How do you define the total rotational energy of an object? This problem arose when I was applying a conservation of energy argument to a mechanics problem, (a spinning coin on a table) and wasn't sure how to define the total rotational energy of the coin. At first I defined it's total rotational energy as about the axis that runs perpendicular to the table and through the center of mass of the coin, and yet it doesn't seem like this is the total rotational energy. 

For example, I can take a cube and spin it about the y axis, and then spin it about the z axis as well. In essence, it seems to me as if the cube is spinning about two different axis at the same time. In this case, to define the total rotational energy wouldn't I have to take the rotational energy about the y axis, and add it to the rotational energy about the z axis? Or do I only need one axis to define the total rotational energy of an object?
 A: Let's do this using angular momentum as a vector. This should clear up the question on using both axes separately or one new one.
The spinning around the y-axis will give an angular momentum in the y-direction: $\vec{L_{y}} = \hat{y} L_{y}$, while the spinning around z-axis gives angular momentum in the z-direction: $\vec{L_{z}} = \hat{z} L_{z}$. We can get the net momentum by addition of these two vectors:
$\vec{L} = \vec{L_{y}} + \vec{L_{z}} = \hat{y}L_{y} + \hat{z}L_{z}$.
We will manipulate this to give us magnitude and a unit vector:
$\vec{L} = (\hat{y} L_{y} + \hat{z} L_{z} ) \frac{\sqrt{L_{y}^{2} + L_{z}^{2}}}{\sqrt{L_{y}^{2} + L_{z}^{2}}} = \sqrt{L_{y}^{2} + L_{z}^{2}} \left( \frac{\hat{y}L_{y} + \hat{z}L_{z}}{\sqrt{L_{y}^{2} + L_{z}^{2}}} \right)$.
Here we have the new angular momentum magnitude and also the new axis it is spinning around. The energy will be:
$E = \frac{|L|^{2}}{2 I} = \frac{L_{y}^{2} + L_{z}^{2}}{2 I}$,
with $I$ being the moment of intertia. Note that we actually could have just added the energies from the different axes and we would have gotten the right answer, even though the object is truly spinning around only one new axis.
