Rotational kinetic energy of a body

Question

How to find total kinetic energy of a body which is rotating about multiple axes at the same time? For example consider a wheel with a shaft, and that the shaft is hinged. So the wheel is rotating about the shaft and the shaft itself is simultaneously rotating about the hinge. So we can say that the wheel is rotating about two axes : one along the shaft and another about the hinge. another example would be a gyroscope.

Answer 1

At any moment, the object is only rotating about one axis. You are free to break this angular velocity up into components however you like, including one along the shaft and one through the hinge, but that does not mean that the object is fundamentally rotating about two axes at once. In this sense, angular velocity is analogous to ordinary (translational) velocity: you can break the velocity down into northward, eastward and upward components, but that does not mean the object has 3 velocities.

In general, to get the rotational kinetic energy, $T_{rot}$, from the angular velocity $\vec{\omega}$, you need to know the inertia tensor, $I$, which is a 3-by-3 matrix: $$T_{rot}=\frac{1}{2}I_{ij}\omega_i\omega_j,$$ where summation over the indices $i$ and $j$ is implied.