2. Students are investigating the relationship between the torque applied to a rotating object and its subsequent motion. The students build the apparatus shown, which consists of a vertical pipe of radius r connected to a horizontal pipe at the top. An electromagnet inside the top pipe holds two heavy spheres at the center of rotation. The vertical pipe can rotate on an axis fixed to a table with no friction. Students wrap several turns of string around the vertical pipe and pass the string over an ideal pulley so that the free end of the string connects to a hanging mass m. Magnet Spheres Radius = r C m (a) When the system is released, the mass falls with constant acceleration. The tension in the string is T and the rotational inertia of the apparatus is I. Derive expressions for the following as functions of time t in terms of I, T, and the radius r of the vertical pipe. i. The angular momentum L of the rotating apparatus ii. The kinetic energy K of the rotating apparatus iii. The speed at which the mass m falls (b) In a third experiment, the magnet is initially on and the spheres are in the center. At time t = 0, the apparatus and hanging mass are released from rest. At time t = 3 seconds, the magnet is turned off and the spheres move to the ends of the horizontal pipe and the apparatus is allowed to continue rotating. The downward speed v of the mass is shown in the graph as a function of time t. i. Explain why the graph has a greater slope during the interval 0 to 3 seconds than during the interval 4 to 6 seconds. ii. 1 2 3 4 5 (s) 14 Explain why the graph decreases in value just after time t = 3 seconds.