An 8-kg mass is attached to a spring hanging from the ceiling and allowed to come to rest. assume that the spring constant is 40 n/m and the damping constant is 4 n-sec/m. at time =0, an external force of =4cos(+) is applied to the system. formulate the initial value problem describing the motion of the mass and determine the amplitude and period of the steady-state solution.

The center of a long frictionless rod is pivoted at the origin, and the rod is forced to rotate in a vertical plane with constant angular velocity \omega. write down the lagrangian for a bead of mass m threaded on the rod, using r as your generalized coordinate, where r, \phi are the polar coordinates of the bead. (notice that \phi is not an independent variable since it is fixed by the rotation of the rod tobe \phi = \omega * t) 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 m* \omega ^2 *r.