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an experiment to disprove the statement--"frictional force is irrespective of the surface area in contact." take a x rs note. fold it in a half and put it in the pocket of a shirt. then invert the shirt. lets assume it doesn't fall. now, take it out, fold it again and repeat the experiment. after a certain no. of folds, we can see that the bank note falls off. in our experiments, we have changed nothing but the surface area. and the frictional force has changed. voila!

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Friction does depend on surface area. It also depends on the shape of the contacting surfaces. The standard description of friction is that $F_f = \mu F_N$ where $F_f$ is the friction force, $\mu$ is the coefficient of friction (static or kinetic as needed) and $F_N$ is the normal force. This is called 'Coulomb Friction', which is generally a very effective relation. In general, however, the friction coefficient is actually only a 'constant' to leading order. It can be a function of surface area, i.e. $\mu = \mu(A)$.

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Congratulations on doing the experiment!

What is probably happening is that the folded note is unfolding, like a spring, and gripping the sides of the pocket. You could try it with a piece of paper that you cut in half (probably not a banknote)

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The pocket is not a rigid surface. It can bend outwards due to the pressure increase, which happened by decreasing the area of contact (to keep the force constant). The pushing of the folded note outwards due to the folding greatly contributes to this, thus allowing the note to fall.

Note here that the friction force is not perfectly parallel to the surface of contact.

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