Rope On Cylinder

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?

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.

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