# Rolling wheel problem

Gold Member
Suppose there is a hard wheel rolling on a flat surface with friction, will the wheel keep on rolling with constant velocity or stopped by the friction?

If it keeps on rolling, it seems that there is always a friction action against its rotation direction. But where does the torque come from if there is only a single force of friction(i.e. no couple is form)?

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Doc Al
Mentor
Suppose there is a hard wheel rolling on a flat surface with friction, will the wheel keep on rolling with constant velocity or stopped by the friction?
Is the surface horizontal? What kind of friction is acting?

Gold Member
It's horizontal and the surface is flat but rough in microscale.

Doc Al
Mentor
It's horizontal and the surface is flat but rough in microscale.
Assuming the wheel is rolling without slipping, and that you are ignoring rolling friction, there would be no static friction acting on the wheel.

(In real life you cannot ignore rolling friction, which will eventually dissipate the energy of the wheel.)

Gold Member
Thanks. If there is rolling friction, how can a torque be form to reduce the angular velocity? It seems there is only a single force parallel to the surface.

Doc Al
Mentor
Thanks. If there is rolling friction, how can a torque be form to reduce the angular velocity? It seems there is only a single force parallel to the surface.
Why can't a single force exert a torque?

Gold Member
There should be two forces in opposite direction to form a couple. Is that right?

Doc Al
Mentor
There should be two forces in opposite direction to form a couple. Is that right?
Sure, to form a couple. But why do you need a couple?

And if a pure couple were applied, how could the wheel slow down without a net translational force?

Gold Member
A torque always involve opposite forces. I just wonder where the other force comes from it there is only a contact point.

Doc Al
Mentor
A torque always involve opposite forces.
Nope. A single force is all you need to create a torque.

I just wonder where the other force comes from it there is only a contact point.
Good question. Luckily, you do not need another force and there is none.

Gold Member
Are there any other example that a single force composes a torque? It seems that in most cases when you rotate something there is a pivot point that provide an opposite force.

jtbell
Mentor
Imagine a stick lying on a sheet of frictionless ice. Poke it horizontally at one end, perpendicular to its length. It starts to rotate.

A.T.
If there is rolling friction, how can a torque be form to reduce the angular velocity? It seems there is only a single force parallel to the surface.
No, the force from the surface is not parallel to the surface. Rolling resistance comes from deformation of the wheel and/or surface at the contact patch. The center of pressure shifts forward so the force from the surface creates a torque opposite to the rotation.

Are there any other example that a single force composes a torque?
Any single force creates a torque, around any point that is not on the line of action of that force. For example: The thrust force of a single rocket engine will make a spaceship spin, if it doesn't pass exactly through the center of mass, of the spaceship.

Doc Al
Mentor
Are there any other example that a single force composes a torque? It seems that in most cases when you rotate something there is a pivot point that provide an opposite force.
A couple is needed only if you want to create a torque without also creating a net force. Any applied force will create a torque about a body's center of mass (unless it acts on a line intersecting that center of mass).

Example: Imagine a stick lying on a frictionless table. Give it a smack perpendicular to one end. The stick will rotate about its center of mass as well as translate.

Another example: Imagine a ball rolling without slipping down a rough incline. A static friction force pointing up the incline will act at the contact point. That force exerts a torque on the ball that increases its rotational speed, while at the same time acting to slow down the ball's linear acceleration.

Dale
Mentor
Are there any other example that a single force composes a torque? It seems that in most cases when you rotate something there is a pivot point that provide an opposite force.
You will get a torque about the center of mass any time you have a force such that the force does not go through the center of mass. That happens regardless of the number of forces involved.

In addition to the other examples given, consider a game of pool/billiards, but played on a frictionless table. If you strike the cue ball through the center then it will slide, but if you strike it off-center then it will slide and spin.

Gold Member
Example: Imagine a stick lying on a frictionless table. Give it a smack perpendicular to one end. The stick will rotate about its center of mass as well as translate.
I have no doubt the stick will rotate in this situation. Is the torque equal to force times distance between the end and the center of mass? It seems that in the case of two or more forces acting on a body, the magnitude of moment is the same wherever you take the moment.

Perhaps it is a stupid question, why a single force always produce a torque about center of mass (even there is no gravity acting)? In many calculations, there are opposite forces and the moment can be taken about any point. It seems the choice of point become limited to the center of mass if there is only one single force.

Doc Al
Mentor
I have no doubt the stick will rotate in this situation. Is the torque equal to force times distance between the end and the center of mass?
Yes. Torque about a point is defined as $\vec\tau = \vec{r}\times\vec{F}$.

It seems that in the case of two or more forces acting on a body, the magnitude of moment is the same wherever you take the moment.
An interesting property of a couple (where the net force is zero) is that the moment is the same about any point.

Perhaps it is a stupid question, why a single force always produce a torque about center of mass (even there is no gravity acting)? In many calculations, there are opposite forces and the moment can be taken about any point. It seems the choice of point become limited to the center of mass if there is only one single force.
Depending on your purpose, you can take torques about any point. It's just that the center of mass is very useful in describing the motion of an object.

• 1 person
Gold Member
Assuming the wheel is rolling without slipping, and that you are ignoring rolling friction, there would be no static friction acting on the wheel.
In what circumstances a rough flat surface's friction can be ignore? Is it like a gear rolling on a surface have teeth match exactly that of the gear?

A.T.
In what circumstances a rough flat surface's friction can be ignore?
When slippage and deformation are negligible.

Is it like a gear rolling on a surface have teeth match exactly that of the gear?
Yes.

• 1 person
A.T.
In what circumstances a rough flat surface's friction can be ignore?
In my previous post I assumed you mean a freely rolling wheel, not one that is driven or braked via the axis. In those cases an efficient wheel (no slippage and deformation) will still have horizontal static friction.

Gold Member
When slippage and deformation are negligible.
In the ideal rolling case, are there any friction between the wheel and the road? Since there should be no relative motion between the wheel and the road I wonder whether there will be any force between them.

If there is friction that can slow down the wheel, is it the static friction? It seems if there is a force there must be a torque according to our previous discussion, no matter it is called static friction or rolling friction.

Gold Member
Example: Imagine a stick lying on a frictionless table. Give it a smack perpendicular to one end. The stick will rotate about its center of mass as well as translate.
Another question I would like to ask is, consider the free body diagram there is a translational force in the direction of the smack, why isn't the whole stick move translationally according to the force provided? From your example it seems that there is a portion of force used to cause rotational motion and the other portion causes translational motion.

What factors determine the the fraction of force that is used to move the stick translationally and that used to cause the torque?

Gold Member
I think I got the solution of the first question about ideal rolling case. For a wheel rolling freely, if there is no slippage and deformation, there is no force between the wheel and the road and the wheel just keep on rolling forward.

In real case there is deformation of the wheel, some part the wheel contacting the road is squeezed forward relative to the center of mass so that the normal reaction from the ground is not in-line with the center of mass. Therefore the weight mg and the normal reaction form a couple and the torque slows down the wheel.

A.T.
In the ideal rolling case, are there any friction between the wheel and the road? Since there should be no relative motion between the wheel and the road I wonder whether there will be any force between them.
There can be static friction without slippage, for example if a torque is applied via the axis. But for ideal free rolling it is zero.

A.T.
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