Question

A string is wrapped on the surface of a solid cylinder as shown in the figure. If the cylinder is released, then find the acceleration of the centre of mass of the cylinder and the tension in the string respectively. Mass of the cylinder is $M$ and radius is $r$. (No slipping occurs between any contact points)

• A. $\frac{2g}{3},Mg$
• B. $\frac{g}{2},\frac{2Mg}{3}$
• C. $\frac{2g}{3},\frac{Mg}{3}$
• D. $\frac{4g}{3},\frac{Mg}{3}$

Answer 1

The correct option is C $\frac{2g}{3},\frac{Mg}{3}$


Let $T$ be the tension in the sting and $a$ be the acceleration of the centre of mass of the cylinder.

For translational motion :
$Mg-T=Ma.....\left(1\right)$

For rotational motion :
${\tau }_0=I\alpha$
(About the axis of rotation O)
$\Rightarrow Mg\times 0+T.r=\frac{M{r}^2}{2}\left(\frac{a}{r}\right)$
$(\because a=r\alpha )$
$\Rightarrow T=\frac{Ma}{2}.....\left(ii\right)$
From eq (1) & (ii),
$Mg-\frac{Ma}{2}=Ma$
$\Rightarrow a=\frac{2}{3}g$
& $T=\frac{Ma}{2}=\frac{M}{2}.\frac{2g}{3}=\frac{Mg}{3}$