Could anyone give me a mathematical proof that Friction $=μN$, where $N$ is the normal force
And if the maximum value of the static friction is μN, does it mean that if we push a block with force equalling μN, the static friction will become a kinetic friction and the block starts moving with constant velocity?
Also, from Newton's second law when 2 equal forces act on a block in different directions, that means that the block is at rest or moving with constant velocity but if we tried to push a block at rest with different amount of forces less than μN, the block won't move because we didn't over the friction force value so does it mean that the static friction always balance the applied force as long as the applied force is less than μN? Is that something that I understood incorrectly?
Edit: Also, I know that force (net force I should say ) is the rate of change of momentum and from this equation newton's second law was derived in it's " famous" form (f=ma) but is there a proof that the rate of change of momentum = net force ??