# Why does a cuboid spin stably around two axes but not the third?

Let $C$ be a cuboid (rectangular parallelepiped) with edges of lengths $a < b < c$.

Consider an axis that passes through the centers of two opposite faces of $C$. There are three such axes, one passing through the centers of the $a$–$b$ faces, one through the centers of the $a$–$c$ faces, and one through the centers of the $b$–$c$ faces.

Someone told me many years ago that if you throw the cuboid in the air and spin it around the $a$–$b$ or the $b$–$c$ axis, rotation will be stable in the sense that the rotation will tend to return to its original axis if it is slightly perturbed. But they said that a cuboid spinning around its $a$–$c$ axis is unstable, in that any small deviation in the axis of rotation will tend to be magnified over time.

I have tried to verify this by tossing various cuboid objects, such as Zippo lighters, cell phones, and wooden blocks; it appears to be true.

My questions are:

1. Did I describe this correctly? If not, what's the right description?
2. What is the mathematical explanation of this phenomenon?
3. Is there an intuitive explanation?

## 1 Answer

I like your description of this cool bit of unintuitive physics. I find the best balance of $a$ to $b$ to $c$ to cost of the object involved is best for a (boxed) pack of playing cards.

The mathematical explanation for this (see also Wikipedia) is that when considered in the principal axis frame (i.e. the frame of reference that rotates with the body and whose axes are the principal axes of inertia of the body), the motion can be described by the angular velocity $\vec{\omega}$ and the angular momentum $\vec{L}=I\vec{\omega}$, and in the absence of external torques it must conserve the magnitude of the angular momentum, $$L^2=L_1^2+L_2^2+L_3^2$$ (though not its direction since the frame is noninertial), and the rotational energy, $$E=\frac{L_1^2}{2I_1}+\frac{L_2^2}{2I_2}+\frac{L_3^2}{2I_3}.$$

The motion is then constrained to move along the intersections of an ellipsoid and a sphere: These curves are closed ellipses, or nearly so, close to the axes with the smallest and largest moments of inertia, but they are locally hyperbolae close to the middle one. Hence the instability.