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I am a bit confused about the relation between rolling resistance and static friction.

I have often heard that it is the static friction that lets the wheel roll. Consider the following two cases:

a) A tire of a car of mass $m$ moving with positive acceleration $a$ on a street
b) A wheel of a moving wagon (of mass $m$) which is pulled by a string (with force $F_\mathrm{string}$).

  1. How do the free body diagrams look like? (including static friction, rolling friction and friction forces at the axis)
  2. What is the resulting force accelerating the center of mass of the wheel in case b) (in dependence of the frictional coefficients and $m$)
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Please make your question more clear. Is it "What is the difference between rolling friction and static friction on a wheel?" or something else? Like this it is looks too much like a do-my-homework question. –  Maksim Zholudev Jan 9 '12 at 9:17
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I'm not sure I fully understand what you're asking, but rolling resistance and static friction are very different.

Rolling resistance tends to be a catch all term for the energy dissipated in the many moving parts as a vehicle moves. Most of this is probably viscous drag due to oil in the bearings, gearbox etc.

Static friction is the force required to make two surfaces slide over each other, but as long as the surfaces remain static and don't skid there is no energy dissipated.

Take your example of towing a car. Suppose you tow it at a constant speed for 1 metre and suppode you have to pull with a force of 100N to do this, then the work you've done is just force times distance or 100J. Since the car was moving at a constant speed no energy was used to accelerate it, so the 100J went into heating up the oil in the bearings and gearbox etc. It's this energy dissipation that is responsible for the "rolling resistance" of 100N.

The static friction in this example is between the tyres and the road. However as long as the tyres don't skid no energy is dissipated so the static friction doesn't affect the force you feel as you try to tow the car. If you reduce the friction between the tyres and the road, e.g. tow the car on wet ice, then at some point the tyres will start skidding instead of gripping the road. When this happens the wheels don't rotate and effectively the car behaves as a single solid object. Now the energy is dissipated in the contact patch between the tyres and the road and the force you need to tow the car at a steady speed depends on how much energy is dissipated. The slipperier the surface the less energy is dissipated so the less force is needed to tow the car.

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