Exponential, wow. What's the physics behind that?
Cheers, Wayne
A long time ago I was walked through the derivation, but I can't pull it out of my hat.
The first friction approximation that sliding friction is simply <coefficient of friction> * <normal force>, and doesn't depend upon the area through which the force acts.
Next you ask 'what is the side force when a cable changes direction?' Ignore a continuous curve, just look at a turn at a single point. It is pretty clear for a turn around a single point is just the vector sum of the two cable tensions on either side of that bend. This gives the basic point: at a bend the friction force depends on the tension of the cable on either side of the bend. Note the approximation that the weight of the cable is being ignored.
Next step: to pull a cable the pull force at any point must exceed the friction at that point. Looking at the bend, the two cable tensions are: the tension on the entry side of the bend (call it A) and the tension on the exit side (the side you are pulling toward, call it B) of the bend. B = A + <bend friction> and <bend friction> is a function of A and B. The greater A is, the greater the side force and thus the greater the friction adder.
To finish the derivation you'd need to treat a continuous curve as a series of point bends, and then take the limit as the number of bends is made infinite while the angle of each is made flat. I'll leave that as an exercise for the student (meaning I don't remember how to do it
and point you to the answer:
http://www.polywater.com/techtalk1.pdf
For a bend: Tout = Tin * e^(<mu><theta>) where <mu> is the coefficient of friction and <theta> is the angle of the bend in radians.
The exponential nature of the friction going around a bend is put to good use controlling things like the speed of ropes under load, or the winches on boats; the rope is wrapped around the winch, and the friction between the rope and winch is adjusted by small forces one length of rope, with much larger forces being controlled on the other side of the winch.
-Jon