Question

Two thin planks are moving on four identical cylinders as shown. There is no slipping at any of the contact points. Calculate the ratio of angular speed of upper cylinders to lower cylinders

Answer 1

Since there is no slipping between contact points, motion of the cylinders is as shown in the figure.
Let $R$ be the radius of the cylinders.
For upper cylinder,
$v_1+R{\omega }_1=2v........\left(1\right)$ [top]
${\omega }_{1}R-v_1=v.......\left(2\right)$ [bottom]

For lower cylinders,
$v_2+R{\omega }_2=v......\left(4\right)$ [top]
$v_2=R{\omega }_2.........\left(3\right)$ [bottom]

From $\left(1\right),\left(2\right),\left(3\right)$ and $\left(4\right)$, we get

${\omega }_2=\frac{v}{2R}$ and ${\omega }_1=\frac{3v}{2R}$

$\therefore \frac{{\omega }_1}{{\omega }_2}=\frac{3}{1}$