Undergrad Energy of spinning objects as axis of rotation moves

[Trollfaz]

Imagine an object, e.g throwing knife, spins in the air but not forced to rotate about a particular axis, i.e no rod impaling it and forcing it to spin about the rod. Then the axis of rotation converges to it's center of mass (CM) to minimize I. But there's nowhere for it's rotation energy to go assuming no resistive forces of the medium.
$$E=\frac{1}{2} I \omega^2= k$$
Both I and $\omega$ are functions of t and
$\frac{dI}{dt}<0$
So angular velocity increases?
$$\frac{dE}{dt}=\frac{dI}{dt}\omega^2+ 2I\omega\frac{d\omega}{dt}=0$$
$$\frac{d\omega}{dt}=-\frac{dI}{dt}\omega^2/2I\omega>0$$
And if we know the rate of change of I we can solve this differential equation to find how angular velocity evolves

 

[vanhees71]

Your expression for the energy only describes the rotational energy of the body around the given axis but not the translational energy of the body as a whole. It's most simple to choose the center of mass as the body-fixed reference point. Then the total kinetic energy of the rigid body reads
$$E_{\text{kin}}=T=\frac{M}{2} \dot{\vec{R}}^2 + \frac{1}{2} \vec{\omega} \hat{\Theta} \vec{\omega},$$
where $M$ is the total mass and $\hat{\Theta}$ is the tensor of inertia around the center of mass of the body.