Why the bottom of a rolling wheel has zero velocity

[BadgerBadger92]

I have heard on several sites that the bottom of a wheel or ball that’s rolling is not moving at all. How does this work? I can’t quite wrap my head around this one lol

 

[hutchphd]

I have heard on several sites that the bottom of a wheel or ball that’s rolling is not moving at all. How does this work? I can’t quite wrap my head around this one lol

Of course you should say "moving relative to the ground". If the cart on which the wheel is attached is moving at speed v then the wheel will rotate such that the bottom of the wheel is at 0 and the top of the wheel moves at 2v (relative to ground) ,assuming unimpeded rotation.

 

[Parthib Roy]

I have heard on several sites that the bottom of a wheel or ball that’s rolling is not moving at all. How does this work? I can’t quite wrap my head around this one lol

the bottom of the wheel acts as a instantaneous axis of rotation ,so as to simplify the part of the wheel in contact with the ground acts as a hinge on which the top of the wheel rotates

 

[Ibix]

I have heard on several sites that the bottom of a wheel or ball that’s rolling is not moving at all. How does this work? I can’t quite wrap my head around this one lol

If it were moving relative to the ground it would be sliding.

Think about what the wheel looks like if you've lifted the car up on jacks and then started it. The bottom of the wheel is moving backwards, right? Now if the car is going forwards, and the bottom of the wheel is going backwards, the velocities might cancel - and in fact they must cancel or the wheel would be slipping against the ground.

 

[berkeman]

I have heard on several sites that the bottom of a wheel or ball that’s rolling is not moving at all. How does this work? I can’t quite wrap my head around this one lol

Have a look at this article about the Cycloid motion of a point on a wheel:

https://en.wikipedia.org/wiki/Cycloid

https://upload.wikimedia.org/wikipedia/commons/6/69/Cycloid_f.gif

 

[BadgerBadger92]

Have a look at this article about the Cycloid motion of a point on a wheel:

https://en.wikipedia.org/wiki/Cycloid

https://upload.wikimedia.org/wikipedia/commons/6/69/Cycloid_f.gif

View attachment 372134

Do you know of any good videos about this? I learn best visually and from videos.

 

[berkeman]

Do you know of any good videos about this? I learn best visually and from videos.

I did this Google Videos search and got good hits:

 

[Herman Trivilino]

I have heard on several sites that the bottom of a wheel or ball that’s rolling is not moving at all. How does this work? I can’t quite wrap my head around this one lol

It's not moving only at the instant it makes contact.

 

[haruspex]

I have heard on several sites that the bottom of a wheel or ball that’s rolling is not moving at all. How does this work? I can’t quite wrap my head around this one lol

"not moving at all" may be misleading. Its instantaneous velocity is zero, but it is accelerating upwards.

 

[A.T.]

"not moving at all" may be misleading. Its instantaneous velocity is zero, but it is accelerating upwards.

@BadgerBadger92
It's a bit like an object thrown vertically upwards, when it's at the highest point of its trajectory:
- vertical velocity passes though zero at some time point, but doesn't remain zero for any finite time interval. The vertical acceleration is non-zero all the time

For the point on the outer wheel it's similar but in two dimensions:
- vertical velocity passes through zero at the lowest position, but vertical acceleration is non zero
- horizontal velocity touches zero at the lowest position, while horizontal acceleration passes through zero
Again, only at some timepoint, not for any finite time interval.

Also note that this is completely frame dependent. You can choose a reference frame to have any instantaneous point of rotation you want.

 

[tech99]

Another interesting case is the bottom point of a train wheel, which is travelling backwards.

 

[sophiecentaur]

Do you know of any good videos about this? I learn best visually and from videos.

There are a number of fairground rides which combine circular motions to give this sort of effect. I seem to remember the Octopus ride ?? Every so often you were stationary relative to a spectator and then you were whipped away. (Really unpleasant for an adult.)
Better than a video but could make you unwell.

 

[wrobel]

Let's discuss some details. Assume that two bodies are in contact at point $A$. Actually, we have three different points here:

1. the point $A_1$ of the first body at which it contacts the second body;
2. the point $A_2$ of the second body at which it contacts the first body;
3. the geometric point of contact, $A$.

Instantaneously, $A_1=A_2=A$, but in general, each of these points can have its own velocity.

By definition, the bodies are in contact without slipping if $\boldsymbol v_{A_1}=\boldsymbol v_{A_2}$.

Spoiler: A remark beyond the high-school level

In 3D, in general, such an equality leads to a nonholonomic constraint.

 

[Nugatory]

Another interesting case is the bottom point of a train wheel, which is travelling backwards.

That one confused me for a moment, and then I realized why it had to be specifically a "train" wheel - the flange extends below the bearing surface and the top the rail.

 

[Ibix]

That one confused me for a moment, and then I realized why it had to be specifically a "train" wheel - the flange extends below the bearing surface and the top the rail.

A literal edge case.

 

[sophiecentaur]

This thread looks like a job for an appropriate Spirograph setup.

 

[rcgldr]

A point on a wheel follows a cycloid path where the point is only instantaneously motionless when it is in contact with the ground. The term "bottom" of a wheel would more appropriately describe the point of contact between wheel and ground which moves a velocity v, with the outer surface of the wheel "flowing" through the "bottom" of the wheel. This view equates the idea of "bottom of wheel" with "contact patch or contact point".

 

[tech99]

A point on a wheel follows a cycloid path where the point is only instantaneously motionless when it is in contact with the ground. The term "bottom" of a wheel would more appropriately describe the point of contact between wheel and ground which moves a velocity v, with the outer surface of the wheel "flowing" through the "bottom" of the wheel. This view equates the idea of "bottom of wheel" with "contact patch or contact point".

Is this definition universal, so it will hold if the wheel has a flange protruding below the contact surface, as on a train?

 

[A.T.]

The term "bottom" of a wheel would more appropriately describe...

As @wrobel mentioned, there is ambiguity in the terminology which often leads to confusion:

Actually, we have three different points here:

1. the point A1 of the first body at which it contacts the second body;
2. the point A2 of the second body at which it contacts the first body;
3. the geometric point of contact, A.

Instantaneously, A1=A2=A, but in general, each of these points can have its own velocity.

It's important to be clear about that distinction. For example it's the velocities of A1 and A2 that determine the rate of work done on the bodies by the contact forces, not the velocity of A.

 

[haruspex]

Is this definition universal, so it will hold if the wheel has a flange protruding below the contact surface, as on a train?

To put it more rigorously, if two bodies are wholly or partly in rolling contact then at each instant a point on one body that is in such contact with a point of the other has the same velocity as that other point.
The flange of a wheel on a moving train is never in rolling contact with the rail line.

 

[wrobel]

Spoiler: Acceleration is a more interesting story.

Assume that a convex rigid body rolls without slipping on a fixed surface. Let $\boldsymbol \omega$ stand for the angular velocity of this body and $\boldsymbol v$ is the velocity of the contact point. Then the acceleration of the point of the rigid body by which it contacts the surface is $\boldsymbol a=-\boldsymbol\omega\times\boldsymbol v$