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# A juggler is juggling a rod one end of which is burning

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A juggler is juggling a uniform rod one end of which is coated in tar and burning. He is holding the rod by the opposite end and throws it up so that, at the moment of release, it is horizontal, its Center of Mass is traveling vertically up at speed vo and it is rotating with angular velocity wo. To catch it, he wants to arrange that when it returns to his hand it will have made an integer number of complete rotations. What should vo be, if the rod is the have made exactly n rotations when it returns to his hand?

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#### Solution Preview

Let's first find out how much time it takes for the CM of the rod to return at the height of the hand as a function of v0. We can reason as follows. The time it takes to reach the maximum height is the same as the time it will take to fall back from that maximum height to the height of his hand. At maximum height ...

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