When does the angular momentum point in a different direction from the angular velocity?

Question

I read this somewhere:

$$\mathbf{L} = \tilde{\mathbf{I}}\mathbf{\omega}$$

In general, the angular momentum vector, $\mathbf{L}$, obtained from Equation above, points in a different direction to the angular velocity vector, $\mathbf{\omega}$. In other words, $\mathbf{L}$ is generally not parallel to $\mathbf{\omega}$.

I don't quite get it: when is the angular momentum not parallel to the velocity? Because judging from the above equation, it should be parallel, since moment of inertia is a scalar?

Answer 1

The moment of inertia is a rank 2 tensor not a scalar.

You'll commonly see it written as a scalar, but this is because by choosing your axes to line up with the principal axes of the object the matrix representing the moment of inertia can be diagonalised:

$$ {\bf I} = \left( \begin{matrix} I_{00} & 0 & 0 \\ 0 & I_{11} & 0 \\ 0 & 0 & I_{22} \end{matrix} \right) $$

So if the rotation is about, for example, the $0$ axis you get:

$$ \vec{L} = \left( \begin{matrix} I_{00} & 0 & 0 \\ 0 & I_{11} & 0 \\ 0 & 0 & I_{22} \end{matrix} \right) \left( \begin{matrix} \omega \\ 0 \\ 0 \end{matrix} \right)$$

or:

$$ \vec{L} = I_{00} \vec{\omega} $$

where $I_{00}$ is indeed a scalar. However this is a special case and whenever you see the moment of inertia given as a scalar you'll find this applies to only one axis of rotation and that axis is one of the principal axes.