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.