Featured Problem

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Rotating tube 12/27/2009 with solution (ws)

A particle of mass m is placed in a smooth uniform tube of mass M and length l.  The tube is free to rotate about its center in a vertical plane.  The system is started from rest with the tube horizontal and the particle a distance ${\displaystyle r_{0}}$ from the center of the tube.

For what length of the tube will the particle leave the tube when ${\displaystyle \dot{\theta}=\omega}$ is a maximum and ${\displaystyle \theta=\theta_m}$?  Your answer should be in terms of ${\displaystyle \omega}$ and ${\displaystyle \theta_{m}}$.