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 $\ell$ from the cylinder holds on end of the string and pulls the cylinder towards him as shown in figure. There is no slipping anywhere. The length of the string passed through the hand of the man while the cylinder reaches his hands is $n \times \ell$. Find $n$.

Answer 1

$T \times 2 R =\frac{3}{2} mR ^{2} \alpha$
$ \alpha =\frac{4 T }{3 mR } $
$\alpha R =\frac{4 T }{3 m } $
$ a =\alpha R $
$ a =\frac{4 T }{3 m } $
$ a _{ A } =\frac{8 T }{3 m } $
$ \theta=\frac{1}{2} \alpha t ^{2} $
length of wire unwind $=R \theta$
$= R \times \frac{1}{2} \times \frac{2 \ell}{ a }$
through hand $=\ell+R=2 \ell$