## ProblemEdit

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 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 .

## SolutionEdit

Find the Lagrangian and Euler-Langrange equation of motion

(where is the moment of inertia of the tube about its center of mass).

so

At the moment when is at a maximum:

so

or

where is the radial component of velocity when . can be found by conservation of energy, namely

subbing

Therefore,

or

which is a quadratic equation in , which can be solved for in terms of *g*, *m*, *M*, , and