In the year 2220 a new ring-shaped Space Station is being designed that will provide artificial gravity by spinning about its axis. The specifications call for a centripetal acceleration at the edge of the ring that matches Earth's surface gravitational acceleration (9.8 m/s2). If the diameter of the Space Station is 2 km, what is the angular speed of the ring

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.