$\vec {v} A=0$

$\Rightarrow v-wR0\Rightarrow V=wR$

$\overrightarrow{v}B=v+wR=2v$

velocity of string unwrapping from the cylinder $=$ vel of $B$ w.r.t. center of mass

$=\overrightarrow{v}B/0$

$\overrightarrow{v}B/0=2v-v=v$

Distance traveled by the cylinder during its motion $={\int }_0^{t}v.dt=l$

where $t$ is time of motion

So,

Distance (length ) of the string passed through man's hand $={\int }_0^t\overrightarrow{v}B/0.dt$

$={\int }_0^{t}v.dt=l$

Option $A$ is correct.