# How to derive friction laws?

How do I derive the laws:

$$F_s = \mu_s N, F_k = \mu_k N$$

For static and kinetic friction? It was one of the very first things I was ever taught in a physics lesson, and yet to this day I still don't know why this approximation holds. Consequently, when trying to explain and teach this law, I could do little better than say 'It's just something you have to accept', and 'It's sort of like magic'.

My understanding of the frictional force is so bad that I don't even understand WHY a frictional force even exists in the first place! I believe the frictional force originates from the electrostatic force, but why does the electrostatic force oppose motion? Atoms are made of both electrons and protons, and if neither object is charged I would have believed there to be NO such force between the two objects. Is it fundamentally a quantum mechanical effect?

I have read the answers to this question, but they only seem to say that 'the law fits the data pretty well', without explaining WHY it should depend on the Normal force, and WHY we can assume certain things about the frictional force. Any insight would be appreciated.

• Have a look at friction on atomic scale and Friction force and contact area. Although neither of these is an exact duplicate of your question I think between they do provide the answer you're looking for. Commented Dec 11, 2015 at 8:01
• The section of this lecture from UVA labeled "Quick Review of Friction Between Solids" also provides a good answer Commented Dec 11, 2015 at 13:56
• My impression is that this problem is too complicated to be addressed from first principles at the moment. There are many heuristic arguments, many of which are compelling, but none of which are satisfying. I made a brief survey some years ago and concluded at that time that theoretical efforts generally rely on some simple heuristic claim or approximation chosen to get the result to match experiment. Things may be different. Most problems are too hard to solve, and physics gives up on some of them and gives them names like "chemistry" and "biology". Commented Dec 11, 2015 at 14:31