BadgerBadger92 said:

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.

BadgerBadger92 said:

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.

BadgerBadger92 said:

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:

BadgerBadger92 said:

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.

BadgerBadger92 said:

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.

haruspex said:

"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.

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

BadgerBadger92 said:

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.

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}##.

Nugatory said:

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.

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

rcgldr said:

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?

tech99 said:

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.