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A.T.
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Sorry, I meant to reply to him.PeroK said:I think everyone except @andrewkirk gets that!
Sorry, I meant to reply to him.PeroK said:I think everyone except @andrewkirk gets that!
This is a frame of reference issue. If the frame of reference moves at the same speed as the axle (consider a bicyclist observing the bikes wheels rotating), then the wheel rotates about the axle and it's the ground that is moving linearly (the axle has zero velocity relative to this frame).andrewkirk said:A crucial insight from that article is that, when a wheel rolls along a flat surface, its axis of rotation is through the instantaneous point of contact with the ground, not through its axle.
What makes you think that?PeroK said:It sounds like you are saying that a cycloid is geometrically impossible?!
andrewkirk said:What makes you think that?
I agree with all that as far as the 'refute' sentence. But description of the generation of a cycloid does not require making assumptions that the circle is rotating around the point of contact, whereas the description of commencement of rolling motion that is currently under discussion does.Nugatory said:Every point on the rim of a translating and rotating ideal rigid wheel describes a cycloid , and there is a straight Iine parallel to the direction of motion such that all the cusps of all the cycloids lie on that line and all other points lie on the same side of the line. That appears sufficient to refute your original argument that some point must lie below the floor - so if we accept that argument we must reject some properties of cycloids, or vice versa.
Indeed. The wheel never rotates for any finite amount of time about any single fixed point.andrewkirk said:It occurs to me that perhaps part of the problem lies in the phrase 'the circle is rotating around the point of contact'.
andrewkirk said:I've been thinking about rolling motion, helped by @kuruman's excellent Insights article on the topic.
A crucial insight from that article is that, when a wheel rolls along a flat surface, its axis of rotation is through the instantaneous point of contact with the ground, not through its axle.
Thinking about this, I reached a tentative conclusion that a necessary condition for the commencement of rolling to be possible is that either the wheel not be a perfect circle as it sits on the floor OR the ground deforms under the weight of the wheel. Otherwise the wheel could not rotate around the point of contact without part of the wheel going below the floor.
The problem is easily solved by replacing the circular wheel by a regular polygon of n sides. No matter how large n is, there will always be one or two vertices in contact with the floor and, on application of a suitable force to the wheel, it can always pivot around the grounded vertex closest to the forward direction of the applied force. It pivots around that until the next vertex hits the ground, then it starts to rotate around that vertex instead.
A more realistic depiction that makes the commencement of rolling possible is to consider the wheel's shape as a circle with a portion of the bottom part chopped off, along a chord so that the contact with the ground is a line segment (the chord at which the excision takes place) rather than just a point. This allows the wheel to rotate around the foremost part of the chord without 'running into the ground'.
So in practice, the commencement of rolling is possible because the wheel will deform under its own weight (plus that of any load on the axle) to create a flat contact region that allows rolling. With a pneumatic tyre, this is easy to imagine. But even with a metal railway or tram wheel there will be some deformation of the wheel to create a small flat contact patch.
My theory (speculation, rather) is that, without that deformation, the commencement of rolling motion would be impossible.
This would all be much clearer and make much more sense with some diagrams, and I am proposing to make some, and maybe write a short note about this idea if it turns out not to be misconceived. But before I do that, I'd be interested to hear from anybody that has thought deeply about the commencement of rolling motion, if they think the idea is daft because I've missed some important feature, or alternatively if they think it is correct. Perhaps somebody has already written about this. If so it would be good to get a link to that.
Thank you
EDIT 6 Nov 2017: I realized after some of the discussion below that the real difficulty was not in explaining rolling motion, but in explaining the commencement of rolling motion by application of a force to the wheel. That is corrected in later posts below. But in order to avoid wasting the time of newcomers to the thread, who don't have time to read the whole thing, and who otherwise might spend the time to help by writing explanations of the constant-speed motion of a translating, rotating circle - which is already clearly understood, I have added words in this green font above to make clear that it is only the commencement of motion that is under investigation.
It is natural to see parallels with that problem, which is from Zeno of Elea (not to be confused with Zeno of Citium, who was the founder of Stoicism). Zeno wrote many paradoxes, most of which appeared to suggest that motion was impossible. Calculus - which was not invented until about more than two millenia after Zeno - can resolve all of those paradoxes.poor mystic said:Hi :-)
This question reminds me of the problem that ancient mathematicians had with Achilles and the Tortoise, because their problem required the expansion of mathematical understanding to encompass the calculus - in other words the paradigm within which they attempted to get a grip on the problem was unsuitable for such work.
Nowadays, a philosopher could point out that the rate of progress is the pertinent fact in the discussion, and easily demonstrate that Achilles must win the race on that basis. I think that the correct approach here is to admit that the centre of rotation is at a height r with position also mapped at the tangent point on the ground.
A frame of reference issue. Consider the case where the wheel's axis is the frame of reference, for example a wheel rolling on a treadmill that is moving at constant speed, using the ground as a frame of reference, and with the wheel's axis having zero velocity relative to the ground. The wheel constantly rotates about it's fixed (wrt ground) axis.jbriggs444 said:The wheel never rotates for any finite amount of time about any single fixed point.
andrewkirk said:But I don't think that calculus is the answer to this problem of rotation. I am now fairly sure that the difficulty in this case (I wouldn't call it a paradox) is that saying things like 'the wheel is rotating around the point of contact with the ground' lead one (or at least it leads me) to think that rotation by some nonzero angle around that point occurs. But in fact there is no rotation through any nonzero angle about that point. Rather, the statement that it is 'rotating about that point' is simply saying something about the relationships between the instantaneous velocities, relative to that point, of all the points in the wheel at an instant in time.
rcgldr said:Consider the case where the wheel's axis is the frame of reference, for example a wheel rolling on a treadmill that is moving at constant speed, using the ground as a frame of reference, and with the wheel's axis having zero velocity relative to the ground. The wheel constantly rotates about it's fixed (wrt ground) axis.
jbriggs444 said:The wheel never rotates for any finite amount of time about any single fixed point.
rcgldr said:A frame of reference issue. Consider the case where the wheel's axis is the frame of reference, for example a wheel rolling on a treadmill that is moving at constant speed, using the ground as a frame of reference, and with the wheel's axis having zero velocity relative to the ground. The wheel constantly rotates about it's fixed (wrt ground) axis.
I only mentioned that as an example where a perfectly circular wheel does rotate about a fixed point for a finite amount of time, if the wheels axis is chosen as the basis for a frame of reference.Mister T said:But that point is the center of the wheel, not the point where the rim contacts the treadmill's surface. The rotation you describe occurs for finite nonzero amounts of time, whereas the other doesn't.
No, because the contact point is also moving along the circumference of the wheel. Look at your other drawings, you allowed the contact point to move in order for the wheel to stay above the floor in those cases, so why not for the circular wheel as well?andrewkirk said:We see that, if the wheel rotates around the foremost point of contact in the direction of travel (marked) through a non-zero angle, it all works OK for the polygon and the flattened wheel, but not for the perfect circle, which has to go below the floor.
andrewkirk said:Here are three pictures, showing a wheel rotating around a point of contact with the ground. The first is a perfectly circular wheel,
It's that part I do not get, and I have the following comment on the argument.andrewkirk said:but not for the perfect circle, which has to go below the floor.
Note that your polygon will go below the floor too, when you fail to shift the rotation center forward (rotate around the previous corner). This is what you do with the circle here.andrewkirk said:We see that, if the wheel rotates around the foremost point of contact in the direction of travel (marked) through a non-zero angle, it all works OK for the polygon and the flattened wheel, but not for the perfect circle, which has to go below the floor.
andrewkirk said:I promised some pictures earlier, and I've finally made them. Here are three pictures, showing a wheel rotating around a point of contact with the ground. ...snip..
View attachment 215189 View attachment 215190 View attachment 215191
CWatters said:The assertion that a perfectly stiff wheel cannot roll on a stiff floor is also contrary to experience, in that rigid wheels on rigid surfaces generally have a lower rolling resistance than softer ones.
Even on a frictionless surface you could generate rolling by applying the right torque and force combination.Mister T said:But it's friction that initiates and maintains the roll
Physically you cannot have perfectly rigid bodies at all. But for many applications it's a acceptable simplification.Mister T said:Mathematically, yes, you can have a perfectly rigid circle rotate without slipping along a perfectly straight line, but physically you cannot.
What is acceptable depends on the application.Baluncore said:if a square wheel is unacceptable then all polygons must be unacceptable
Sure, everybody knows that, but I don't see it as having any bearing on the problem. The problem is easily solved by simply interpreting the statement that 'the wheel is rotating around the contact point' to be the statement about relationships of instantaneous linear velocities of different points on the wheel that was made in post 43.A.T. said:The contact point of a rolling wheel is not stationary w.r.t. ground
The "problem".andrewkirk said:Sure, everybody knows that, but I don't see it as having any bearing on the problem.