Infinite coefficient of friction

Seems nonsensical. Wouldn't an infinite coefficient of friction say that you can't move an object at all?

 

It just would mean there's no slippage while the ball is rolling. Infinite friction isn't required. You just need enough friction combined with enough speed and the related centripetal force to keep the ball rolling and not sliding, even when at the top of the loop.

 

rcgldr is correct- an infinite coefficient of *static* friction simply means there is never any sliding, no matter how fast the ball is rolling. Static friction does no work.

That said, an infinite coefficient of static friction is not *required*- all that's needed for the ball (or any object, sliding or not) to complete a vertical loop is that the normal force does not vanish.

 

Infinite friction

A few comments and clarifications from a self-declared relative expert
(My PhD was about friction, I have written a text book about mechanics):

1) When proposing or modeling problems where there could conceivably be slip, people often say "assume infinite friction". This doesn't mean that the problem solution depends on infinite friction, it only means you are supposed to assume there is no slip. For most such problems, like this roller coaster one, there is some finite value of friction that is big enough to make there be no slipping, in this case just rolling. You just "assume infinite friction" so as not to worry about it. They should better say, probably, assume no sliding.

2) For the real experts: Actually infinite friction does not always imply no sliding. It only implies that the normal force be zero when there is sliding. There are several mechanics problems where the solution is sensible when the friction coefficient is infinite, the sliding force is finite, and the normal force is zero. A great classic one is the rolling of a wheel on a journal bearing. Even when the journal bearing has infinite coefficient of friction the resistance to rolling is finite. That one example is in my book (google Ruina book for a pdf, in box 4.6. Please cite Ruina/Pratap text if you use this.).

 

You are assuming that all normal force comes from the normal component of gravity (which would be true, for a ball rolling down a constant incline). This also explains your result - if all the normal force comes from the normal component of gravity, the normal force will be zero for a vertical slope, thus the coefficient of friction would have to be infinite (and it would have to be negative during the top part of the loop, where the angle is greater than 90 degrees by the same logic). This is clearly nonsensical, since we know that a ball can traverse a loop without slipping, and an infinite friction coefficient does not exist.


In the case of a loop however, some of the normal force comes from the fact that the ball is experiencing acceleration normal to the surface due to the curvature of the path. Thus, your analysis is not relevant to the situation (since a ball going around a loop is an entirely different scenario from a ball rolling down a constant incline).