Answer
Every second, the cylinder should make $0.85$ revolutions.
Work Step by Step
The clothes will lose contact with the wall when there is no normal force exerted on the clothes by the wall. In other words, the clothes' weight provides all the centripetal force: $F_c=W$
In the figure below, we see that the component of the weight that provides $F_c$ at the angle given is $mg\cos20$. Therefore, $$mg\cos20=F_c=\frac{mv^2}{r}$$ $$g\cos20=\frac{v^2}{r}$$ $$v=\sqrt{rg\cos20}$$
We have $r=0.32m$ and $g=9.8m/s^2$. So $$v=1.717m/s$$
Now we find the period $T$: $$T=\frac{2\pi r}{v}=1.17s$$
So the cylinder should finish $1 rev$ every $1.17s$, meaning every second, the cylinder should make $1/1.17=0.85$ revolutions.
