Question

A cylindrical rod of mass $M$,Length $L$ and radius $R$ has two cords wounds around it whose ends are attached to the ceiling as shown. The rod is held horizontally with the two cords vertical. When the rod is released cords unwind the rod rotates.Find the tension in the cords as they unwind?

Answer 1

$mg-2T=ma$......(i)(linear)
$2TR=T\alpha$ (rotational)
$2TR=\frac{M{R}^2}{2}\alpha$.......(ii)
from equation (i)
$a=\frac{mg-2T}{m}$
$R\alpha =\frac{mg-2T}{m}$
$\alpha =\frac{mg-2T}{Rm}$
substitution is eq(ii)
$2T=\frac{MR}{2}\frac{(mg-2T)}{2m}$
$2T=\frac{mg}{2}-T$
$3T=\frac{mg}{2}$
$T=\frac{mg}{6}$