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I have been thinking about this question for a long time and I din't find an explanation in the web for it

lets say that we a rotating spaceship in the shape of cylinder, that is rotating in an angular velocity $w$. and the astronaut inside the spaceship wants to run in a way such that the angular velocity rises in "Linear" way with time.

I can't grasp the intuition behind the answer that is suggested in one of the books, which says that the astronaut must run in a circular way on the circumference in a way that is opposite of the rotating of the spaceship, and in a velocity that gets higher in a Linear way with time.

I tried to write the mathematical equations for angular momentum in order to find some relation. but couldn't. I would be thankful for any attempt to clarify this point for me.

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    $\begingroup$ Your question is not clear. First, what is the axis of rotation? You have two options for a cylinder. Second, I do not really get where you are going with linearly increasing anglar velocity. $\endgroup$ – Bernhard Feb 22 '15 at 15:50

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