If I am an astronaut on a spacewalk in zero-g, and I reach out and hold onto a pole (sticking out from a large space station) on front of me (with the pole parallel to my head-to-feet axis), and then I twist the pole in the direction of my fingers (around the poles long axis) I should be able to rotate myself around the pole, right? Now what happens when I let go of the pole... Do I start to rotate around my CoM, and also gain a translation at a tangent to where I let go?
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Yes and yes, whatever linear and angular momentum you have immediately before you let go will continue indefinitely after you let go. Basically Newton's first law.
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$\begingroup$ The post-pole rotation about CoM is a little tricky - there might be some "wrist action" or acrobatic maneuver to rotate around the pole without gaining any CoM angular momentum. $\endgroup$– uhohCommented Jan 6, 2017 at 8:14
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1$\begingroup$ @uhoh: Not in a spacesuit :) The way to go about it would be to tuck your legs, and as you rotate wrists around the pole, counter-rotate your shoulders so that your body retains the same orientation. Good luck tucking your legs far enough for feet to pass over the pole in a spacesuit though. OTOH tuck your legs and lean onto the pole at your waist level and you might gain rotary momentum without linear. $\endgroup$– SF.Commented Jan 6, 2017 at 14:10
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$\begingroup$ You can cope with the unwanted angular momentum by continuously rotating your arms/legs to keep your head and body stationary, though you may get tired after a while. Linear momentum, however, is more of a problem and would require chopping off one or more limbs and throwing them away. $\endgroup$ Commented Jan 10, 2017 at 3:16