Angular momentum and Conservation of energy [duplicate]

Question

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.

Answer 1

The ballerina had to do work to pull in her arms. The necessary energy was supplied by biological processes, and ultimately from what she ate.

Answer 2

But if no net torque is acting, then where is this energy coming from?

The energy comes from some internal source of energy. For the case of a ballerina it comes from chemical energy in the muscles. For other cases it could be an internal battery or spring or something similar.

Note: It doesn’t have to be a ballerina dancer, theoretically it can be any rigid body.

Actually, it must not be a rigid body. A ballerina is not rigid and rigid objects will not exhibit this behavior. Only a non-rigid object can reduce or increase the KE by changing some other internal energy.

Answer 3

In a more general note, kinetic energy can be changed by internal forces (or internal torques). This is a matter of confusion sometimes as this behavior is different from that of momentum which can only be changed by external forces (or externl torques). The confusion may come from the fact that the dynamics of a single, rigid object is somehow "extended" to the dynamics of systems of bodies. Many introductory physics courses do not treat systems of bodies. For a single, rigid body it is true that only external forces can change the KE because the internal forces cannot do work (rigid = no relative displacement). This is not the case when we have a system were the internal forces can do work, like in the case of a deformable system.

Answer 4

$$\omega(dI) + I(d\omega) = 0$$ $$0.5 I\omega^2 = KE $$ $$\frac{d(KE)}{dt} = -\omega^2 \frac{dI}{dt}$$

this shows KE decreases with increase in MI and decrease in $\omega$.

Suppose there is a rotating disc of mass $M$ rotating with angular velocity $\omega$ and we slowly add sand on its boundry $L$ will remain conserved since there would be relative motion between sand particles and the disc energy would be lost as heat.