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I understand that it can't be placed anywhere on the radius because it doesn't vary with it ( and so of course it doens't make sense to place it anywhere else on the plane), but why do we place it there?

I undertand all the advantages that came after $$\vec v = \vec\omega\times\vec r$$ $$\vec a = \vec\alpha\times\vec r + \vec\omega\times\vec v$$

but I can't see what's before.

PS. I know there is another question opened: Direction of angular velocity but I don't understand the answers given there (it's stated that angular velocity points in every direction of the circumference's plane!?) and I can't comment yet so please don't close this one. Thank you.

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Other parts than center, have both angular and linear velocities. Center only has angular one, with respect to reference frame which is center axis. – huseyin tugrul buyukisik Jun 30 '13 at 14:25
I apologize domenicop, but realized I supplied you with a phony explanation! I've accordingly deleted my answer, and I apologize for misleading you. Real answer is on the way, unless someone beats me to it. – David H Jul 1 '13 at 5:20
@huseyintugrulbuyukisik why do you say that other parts have both velocities and the center has not. Could you please elaborate better? – doplumi Jul 9 '13 at 21:39
@DavidH I liked your answer. I don't think it's what I could say to my prof if asked at an exam, but it made calm the chaos in my head. It was a little i-don't-know-were-to-put-it-so-I-put-it-there though :) Have you elaborated a new answer by now? :) – doplumi Jul 9 '13 at 21:46

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