Aproton, initially traveling in the +x-direction with a speed of 5.05×10^5 m/s , enters a uniform electric field directed vertically upward. after traveling in this field for 3.90×10^−7 s , the proton’s velocity is directed 45° above the +x-axis. what is the strength of the electric field?

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.