IF YOU SPIN a certain type of stone in the "wrong" direction, it will quickly stop, rattle up and down for a few seconds and then spin in the opposite direction. Going in the "right" direction, however, it will usually spin stably. The stone is apparently biased toward one direction of spin. It will even develop a spin in that direction if you just tap one end downward. The rocking of the stone caused by the tap is quickly converted into a spin.

These curious stones were originally called celts because their behavior was discovered by archaeologists studying the prehistoric axes and adzes so named. I imagine the spin reversal was first noticed by an archaeologist idly spinning one of the celts on a table. My introduction to the stones came some years ago when I read Harold Crabtree's delightful book An Elementary Treatment of the Spinning Tops and Gyroscopes. Among the many spinning tops, hoops and other toys he described were the mysterious celts, but he did not indicate why they exhibited a bias for one direction of spin. Recently two people have rekindled my interest in the stones. Nicholas A. Wheeler of Reed College has been considering doing an analytical investigation of them. (If he does one, it will be to my knowledge the first study of the stones since the 1 9th century.) A. D. Moore of the University of Michigan has sent me several of the hundreds of celts he has made out of dental stone, spoons and other materials. Moore has dubbed his stones rattlebacks, a term I shall employ here.

Most rattlebacks have a smooth, ellipsoidal bottom. The top can be flat, hollowed out or ellipsoidal. The spin bias apparently results from the shape of the bottom and the distribution of the mass with respect to the axis of spin. The basic rattleback in Moore's collection has a flat rectangular top. The key feature of the design is that the long axis of the ellipsoid is aligned at an angle of from five to 10 degrees to the long axis of the rectangular top. This skewing (and perhaps some subtle shaping of the ellipsoid) introduces the bias to the spin of the stone but in a way that may be difficult to see in detail.

Most of the rattlebacks Moore has sent me spin clockwise as seen from above. When I spin such a rattleback clockwise it rotates rather smoothly until friction eventually stops it. When I spin it counterclockwise, it first turns through a few revolutions but then stops as its ends begin to rock up and down. Soon it starts to spin clockwise until friction halts it. If I tap the end of the stone when it is stationary, the ends chatter up and down for a few seconds and then the stone begins to turn clockwise. Again it is friction with the table that eventually stops the stone.

Some of the rattlebacks display a second reversal, from clockwise to counterclockwise, near the end of the spinning just as the friction is about to remove the last of the stone's energy. The second reversal is usually not as strong as the first and does not display the same type of vertical oscillation. Instead the stone rocks from side to side rather than longitudinally. A few of the rattlebacks display strong reversals with either direction of spin.

Moore forms most of his rattlebacks with dental stone, which he polishes and varnishes. Once he has made a good one (a feat that requires both experimentation and luck) he makes a mold of it for duplication. Some of his rattlebacks are made from the bowl of a spoon that he has detached from its handle. The bowl is glued to a flat rectangular piece of metal or Plexiglas with the long axis of the bowl's ellipsoid at an angle to the long axis of the rectangle. Pennies are glued on top of some of the rectangular pieces to add weight and to change the rattleback's moment of inertia.

One type of rattleback looks like half an egg. Across the flat top Moore tapes a brass rod at an angle to the long axis. Another type is similar to the basic design except that the inside is hollowed out to resemble a boat. Still another type has central terraces built on top of a rattleback of the basic design. Moore's favorite rattleback must certainly be the half egg. He once engaged in a contest with C. L. Stong, who conducted this department for almost two decades, to determine which of them could get the most back turns out of a rattleback. Moore won with the half egg, which made more than 15 back turns before it stopped.

When I examined the rattlebacks, I found that the basic kind exhibited the strongest reversal (counterclockwise to clockwise), which is followed by a slight second reversal as the stone nears the end of its energy. The rattleback with the pennies and the one with the terraces performed about as well as the basic type. The half egg did not spin well, tending to wobble so much that the ends of the brass rod scraped the table and thus drained energy from the egg. Moore is not sure why this former winner now does so poorly, but he suspects that a hairline crack on the bottom surface may be the cause.

The most fascinating rattleback is the boatlike one because it displays strong reversals in each direction of spin. When I spin it counterclockwise, it rotates for part of a revolution, stops, wobbles briskly up and down and then reverses for several revolutions. Then it stops spinning again, rocks from side to side and begins to spin counterclockwise once more. The performance continues until the stone runs out of energy.

What causes the reversals of spin? Although a complete mathematical explanation is difficult, careful observation reveals a few clues. When a rattleback of the basic design begins to reverse from a counterclockwise rotation, its longitudinal oscillations appear to be approximately around the short axis of the ellipsoid because the points marked C and D in the illustration on page 176 vibrate up and down more than the points marked A and F. (I think the oscillation is actually around another important axis, one of the stone's principal axes that is almost parallel to the short axis of the ellipsoid.) Somehow the counterclockwise rotation excites an instability of this kind.

When the rattleback executes its second and milder spin reversal, it appears to oscillate approximately around the long axis of the ellipsoid, sending points. G and H up and down with more amplitude than other points. (Again, theoretically the actual axis is a principal axis closely aligned with the long axis.) Apparently the clockwise spin excites this type of instability.

I can easily set up either type of oscillation in an initially stationary stone by pressing down on an appropriate part of the stone and then letting go. When I do this at point A, not much happens. The rattleback oscillates for a short time and may turn counterclockwise slightly. At B a similar action results in more pronounced oscillations and a clockwise rotation, but the performance is still far from dramatic. Livelier effects are obtained with a press and release at point C; the longitudinal oscillations are vigorous and the stone is quickly propelled into a clockwise rotation. The oscillations are identical with the ones I see when the spin reverses from counterclockwise to clockwise.

When I press point G and release it, the stone rocks from side to side and soon begins to spin counterclockwise. Thus when I excite the vertical oscillations at G, I am duplicating the second type of spin reversal (of the initial clockwise rotation). Similar results are obtained with a press and release at other points between G and A.

The transverse oscillations are much slower than the longitudinal ones, but both types are so fast that I cannot follow the motion easily. Moreover, once the oscillations start the stone begins to turn and my perspective is altered. To follow the oscillations better I tried tapping a rattleback I had placed on a strip of tape that had its sticky side up. The tape kept the stone from rotating, but of course it may also have interfered with the stone's natural motion. Although the stone seemed to oscillate primarily around a single axis, it also gave the appearance of rolling out of the primary plane of oscillation. I do not know if such a feature is essential to the reversals of spin.

The first scientific investigation of spin-reversing stones was made by G. T Walker (no relation) in 1896. Some of his results can still be found in reprints of old books dealing with rotational mechanics. Usually the description of the stones is put in the chapter on asymmetrical tops, that is, tops that are not symmetrical around the vertical axis. Theoretical discussions of symmetrical tops are challenging enough, but the theory of asymmetrical tops might be called mind-spinning.

A successful rattleback has several key features. One has to do with the misalignment of the stone's principal axes and the axes of the ellipsoid on the bottom. A principal axis of an object is an axis about which the object could be rotated freely with no rotation about either of the other two principal axes. The three mutually perpendicular axes are usually found along the axes of symmetry of many common objects. If the ellipsoid were aligned with the rectangular top in Moore's basic rattleback, the principal axes would be easy to find because they would coincide with the axes of symmetry. One axis would be vertical and the other two would be parallel to the short and long axes of the ellipsoid. They would all cross at the center of mass of the stone.

A good rattleback's ellipsoidal bottom is not aligned with the rectangular top. The vertical axis is still a principal axis, but the other two axes are now shifted around the vertical one in the direction in which the ellipsoidal bottom has been shifted out of alignment with the rectangular top. One of these principal axes, call it axis 2, is somewhere between the long body axis of the rectangular top and the long axis of the ellipsoid. The other one, axis 3, is somewhere between the corresponding two short axes and is perpendicular to axis 2. Their exact locations depend on the relative mass distributions of the rectangular top and the ellipsoid. The misalignment of the ellipsoid from the principal axes is an essential feature in the reversal of spin.

A second important feature of a well-trimmed rattleback is that the radius of curvature is different along the two axes of the ellipsoid. (A large radius of curvature corresponds to a surface with a small amount of curvature.) If the bottom were perfectly spherical, the rattleback would not reverse its spin.

It is also important that the mass distribution of the stone be different for the two principal axes. The function giving the distribution of mass with respect to a particular axis is called the moment of inertia. Consider principal axis 3. When one of Moore's rattlebacks oscillates up and down around that axis, as it does during the strong type of spin reversal, the moment of inertia for the oscillation is relatively large because some of the mass is at a relatively large distance from the axis. Suppose the stone oscillates around principal axis 2, as it does during the second, weaker type of spin reversal. The moment of inertia for that oscillation is relatively small because most of the mass is relatively close to the axis. This difference in the moment of inertia of the two horizontal principal axes is essential to the success of a rattleback.

In sum, Moore's basic rattleback has the following key features. The long axis of the ellipsoid (the one with the large radius of curvature) is shifted counterclockwise (as seen from above) from principal axis 2, which has a small moment of inertia. This arrangement biases the rattleback for a strong spin reversal from counterclockwise to clockwise and a weaker spin reversal in the other direction. What would happen if the long axis of the ellipsoid were shifted in the other direction? Would the strong and weak spin reversals be in the opposite directions? Although G. T. Walker's theoretical work predicts the behavior, I shall leave the questions for you to answer with suitably carved rattlebacks.

How do the key features of Moore's rattleback cause the stone to reverse spin? Primarily they make it unstable against small perturbations of the spin around the vertical axis. If the bottom were spherical or the ellipsoid were aligned with the principal axes, any small perturbation would have a small effect on the spin and would not cause noticeable oscillations. With a properly shaped stone, however, small perturbations from the initial spinning of the stone or from the tabletop generate oscillations that rapidly grow in amplitude. Which of two types of instability is excited depends on the direction of spin and the design of the stone.

Suppose a rattleback of the basic shape is spun counterclockwise. Because of its design any small perturbation causing an oscillation around principal axis 3 is rapidly enhanced. The frictional forces acting on the stone during the oscillation stop the spin and then initiate an opposite one. Once the reverse spin has begun, the friction forces stop the oscillation. Hence a spin in the counterclockwise direction allows one type of instability-the oscillation around principal axis 3-to grow large quickly, and then the frictional forces accompanying the oscillations reverse the spin.

If the stone is initially spinning clockwise, another instability is encouraged, one that enhances oscillation around the other horizontal principal axis. Again the design of the rattleback allows this instability to grow in amplitude exponentially. Frictional forces during this kind of oscillation cause the clockwise spin to stop and then reverse.

I wanted to investigate how the behavior of a basic rattleback would change if I altered one of the key design features. The only feature I could change easily was the relative moment of inertia for the two horizontal principal axes. To alter it I taped a pencil across the top of a rattleback, first parallel to the length of the rectangle and then parallel to the width. Each time I was careful to balance the pencil so that the rattleback would spin on the same area on the ellipsoid.

With the pencil lengthwise the rattleback behaved generally as it had before because the pencil merely increased the difference in the moments of inertia around the principal axes. With the pencil parallel to the width the stone did not reverse spin or show any of the characteristic instabilities. By placing the pencil across the width of the rectangle I was increasing the moment of inertia around principal axis 2 without much changing the moment of inertia around principal axis 3. Apparently I increased the one enough to make it comparable to the other. When the moments of inertia are much the same, the oscillations due to small perturbations do not grow exponentially with time and therefore remain small. In the absence of large oscillations around the principal axes the frictional forces from the tabletop cannot stop the rattleback and then reverse its spin.

A moment of inertia can be varied in a more controlled way by mounting long stove bolts on a rattleback. Extend a bolt in each direction along a principal axis and screw several nuts on the ends of the bolts. The moment of inertia is changed by moving the nuts closer to the center of the rattleback or farther away from it. (The stone must still be balanced.) With this arrangement I was again able to eliminate the spin reversals when I equalized the moments of inertia for the two horizontal principal axes. What would happen if one adjusted the nuts to make the moment of inertia around principal axis 2 larger than the one around principal axis 3? Would spin reversal appear again but with the directions of the strong and weak reversals interchanged?

Moore's half-egg rattleback has a brass rod by which the moments of inertia can be altered. (Repositioning the rod also shifts the principal axes somewhat.) When I tape the rod across the short width of the flat top, the egg spins stably in each direction. With the rod along the longer axis of the flat top instabilities and spin reversals appear. Normally the rod is positioned at a small angle to the longer axis of the flat top, as is shown in the illustration below. The half egg reverses weakly in each direction of spin. With the rod mounted in this way the half egg is also sensitive to a tap on its perimeter. The direction of the resulting spin depends on the location of the tap. The illustration indicates the possibilities.

G. T. Walker's analysis predicts that the frequency of oscillation is higher around principal axis 3 than it is around axis 2, a point easily verified with a basic rattleback. His equations also predict that neither instability will arise if the spin frequency is higher than the oscillation frequency. With Moore's rattlebacks my attempt to eliminate the instability for the strong reversal (the one for a counterclockwise spin) is impractical because the vigor of my initial twist inevitably introduces instabilities. I can easily spin the rattleback clockwise, however, with a spin frequency greater than the oscillation frequency around principal axis 2. As far as I can tell the oscillation does not appear until friction has slowed the stone, presumably to a spin frequency lower than the oscillation frequency.

I played with Moore's rattlebacks on a smooth Formica top and a few other surfaces. I have not investigated the effects of the friction coefficient between the rattleback and the surface on which it is spun. If the friction is too high, the spinning should be eliminated so quickly that no spin reversal develops. If the friction is too low, the stone will reverse spin with difficulty because the reversal requires friction.

You can easily carve rattlebacks from dental stone or wood. Adjust the shape of the bottom until the rattleback works properly. You may be able to think of variations of the basic design. For example, you could investigate the effect of weight distribution by taping small weights or metal rods on the top of the rattleback. By making a number of rattlebacks you could determine how the orientation of the ellipsoid on the bottom affects the spin reversal. If you are out to win a contest for the number of spin reversals of a rattleback, look for an optimum angle between the long body axis and the long axis of the ellipsoid. You might also try to build a rattleback big enough to ride on. If you could make one with a vigorous spin reversal, riding it would be like riding a bucking bronco.

I turn now to the Tippe Top, which also reverses unexpectedly. Such a top usually has a spherical bottom and a central stem. It is spun by sharply rolling the stem through the fingers and dropping the top on a flat surface. The surprise is that the top spins on its spherical end for only a few seconds and then, turning upside down, spins on its stem. The motion appears to violate the law of the conservation of energy because the top seems to raise its center of mass (which is in the spherical section) without outside help.

The top has long fascinated observers, including several distinguished physicists and mathematicians. In a recent paper Richard J. Cohen of the Massachusetts Institute of Technology describes how William Thomson (the eminent physicist better known as Lord Kelvin) spent his time spinning smooth stones on the beach instead of preparing for his mathematical examination at the University of Cambridge. Later Niels Bohr who developed the first modern mode of the hydrogen atom, became similarly entranced with the mechanics of the Tippe Top.

The surprising inversion of the top arises from friction between the top and the surface on which it spins. Suppose it spins with its stem tilted away from the vertical axis, as is shown in the illustration below. Because it is spinning it develops a sliding friction force (perpendicular to the plane of the page in the illustration). This force creates a torque about the center of mass, causing the top to invert.

You could make a Tippe Top by cutting off part of a solid rubber ball and inserting a bolt in the flat surface. A typical school ring with a smooth stone displays the same type of inversion. Spin the ring on the stone. A hard-boiled egg that is spun flat will rise to spin on one end. Indeed, spinning an egg is one way to ascertain whether it is hard-boiled; a fresh egg will not stand up because of the sloshing of its internal liquid.

AT this year's International Science and Engineering Fair in San Antonio, Tex., several projects dealt with topics discussed recently in this department. In the physics section David Kirk of Edmond, Okla., sought to determine whether initially hot water might under some circumstances freeze faster than initially warm water. He found, as I did, a great deal of variation, but his results seemed to indicate that with open top containers the effect can be demonstrated. Joella Carr of Winchester, Tenn., studied the floating of water drops above the surface of a body of water, confirming that vibration of the bulk water prolongs the lifetime of the drops for 10 minutes or so.

In the chemistry division two students investigated oscillating chemical reactions. Helen Runnels of Baton Rouge, La., studied the periodicity of the Belousov reaction, finding that by increasing the cerium concentration from 0 to 10^-6 molar she was able to increase the period of the reaction threefold. She also found that contaminations of chloride bromide or iodide stopped the oscillation. Her indicator was nitroferroin. William Crooke of Pensacola, Fla., investigated the chemical waves in a similar reaction. A booklet containing abstracts of all the projects at the fair can be obtained for $2.50 from Science Service, 1719 N Street NW, Washington, D.C. 20036.

Bibliography

AN ELEMENTARY TREATMENT OF THE THEORY OF SPINNING TOPS AND GYROSCOPIC MOTION. Harold Crabtree. Longmans, Green and Co., 1909.

A TREATISE ON GYROSTATICS AND ROTATIONAL MOTION: THEORY AND APPLICATION. Andrew Gray. Dover Publications, Inc., 1959.

MECHANICS: LECTURES ON THEORETICAL PHYSICS, VOL. 1. Arnold Sommerfeld. Academic Press, 1964.

THE TIPPE TOP REVISITED. Richard J. Cohen in American Journal of Physics, Vol. 45, No. 1, pages 12-17; January, 1977.

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