I Is There a Better Model for the Relationship Between Friction and Velocity?

[erobz]

What’s your point? Why is what you said different from what I said?

We are talking about different functions.

 

[Frabjous]

In that case I would say that μ_k is not defined at w=v.

 

[erobz]

In that case I would say that μ_k is not defined at w=v.

In the drag type model, we could have anything happening to $\beta$ as $f_r \to 0$ continuously. In the constant force model $f_r$ is taking a quantum leap to $0$ when $v = w$. Pick your poison.

 

[Frabjous]

In the drag type model, we could have anything happening to $\beta$ as $f_r \to 0$ continuously. In the constant force model $f_r$ is taking a quantum leap to $0$ when $v = w$. Pick your poison.

I would say that drag is a fluid model and is not applicable to simple solid-solid friction. You are using the wrong physics.

 

[erobz]

I would say that drag is a fluid model and is not applicable to simple solid-solid friction. You are using the wrong physics.

Ok, but the constant force seems equally unclear unless we accept quantum jump models in classical mechanics. Clearly there is something happening near $v = w$ that is not commonly understood?

 

[Frabjous]

Ok, but is the constant force seems equally unclear unless we accept quantum jump models in classical mechanics.

Experimentally it fits the data. Eventually one is going slow enough so that distances between asperities matter will begin to matter, but those are not macro features. I hope that when you say “quantum” you are actually mean “discrete”

 

[erobz]

Experimentally it fits the data. Eventually one is going slow enough so that distances between asperities matter will begin to matter, but those are not macro features. I hope that when you say “quantum” you are actually mean “discrete”

Yeah, discrete. I was using quantum to highlight the point about a distinct "jump".

 

[erobz]

I'm not trying to annoy people with these inquiries, I'm just trying to fill some (personal) gaps.

 

[jbriggs444]

Ok, but the constant force seems equally unclear unless we accept quantum jump models in classical mechanics. Clearly there is something happening near $v = w$ that is not commonly understood?

Chattering, creeping, tires squealing, blocks tumbling -- things not contemplated by the freshman model.

In the real world, there are no such things as a rigid bodies or uniform surfaces. As you look closer and closer things look messier and messier. That is an important life lesson which applies to a host of different things.

As H.L. Menken wrote, "Explanations exist; they have existed for all time; there is always a well-known solution to every human problem—neat, plausible, and wrong."

In the freshman model, the decelerating force of friction is given by the normal force multiplied by the coefficient of kinetic friction all the way until relative motion stops. At which point the frictional force drops to zero. For a situation of decelleration to a relative stop, the coefficient of static friction never enters in at all.

In the real world, the block does not have a single velocity. It is not a rigid body. In the real world, portions of the contact surface can be catching on the belt while other parts of the contact surface are still sliding. The parts of the surface that catch lag behind the parts of the surface that slide. Stresses build within the material until the lagging parts catch up and, perhaps, surge ahead while other parts are now catching.

The shear force of friction may not be uniform across the contact surface. The normal pressure may not be uniform either. Certainly, normal pressure will be non-uniform if we account for torques. Vibrations and distortions during chattering may feed back and amplify themselves. The relative velocity between the surfaces may not be constant throughout the contact surface and may not match the velocity of the block's center of mass relative to the belt.

There is plenty of unpredictability and wiggle room so that any simplistic and deterministic model is sure to be inaccurate.

However, all is not lost. If we can measure how stiff the block is, we can calculate how much deflection (strain) would correspond to the difference between static and kinetic friction (stress). We can place reasonable bounds on just how bad the freshman model is likely to be.

Or we can run the experiment.

 

[erobz]

Chattering, creeping, tires squealing, blocks tumbling -- things not contemplated by the freshman model.

In the real world, there are no such things as a rigid bodies or uniform surfaces. As you look closer and closer things look messier and messier. That is an important life lesson which applies to a host of different things.

As H.L. Menken wrote, "Explanations exist; they have existed for all time; there is always a well-known solution to every human problem—neat, plausible, and wrong."

In the freshman model, the decelerating force of friction is given by the normal force multiplied by the coefficient of kinetic friction all the way until relative motion stops. At which point the frictional force drops to zero. For a situation of decelleration to a relative stop, the coefficient of static friction never enters in at all.

In the real world, the block does not have a single velocity. It is not a rigid body. In the real world, portions of the contact surface can be catching on the belt while other parts of the contact surface are still sliding. The parts of the surface that catch lag behind the parts of the surface that slide. Stresses build within the material until the lagging parts catch up and, perhaps, surge ahead while other parts are now catching.

The shear force of friction may not be uniform across the contact surface. The normal pressure may not be uniform either (it certainly will not be if we account for torques). The relative velocity between the surfaces may not be constant throughout the contact surface and may not match the velocity of the block's center of mass relative to the belt.

There is plenty of unpredictability and wiggle room so that any simplistic and deterministic model is sure to be inaccurate.

However, all is not lost. If we can measure how stiff the block is, we can calculate how much deflection (strain) would correspond to the difference between static and kinetic friction (stress). We can place reasonable bounds on just how bad the freshman model is likely to be.

Or we can run the experiment.

After @Frabjous mentioned about the microscopic scale, I was thinking that the rough belt ( a jagged surface) is applying little impulses to the rough box (another jagged surface) as is slide past it. At some speed these impulses are plenty enough to just imperceptibly raise the COM of the package as the interlocking surfaces slide over(under) each other. However, at some point $v$ close to $w$ these impulses can't supply that vertical force ( acting through the "micro normals"), so the surfaces just stay locked relative to each other.

Something like this picture. The impulse delivered to the upper block from lower section of belt would have to be such that the box slides over the microscopic hills and valleys of the belt. Once it fails to deliver sufficient impulse as $v \to w$ they stay locked relative to each other. Basically, the box c.o.m. is oscillating vertically on a microscopic scale as the belt slides underneath it.

 

[.Scott]

The frictional force is the only force acting on the box (in the direction of motion). If $\mu_k$ is tending to some non-zero $\mu_s$ ( as you say) then by some other mechanism the frictional force must be tending to zero as $v \to w$

You are stumbling on semantics.
In the Freshman model, $\mu_k$ and $\mu_s$ are constant for an given experimental set-up.
For your setup, both can be used to directly calculate a maximum frictional force (max=$N\mu$): Not the force, just a limit to the force, the maximum.
Also: For $\mu_k$, the direction of this maximum force will always be opposite to the direction of the relative motion of the surfaces. For $\mu_s$, the direction of this maximum force will always be opposed to the applied force.

The motion of the belt is clouding the issue.
If we attached the box to a string and allowed it to bounce back and forth across the surface of the belt, then when the box reached its furthest extend to the left or right, the magnitude of the frictional force would stay the same, but the direction of the frictional force would reverse.

In this situation, I think you can see what that "other mechanism" is - it's the direction of the relative movement of the box and the belt. When that motion goes to zero, the situation abruptly changes. It is no longer possible for the direction of the force to be opposite the relative motion because there is no longer any relative motion.

 

[erobz]

In the Freshman model, $\mu_k$ and $\mu_s$ are constant for an given experimental set-up.
For your setup, both can be used to directly calculate a maximum frictional force (max=$N\mu$): Not the force, just a limit to the force, the maximum.

I respectfully disagree. As the belt slides underneath the box, the net force acting on it is $\approx \mu_k N$. When $v \approx w$ the force of friction( the net force acting on the box) abruptly changes to $0$ from $\mu_k N$. The package is no longer accelerating. There is no longer any net force (from friction) acting on the box. The force of static friction acting on the box is $0$ in this condition. If the package were already frictionally interlocked with the belt $v = w$, and the belt then started to accelerate at $a$, the force of static friction could grow from $0$ up to some maximum ( depending on $a$ ) such that it satisfies $f_r = m a \leq \mu_s N$.

 

[.Scott]

I respectfully disagree. As the belt slides underneath the box, the net force acting on it is $\approx \mu_k N$. When $v \approx w$ the force of friction( the net force acting on the box) abruptly changes to $0$ from $\mu_k N$. The package is no longer accelerating. There is no longer any net force (from friction) acting on the box. The force of static friction acting on the box is $0$ in this condition. If the package were already frictionally interlocked with the belt $v = w$, and the belt then started to accelerate at $a$, the force of static friction could grow from $0$ up to some maximum ( depending on $a$ ) such that it satisfies $f_r = m a \leq \mu_s N$.

I agree. I never said otherwise. I suspect you misinterpreted what I meant by "your setup". I only meant the materials you had and how they were set up before you started the experiment. I did not mean the actual placement of the box on a moving belt. I was only trying to emphasize that the values for $\mu_k$ and $\mu_s$ existed before the experiment was started.

 

[erobz]

My issue is not with how the "freshman model" is supposed to work, it is with how it doesn't work quite work very near $v= w$ as proposed. Besides all the other realities @jbriggs444 sites. I think from a microscopic perspective what I was trying to convey in post#60 has some merit. We can imagine part of the belt intermittently slamming into the block at relative velocity $(w-v)$, providing an impulse to the box which is sufficient for the box to climb out of the microscopic gravitational wells. There is relative slipping. However, as $v \to w$ these impulses become insufficient for the box /belt to climb out of the microscopic gravitational wells. The box slides into a well and they stay put relative to each other. Then if the belt were accelerated $\mu_s$ could manifest as the amount of lateral force required to climb out of the wells.

I admit it's a bit "turtles all the way down" until it's all just electrostatics, but is there any reasonable model in between "it jumps to zero inexplicably at $v = w$ and Columbs Law for the molecules on the "touching surfaces""?