Paradoxes of the Coulomb friction

[zwierz]

From Painleve we know that Coulomb's law of friction being applied to rigid bodies systems may produce contradictions. Painleve constructed several examples of such contradictions, so called Painleve's paradoxes, see [Painleve P. Leçons sur le frottement. P.: Hermann, 1895]. Those examples are somewhat complicated and contain big formulas.

I would like to propose a completely trivial paradox of Coulomb's friction.

A cart moves from left to right on a horizontal road. Over the cart there is a pendulum with a fixed hing $O$ and a homogeneous rod $OA$ of mass $m$. Rod's end $A$ rests on the cart such that the angle between the rod and the vertical is equal $\alpha$.
Let $N$ be a normal reaction force that acts on the rod from the cart and $F=\gamma N$ be a force of friction applied to the rod; $\gamma$ is a coefficient of friction.
Applying the law of torques about the point $O$ we get
$$N=\frac{mg\sin\alpha}{2(\sin\alpha-\gamma\cos\alpha)}.$$ Thus if $\tan\alpha<\gamma$ then $N<0$ and the cart attracts the rod. That is impossible.

 

[Dale]

Thus if $\tan\alpha<\gamma$ then $N<0$ and the cart attracts the rod. That is impossible.

Why? It just means that the rod can neither sink into the cart nor leave the cart. I can think of possible ways to accomplish that.

 

[zwierz]

It means that the reaction N is directed downstairs and thus all the forces rotate the rod in the same direction (counterclockwise) but the rod remains at rest. Contradiction.

 

[Dale]

all the forces rotate the rod in the same direction (counterclockwise)

Doesn't F rotate clockwise in this case?

 

[Nidum]

Isn't this just a simplified linear version of a sprag clutch ?

With suitable geometry these mechanisms allow free motion in one direction and strongly retarded or stopped motion in opposite direction .
.

 

[zwierz]

Doesn't F rotate clockwise in this case?

the force of friction acts in the opposite direction of relative velocity. The relative velocity is the same for both cases

 

[A.T.]

Thus if $\tan\alpha<\gamma$ then $N<0$ and the cart attracts the rod. That is impossible.

In reality this means: If friction is too high, the thing will lock or break.

 

[zwierz]

we can also replace the rod with a very strong spring

 

[A.T.]

we can also replace the rod with a very strong spring

A spring will compress and rotate counter clockwise, so no contradiction.

 

[zwierz]

Sure, I just proposed one of the ways to change the model such that the contradiction vanishes.
In reality the rod will likely begin to jump