Dynamics - rope attached to an object

[itsumodoori]

Homework Statement

(Halliday, Resnick, Krane - Physics, Fifth Edition. Chapter 5, Problem 11.)

A massless rope is tossed over a wooden dowel of radius $r$ in order to lift a heavy object of weight $W$ off of the floor. The coefficient of sliding friction between the rope and the dowel is $\mu$. Show that the minimum downward pull on the rope necessary to lift the object is

$$F_{\text{down}} = We^{\pi\mu}$$.

Homework Equations

$$\Sigma \vec{F} = m \vec{a}$$

$$f = \mu N$$

The Attempt at a Solution

Note that in order to lift the object, the magnitude of the tension $T$ in the rope must be more than or equal to the weight of the object. (That is, $T \geq W$.) So the rope is pulled on one end by a force of magnitude $T$ and on the other by a force of magnitude $F_{\text{down}}$. The dowel exerts a normal force $\vec{N}$ on the rope, and the magnitude of the friction between the rope and the dowel is given by $f = \mu N$.

The problem is that I have no idea how to deal with this normal force. If I draw a diagram where $\vec{f}$ opposes the motion of the rope, I end up with $\vec{f}$ and $\vec{T}$ pointing in the opposite direction as $\vec{F}_{\text{down}}$, but $\vec{N}$ is perpendicular to all of those forces. The rope is obviously not moving in the direction of $\vec{N}$, so it seems that some unknown force is balancing the normal force out.

 

[Doc Al]

Hint: Analyze forces acting on a small segment of the rope. You'll need to set up a (simple) differential equation and integrate to find your answer.