# What make the bottom portion of a wheel in rolling motion move?

As I just learn about the rolling motion which is the combination of pure translation and pure rotation. The top portion of the rolling body has the speed of double speed at the center of the object while the bottom one has no speed

Also, it can be view as the rotation that take the axis at the bottom portion

However, comparing both of those 2 explaination with the thing I see in real life, in this case, the rolling wheel, and in dynamical analysis, why the bottom having no speed can move?

For example, a rolling wheel on the street, let say the bottom of the wheel has no speed but the wheel is a rigid body and as time gone by, it move a distance so the bottom must have speed to move but as the definition the bottom have no speed which mean it is stationary so what make its displacement if there is no speed

• You are following Principles of Physics by Walker, Resnick, Halliday. There it is explained lucidly.
– user36790
Commented Oct 23, 2014 at 3:26
• Here we are talking about the concept of instantaneous speed. The bottom at any instant does not move, but the next instant is not at the bottom any more. Commented Oct 23, 2014 at 5:14
• @aukxn For homework you should write down the equation that a point on the outside of tire follows. Hint: The x and y will have something to do with sin and cos functions. Commented Feb 14, 2023 at 19:56

The bottom of the wheel is a different part of it at every moment. If you follow a particular point on the wheel, you'll see it moves down and slows in forward motion until it touches the surface at zero speed and immediately starts to move up and accelerate forward again. Up to twice as fast at the top to catch up and get on the forward side again and then it descends and decelerates and the cycle repeats.

This animation from Wikipedia shows the path taken by arbitrary point on a wheel quite well:

The path is called Cycloid.

In reality the bottom of the wheel indeed does not move. Otherwise it would be skidding and have a poor grip on the surface since static friction (force preventing two surfaces from starting to move relative to each other) is higher than dynamic friction (force opposing skidding of two surfaces).

• can you give more detail why the bottom skid or slide but not move while the pictucture you show me having the bottom point move is the same as the redline state that it move from bottom to the top, if it skid so the motion is not the rolling anymore? Commented Oct 23, 2014 at 8:00
• @aukxn: It is not skidding. I said Otherwise it would be skidding. If it skids, the motion is not rolling any more, but that does not mean the wheel is not rotating at all. I should probably delete the paragraph if it is confusing to you. Commented Oct 23, 2014 at 8:46

This can be answered in two different frames of reference.

If we look at this problem from the perspective of an outside observer, then as the wheel moves forward, the bottom of the wheel doesn't move at all and the top of the wheel moves at twice the speed of the wheel itself. In this frame, as the wheel moves forward, the centripetal force provided by the structure of the wheel makes the motionless bottom point start to move and the fast top point start to slow. Every point on the rear half of the wheel is pulled along by the structure of the wheel and speeds up. Every point on the front half of the wheel is pulled back by the structure of the wheel and slows down.

If we look at this from the perspective of the moving wheel itself, then we find nothing out of the ordinary. If the wheel is actually moving at a speed, v, then in its frame there is nothing more than simple circular motion. The wheel appears to be rotating normally with the bottom moving at -v and the top moving at +v. The reason the bottom begins to accelerate in the positive direction is due to centripetal force, just like in any circular motion.

The second perspective I provide seems a bit simplistic, but it's still true. The bottom point accelerates in the positive direction due to centripetal force. This is also the case in the frame of an outside observer. The bottom, non-moving point accelerates forward because of centripetal force.

• Should be added that of course in the reference frame of the wheel the ground moves at $-v$, so the bottom of the wheel is still not moving relative to the ground. Commented Oct 23, 2014 at 15:28
• @JanHudec how is that relevant or helpful?
– Jim
Commented Oct 23, 2014 at 15:32

The problem is that "the bottom of the wheel" is not a specific physical part of the wheel. It is a role or description that applies to each part of the wheel as it moves around the axle.

You could just as well wonder how the top of the wheel can travel twice as fast as the wheel, and still stay connected. The answer is that the double speed "top of the wheel" is just the slow "bottom of the wheel", catching up, and getting momentarily ahead...

The images show the addition of the "v" at the bottom point with respect to the bicycle + the "v" of the bicycle with respect to the ground which results with the "v" of the bottom point with respect to ground.

v of the bottom point with respect to the bicycle = -v sense it seems to be running along with the ground, (we know that ground isn't moving) but from the bicycle's view it does, try and ride a bicycle and look at the back-view, you'll see how the ground is somehow running in the opposite direction as you.

v of bicycle with respect to the ground = v, just literally imagine that you're a poor guy who's been smashed straight by the bicycle while still being able to see the wheel, you'll find out that the bicycle is running straight forward, in other means in the positive direction.

Finally add the two results of the two perspectives, = 0 The "v" of the bottom point "with respect to the ground" is "instantaneously" at rest.

Answering your last question about the fact that the bottom point must have some sort of speed to be able to cut a distance, it is simple, just literally imagine that the ground is moving against the wheel, so it somehow arrests every point that comes into contact with it by the force of friction, that's why you see the traces of truck tires on the road.

The ground and the wheel are moving in opposite directions and the bottom moves in an instant with the ground "taken by the ground" not moving by self.