Rolling resistance and static friction 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 $F_\mathrm{string}$). 


*

*How do the free body diagrams look like? (including static friction, rolling friction and friction forces at the axis)

*What is the resulting force accelerating the center of mass of the wheel in case b) (in dependence of the frictional coefficients and $m$)

 A: 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.
A: Basically, there are three types of friction (going from smallest to largest): Rolling, Kinetic, and Static. The way that two surfaces are interacting with each other determines which type of friction that is at work at any given time. 


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*If you have a ball rolling on the floor (without sliding), the floor is applying Rolling Friction to the ball. The point of contact between both surfaces is constantly changing due to the ball's roll. That any point of contact the forces are pressed against each-other creating a bond between the two surfaces. As the point of contact changes, the previous point of contact is pulled apart (like a sticky-note). Breaking the bond requires energy, this energy draw is what we know as Rolling Force. 

*If you have a block sliding across the floor, the floor is applying Kinetic Friction to the block. In this case, the point of contact is only changing for the floor's surface. The point of contact is still changing in this case, so the two surfaces don't really have time to form a real solid bond. Yet, just like with the ball, you are constantly breaking the bonds between the two surfaces as the block slides across the floor. Breaking the bonds requires energy, and this energy draw is what we know as Kinetic Friction. What makes this bond stronger than the case of Rolling Friction is that the two surfaces are rubbing against each-other. Under a strong enough microscope, this would look like you are trying to push two mountain ranges through each-other. This isn't really something that happens on it's own so you need to force them to through each-other, which takes energy. In the case of Rolling Friction, you weren't forcing the two mountain ranges to go through each other.

*Now, if the block was just sitting on the floor (no sliding) and something were to bump into the block, the force the floor applies to the block to stop it from sliding would be Static Friction. In this case, the fact that the point of contact between both surfaces wasn't changing gave the the bond between the two surfaces time to strengthen (up to a maximum point). This stronger bond would require more energy to break then the ones that are constantly forming during the case of Kinetic Friction.
I think what mean about about Rolling Friction being related to Static Friction is that Static Friction is what's keeping the rolling object from sliding in the direction of the roll. To be honest, I would think that it would be more like Kinetic Friction that's keeping a rolling object from sliding because the fact that the contact point is constantly changing should stop the two surfaces from forming bonds that are as strong as those formed during an unchanging point of contact between two surfaces. 
You could try testing this by rolling a weighted block car abound a center point of a piece of butcher paper. The car should have a constant turn radius that is built in, and you should measure it. As the car is rolling, you can pull it from the center point of the turn and find out how much force it takes to case it to slide at a constant rate. Then you can try this again, but with a max force-gauge to tell you if the force needed to get a rolling object to start sliding is greater than the force needed to keep it sliding.
