Question

A string is wrapped around a cylinder of mass $M$ and radius $R$. The string is pulled vertically upwards to prevent the centre of mass from falling as the cylinder unwinds the string. Find the tension in the string.

Answer 1

Given: a string is wrapped around a cylinder of mass M and radius R. The string is pulled vertically upwards to prevent the centre of mass from falling as the cylinder unwinds the string.

To find the tension in the string.

Solution:

The equation of motion is

$ma=mg-T.............\left(1\right)$

And,

$\tau =TR=I\alpha =\frac{1}{2}m{r}^2\frac{a}{R}⟹T=\frac{1}{2}ma............\left(2\right)$

Substituting eqn(2) in eqn(1), we get

$ma=mg-\frac{ma}{2}⟹a+\frac{a}{2}=g⟹a=\frac{2}{3}g$

So the tension in the string is, $T=\frac{1}{2}ma=\frac{1}{2}m\times \frac{2}{3}g⟹T=\frac{1}{3}mg$