Hello friends:

Recently I've been studying the physics of the rotational motion of rigid bodies. I understand that rotational kinetic energy is expressed as:

E_{K}= ½ mω^{2}R^{2}

Where m is the mass of the rotating point-mass body, ω is the angular velocity, and R is the radius of the circular path.

I also understand that angular momentum is expressed as:

L = mR^{2}ω

According to the Law of the Conservation of Angular Momentum, if no resultant torque acts on this system, then angular velocity remains constant. If angular velocity increases, then the radius of the circular path must decrease enough to disallow any change in angular momentum. For example, if a 2-kilogram point mass rotates at 1 radian per second on a circular path of .707 meters, the angular momentum L = 1. (I will omit units for the sake of brevity.) If the same point-mass rotates at 10 radians per second, then the radius R must decrease to 0.224 meters to allow L to remain equal to 1.

Now here's the problem I'm running into. I calculated this point-mass's initial kinetic energy as 0.5 Joules. Using the final values of ω = 10 rad/s and R = 0.224 meters, the kinetic energy increases to 5 Joules. How is this increase in energy possible? Where is the extra 4.5 Joules of energy coming from?

I'm making a mistake somewhere. Can anybody here explain this paradox to me?

Thanks!

Jagella