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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 ??

Norhan Ihab
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1 Answers1

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1 "Could anyone give me a mathematical proof that Friction =μN, where N is the normal force" As Time4Tea has commented, $F_\text{max}=\mu N$ with constant $\mu$ is an empirical law. In my experience it is not always obeyed accurately. Attempts to give the law a theoretical basis are hampered by the 'messy' origin of the frictional force (mainly microscopic roughnesses on surface A digging in to surface B and vice versa). In my opinion, writers of elementary applied mathematics textbooks give the law more respect than it deserves.

2 "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?" Not usually with constant velocity; the block will usually get faster and faster if you continue to apply the same force that was needed to overcome static friction. This is because, once the block is moving, $\mu$ decreases. This is to be expected, perhaps you'll agree, from the surface roughnesses model. We talk about the 'coefficient of dynamic (or kinetic) friction' being less than the 'coefficient of static friction'.

  1. re your last paragraph… You understand well.
Philip Wood
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