According to many classical physics textooks, friction force is independant of the surface area? But intuitively, if we increase the surface, we feel that it requires a larger force to push the object. Can anyone explain why?
The pressure (force/area) decreases. Think of small square cells. The frictional force of each is proportional to the pressure. Their frictional forces add. Then integrate over the entire surface.
Friction is the force exerted by the surface. Less friction if the surface is smooth and high friction if the surface is rough. This friction is also the opposition to where you want to push or pull the object.
actually when the surface area increases, the firiction force also increases but this increase is so small that we can neglect it if we are dealing with relatively "small" systems. however, if you consider very very big systems which is experiencing friction force, then you have to consider the surface area of particle if you wanna make more "sharp" calculations.