Question

The arrangement shown in figure consists of two identical, uniform, solid cylinders, each of mass $m$, on which two light threads are wound symmetrically.
Find the tensions of each thread in the process of motion. The friction in the axle of the upper cylinder is assumed to be absent.

Answer 1

Let the lower cylinder go down by $x$ when the upper cylinder $A$ rotates through $\theta$ and the lower one rotates through $\theta$'. Since both cylinder unwind themselves, therefore
$x=R(\theta +{\theta }^{\prime })$
We can write $a=R(\alpha +{\alpha }^{\prime })$
where $a=$ downward acceleration of $B$
$\alpha =$ angular acceleration of $A$
${\alpha }^{\prime }=$ angular acceleration of $B$
Considering the rotational motion of $A$
$\tau =2TR=\left(\frac{1}{2}m{R}^2\right)\times \alpha \Rightarrow \alpha =\frac{4T}{R}$
Considering the rotational motion of $B$
$\tau =2TR=\left(\frac{1}{2}m{R}^2\right)\times {\alpha }^{\prime }\Rightarrow {\alpha }^{\prime }=\frac{4T}{R}$
Considering the downwrd motion of $B$,
$mg-2T=ma$
$mg-2T=mR(\alpha +{\alpha }^{\prime })=mR(\frac{4T}{R}+\frac{4T}{R})$
$mg-2T=8T\Rightarrow T=\frac{1}{10}mg$