# Rolling wheel problem

1. May 19, 2014

### kelvin490

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)?

2. May 19, 2014

### Staff: Mentor

Is the surface horizontal? What kind of friction is acting?

3. May 19, 2014

### kelvin490

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

4. May 19, 2014

### Staff: Mentor

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.)

5. May 19, 2014

### kelvin490

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.

6. May 19, 2014

### Staff: Mentor

Why can't a single force exert a torque?

7. May 19, 2014

### kelvin490

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

8. May 19, 2014

### Staff: Mentor

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?

9. May 19, 2014

### kelvin490

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

10. May 19, 2014

### Staff: Mentor

Nope. A single force is all you need to create a torque.

Good question. Luckily, you do not need another force and there is none.

11. May 19, 2014

### kelvin490

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.

12. May 19, 2014

### Staff: 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.

13. May 19, 2014

### A.T.

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.

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.

14. May 19, 2014

### Staff: Mentor

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.

15. May 19, 2014

### Staff: Mentor

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.

16. May 20, 2014

### kelvin490

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.

17. May 20, 2014

### Staff: Mentor

Yes. Torque about a point is defined as $\vec\tau = \vec{r}\times\vec{F}$.

An interesting property of a couple (where the net force is zero) is that the moment is the same about any point.

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.

18. May 20, 2014

### kelvin490

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?

19. May 20, 2014

### A.T.

When slippage and deformation are negligible.

Yes.

20. May 20, 2014

### A.T.

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.