# Rope On Cylinder

**Classical Mechanics**Level 5

A thin, inextensible rope is placed on a cylinder as in the figure above. The radius is \(R=1\text{ m}\). The friction coefficient between the rope and surface is \(\mu =0.5\). The total length of the rope is \(L=10\text{ m}\). What is the maximum ratio \(\dfrac{y}{x}\) so the rope won't start to slip?

Give your answer to 2 decimal places.

**Hints**:

After writing the equilibrium equations make the following assumptions:

\(\sin \dfrac{d\theta }{2}\approx \dfrac{d\theta }{2}\).

\(\cos \dfrac{d\theta }{2}\approx 1\).

You will also encounter a term of the form \(d\theta dT\) which you have to neglect.

There is gravity, but the mass of the rope and the gravitational acceleration will eventually simplify.