I know the kinetic energy of a rigid object is
\begin{align}\tag{$1$}
KE = \frac{1}{2}mv^{2} + \frac{1}{2}I\omega^{2}
\end{align}
where $v$ is the velocity of the center of mass of the object, $\omega$ is its angular velocity, and $I$ is the moment of inertia about its center of mass.
Now the things is, shouldn't the moment of inertia be specified about an axis as opposed to a point? I can understand that the axis has to be through the center of mass, but which direction ought it to be oriented? If there is a specific axis we have to use, how do we calculate this axis for an arbitrary body undergoing arbitrary motion?
For an example, consider a uniform solid cylinder (radius $r$, height $h$) rolling without slipping at a constant velocity $v > 0$. I could consider the axis along the axial direction of the cylinder through the center of mass and obtain
$$ I = \frac{1}{2}mr^{2}. $$
We can consider $(1)$ with $\omega = v/r\ne 0$ with no issues. But couldn't I also consider a perpendicular axis oriented, say, vertically and through the center of mass? In that case,
$$ I' = \frac{1}{12}m(3r^{2} + h^{2}). $$
This is clearly different, and as the cylinder rolls, I would expect the angular velocity to be $\omega\,' = 0$. Wouldn't this change the result in $(1)$?
Best Answer
If you ran next to the center of mass, it would be stationary in your frame of reference. You would see the object rotating around some axis through the center of mass. That is the axis you should use.