Can anyone tell me why friction does not depend on the area of contact??
Total Friction does depend on the area of contact. The coefficients of friction do not.
The short version is:
If two surfaces is pushed "hard" together they will tend to stick, and the local tangential force will typically be proportional to the PRESSURE acting locally.
The average pressure on the surface ought clearly be N/A, where N is the normal force, whereas A is the contact area.
The LOCAL tangential force (acting on a surface element "dA") can therefore, on average equal to kN/A*(dA), where k is a dimensionless constant.
Summing up the local frictions, gives us the total friction as:
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