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A Paradoxes of the Coulomb friction

  1. Mar 16, 2017 #1
    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.
    fa2ec27c4008.png
    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. jcsd
  3. Mar 16, 2017 #2

    Dale

    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.
     
  4. Mar 16, 2017 #3
    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.
     
  5. Mar 16, 2017 #4

    Dale

    Staff: Mentor

    Doesn't F rotate clockwise in this case?
     
  6. Mar 16, 2017 #5

    Nidum

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    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 .
    .
     
  7. Mar 16, 2017 #6
    the force of friction acts in the opposite direction of relative velocity. The relative velocity is the same for both cases
     
  8. Mar 16, 2017 #7

    A.T.

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    In reality this means: If friction is too high, the thing will lock or break.
     
    Last edited: Mar 16, 2017
  9. Mar 16, 2017 #8
    we can also replace the rod with a very strong spring
     
  10. Mar 16, 2017 #9

    A.T.

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    A spring will compress and rotate counter clockwise, so no contradiction.
     
  11. Mar 16, 2017 #10
    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
     
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