A perfectly stiff wheel cannot roll on a stiff floor?

[CWatters]

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.

 

[rcgldr]

Part of the issue here is the point of contact only exists at a specific point for an instant in time. Consider the limit of an n-sided polygon where the center of the polygon moves in the positive x direction at some average velocity v. The point of contact advances as instances in time, but the average rate of advance is the same as the average velocity of the center of the polygon. As the number of sides of the polygon approaches infinity, the average rate of the advance of contact point continues to be the same as the average velocity of the center of the polygon, and the limit of this case is a perfect circle, in which case the point of contact has the same velocity as the center of the circle (both are now moving in a straight parallel lines and at a constant velocity).

My perspective of this is to view the point of contact as a moving point, being the point that the circle is currently in contact with a flat surface, not as a fixed point on the surface of the wheel or as a fixed point on the surface of the flat surface. The point of contact has a velocity, in the same manner that vehicle physics consider a contact patch to be moving at the same speed as the vehicle. Having point of contact with a velocity will not change the perspective of that point for an instant in time.

 

[CWatters]

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

You have shown the point of contact of the polygon on the right hand side. What happens when the polygon changes direction? Presumably you would move it to the left?

Why do you allow the point of contact to move around the perimeter of the polygon but not move around the perimeter of the circle?

 

[Baluncore]

If a polygon can be used as a rolling wheel model, then consider using a square wheel, if a square wheel is unacceptable then all polygons must be unacceptable. Some concepts are useful and some are distractions.

 

[Herman Trivilino]

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.

It's true that the more rigid the materials, the lower the rolling resistance. But it's friction that initiates and maintains the roll, and you cannot have that unless the surfaces deform.

Mathematically, yes, you can have a perfectly rigid circle rotate without slipping along a perfectly straight line, but physically you cannot. I'm not sure which of these points is being debated in this thread.

 

[A.T.]

But it's friction that initiates and maintains the roll

Even on a frictionless surface you could generate rolling by applying the right torque and force combination.

Mathematically, yes, you can have a perfectly rigid circle rotate without slipping along a perfectly straight line, but physically you cannot.

Physically you cannot have perfectly rigid bodies at all. But for many applications it's a acceptable simplification.

 

[A.T.]

if a square wheel is unacceptable then all polygons must be unacceptable

What is acceptable depends on the application.

 

[andrewkirk]

The contact point of a rolling wheel is not stationary w.r.t. ground

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.

It only remains a problem if we want to interpret the statement as meaning that there is a rotation through a nonzero angle around that point. If we want to make that interpretation, I don't see how replacing the stationary point by the locus of contact points over time helps. I don't even know what it would mean to say that the wheel rotates through a nonzero angle around that locus. Nor can I see any practical benefit to the theoretical work that would need to be done, defining frames of reference etc, to give meaning to that statement.

 

[OCR]

...any bearing...

Lol... nice pun. .

 

[A.T.]

Sure, everybody knows that, but I don't see it as having any bearing on the problem.

The "problem".

 

[russ_watters]

If a polygon can be used as a rolling wheel model, then consider using a square wheel, if a square wheel is unacceptable then all polygons must be unacceptable.

Not that I want to wade into a thread that has probably run its course, but I think it should be obvious that the more "gons," the smoother the roll.

 

[kuruman]

Not that I want to wade into a thread that has probably run its course, but I think it should be obvious that the more "gons," the smoother the roll.

Maybe.
https://www.sciencenews.org/article/riding-square-wheels

 

[Andrew Mason]

For what it is worth: the problem assumes an ideal, non-real, hypothetical situation but relies on an very real, non-ideal and inherently imperfect phenomenon (friction) to still be operating and suggests that this leads to a contradiction. Of course, it does.

The premise that rolling can occur with the wheel and surface can be in contact over an arbitrarily small distance is not correct. The wheel rolls because of friction and friction requires that the two surfaces overlap somewhat, like two gears meshing together. It is just that this occurs at a microscopic level. If the two surfaces were absolutely smooth down to the molecular level there would be no static friction and, therefore, no rolling at all i.e. no lateral force at the contact point and, therefore, no pivot point.

If one makes the wheel and surface harder and the wheel more perfectly round and the surface more perfectly flat, the contact area is reduced which means that the pressure over the contact area increases. At some point, the pressure exceeds the yield pressure for the material that the wheel and/or surface are made out of so the material breaks down (until the surface area of "contact" increases and the pressure decreases to below the yield pressure).

In summary: no matter what you make the wheel and surface out of, one can never reach an arbitrarily small contact area. But even if you could make them out of some idealised material that does not exist, you would not have any friction and, therefore, no rolling.

AM

 

[Dr.D]

Is this question not very close to the matter of how many angels can dance on the head of a pin?

 

[Herman Trivilino]

@Andrew Mason. You are correct. For the case of what we normally refer to as rolling motion, the surface is supporting the wheel. You can't have that support force without deformation. If you want to look at the limit as the support force approaches zero and just have a rotating wheel next to a flat surface such that the tangential speed of the wheel rim matches the speed of the wheel's center relative to the surface then you can have zero deformation, but that is not what we normally refer to as rolling motion.

 

[PeroK]

The wheel rolls because of friction and friction requires that the two surfaces overlap somewhat, like two gears meshing together. It is just that this occurs at a microscopic level. If the two surfaces were absolutely smooth down to the molecular level there would be no static friction and, therefore, no rolling.

But even if you could make them out of some idealised material that does not exist, you would not have any friction and, therefore, no rolling.

AM

A wheel does not require friction to roll. It will keep rolling through conservation of angular momentum.

The rolling could be initiated by any torque.

 

[Baluncore]

If the two surfaces were absolutely smooth down to the molecular level there would be no static friction and, therefore, no rolling at all i.e. no lateral force at the contact point and, therefore, no pivot point.

In summary: no matter what you make the wheel and surface out of, one can never reach an arbitrarily small contact area. But even if you could make them out of some idealised material that does not exist, you would not have any friction and, therefore, no rolling.

Since when has contact area and not force been important to friction? If a point or line contact forms, the chemical bonds between the two surfaces will stick the two particles or objects together and result in friction, hence torque and rotation.

 

[andrewkirk]

A wheel does not require friction to roll. It will keep rolling through conservation of angular momentum.

The rolling could be initiated by any torque.

The question of how to initiate, for a wheel on a frictionless surface, rolling motion identical to what could occur on a frictionful surface, is interesting. Most pushes on a part of the wheel, or a stiff, weightless handle attached to it, would initiate a translating, rotating motion that did not match any frictionful rolling pattern. It is necessary for the wheel's angular velocity $\omega$ to relate to the linear velocity $v$ of the wheel's centre by the equation $v=\omega R$, where $R$ is the radius of the wheel. For the motion to always match a rolling motion, it is necessary that $\dot v=\dot\omega R$ at all times.

On my calcs, if a force is applied at angle $\alpha$ counter-clockwise of vertical, at polar coordinates $(r,\theta)$ relative to the axle (with $\theta$ being measured as angle to counter-clockwise of the vertical), the following equation must be satisfied
$$I\sin\alpha = rRm\cos(\alpha-\theta)$$
where $I$ and $m$ are the moment of inertia and mass of the wheel.

If we are applying the force to a handle that is at a fixed distance $r$ from the axle, we would need to continuously vary the angle $\alpha$ of our push in order to maintain the motion as rolling-like. This gives $\alpha$ as a function of $\theta$.

Alternatively, if we fix the direction of the applied force as always horizontal, the radius at which it must be applied will vary with $\theta$, being at a minimum when it is applied at a point above the axle ($\theta=0$) and increasing without limit as $\theta\to\pi/2$.

The size of the force makes no difference. It cancels out of all the equations.

 

[PeroK]

The question of how to initiate, for a wheel on a frictionless surface, rolling motion identical to what could occur on a frictionful surface, is interesting. Most pushes on a part of the wheel, or a stiff, weightless handle attached to it, would initiate a translating, rotating motion that did not match any frictionful rolling pattern. It is necessary for the wheel's angular velocity $\omega$ to relate to the linear velocity $v$ of the wheel's centre by the equation $v=\omega R$, where $R$ is the radius of the wheel. For the motion to always match a rolling motion, it is necessary that $\dot v=\dot\omega R$ at all times.

On my calcs, if a force is applied at angle $\alpha$ counter-clockwise of vertical, at polar coordinates $(r,\theta)$ relative to the axle (with $\theta$ being measured as angle to counter-clockwise of the vertical), the following equation must be satisfied
$$I\sin\alpha = rRm\cos(\alpha-\theta)$$
where $I$ and $m$ are the moment of inertia and mass of the wheel.

If we are applying the force to a handle that is at a fixed distance $r$ from the axle, we would need to continuously vary the angle $\alpha$ of our push in order to maintain the motion as rolling-like. This gives $\alpha$ as a function of $\theta$.

Alternatively, if we fix the direction of the applied force as always horizontal, the radius at which it must be applied will vary with $\theta$, being at a minimum when it is applied at a point above the axle ($\theta=0$) and increasing without limit as $\theta\to\pi/2$.

The size of the force makes no difference. It cancels out of all the equations.

a) initiate linear motion by a horizontal force through the centre.

b) initiate rotation by a pair of equal and opposite horizontal forces.

It's clearly and trivially possible to accelerate a wheel to any velocity without rotation and to any angular velocity without linear motion. And, therefore, to have any desired combination of the two.

You need to clear your head!

 

[andrewkirk]

a) initiate linear motion by a horizontal force through the centre.

b) initiate rotation by a pair of equal and opposite horizontal forces.

That is a set of three separate forces, not a single torque, which is what your post above says can initiate rolling motion.

The point is that it has to be a very special torque, not just any torque, to initiate rolling-like motion on a frictionless surface without applying force at multiple points, and I find it interesting to investigate what the nature of that specialness must be.

 

[rcgldr]

If we are applying the force to a handle that is at a fixed distance $r$ from the axle, we would need to continuously vary the angle $\alpha$ of our push in order to maintain the motion as rolling-like. This gives $\alpha$ as a function of $\theta$.

What about applying a horizontal force that always remains horizontal and at a fixed distance above the center of mass? For example, imagine a solid rim or wheel, squeezed between two wheels with vertical axis, and those wheels used to apply a continuous force that remains horizontal and a fixed distance above the center of mass as the target wheel accelerates.

 

[Likith D]

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.

It only remains a problem if we want to interpret the statement as meaning that there is a rotation through a nonzero angle around that point. If we want to make that interpretation, I don't see how replacing the stationary point by the locus of contact points over time helps. I don't even know what it would mean to say that the wheel rotates through a nonzero angle around that locus. Nor can I see any practical benefit to the theoretical work that would need to be done, defining frames of reference etc, to give meaning to that statement.

I love your interpretation of the rolling motion!
It made me think a lot more about it in many different ways...
I think the most helpful perspective, I found is this;
Think of a point moving in a trochoid...
Now, think of a bunch of points moving in different torchoids (infinitely many such points, actually)... Great, so now we can twiddle with the variety of curves under "trochoid" and place them in a so-and-so position Such that... when we play the entire motion (of the infinite points we have setup) it mimics a rolling circle on a flat plane very perfectly!
So we successfully synced the trochoid movement of all the infinite points to perfectly mimic the rolling circle on a flat plane!
And also from our knowledge about trochoids, none of the points ever make it through the floor they are rolling on...
And lo! we have created a "rolling" motion of a circle without damaging a perfectly smooth (flat) floor!

I think that humans have a better (and a more agreeable) intuition of understanding resultant of infinitely many things happening (like integrating the motion of infinite number of particles) rather than conceive the happenings at an infinitesimal time gap regarding geometry at infinitesimally small space scale (infinitely zoomed up vision of geometry of stuff/space).
I can say this from my attempts to REALLY believe that perpendicular force does not change the speed of a particle, as a interpretation of it got me believing otherwise... even though the cold hard math for uniform circular motion by differentiating position vectors clearly said otherwise (it was only later that I found a way to drill the geometrical intuition for that result into me - which I won't talk about here)
I digress
Hope that help! Please do tell me your opinion!

 

[PeterO]

If you are thinking about rolling motion, and buoyed by someones article on the topic and reach the conclusion that something cannot start rolling, but we know a wheel CAN start rolling, then either you are either thinking incorrectly or have mis-interpreted the article.
Even if you can come up with a mathematical analysis that shows the wheel cannot start rolling (but we know it can) - something is wrong with your maths.
Peter

 

[Clausen]

This all seems nonsense to me? Time to retire this thread?

NO! This is not nonsense. It is a perfectly valid question and you don't close a thread just because you don't agree with the question posed.

 

[Clausen]

This is rotation around a point stationary w.r.t. ground. The contact point of a rolling wheel is not stationary w.r.t. ground.

That is exactly the point! A rolling wheel will not have a contact point that is stationary wrt the ground. The OP is asking about a perfectly rigid wheel on a perfectly rigid surface. In other words, there is no deformation of the wheel or the surface, and the wheel is only making contact with the surface at a single point.

If the wheel rotates, the adjacent point of the wheel that is turning through the rotation can only set down in the exact same spot on the surface as the point that is lifting off of the surface and this follows for all points on the wheels circumference.

IIn such a case, the wheel spins in place, but does not, and cannot roll, unless there is deformation.

This should be common sense!

Of course, there is no such thing as a perfectly rigid wheel on a perfectly rigid surface, but this still serves as an interesting thought experiment about what is involved in a wheel rolling on a surface; there must be some deformation, however small, for the wheel to roll, otherwise it just spins in place.

 

[jbriggs444]

Even if you can come up with a mathematical analysis that shows the wheel cannot start rolling (but we know it can) - something is wrong with your maths.

Rather than saying that something is wrong with the maths, there might be something wrong with the translation between math and the real world. The model of a perfectly circular and perfectly rigid "wheel" interacting with a perfectly flat and perfectly rigid "road" fails to accurately reflect the behavior of a real world wheel on a real world road. Real world wheels are neither rigid nor circular. Real world roads are neither rigid nor flat.

An argument that a perfectly rigid and perfectly circular wheel on a perfectly rigid and perfectly flat road would experience no friction and therefore never start rolling is plausible and is not falsifiable by physical experiment -- we have no way to perform a real world test. However, that is not the argument made in post #1.

An argument that the motion of a perfectly rigid and perfectly circular wheel about an instantaneous axis of rotation at a perfectly rigid and perfectly flat road surface must involve interpenetration of the wheel with the road can be made. That argument is also not falsifiable by experiment. It is falsifiable by careful examination of the mathematics.

 

[jbriggs444]

If the wheel rotates, the adjacent point of the wheel

There is no such thing as an "adjacent point" on a wheel. [By "adjacent point", I expect that you refer to two points next to each other on the wheel's surface].

 

[Clausen]

Yes, of course that is exactly what I meant.

 

[jbriggs444]

Yes, of course that is exactly what I meant.

It is a provable property of the real numbers (and, accordingly, of points on the circumference of an ideal wheel) that between any two distinct points there is a point between them. It follows that there is no such thing as a pair of "adjacent" points.

 

[Eric Bretschneider]

I find this rather lengthy thread curious. If your frame of reference is the flat surface then the wheel rotates about the contact point. Thus intuition says that the wheel can not be perfectly rigid or it would deform the flat surface and not roll.

If you choose your frame of reference as the center of the wheel then you have an entirely different situation. The flat surface moves and the wheel can be perfectly rigid. The wheel only rotates about its center.

I think its great to look at problems from different viewpoints, but sometimes a little flexibility in thinking makes solving the problem much simpler.

Take for example Zeno's paradox. If I want to cross a 10ft room, I first take a (big) step that covers half of the distance to the other side, then another step that covers half the remaining distance and then another step that covers half the remaining distance and so on. It will take an infinite number of steps to get to the other side of the room. There is the mathematical argument that you can sum an infinite series and get a finite number.

There is also the more mundane argument that if my intent was to go twice the distance then my first step would take me to the other side of the room. Zeno allows you to travel half the distance regardless of what the distance is. Change your frame of reference and Zeno's paradox disappears - actually the paradox is that it is not self consistent.