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I imagine that the first ball would strike the rest as normal, but what would the last ball do, without gravity to swing it back?

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You might have to throw the first ball rather than just letting go. – Henry Sep 11 '11 at 22:51

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It would rotate around through most of a 360${}^{\circ}$ arc, until it collided with the second to last ball on its way back around. Then the whole thing would end up mucked up.

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You need to insure that the frame is heavy with respect to the balls to insure that this happens neatly, of course. – dmckee♦ Sep 11 '11 at 23:54
Would a rotating space station fix this? Could you design a zero G Newtons cradle? – Pureferret Jan 3 '12 at 18:38

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