A person carries a plank of wood 1.9 m long with one hand pushing down on it at one end with a force F1 and the other hand holding it up at 49 cm from the end of the plank with force F2. If the plank has a mass of 14 kg and its center of gravity is at the middle of the plank, what are the magnitudes of the forces F1 and F2? (The distance of 49 cm is measured from the location of F1.)

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.