A26.0 kg child plays on a swing having support ropes that are 2.40 m long. a friend pulls her back until the ropes are 45.0 ∘ from the vertical and releases her from rest. what is the potential energy for the child just as she is released, compared with the potential energy at the bottom of the swing? how fast will she be moving at the bottom of the swing? how much work does the tension in the ropes do as the child swings from the initial position to the bottom?

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.