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If a pendulum is initially at its unstable equilibrium position, how large an initial angular velocity is necessary for it to go completely around?

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Any non-zero velocity will do it, if you assume no friction or other dissipation forces. By conservation of energy it will reach the initial position after one turn with that same speed, and will keep rotating around forever. The maximum speed will be at the bottom, and the smallest speed at the top.

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if it is not a rod what do you mean by "its unstable equilibrium position". Please clarify. – Wolphram jonny Feb 19 '13 at 3:36
I guess OP should clarify the question.Is it a solid/rod pendulum or a string pendulum? – Maesumi Feb 19 '13 at 3:41
Maesumi, I just realized that your comment didnt come from the OP. I am not sure why you stated that with such certainty! just for the records, there is no such thing as a "unstable equilibrium position" for a string pendulum! – Wolphram jonny Feb 19 '13 at 3:44
sorry, I realize that now! – Maesumi Feb 19 '13 at 3:49
It's a rod pendulum – Primer Feb 19 '13 at 5:26

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