A bullet of mass m 1 is fired into a rod of length L and mass m 2 which is pivoted on one end and rests on a frictionless horizontal surface. The bullet's velocity v → i is horizontal, perpendicular to the rod, and a distance 3 L 4from the pivot. The bullet passes through the rod and continues moving in a straight line. Its kinetic energy after emerging from the rod is 1 2 of its initial kinetic energy. Determine the angular speed of the rod, about the pivot point, immediately after the bullet passes through. Express your answer in terms of m 1 , m 2 , v i , L, and constant(s).

As you are trying to move a heavy box of mass m, you realize that it is too heavy for you to lift by yourself. there is no one around to , so you attach an ideal pulley to the box and a massless rope to the ceiling, which you wrap around the pulley. you pull up on the rope to lift the box. use g for the magnitude of the acceleration due to gravity and neglect friction forces. once you have pulled hard enough to start the box moving upward, what is the magnitude f of the upward force you must apply to the rope to start raising the box with constant velocity? express the magnitude of the force in terms of m, the mass of the box.