# Infinite coefficient of friction

• jontyjashan

#### jontyjashan

If we set a ball rolling on a loop-the-loop track,why do we say that we need infinite friction coefficient for the ball to complete the loop. This seems illogical because f=kN. Since f and N are finite how can k be infinite ?

If we set a ball rolling on a loop-the-loop track,why do we say that we need infinite friction coefficient for the ball to complete the loop.
Where did you see this statement?

If we set a ball rolling on a loop-the-loop track,why do we say that we need infinite friction coefficient for the ball to complete the loop. This seems illogical because f=kN. Since f and N are finite how can k be infinite ?

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

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.

But if we look at the friction required on inclined plane for rolling it is some function of tan x . So if we look at the rightmost point of the loop at the height of radius the coeff would be some function of tan90 which is infinite

But if we look at the friction required on inclined plane for rolling it is some function of tan x .
Why do you think this?

I don't think this. Physics says this and it can be proved easily using concept of pure rolling and solving some eqns

Physics says this and it can be proved easily using concept of pure rolling and solving some eqns

Just try it once yourself and btw you can find it in any good standard physics book.

Just try it once yourself and btw you can find it in any good standard physics book.
Nonsense. If it's so simple, just show us. (I suspect you are mixing up a few concepts or situations.)

Let f be friction.
Then mgsinx-f=ma(linear)
Also equating torque we get
fR=I@(angular acceleration)
and a=R@ for pure rolling
Solving this v get
f=Ia/R^2
And a = gsinx/1 + I/MR^2

So finally we have minimum friction coefficient for pure rolling is tanx/1 + MR^2/I

Let f be friction.
Then mgsinx-f=ma(linear)
Also equating torque we get
fR=I@(angular acceleration)
and a=R@ for pure rolling
Solving this v get
f=Ia/R^2
And a = gsinx/1 + I/MR^2

So finally we have minimum friction coefficient for pure rolling is tanx/1 + MR^2/I
OK, I see what you're saying now. That's for rolling down an incline. Note that there is no acceleration normal to the surface, so only the normal component of gravity is available to provide the normal force. So if you have too steep an incline there will not be enough friction to produce rolling without slipping. (Another way of saying that is to claim an 'infinite' coefficient is required, which is clearly unphysical.)

My apologies for not seeing what you meant right away! Things are different in the loop, since there's a minimum speed to maintain contact and there's acceleration normal to the surface. The normal force is not equal to the normal component of the weight.

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Still my query is not solved

Still my query is not solved
Your statement about needing an infinite coefficient of friction was based on rolling without slipping down an incline (with infinite slope). The loop situation is different as there is acceleration normal to the surface.

Infinite friction

A few comments and clarifications from a self-declared relative expert
(My PhD was about friction, I have written a textbook 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.).

Let f be friction.
Then mgsinx-f=ma(linear)
Also equating torque we get
fR=I@(angular acceleration)
and a=R@ for pure rolling
Solving this v get
f=Ia/R^2
And a = gsinx/1 + I/MR^2

So finally we have minimum friction coefficient for pure rolling is tanx/1 + MR^2/I

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).

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