Celtic wobble stone

This at first sight quite inconspicuous stone has fascinated people already hundreds of years ago. It was usually found in riverbeds or at the sea as a round-washed ellipsoidal body, which looks symmetrical from the outside, but has an inhomogeneous — i.e. not uniform — mass distribution inside. By the way, it doesn’t take much effort to recreate the strange behavior of this stone. If you have an old spoon at home that you no longer need, you can bend the handle over the ladle so that it balances on the round bottom. Depending on which side it sticks out more, you will have a preferred direction of rotation. If you now bring the spoon to rotate on a flat surface, you will see that in the preferred direction of rotation it simply comes to a stop slowly due to friction. In the other direction, however, the speed of rotation decreases faster and the energy of the rotation leads to a rocking motion. This rocking motion in turn transforms into a rotating motion, and in the opposite direction! So the spoon has reversed its angular momentum by itself. Celtic soothsayers have used this apparent obstinacy to make decisions or to interpret the will of the gods. Whether this was done out of ignorance or with ulterior motives is not immediately comprehensible to us, because depending on how you ask the question, the stone naturally always gives you the answer you want. However, that no higher will or a cosmic power hides behind it, we will see in the following.

Figure 1: Bent spoon as “wobble stone

And now … the mathematics of it:

Every body with mass has a center of gravity and its three main axes of inertia pass through it. These are the axes on which you could clamp it and let it rotate without it “wobbling”. In the case of car wheels, for example, the inhomogeneity of the tires due to the manufacturing process is subsequently compensated for with small weights. The main axes are always perpendicular to each other and they coincide with the symmetry axes in symmetrical bodies. The axes of the Celtic wobble stone do not lie on each other, hence the asymmetry in its movement. Only the rotation around the vertical axis is stable, where it “runs ahead” of the horizontal ones. In the other direction, the rotation is unstable. There is some equilibrium, but this is much like balancing a soccer on the point of a needle. The slightest bump is enough to throw the system off balance. In the case of the Celtic Wobble Stone, this push is caused by the frictional force at the point of contact. Indeed, if there were no friction, the rotation would be stable in both directions. What happens now is that the energy of the rotation flows into the rocking motion. From here on, it is now not surprising that with each up and down movement, the stone continues to tilt in the direction on which the additional mass lies. After only a short time, the rotation in the preferred direction of the rocking motion has extracted all the energy and the stone rotates in exactly the same way as if it had been given the rotational impulse in the same direction, only now, of course, the energy is missing, which in the meantime has been “destroyed” by friction.

The exact principle, why the stone behaves now in such a way and not differently, is hidden in complicated equations, but it would be still interesting to mention that in the meantime also stones were manufactured, which are unstable in both directions. Depending on how cleverly you apply the weights, the frequency ranges in which the rotation becomes unstable shift. For example, if you rotate your spoon very fast in either direction, nothing happens at all. Only when the rotation frequency comes into the range of the rocking frequency, the asymmetry becomes noticeable. If you exploit this behavior, you can build many different frequency ranges into one and the same stone. There are already some that can reverse their angular momentum up to five times. However, these are difficult to manufacture, since the substrate must also be well chosen for this, so that there is still enough rotational energy after the first change of direction.


Literature

Kuyper, F.: Klassische Mechanik, 9. Auflage, Berlin, 2010.