Frictionless cylinder yet chain wraps around it

[pitchharmonics]

Could a chain wrap around a frictionless cylinder?

My theory is that for a chain to wrap around a cylinder, there has to be a certain amount of friction.

 

[Duarh]

Well, two possibilities. It's likely that they meant that the _axle_ of the cylinder is frictionless - that is, that the cylinder is free to turn on its axis. It seems that usually in problems with cylinders & chains the assumption is that there is sufficient friction on the _surface_ of the cylinder - that is, that the chain doesn't slide, but "rolls" with the cylinder. The second possibility is that they actually meant that the surface is frictionless, in which case, of course, the chain would just slide around the cylinder without moving the cylinder. I find that rather unlikely, though.

 

[maverick280857]

You might be interested in this. There's something called belt friction (usually taught in a statics course). If the coefficient of static friction for a pulley-belt interface is $$\mu_{s}$$ and the tensions on the pulley are (different) $$T_{1}$$ and $$T_{2}$$ and the angle of wrap is $$\beta$$ then the larger tension (say T2) is related to T1 by

$$T_{2} = T_{1}e^{\mu_{s}\beta}$$

Cheers
Vivek

 

[Duarh]

Well, how can the tensions be different if there is no net torque on the axle (the chain is not accelerating)? Also, I don't get the impression this is an advanced statics problem.

Ah, right, this is an abbreviated version of the post. The full problem referred to by the poster is, I believe, described in a thread further down that has something about yanking out hair in the title. If you have insights on that problem or if you can confirm my conclusions there, do post, because the problem seems fairly weird to me the way it's set up.

 

[maverick280857]

Well, how can the tensions be different if there is no net torque on the axle (the chain is not accelerating)? Also, I don't get the impression this is an advanced statics problem.

Ah, right, this is an abbreviated version of the post. The full problem referred to by the poster is, I believe, described in a thread further down that has something about yanking out hair in the title. If you have insights on that problem or if you can confirm my conclusions there, do post, because the problem seems fairly weird to me the way it's set up.

As far as mathematics and principles are concerned, there is no difference between a chain and a rope. Both are modeled as continuous mass distributions and are totally equivalent. There are two possibilities as has already been pointed out earlier: (1) the cylinder can rotate about an axis passing through the center of mass of the cylinder without friction in the chain-cylinder interface (if this is to be believed then it is equivalent to saying that the chain "carries" the cylinder with it...there is no relative motion of the chain and the pulley); (2) the cylinder can rotate about the axis with friction between the chain and the cylinder.

In real life, there has to be friction for wrap. Clothesline tacks are a common example. pitchharmonics says "My theory is that for a chain to wrap around a cylinder, there has to be a certain amount of friction." A certain amount of friction indeed...but where do you want it to be?

Cheers
vivek

 

[Duarh]

My impression is that the tacit assumption in these problems is almost always that there is sufficient friction on the surface of the cylinder for whatever tension force exerted by the rope/chain to be matched by an identical static friction force, so that an explicit mention of 'frictionless' usually applies to the axle.

That's because a chain wrapped around a fixed-axis cyllinder cannot really by any conceivable mechanism 'carry the pulley with it' - make the cyllinder turn - unless it exerts a contact (frictional) force on it. So, if 'frictionless' actually referred to the surface of the cylinder, most 'frictionless' problems would become fairly meaningless since you could just as well have a rigid, smooth stick in place of the cylinder for all the difference that it made (not to mention that the problems would then be soo aphysical)

 

[maverick280857]

Yes and so I think a better way (if one has time of course) would be to analyze these problems with friction and then specialize to the "ideal" cases.

Cheers
vivek