Suppose if a ballerina dancer was rotating at a constant angular velocity with her arms wide open. Then she retracted her hands which changed her moment of inertia and increased her angular velocity. No external torque was acted upon in this Process. That means the angular momentum is conserved. So by the above calculation, that means her kinetic energy increases.
But that doesn’t abide by the Conservation of energy. What is happening here?
We can see that if the moment of inertia decrease than the angular velocity increases. But that means the kinetic energy overall increases. But if no net torque is acting, then where is this energy coming from?
Note: It doesn’t have to be a ballerina dancer, theoretically it can be any rigid body.