Question

A string is wrapped several times round a solid cylinder and then the end of the string is held stationary while the cylinder is released from rest with an initial motion. The acceleration of the cylinder and tension in the string will be

• A. $\frac{2g}{3}and\frac{mg}{3}$
• B. $gand\frac{mg}{2}$
• C. $\frac{g}{3}and\frac{mg}{2}$
• D. $\frac{g}{2}and\frac{mg}{3}$

Answer 1

The correct option is A $\frac{2g}{3}and\frac{mg}{3}$

Let the mass of cylinder is m and radius of it is r.

so, moment of inertia of cylinder , $I=\frac{1}{2}m{r}^2$

torque = moment of inertia * angular acceleration

force * displacement $=\frac{1}{2}m{r}^2\times \frac{a}{r}$

we know, linear acceleration, a = r * angular acceleration

$T\times r=\frac{1}{2}mr\times a$

$T=\frac{1}{2}ma$ ......(1)

now, net force = ma

weight - tension = ma

or, $mg-\frac{1}{2}ma=ma$

or, $mg=\frac{3}{2}ma$

or, $a=\frac{2g}{3}$

hence, tension in the string $=\frac{1}{2}ma=\frac{mg}{3}$ and acceleration $=\frac{2g}{3}$