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If you'll go back to HS physics textbooks, you'll see that the force caused by friction does not actually have a term for the area of the things that are in contact. Now I realize that the real world does not quite work the way highschool physics texts say they do (or how could we corner at more than 1 G?) but I believe that the pad area does not have that much to do with the actual stopping power.
I think you are on to something. (THE REAL WORLD)
Force caused by friction?
I feel that you mean in this instance the force that CAUSES the friction.
When you are going back to those basics, then you would be referring to using the coefficient of friction between two surfaces (unitless).
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Laws of Friction for Dry Surfaces:
1. Friction between two given bodies is directly proportional to the pressure; the coefficient of friction is constant for all pressures.
2. The coefficient and amount of friction for given pressures are independant of the area of contact.
3. The coefficient of friction is independant of the relative velocity, although static friction is greater than kinetic friction.
The preceeding laws are only approximately true. The coefficient of friction is slightly greater for small pressures upon large areas than for great pressures upon small areas. The coefficient of friction decreases as the speed increases.
Eshbach's Handbook of Engineering Fundamentals
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They don't necessarily reveal in HS Physics that some of the laws they teach are not 100 percent accurate, but more rather an approximation.
The reason friction brakes work is that they convert kinetic energy (motion) into heat energy. The more surface area the brakes (pads, rotors, calipers) have, the better they are at dissipating the heat (real basic heat transfer).
just my 0.02 and I'm done on this topic.