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

[.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""?