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A juggler is juggling a uniform rod one end of which is

Chapter 3, Problem 3.34

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QUESTION:

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 CM is traveling vertically up at speed \(v_o\) and it is rotating with angular velocity \(\omega_{\mathrm{o}}\). 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 \(v_o\) be, if the rod is to have made exactly n rotations when it returns to his hand?

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QUESTION:

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 CM is traveling vertically up at speed \(v_o\) and it is rotating with angular velocity \(\omega_{\mathrm{o}}\). 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 \(v_o\) be, if the rod is to have made exactly n rotations when it returns to his hand?

ANSWER:

Step 1 of 2

The vertical displacement of the rod is given as,

\(y=v_{0} t-\frac{1}{2} g t^{2}\)

Here, \(v_{0}\) is the initial velocity, \(t\) is the time and \(g\) is the acceleration due to gravity.

The vertical displacement of the rod is zero when it comes back to the point of projection i.e., \(y=0\)

\(\begin{aligned} 0 & =v_{0} t-\frac{1}{2} g t^{2} \\ v_{0} t & =\frac{1}{2} g t^{2} \\ t & =\frac{2 v_{0}}{g} \end{aligned}\)

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