The Amazing Rattleback!
From New Scientist, 26 July 1997
Try to spin it clockwise. Watch it spin, shake, and
rattle, and slow down.
Then, watch it spin in the opposite direction!
In Wales, it is called a rebellious celt. To us, it's the rattleback. "It" is a long, thin plastic toy with a base shaped like the hull of a boat. When you spin it one way, it turns a few times before the ends start to rattle up and down. The more it wobbles, the slower it rotates - until it stops spinning altogether. Finally, it starts to spin in the opposite direction. What could possibly cause this?
The first attempt to analyze rattlebacks was around a century ago. In the mid-1980s, two detailed mathematical analyses were done: one by Hermann Bondi (then Master of Churchill College, Cambridge) and the other by Mont Hubbard (Professor of Mechanical Engineering at the University of California, Davis).
Bondi and Hubbard agreed that the rattleback's amazing behavior needs three main ingredients. First, the curved base must have two different radii -- one long radius for the lengthwise curve and one shorter radius for the tighter curve across its width. Next, the symmetry axes of the rattleback must be twisted slightly from its principal axes of inertia (see upper part of figure).
Any rigid object has three principal axes of inertia. They sit at right angles to each other and if you spin the object about one of them, there is no tendency to rotate about the other two.
Finally, there must be a different distribution of mass about each of the two horizontal axes of inertia -- a long, thin shape, say.
Given these characteristics, the mathematics of mechanics predicts how the rattleback should behave. The trouble comes in understanding these equations in physical terms - what the mathematics actually represents. "It's only clear through the equations," says Hubbard. "I don't intuitively understand it."
We can get some idea about it's physics by looking halfway through the rattleback's run, while it is rocking back and forth (see lower part of figure). Friction acts horizontally (at the point of contact between the rattleback and the tabletop) to prevent the rattleback from slipping. One part of this frictional force creates a torque that tends to rotate the rattleback about its vertical axis. Since friction always acts in opposition to the motion, this part of the frictional force acts in the direction opposite to the spin direction the rattleback had. That is, it rotates the rattleback in the opposite direction to the way it started.
"To make it more complicated, the point of contact is moving all the time and the torque changes," says Hubbard. If the inertial and symmetry axes of the rattleback were the same, the average torque over a single oscillation would be zero. But for the rattleback, there is a net torque in one direction. And it is this that reverses the angular momentum, according to Bondi.
Another way to understand how the rattleback works is in terms of energy. According to the math of it, each direction of spin is linked to a different mode of oscillation: if clockwise rotation feeds the pitching oscillation, then counter-clockwise spin would feed a side-to-side, or rolling, oscillation. So, when the rattleback spins clockwise, any tiny pitching oscillation will grow exponentially. It feeds off the rotational energy which slows down the spin. "But even when there is zero spin, the torque still acts," says Hubbard. So the direction of spin changes. Now beginning to spin counter-clockwise, the energy stored in the pitching oscillation feeds into the counter-clockwise rotation (and the rolling oscillation). Some rattlebacks can reverse spin directions again by trading rotational energy with the rolling oscillation.
We may have to wait for a better physical description of how the rattleback behaves. Still, after a hundred years of probing the toy, it's unlikely that scientists will stop now. "The thing about scientists is that they really like toys," says Brian Pippard of the University of Cambridge. "They have an interest in anything that looks odd. And they're not happy until they can describe how it happens."
Adapted and expanded from the New Scientist, 26 July 1997 by 4Physics