# I Confusion About Wheels and Friction

1. Jan 17, 2017

### QQB

If a wheel is freely rolling on the ground (not torque driven, but by a force acting on its center of gravity) towards the right, the friction force on the bottom of the wheel would be acting towards the left.

What if the friction force was equal to the force acting on the wheel's COG? Then the wheel would be translating with a constant velocity, yet its angular velocity would be increasing... and that's not possible.

I'm sure my thinking's wrong somewhere, but I don't know where.

2. Jan 17, 2017

### A.T.

The "what if" part.

3. Jan 17, 2017

### Staff: Mentor

As @A.T. points out, the problem is the assumption that the two forces can be equal in magnitude. In fact, what you did is called proof by contradiction. You assumed something, showed that it leads to a contradiction, and therefore proved the assumption is false.

4. Jan 17, 2017

### QQB

I'm afraid I don't see why the forces cannot be equal in magnitude (except by proof by contradiction). Is it just one of those facts of the universe, or do I have a poor understanding of the situation?

If the two forces were instead replaced by strings in tension (maybe the wheel is lying down on some friction-less plane), I can visualize that indeed the translational velocity would remain constant while the wheel spins faster and faster. Or is that wrong too...?

5. Jan 17, 2017

### rcgldr

Imagine that the wheel is on an long treadmill surface that is accelerating to the right, exerting a friction force F to the right, and that an accelerating string attached to a frictionless hub is also exerting a force F to the right. What is the total force to the right on the wheel? What is the net torque applied to the wheel?

6. Jan 17, 2017

### Stephen Tashi

You mean equal in magnitude? Forces are vectors.

There is the distinction between "free vectors" and "bound vectors". The situation that you wish to consider is one where the frictional force and the force on the center of mass would sum to zero if they were treated as "free vectors" - i.e. as forces applied to the same point.

There is a difference between the practical situation and the theoretical situation.
In a theoretical model, if we roll a wheel along a flat surface, it will keep rolling at a constant velocity after we stop pushing it. The frictional force between the surface of the wheel and the flat surface drops to zero after we let the wheel roll "on its own".

Imagine a wheel rolling along a flat horizontal surface with some applied torque that makes it "burn rubber" - i.e. attempts to accelerate the wheel's angular velocity.. The force of friction upon the wheel points in the direction of the wheels translational velocity. Imagine a wheel rolling along a horizontal surface with some applied torque that attempts to "slam on the brakes" - i.e. deaccelerate the wheel's angular velocity. The force of friction upon the wheel points in the opposite direction to the wheels translational motion. We can appeal to a continuity argument to say that if the wheel's angular velocity is neither accelerating or deaccelerating then the frictional force is zero.

If we imagine a situation where a constant force of F is applied to the center of gravity of the wheel and a constant force of -F is applied to the wheel where it meets the flat surface, then the angular velocity of the wheel would accelerate. How can we imagine that happening? We can imagine there are small gremlins who live in the flat surface that grab the bottom of the wheel and try to spin it faster as its moves overhead. But to exert force to spin it, the gremlins would have to move their hands faster than the tangential velocity of the place where they were touching the wheel. If they just matched the velocity of their hand with the velocity of the place they were touching, they couldn't exert any force on it.

7. Jan 17, 2017

### malemdk

The wheel experiences rolling resistance that is counter to the driving force ,it is a reactive force , so that it cannot accelerate wheel

8. Jan 17, 2017

### A.T.

If the tangential string is unwinding from the wheel, then it's like rolling (same kinematic constraint).

The tangential string would go slack, or it would accelerate relative to the wheel, so not like rolling at constant speed anymore.

Last edited: Jan 17, 2017
9. Jan 17, 2017

### Staff: Mentor

Well, you can calculate the relationship between the axle force and the friction force and prove it that way, but frankly your proof by contradiction is more elegant.

It feels weird to try to convince someone that their own line of reasoning is right. Usually I am contradicting them.

10. Jan 17, 2017

### malemdk

I agree with Dale

11. Jan 18, 2017

### Staff: Mentor

So you can derive it with Newton's second law, the rotational form of Newton's second law, and the no slipping condition written as $a=r\alpha$

12. Jan 18, 2017

### QQB

Ohhh, after trying to piece together what everyone said and thinking about it some more, I remember the relationships now. I forgot it when I thought of the situation in my question!

Thanks for everyone's help!