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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 ??

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    $\begingroup$ Some of the answers to this question may help: physics.stackexchange.com/q/154443. There is no mathematical proof of why friction = $\mu N$ - it's a simple model based on empirical measurements, not derived from fundamental axioms. $\endgroup$ – Time4Tea Jul 30 '18 at 21:09
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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.
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  • $\begingroup$ yeah I agree that making a law based on experimental physics is not a good thing at all regarding my 2nd and 3rd questions, does it mean that static friction coefficient is always changing to balance the applied force as long as the applied force is less than the static friction force ?!! and if we have applied force = static friction then the static friction coefficient becomes at the maximum value and then become kinetic friction?! although forces are considered as interactions between objects, I find these interactions very strange or I might have understood something wrong $\endgroup$ – Norhan Ihab Jul 31 '18 at 9:55
  • $\begingroup$ (1) "does it mean that static friction coefficient is always changing to balance the applied force as long as the applied force is less than the static friction force ?" The coefficient itself doesn't change, as $\mu N$ gives the $maximum$ static frictional force that the surfaces can supply. For smaller forces than $\mu N$ the frictional force itself changes, so as always to be equal and opposite to the applied force (parallel to the surfaces). 2. "if we have applied force = static friction…" This will always be so if the applied force is less than $\mu N$, so I think you need to reword. $\endgroup$ – Philip Wood Jul 31 '18 at 11:11

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