Friction and Contact Area
It is an empirical fact for many materials that F = μN, where F is the frictional force for sliding friction, μ is the coefficient of friction, and N is the normal force. There isn't a simple derivation for sliding friction's independence from area.
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“We have got onto slippery ice where there is no friction and so in a certain sense the conditions are ideal, but also, just because of that, we are unable to walk. We want to walk so we need friction. Back to the rough ground!”
—Ludwig Wittgenstein (1889–1951)
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