Question

A load of mass $m$ is attached to the end of a string wound on a cylinder of mass $M$ and radius $R$. The string passes round a pulley.
Find the acceleration of the cylinder $M$ when it rolls.

Answer 1

Let a be acceleration,

$mg-T_m=ma$

$T_M=Ma$

From rotation of pulley

$I\alpha =\tau =(T_m-T_M)r$

But $a=r\alpha$

$\frac{Ia}{r}=[M(g-a)-ma]r$

$a=\frac{Mg}{M+m+\frac{I}{r^2}}$