# Friction of a rotating wheel on non-uniform terrain

1. Jul 17, 2013

### wuphys

I have a conceptual issue with wheel friction that has been bothering me for a while. Consider the wheels on a car set to cruise control such that they rotate with constant angular velocity. Neither the wheel nor the ground are deformable (so we can ignore rolling friction) and the wheels slip while rolling. I understand that if the car is driving on a constant slope and the wheels are slipping that the friction between the wheels and surface is dynamic. But is this still true on a bumpy surface where the car has to constantly accelerate to maintain angular velocity? If the wheel is accelerating, then doesn't there have to be at least some static friction between the ground and the wheel?

Any help would be greatly appreciated. Thanks!

Edit: I guess what the question boils down to is: if a wheel is rotating and slipping but also being acted upon by an external force (a car accelerating), then is the friction static or dynamic?

Last edited: Jul 17, 2013
2. Jul 18, 2013

### Andrew Mason

μ
Why do you say the wheels slip while rolling? What do you mean when you say that the car has to constantly accelerate to maintain angular velocity?

If a wheel is in a state of constant slipping, the friction force is determined by the co-efficient of kinetic friction (i.e. F = μkN). That is one reason you lose steering control when you slam on the brakes on ice. Static friction can only occur if there is no relative movement between the tire surface and road surface.

AM

Last edited: Jul 18, 2013
3. Jul 18, 2013

### jbriggs444

Rather than contemplating non-uniform terrain, how about simplifying. You have a rigid wheel that is rolling without slipping on a smooth, rigid, horizontal surface when it encounters a rigid point-like bump at some elevation above the road surface.

Before this impact can be analyzed, we must have information on at least two important parameters: Coefficient of restitution and coeffecient of friction.

Coefficient of restitution:

When a rigid wheel strikes a rigid surface, does it bounce off? Or does it stick? If all you are told is that the wheel and surface are rigid, the answer is indeterminate. You have an infinite force over a zero distance and a zero interval.

Let us assume a coefficient of restitution of zero and a coefficient of friction of mu. And let us not worry for the moment about the distinction between static and dynamic friction.

When the wheel strikes the bump, its radial velocity toward the point is zeroed out. Its tangential velocity around the point is unaffected. This results in an impulsive change in momentum. The net radial impulse can be easily calculated based on the wheel's mass, impact speed and angle of impact.

Multiply by the coefficient of friction and you have the maximum available tangential impulse due to the impact.

The tangential velocity of the wheel's center-of-gravity about the bump prior to impact is known. The required tangential velocity of the wheel's center-of-gravity about the bump post-impact in order to roll without slipping is calculable. If the delta exceeds the maximum available tangential impulse due to impact then the wheel slips. Otherwise, it rolls without slipping.