Mass M1, of 5.00 kg, is pressed against a spring, of spring constant = 770 N/m, as shown above. Once compressed, the front of the mass sits a distance 1.25 m from the bottom of an inclined plane. The incline’s angle of elevation measures 36.5°. A) B) C) D) E) What is the potential energy stored in this system before M1 is released? What will be the maximum kinetic energy of the mass once M1 is released? What will be the maximum velocity of the mass once M1 is released? How far up the incline will M1 move before it comes to a stop? What is the measure of the work done on M1 by the spring?

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.