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 ω. Write down the Lagrangian for a bead threaded on the rod, using r as your generalized coordinate, where r, φ are the polar coordinates of the bead. (Notice that φ is not an independent variable since it is fixed by the rotation of the rod to be φ = ωt.)
a. 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.
b. Explain your results in terms of the centrifugal force mω²r.