A small bead of mass, m, slides freely inside a smooth tube with length 2L. The tube is spun about its center with constant angular speed, \theta\dot. Imagine that with the tube spinning, the bead is placed at its center and given an initial velocity of v_0 along the tube.

1. Write down Newton's laws in polar coordinates and use them to find the radial position of the bead as a function of time.
2. Suppose that the bead exits the tube after it completes exactly one rotation. Find the velocity of the exiting bead, v, expressed in terms of its radial and tangential components.