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Originally Posted by suneel112
if the situation is completely free of friction, then the second sphere will not rotate, and rotation won't even be figured in the picture. If muKn for the ball to the sphere is 0 and muKn for the ball to the other ball is infinite (it transfers energy whenever touching), you assume that all rotational energy is transferred, and it becomes a rotational energy problem. Sorry, I forgot physics (I took AP last year, and I didn't understand rotation to begin with), but I think I=2/5mR^2 (for a sphere) and tau = I alpha. Make energy considerations involving energy, rotational (1/2 * I * omega^2) and translational (K = 1/2 mv^2). If the collision is elastic and the finishing rotational energy for both spheres is constant R1 + R2 = Rinitial, it becomes an easy problem, since K1i + K2i = K1 + K2.
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Even if friction between the balls was infinite, you still would
not have all rotational energy transferred. I presume you're saying that the first ball would stop spinning completely, and the second ball would start spinning in the opposite direction (with some unknown speed). However, since the spin is in the opposite direction, you have clearly violated conservation of angular momentum.
EDIT: Re-reading the post, I see from the last line that you have not in fact assumed that the small ball ends up with zero rotational kinetic energy (although now I'm a bit confused as to what you mean by "all rotational energy is transferred"). In any case, you can't separate rotational and translational kinetic energy like you can with momentum; any time you have friction transferring rotational kinetic energy between the balls, you will also necessarily accelerate the balls sideways, i.e. some of the angular kinetic energy is transferred into translational kinetic energy. Consider the case of two high-friction balls, with the small one spinning very quickly and moving very slowly, and the large one stationary and having an extremely high mass. As the two balls come in to contact, the small ball will jump quickly off to the side, clearly transferring rotational kinetic energy into translational kinetic energy. The net result is that you cannot state that R1 + R2 = R1initial + R2initial, or that K1 + K2 = K1initial + K2initial; you can only state (assuming a perfectly elastic collision) that R1 + R2 + K1 + K2 = R1initial + R2initial + K1initial + K2initial.
Al: I completely agree that it's possible to have cases with straight rebound; however, the question stated that the small ball was spinning and the large ball was not, and I assumed from the way the question was asked that there was some friction between the balls (and that each ball had finite mass and radius). In this situation, you will not have direct rebound, except perhaps in one theoretical case; if you had finite sliding friction, but the balls were so stiff that they remained in contact only instantaneously (i.e. zero time taken for the balls to deform and pop back to their original shape), the impulse delivered by friction would be zero. My answer didn't address the most general case of two balls colliding; it addressed the closest physically feasible version of the problem stated.