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Jul 15, 2021 at 14:27 comment added JEB @00100 Right, I get that. Now model it as suggested. Make the spring "ideal" (massless, no damping), and leave the 2 $m$'s oscillating. Track the COM. Make the eye-lid masses different, if need be. Do whatever you need to to make the COM move. Good luck.
Jul 14, 2021 at 20:58 comment added 00100 My argument is as follows: imagine a cat that blinks in empty space where there is no external forces on it. I claim the center of mass will change, since we can imagine a cat with enough heavey eyelash so that when the cat blinks the center of mass will change. Also, the cat will always be able to do that movement, since it's a result of a chemical reaction that is independent of the situation the cat is in.
Jul 14, 2021 at 20:49 history answered JEB CC BY-SA 4.0