Conceptual issue with rolling sphere and friction.

[mfig]

EDIT: Ooops, the title says sphere, but it is a cylinder. I cannot edit the title... sorry...

I saw this problem in a book and it is really bugging me. It is not a homework problem, I have the fully worked out solution in the book. I just have a conceptual problem.

Given a hollow cylinder of mass M and radius R on a ramp inclined at angle α, what coefficient of friction is necessary for the cylinder to roll down the ramp?
The author points out that there are two forces acting on the cylinder. One is the normal force and the other is the force of friction. The normal force acts perpendicular to the ramp, whereas the frictional force acts in the up-ramp direction, parallel to the ramp.

My problem is this: if we do a torque balance about the center of the cylinder, there is only one torque - the frictional force acting on a lever arm equal to the radius of the cylinder. If this is true, then how come ANY frictional force won't cause the thing to roll? In other words, what is this torque balancing against when it is the ONLY torque? It would seem that no matter how small, a frictional force is the only thing torquing the cylinder and so must cause it to roll. Yet the author says that it will not roll if the coefficient of friction is less than tan(α)/2 - it will only slide down the ramp.

Where is my logic wrong?? Thanks...

 

[oli4]

I suppose the author is talking about rolling 'perfectly' (without sliding at the same time)
Cheers,

 

[mfig]

I suppose the author is talking about rolling 'perfectly' (without sliding at the same time)

I suppose so, but how does this resolve the problem?

 

[oli4]

Maybe I didn't understand your question ?
What I was saying is that, as you say, any friction will cause the cylinder to roll, but thee is a specific friction threshold that will allow the roll to be 'perfect' (the cylinder will not slide, just roll)

 

[mfig]

What I was saying is that, as you say, any friction will cause the cylinder to roll, but thee is a specific friction threshold that will allow the roll to be 'perfect' (the cylinder will not slide, just roll)

Now I understand what you meant. Perhaps that is it. The author seems to put it in terms of either it rolls or it slides. Here is link to the book. The one example is on the bottom of page 248. Here the author isn't quite as explicitly either-or on this example, but there is a follow-up problem on page 319 (problem 11.12), with the answer given on page 584 that sounds definitely like either-or.

http://www.ciberdigital.net/books/Cambridge_UP_-_Classical_Mechanics.pdf


I guess you must be correct though, it is not either sliding or rolling, it is either pure rolling or rolling with sliding if there is any friction whatsoever.

Thanks.