# A Paradoxes of the Coulomb friction

1. Mar 16, 2017

### 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.

2. Mar 16, 2017

### Staff: Mentor

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.

3. Mar 16, 2017

### 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.

4. Mar 16, 2017

### Staff: Mentor

Doesn't F rotate clockwise in this case?

5. Mar 16, 2017

### 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 .
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6. Mar 16, 2017

### zwierz

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

7. Mar 16, 2017

### A.T.

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

Last edited: Mar 16, 2017
8. Mar 16, 2017

### zwierz

we can also replace the rod with a very strong spring

9. Mar 16, 2017

### A.T.

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

10. Mar 16, 2017

### 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