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Amonton's Law of Friction is a very simple concept, first logged on the books in the late 17th century by Guilliame Amonton. (Incidentally, it was first discovered by Leonardo da Vinci, but never popularised until later.) It states, simply put, that the amount of force needed to overcome friction between two surfaces only depends on the types of surfaces being pushed together and the amount of force pressing them together. The actual surface of contact between the two surfaces is not related to the friction force at all; large, small, whatever, you have a two types of surfaces (expressed in a coefficient) and the amount of force between the two. Size Doesn't Matter.

This law was proven and proven again and again in experiments and taught up until the 1930s in physics classes. But then, people came to their senses.

Chances are, the surface you are considering, any surface, in fact, is not perfectly flat. There are many millions of imperfections, crags, teeth and ridges, all smaller than the eye can see or the skin can feel. And when two surfaces come into contact, these tiny geographies will not match up correctly. The mountains on one side will not fit into the valleys on the other. The actual surface of contact is much smaller than what you may think. So the force needed to overcome friction is indeed proportional to the contact area - measured accurately.

This does not erase the usefulness of Amonton's Law. The equation still stands, even though the explanation behind it does not. Any engineer involved in studies of storm and stress (structural engineers, seismologists, etc.) has used Amonton's Law to quickly and accurately approximate friction forces.

Amonton's Law has also been shown to break down in various nanometric scenarios, such as when two surfaces get close enough so that molecular interactions and atomic forces come into play, sometimes attracting the two surfaces together and creating what's known as 'negative load'.

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