The laws of conservation of energy and angular momentum?

Question

If I swing some object attached to a rope around myself in the horizontal plane and I wish to bring it closer, I have to pull the rope. The law of conservation of angular momentum states that the velocety of an object will increase, also, the kinetic energy of the object will increase, so, some work must be done on the object. But how can work be done on an object when the tension of the rope (the centripetal force) is always perpendicular to the object's velocity and therefore cannot perform work. What force accelerates the object?

Answer 1

You are almost correct on your details before the question.

Conservation of angular momoentum states the ANGULAR velocity must change if the radius changes (number of rotations per unit time).

Conservation of kinetic energy is for LINEAR velocities.

Angular velocity=Linear velocity/radius of object's path

You are right on the fact that there is no work done, since the force you applied is perpendicular to the velocity.

Assuming first no work is done on the body, and the radius suddenly shortens, then energy conservation would dictate that linear velocity must remain constant. Since the radius shortened, angular velocity must increase. The effect perceived by the human eye is that the object moves at a faster LINEAR rate (but it is really still the same as before), but is really just spinning faster, and a corresponding larger ANGULAR speed. Kinetic energy is then maintained.

Answer 2

When you are pulling rope, you are pulling the object towards the center through some distance along the radius.

The extrra force you applied over the centripetal force needed to maintain in the circular path at that radius is the cause of the motion of the object toward the center and that force has done some work. Force x distance moved along the radius.

Answer 3

You're correct that when you wind in the object,its kinetic energy increases because it's moving faster. This extra energy comes from the work you do winding it in against the centripetal force in the rope. Remember, if you take an infinitesimally short part of the circle it's swinging in, you're applying a force to prevent it continuing in a straight line, and when you wind it in, that force moves. Force times distance = work done.