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I have been pondering on the nature of the frictional force and its dependency on the area of contact for the past few days and I had already searched for plausible explanations for the same.Although I could gather a few discrete points ,I couldn't get a complete picture of it.

Some of the points I had found online are as follows

**a)**It is independent of area because the area of actual contact(at the microscopic level) is a very tiny fraction of the geometric area of the object. Thus any increase or decrease in geometric area is insignificant.

**b)**Friction is independent of area up to a limit, that limit being the point at which the object starts dipping into the other object.

Based on some literature survey, The following are my deductions on the nature of the frictional force

1)Regarding point 'a', if that is so a thin sheet and a small box of the same mass should have the same friction. But shouldn't an increase in the geometric area result in more chances of actual contacts as well.If so, this should surely increase the friction(because of more interlocking at the newly contacted sites.)

2)The point 'b' seems plausible from a practical point of view but gives no mathematical or cause for why friction exhibits such behavior,in the ideal and in the dipping case.

3)One theory that I derived from point 'a', is that the points of contact can be considered as free bodies with dF(friction) and dN(normal force) acting at area dA. Then the total friction would be the sum of the friction acting at all points. Thus if geometric area increases, the number of such area elements also would increase but with a reduction in the normal force at the elemental area.So this would tend to cancel out the effect in the increase of area and thus friction becomes independent of area. But the problem with this theory is that it assumes that the reduction in normal force and the increase in the number of elemental areas has a connection which exactly cancels them both out.

4)Another theory that I have is based on pressure.It states that

Friction,F=Pressure x Area x coefficient of friction. So when area increases force acting per unit area (pressure)decreases and this would cancel out.

But this does not explain why friction would depend on area when the object starts dipping into the other object and this theory does not reconcile with the point 'a'.

Please comment on this issue with both mathematical and logical explanations.

Thank you :)