A perfectly stiff wheel cannot roll on a stiff floor?

[andrewkirk]

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.

EDIT 2: 18 Nov 2017: There are now diagrams of what this is talking about, in this post.

 

[haruspex]

the wheel could not rotate around the point of contact without part of the wheel going below the floor.

That only appears to be so because it is hard to think in infinitesimals.
If the wheel were to make any non-infinitesimal rotation about the same point then you would be right, but the point of contact continuously shifts to match the rate of rotation, so no conflict arises.

Look at it this way. Suppose a circle radius r moves to the right at speed v, while simultaneously rotating clockwise at rate v/r. Where is its instantaneous centre of rotation? Does any part of it ever move below its lowest initial point?

 

[CWatters]

Imagine you are sitting on the axel looking at the wheel and moving with it. From that viewpoint the wheel appears to rotate about it's centre and the ground moves backwards under it.

 

[DaveC426913]

If the wheel were to make any non-infinitesimal rotation about the same point then you would be right, but the point of contact continuously shifts to match the rate of rotation, so no conflict arises.

Indeed. The point of rotation is always moving. The duration of any point being the axis of rotation is infinitesimally short, before the axis has moved on.

 

[andrewkirk]

I understand that the point of contact moves and that one can try to motivate the analysis by talking about infinitesimals. But these are not mathematically rigorous. What I am interested in is whether there is a rigorous model of the motion of a perfectly stiff wheel on a perfectly stiff floor, or whether one has to assume some degree of deformation - however small, but nonzero - in order for the model to be viable.

 

[kuruman]

The point of rotation is always moving.

Put a mark on the rim of the wheel to identify a specific point. That specific point becomes the point of rotation once every 2πr/v when it is in contact with the surface. At that time it is instantaneously at rest.

On edit: To clarify the last sentence, I meant at rest relative to the surface.

 

[kuruman]

I understand that the point of contact moves and that one can try to motivate the analysis by talking about infinitesimals. But these are not mathematically rigorous. What I am interested in is whether there is a rigorous model of the motion of a perfectly stiff wheel on a perfectly stiff floor, or whether one has to assume some degree of deformation - however small, but nonzero - in order for the model to be viable.

You don't really need a floor or a stiff wheel. Just imagine a wheel of radius R in free space with its center translating with speed V and rotating with angular speed ω = V/R about its center. There could be a floor underneath it or not, it doesn't matter.

 

[DaveC426913]

I understand that the point of contact moves and that one can try to motivate the analysis by talking about infinitesimals. But these are not mathematically rigorous. What I am interested in is whether there is a rigorous model of the motion of a perfectly stiff wheel on a perfectly stiff floor, or whether one has to assume some degree of deformation - however small, but nonzero - in order for the model to be viable.

If a point on the wheel is in contact with the surface, such that any forward rotation would cause a portion of the wheel to overlap, then that point is not going to be the axis of rotation; the axis of rotation is infinitesimally farther forward than that.

 

[haruspex]

But these are not mathematically rigorous.

You think calculus is not rigorous?
Did you you consider the last para in my previous post?

 

[andrewkirk]

You think calculus is not rigorous?

Certainly not. But on a formal level calculus only involves references to limits, not to infinitesimals.

There are rigorous notions of infinitesimals in areas such as the surreal numbers, and in differential geometry. But the first is very much pure mathematics and the second seems like way too much heavy machinery to bring to bear on what should be a simple application of Newton's laws.

Did you you consider the last para in my previous post?

Yes I did, and thank you for that. However it doesn't seem to me to be applicable, as it is not necessarily rolling motion. It could be a star spinning as it moves through space, or a bowling ball spinning as it moves across a frictionless surface. A key feature of rolling motion is that the instantaneous axis of rotation passes through the point(s) of contact with the floor, rather than through the centre of the wheel or ball, as kuruman points out in his article. For a spinning star I don't think one would say that the instantaneous axis of rotation was on its surface. But perhaps it's just a matter of considering different frames of reference.

Thanks to the others for your answers as well. To clarify, I am asking whether the commencement of rolling motion by the application of suitable forces to a 'perfect' wheel on a 'perfect' floor can be formally and rigorously explained using only Newton's laws - as for instance sliding and orbital motion can. I am not claiming that it can't. I'm just hypothesising that maybe doing so requires an assumption that the wheel is not perfectly round or the floor is not a perfect plane - an assumption which is always satisfied in practice. I can't think right now of a system of equations that rely only on Newton's laws, which predict that rolling motion will occur on application of appropriate forces to the wheel, unless we assume those imperfections. If somebody can produce a set that would be brilliant.

 

[haruspex]

on a formal level calculus only involves references to limits

so take limits

it is not necessarily rolling motion

It is indistinguishable from (theoretical) rolling motion. The instantaneous centre of rotation is always at the lowest point, and the locus of the lowest point is a straight horizontal line. According your post #1, that was the combination you had difficulty with.

the commencement of rolling motion

Commencement of rolling is a bit different. It matters where the impulse is applied. But there is a height at which an applied horizontal impulse would initiate rolling motion without any need for friction.

 

[andrewkirk]

so take limits

That's what I'm asking somebody to show me how to do, if they think it can be done. I can't see a way of formalising this problem as an application of calculus, but if somebody else can that would be great.

 

[NFuller]

I don't think I understand your justification for the wheel being required to deform in order to experience circular motion.

It is indeed true that the axis of rotation is the point at which the wheel contacts the floor. There is nothing wrong with this being a literal point, such that the wheel only makes contact on an infinitesimally small area of the floor. You could of course argue that friction would vanish in such a scenario, but that's a matter of arguing the pragmatic rather than the theoretical situation.

I'm not sure if this clarifies anything, but the picture I have in my head is that I first shift to a frame where the wheel is stationary. I would then see the floor as a plane which is constantly tangent to a circle (the wheel) while the contact point between the plane and circle moves along the radius of the circle. Thus, the rotation seen in this frame is the floor roataing about the center of the wheel

 

[haruspex]

That's what I'm asking somebody to show me how to do, if they think it can be done. I can't see a way of formalising this problem as an application of calculus, but if somebody else can that would be great.

An infinitesimal rotation about a fixed point leads to an infinitesimal indentation. Indeed, the indentation is a lower order, being r(1-cos(dθ)) ≈ ½r dθ². Meanwhile, the wheel advances rdθ. The ratio of indentation/advance is ½dθ. Now take limits.

 

[Nidum]

An infinitesimal rotation about a fixed point leads to an infinitesimal indentation

+1

You could create a physics based mathematical model showing the local deformation of wheel and rolling surface as the wheel rolls along . Maybe overkill for this problem but you could take the model to it's limit state when the wheel and surface become infinitely stiff and deformations become zero . Actually doing so though would probably lead to a minefield of absurdities and contradictions .

 

[PeroK]

Thinking about this, I reached a tentative conclusion that a necessary condition for 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.

Theoretically, a perfectly rigid wheel can roll on a perfectly rigid surface . Simply imagine a wheel moving and rotating in space and then introducing a surface at the lowest point of the wheel. This is just simple geometry. Note that there is no need for any interaction between the wheel and surface for rolling to continue. The theorectical point here is that the normal force to support the weight of the wheel can be delivered without deformation.

Practically, of course, there is no such thing as a perfect wheel, let alone a perfectly rigid wheel and all objects will deform to a certain extent. Deformation of some sort would be necessary to maintain the normal force. Moreover, analysing the problem at the molecular/atomic scale will introduce a new perspective in any case; especially if you consider QM. The question of precisely where is the lowest point of the wheel - below a certain scale - becomes absurd.

 

[A.T.]

For a spinning star I don't think one would say that the instantaneous axis of rotation was on its surface. But perhaps it's just a matter of considering different frames of reference.

Of course it is. The instantaneous axis of rotation is frame dependent and has nothing to do with material properties. It's pure kinematics.

 

[andrewkirk]

Theoretically, a perfectly rigid wheel can roll on a perfectly rigid surface . Simply imagine a wheel moving and rotating in space and then introducing a surface at the lowest point of the wheel.

Yes that's correct. There's no problem at all in describing unaccelerated rolling motion using Newton's laws (*well, maybe there is for me, see asterisked para below. But at least it's a different problem). We don't even need to invoke friction. The difficulty arises in describing how the motion commences by application of a force to a wheel sitting stationary on a non-frictionless surface.

It's my fault that I didn't say that in my OP, and I apologise for any inconvenience caused. I'm still exploring these ideas and hadn't got it clear in my mind exactly where the obstacle was. Confusion is commonplace when one starts to question something one has always accepted as simple and obvious.

I'm afraid I'm unconvinced by any of the arguments using limits. I have to apologise (again!) for being pedantic, but I am a pure mathematician by training and inclination and cannot accept an argument that uses limits unless it has a recognised limit theorem to validate it - such as the theorem that $\lim_{x\to a}f(x)+\lim_{x\to a}g(x)=\lim_{x\to a}(f(x)+g(x))$. Properties can disappear when one takes limits - eg integrable functions can become non-integrable and differentiable functions can become non-differentiable - so we can't just assume that when we take a limit all the properties we want remain in place.

Also, limits are operators that are applied to functions, and it is not clear to what function the limits referred to above are being applied. Unless an argument using limits can be presented as a pure application of Newton's laws together with the law of static friction and recognised theorems about limits, it remains an intuition pump - highly valuable for getting a visceral understanding of what is going on, but not qualifying as a proof.

The best limit-based argument I can think of uses regular n-gons, rather than a circle that is allowed to protrude below the floor. We can formally describe (*I think, see below) the instigation of rolling motion of a regular n-gon (n>2) upon application of a suitable impulse, using only Newton's + frictional laws. We can then take the limit as n goes to infinity so that the n-gon asymptotically approaches a circle and say 'look, so an impulse can make a circle start to roll'. But there is no theorem about limits that validates that last step. It's highly intuitive, but non-rigorous. How to validate that step is what troubles me.

Nevertheless I feel that, if we are to find a formal derivation of an equation describing the commencement of 'ideal' rolling motion, it is more likely to be along the n-gon route than by modelling second order deformations of the floor or the wheel and then making them disappear via a limit. The trouble with deformations is that they are very complex to describe, requiring various assumptions about elasticity and so forth. I think an equation describing the motion in terms of deformation and elasticity - to which we subsequently apply limits - would be horribly complex.

* Perhaps the problem is that I'm not sure I even understand how the laws of rotary motion of a rigid object - such things as 'torque = moment of inertia times angular acceleration' - are derived from Newton's laws. When I was taught them we were just given them by fiat, and accepted them because they had such a nice analogous form to the laws of linear motion. But when I try to derive the rotary laws from the linear ones, at first it appears as though I'll have to make all sorts of complicated assumptions about forces between particles that make up a rigid body.

Maybe I need to take a step back and first learn how that is derived before I worry about rolling motion.

Does anybody have a link to a source that carefully derives the laws of rotary motion of rigid bodies from nothing more than Newton's laws?

Thank you

 

[jbriggs444]

upon application of a suitable impulse

If we are trying to construct an argument using limits, why should we begin by imposing an infinite force over a zero time interval instead of taking the limit of a finite force over a small time interval?

 

[andrewkirk]

If we are trying to construct an argument using limits, why should we begin by imposing an infinite force over a zero time interval instead of taking the limit of a finite force over a small time interval?

We shouldn't. I envisaged the impulse being evenly spread over a non-zero time interval.

 

[Nidum]

The wheel does not actually roll for any distance at all around an instantaneous centre of revolution . For even the slightest rotation of the wheel the instantaneous centre moves as well and the relative position of the instantaneous centre of revolution to the centre of the wheel remains the same .

 

[CWatters]

Consider a stationary train wheel. Draw a vertical line from top to bottom through the centre of the axel. A point on the lowest part of the wheel (eg on the flange) will be below the top surface of the track. If you pick a point on that line a bit higher up it can be coincident with the top of the track or above it. I see no problem with "limits" as this point is on a continuum of possible positions. There is nothing "special" about the position level with the top of the track. I don't see how rotating the wheel makes any difference.

 

[btlang]

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 the wheel can always pivot around the grounded vertex closest to the forward direction of motion. It pivots around that until the next vertex hits the ground, then it starts to rotate around that vertex instead.

Should we consider the inverse? As the number of polygon sides decrease, the polygon will require a larger moment of inertia to rotate.

So in practice, 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 is that, without that deformation, rolling motion would be impossible.

Deformation would make rotation more difficult as it would require the center of mass to have a vertical component to its movement rather than a perfectly horizontal vector. Also, the tire deforms to provide bump absorption, not as a necessity for forward motion. Without deformation, the wheel will simply bounce more as it encounters any points higher than the plane of movement.

 

[kuruman]

A point on the lowest part of the wheel (eg on the flange) will be below the top surface of the track.

An interesting feature of all such points below the top of the track is that they have a backward instantaneous velocity relative to the track while all the points on the train above the top of the track have a forward velocity. The points level with the top of the track have zero velocity. Their "special" property could be that they mark the place where the instantaneous velocity of points on the wheel changes direction.

 

[Mister T]

So in practice, 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.

There's a contact patch, although it's not flat.

My theory is that, without that deformation, rolling motion would be impossible.

You can't have the friction you need without it. There are articles about this in the The Physics Teacher, somewhere, as I remember reading them.

But you can easily imagine a wheel spinning about its center, and instead of it rolling across a level surface the surface is instead moving at the same linear speed as the rim of the wheel.

 

[PeroK]

I'm afraid I'm unconvinced by any of the arguments using limits. I have to apologise (again!) for being pedantic, but I am a pure mathematician by training and inclination and cannot accept an argument that uses limits unless it has a recognised limit theorem to validate it - such as the theorem that $\lim_{x\to a}f(x)+\lim_{x\to a}g(x)=\lim_{x\to a}(f(x)+g(x))$.

You might ask yourself whether trying to learn elementary mechanics as though it were pure maths (and with a pure mathematician's mindset) is as fruitless as trying to learn pure maths as though it were mechanics.

One particular aspect that I think is a problem for you is that in pure maths you have everything you need as an axiom or assumption. If we say "let $p$ be a prime number", then there is no ambiguity. It's not like $p$ is an approximation of a prime, where there are bits of $p$ we have to ignore.

But, if we say "imagine a rigid wheel", then that doesn't and can't say everything we are assuming about the wheel - internal molecular forces etc. And, if we apply an impulse to that wheel, it doesn't say exactly how the impulse is applied. Think of the simple case of modelling a ball as a particle being thrown upwards. You could probably come up with a hundred issues and questions about how a force is actually applied to a ball, how we can neglect air resistance; how we can treat the gravitational field as constant; how we can ignore the Earth's rotation, how a ball is not a particle but a non-rigid body? It's not pure maths: you have to pick and choose what you are going to study and what you are going to ignore.

Treating these sort of elementary physics problems as pure maths and trying to reach the stage where you have a complete list of axioms and assumptions is possibly the road to madness.

One example that I think is interesting is when we consider a body as a continuous mass distribution and use integration to determine its moment of inertia, say. But, isn't a body a large number of particles? So, we should be dealing with a large sum and, in fact, the integration and continuity are an approximation! This is the reverse of pure maths, where integration gives the exact answer as the limit of finite sums.

In physics, therefore, we often use calculus to get an approximation of the reality. In fact, your accelerated wheel problem is perhaps just another example of this principle. The basic mathematical model is clear: no deformation. The reality is that there must be deformation, but not because the mathematical model requires it.

Does anybody have a link to a source that carefully derives the laws of rotary motion of rigid bodies from nothing more than Newton's laws?

Thank you

In terms of learning mechanics, you could try "Kleppner & Kolenkow". I have a maths background and I liked it a lot. I thought their style really suited me.

You may also like to watch Feynman's lecture on the relationship between maths and physics:

http://www.cornell.edu/video/richard-feynman-messenger-lecture-2-relation-mathematics-physics

 

[A.T.]

The difficulty arises in describing how the motion commences by application of a force to a wheel sitting stationary on a non-frictionless surface.

You mean because friction applied to a single point (no deformation) would lead to infinite stress in the wheel? Or that contact forces come from EM interaction and require deformation of the atom lattice from its unloaded equilibrium? These are general limitations of the rigid body assumption, not specific to rolling.

 

[jbriggs444]

Otherwise the wheel could not rotate around the point of contact without part of the wheel going below the floor.

Can you justify this contention?

 

[andrewkirk]

Can you justify this contention?

Rotating a circle $\mathscr C$ that is tangent to the floor (line $l$) at point $P$, around $P$, by angle $\alpha>0$, gives a new circle $\mathscr C_\alpha$ that is tangent, at $P$, to a line $l_\alpha$that passes through $P$ at an angle of $\alpha$ to $l$. Hence $l$ is now a secant that cuts $\mathscr C_\alpha$ at both $P$ and another point $Q$ such that $P$ and $Q$ mark out a sector that subtends angle $2\alpha$ at the centre of $\mathscr C_\alpha$ The chord $\overline{PQ}$ of that sector aligns with the floor $l$, so that the part of the circle outside the chord is below the floor.

Sorry about the lack of a diagram. I am too slow at making them.

 

[A.T.]

Rotating a circle that is tangent to the floor (line $l$) at point $P$, around $P$, by angle $\alpha>0$, results in the circle now being tangent, at $P$, to a line $l_\alpha$that passes through $P$ at an angle of $\alpha$ to $l$. Hence $l$ is now a secant that cuts the circle at both $P$ and another point $Q$ such that $P$ and $Q$ mark out a sector that subtends angle $2\alpha$ at the centre of the rotated circle. The chord $\overline{PQ}$ of that sector aligns with the floor $l$, so that the part of the circle outside the chord is below the floor.

This is again a purely geometrical argument, that has nothing to do with starting the rolling and Newtons Laws. What is missing, is that during rolling the instantaneous rotation center is not a fixed point P.

 

[andrewkirk]

This is again a purely geometrical argument, that has nothing to do with starting the rolling and Newtons Laws. What is missing, is that during rolling the instantaneous rotation center is not a fixed point P.

Because it's a line rather than a point, as the wheel has width? Or because of something else? Can you elaborate?

 

[PeroK]

Rotating a circle $\mathscr C$ that is tangent to the floor (line $l$) at point $P$, around $P$, by angle $\alpha>0$, gives a new circle $\mathscr C_\alpha$ that is tangent, at $P$, to a line $l_\alpha$that passes through $P$ at an angle of $\alpha$ to $l$. Hence $l$ is now a secant that cuts $\mathscr C_\alpha$ at both $P$ and another point $Q$ such that $P$ and $Q$ mark out a sector that subtends angle $2\alpha$ at the centre of $\mathscr C_\alpha$ The chord $\overline{PQ}$ of that sector aligns with the floor $l$, so that the part of the circle outside the chord is below the floor.

Sorry about the lack of a diagram. I am too slow at making them.

It sounds like you are saying that a cycloid is geometrically impossible?!

 

[Nidum]

because of something else? Can you elaborate

I tried in #21 . For a continually rolling wheel the position of the instantaneous centre of revolution is continually moving as well

 

[A.T.]

Because it's a line rather than a point, as the wheel has width? Or because of something else? Can you elaborate?

The circle is not rotating a angle alpha around a fixed point P. The instantaneous center of rotation is moving along the line, and along the circumference of the circle, while the circle rotates.

 

[PeroK]

The circle is not rotating a angle alpha around a fixed point P. The instantaneous center of rotation is moving along the line, and along the circumference of the circle, while the circle rotates.

I think everyone except @andrewkirk gets that!

It's really just old wine in a new bottle. A moving particle is not at any point for a finite length of time. An accelerating particle does not have any specific velocity for a finite length of time. A particle moving in a circle is not moving in any given direction for a finite length of time. And, a rolling wheel is not rotating about any given point for a finite length of time.

And, calculus allows us to have these scenarios as a valid mathematical model.