ProblemEditA 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 from the center of the tube.
For what length of the tube will the particle leave the tube when is a maximum and ? Your answer should be in terms of and .
Find the Lagrangian and Euler-Langrange equation of motion
(where is the moment of inertia of the tube about its center of mass).
At the moment when is at a maximum:
where is the radial component of velocity when . can be found by conservation of energy, namely
which is a quadratic equation in , which can be solved for in terms of g, m, M, , and