$1.5.18.$ The rod, with one end pivotally fixed on a horizontal plane, lies on the cylinder. Angular velocity of the rod $\omega$. There is no slippage between the cylinder and the plane. Find the dependence of the angular velocity of the cylinder on the angle $\alpha$ between the rod and the plane.
NO: Before viewing the solution to this problem, I advise you to familiarize yourself with the solution 1.5.17
By the time $t$ the rod will form an angle $\alpha$
Consider the change in coordinate $x = AC$ over a short period of time $dt$
$$ dx = d\left(\frac{R}{\tan (\frac{\alpha}{2})}\right) $$
I share both honors with $dt$
$$ \frac{dx}{dt} = \frac{d\left(\frac{R}{\tan \alpha/2}\right)}{dt} $$
Considering that $\omega = \frac{d \alpha}{dt}$ and $v = \frac{dx}{dt}$
$$ v = \frac{\omega R}{2 \sin ^2 \left(\frac{\alpha}{2}\right)} $$
Let's write down the condition of no slippage
$$ \omega ' R = v $$
From here
$$ \fbox{$\omega ' = \frac{\omega}{2 \sin ^2 \left(\frac{\alpha}{2}\right)}$} $$