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Originally Posted by Venkatesh
Imagine a large sphere at rest, on a frictionless plane, free to both rotate and/or translate. Suddenly a small spinning and translating sphere strikes the large sphere on its horizontal circumference, in line with the center of mass, and then recoils, spinning the other way. Will the large sphere start translating or spinning?
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Can you be a little more precise with the exact question? If you're saying what I think you're saying, the situation is actually impossible, in a couple of different ways.
First impossibility: The small ball comes in with velocity v and angular velocity w, and leaves with velocity -v and angular velocity -w. Although you could find a velocity and angular velocity for the large ball that would satisfy conservation of momentum, you would have violated conservation of energy - note that the small ball leaves with exactly the same energy as before, so any translation or rotation of the large ball would mean that energy would have to come out of nowhere (generally a bad sign).
Second impossibility: The small ball 'recoils'; I'm assuming you mean by this that the small ball leaves along the same direction that it came in on. Even if you accept my first argument and say that the ball leaves with some velocity -kv, with 0 < k < 1 (i.e. the ball reverses direction and leaves with some lesser speed, since some kinetic energy was transferred to the large ball), the situation is still impossible. The tangential force between the two balls that results in a change in angular velocity ALSO accelerates both balls sideways, so the small ball will leave in a direction some angle from the direction it came in on, and the large ball will start moving in another direction entirely.
I have attached some top-down sketches of what would happen (qualitatively); let me know if they aren't clear.
Unfortunately, I think we'd need more information to solve the problem; for instance, if you're trying to solve for the final velocities of both balls (both x and y components) and the final angular velocities of both balls, you have 6 unknowns; with conservation of momentum in x, conservation of momentum in y, conservation of angular momentum, and conservation of energy (assuming a perfectly elastic collision), you only have 4 unknowns.
You could be clever and try to solve for the two components of the impulse between the balls (from which you could easily figure out everything else); however, this implicitly guarantees conservation of momentum, so you're left with conservation of energy being your only equation, and you still can't solve. You'd need to know something else about the problem.
Of course, you could also consider slipping (kinetic friction) between the two balls if you really wanted to be more realistic, but the calculations would probably become pretty hideous.