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In an ideal environment with no friction, in a vacuum, what happens to the velocity of the spin of two spheres spinning in perfect parity at two different velocities when they come into contact?

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I think it would be more interesting if you allowed friction between the spheres. In that case we can still use conservation of angular momentum. So for example if the balls had equal and opposite angular momentum before collision they could both have zero angular momentum after collision.

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Two balls with equal and opposite angular momentum would be achiral, correct? How about if you have two balls which are chiral — spinning in opposite directions, but with the same velocity? – Luke Burns Jan 19 '13 at 18:01
I am not sure it is enlightening to bring chirality into this. I think the result you get just has to be consistent with the total angular momentum, obtained by vector addition, being conserved. – Virgo Jan 20 '13 at 22:26

The rotating surfaces of the spheres would just slide over each other at the instant of contact: no forces perpendicular to the line connecting the centers of the two spheres would exist (i.e. no torque would exist). They would undergo a perfectly elastic collision (no loss of energy, thus no friction), thus conserving angular (they just keep spinning) and linear (elastic collision) momentum.

The angular velocity of each sphere remains constant: the only 'usual' things which can change the angular velocity is exerting torque or changing the mass of the sphere. At contact, colliding spinning spheres which have no friction just don't care about the spinning.

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