The center of a long frictionless rod is pivoted at the

QUESTION:

The center of a long frictionless rod is pivoted at the origin, and the rod is forced to rotate in a horizontal plane with constant angular velocity w. Write down the Lagrangian for a bead threaded on the rod, using r as your generalized coordinate, where r, are the polar coordinates of the bead. (Notice that 0 is not an independent variable since it is fixed by the rotation of the rod to be 0 = wt.) Solve Lagrange's equation for r(t). What happens if the bead is initially at rest at the origin? If it is released from any point ro > 0, show that r(t) eventually grows exponentially. Explain your results in terms of the centrifugal force mw2r.