Rotation in 3D
For a point mass, rotation is simple. For a rigid object (like a molecule), we must account for how the mass is distributed in all three directions. This is captured by the Inertia Tensor (\(I\)), a \(3 \times 3\) matrix.
\[\vec{L} = I \vec{\omega}\]
Notice that the angular momentum \(\vec{L}\) might not point in the same direction as the rotation axis \(\vec{\omega}\)!
Worked Examples
Example 1: Principal Axes
Every object has three "Principal Axes" where the inertia tensor becomes diagonal. If you rotate around these axes, the motion is stable. If you rotate around anything else, the object "wobbles." This is why a spinning football or a planet stays steady only in specific orientations.
The Bridge to Quantum Mechanics
Atomic nuclei and molecules are rigid bodies. Their rotation is governed by the inertia tensor. In Quantum Mechanics, the energy of a rotating molecule is \(\hat{H} = \frac{\hat{L}^2}{2I}\). Because the inertia tensor can have different values for different axes, the energy levels of a "top" (like a water molecule) depend on how it is spinning. This leads to Rotational Spectroscopy, which we use to identify the shapes of molecules in space.