Proof that the Force of Friction is μmg

[luckis11]

Consider the case that when pushing stops the mass stops moving. And that when pushing, it moves with constant speed.

For example, one tells you that F=(some material constant)(momentum)mg or something else like that. Actually with some kind of reasoning I get that it should be F=(momentum)/1sec. I do not want to mension my reasoning, it's a bit messy anyway.

How would you proove any such opinion as wrong and the the truth is F=μmg? From the exersises I have seen it seems that by supposition it is F=μmg. Is this supposition...enough?

PLEASE a link if you are not 100% sure of the proof or if you cannot formulate it.

 

[Andrew Mason]

Consider the case that when pushing stops the mass stops moving. And that when pushing, it moves with constant speed.

For example, one tells you that F=(some material constant)(momentum)mg or something else like that. Actually with some kind of reasoning I get that it should be F=(momentum)/1sec. I do not want to mension my reasoning, it's a bit messy anyway.

How would you proove any such opinion as wrong and the the truth is F=μmg? From the exersises I have seen it seems that by supposition it is F=μmg. Is this supposition...enough?

I'd prefer a link if you are not 100% sure of the proof or if you cannot formulate it.

You cannot prove mathematically that F=μmg. This is determined by observation. It is not a law of physics. It is really only a good approximation.

AM

 

[rcgldr]

For most surfaces, the sliding coefficient of friction decreases slightly as speed increases (even in a vacuum, where trapped air isn't a factor).

 

[russ_watters]

Actually with some kind of reasoning I get that it should be F=(momentum)/1sec. I do not want to mension my reasoning, it's a bit messy anyway.

Yes, force is a change in momentum with respect to time. That's pretty easy to prove mathematically.

 

[luckis11]

You cannot prove mathematically that F=μmg. This is determined by observation. It is not a law of physics. It is really only a good approximation.
AM

How is it prooved that it's a good approximation? That's the proof I am asking for.

 

[Pengwuino]

How is it prooved that it's a good approximation? That's the proof I am asking for.

Experimental observation. "Proof" and "good approximation" don't mix. We just see that if we model friction as a coefficient * Normal force, we get predictions that match up fairly well with experiment. That's all we can say about it. There's nothing deep about it, there is no derivation, it just is what it is.

 

[Dale]

I would say that it is by definition. That is how we define the coefficient of friction.

 

[Andrew Mason]

I would say that it is by definition. That is how we define the coefficient of friction.

Yes, but why is it a constant? That, I think, is the question the OP was asking: he wants proof that the force of kinetic friction is proportional to the normal force.

AM

 

[Studiot]

Here is a simple explanation / derivation for the friction between solid bodies. It accounts for most of the observable characteristics.

The normal force, N between bodies is resisted in contact by deformation of the surface layer(s) of both bodies. The surfaces do not touch at every surface point. Rather there is an initial contact at one highest point. The stress here is very high so the local surface deforms plastically until the next highest point comes into contact, which then deforms plastically until the next.. and so on.

So the actual contact area consists of patches of surface material at yield. The total area of these patches is just sufficient to sustain the stress imposed by the normal force and is proportional to that force. This is why the friction is independent of the gross contact surface area.
All the contacting material is thus stressed to the same level equal to the normal yield strength, S_y of the material.

N = S_y A………..1

N = normal force
S_y = yield strength
A_c = actual contact area

It also accounts for the slight drop from static friction to kinetic friction as the patches are always breaking and reforming as the object moves on. I am not differentiating between the max static friction and the kinetic friction in this analysis.

Now apply a lateral force. This lateral force imposes a shear force across all the patches, at right angles to the normal force.

So the shear force acts across the same area, A_c, as the normal force.

So this lateral force is resisted by the frictional force, F. As the imposed lateral force increases so does the opposing frictional force, always maintaining the condition of being equal but opposite to the imposed lateral force, until the body moves. At this point the frictional force does not increase further. Any increase in lateral force serves to accelerate the body.
The body starts to move when the shear stress imposed by the lateral force exceeds the shear strength of the contact material.

F_max = S_s A_c ………..2

F_max is the maximum friction
S_s = shear strength

Combining equations 1 and 2 leads to

$${F_{kinetic}} = \left( {\frac{{{S_s}}}{{{S_y}}}} \right)N$$

But both the normal strength and the shear strength are constants for any given pair of surfaces so

$${F_{kinetic}} = \mu N$$