I never quite understood why the force of friction between two surfaces is independent of the area of contact between them. It seems to me that it should have a directly proportional relationship, but in physics we always used the equation [math]F_f = F_N \mu [/math] which doesn't take any sort of area into account at all. What's the deal?

The coefficient of friction would include any area effects. In many 1st and 2nd semester physics classes, this isn't covered, because it is a complication. But in general the coefficient of friction is not a constant or a piecewise function (like with static and dynamic friction) -- it is a function of all the different environmental variables. For example, it is usually at least somewhat dependent on the velocity of the obejct in motion, even if that dependence is typically weak which is why assuming a constant value isn't a terrible assumption. If the area of contact became important, then the coefficient of friction would also include that dependence.

 

There is also an issue of remembering that force itself is area independent. That is, if I apply 20 N of force to a ball, it doesn't matter if I am grabbing the entire hemisphere or just touching it with a fingertip. 20 N is 20 N. What is dependent on area is the pressure. The pressure applied by having the ball in my palm and grabbing a hemisphere versus the pressure applied using only a fingertip would be very different, even if the force is the same 20 N in both cases.

I think like bignose said, the friction is independent of are for a certain weight (i.e. 20 pounds). The more area, the more surface area touching, but the less force pushing down per square inch.

 

But friction is also directly proportional to weight/area.

 

So if you decrease the area, you'd be increasing the pressure on that area for a constant weight, thus friction would remain the same.