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I thought up of a strange conundrum. Suppose we have two boxes of equal size and mass moving at equal speeds but opposite directions at each other (in the given inertial reference frame) in space. They have flat surfaces and when they meet each other, they graze each others' surface. See the illustration below.

enter image description here

As you can tell, the kinetic friction will take place as the surfaces graze each other and slow the two boxes down. We can imagine they slow down to a stop, as depicted in the image.

  • Energy is conserved: we know friction converts kinetic energy into thermal energy, causing heat, so there is no problem with energy conservation.
  • Momentum is conserved: the friction forces always come in equal but opposite pairs, and in our scenario we have $p = mv + m(-v) = 0$ before grazing and $p=0$ after grazing.

However, if we take the reference point / origin exactly at the midpoint between their center of masses, then before the contact we have

$$ \vec{L} = (-x\hat{x} + r\hat{y})\times (mv\hat{x}) + (x\hat{x} - r\hat{y})\times (-mv\hat{x}) = -2mv\hat{z} $$

and after contact we have

$$ \vec{L} = 0. $$

This quantity clearly changes, unlike with momentum, and there is nothing it can be converted to, unlike with energy. I don't think friction violates angular momentum conservation, so what is the mistake I have committed and what is the correct way to analyze the scenario?

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  • $\begingroup$ You should try it with air hockey pucks maybe $\endgroup$
    – RC_23
    Jul 8, 2023 at 4:28

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The friction forces attack at the side of each box, not at their center of mass. Which means the boxes would either start to rotate - and your scenario doesn't work - or you would need additional forces that keep the boxes from rotating. In the latter case those additional forces will transfer angular momentum to whatever is causing these forces.

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