SET A COIN ON ITS EDGE on a smooth surface and flick it with your finger. It will spin vertically but soon will start to wobble and tilt until, with a clatter that increases in frequency, it ends up flat on the surface. A cylindrical object such as a bottle can also be made to wobble on its base, but unless the wobble is so strong that it makes the object fall over it remains upright. A bit of experimenting reveals a lot about how such objects wobble and what determines whether a wobbler finishes flat or upright. My analysis is based on a study by Lorne A. Whitehead and Frank L. Curzon of the University of British Columbia that will appear in American Journal of Physics. Whitehead designed an apparatus for studying the phenomenon and initiated the mathematical analysis. I have also drawn on earlier work by Martin G. Olsson of the University of Wisconsin at Madison.

Wobbling has three noteworthy features. First, it is periodic. Second, the object rolls around on its supporting surface without slipping. Third, the center of mass of the object gradually moves downward but does not shift much horizontally unless the object has been started into a clumsy spin.

Wobble lasts for as long as it does because two losses of energy are minimized. The lack of slippage diminishes the loss of energy to friction with the surface on which the object is spinning. The lack of vibration of the center of mass and of the support point eliminates an additional loss of energy.

Figure 2 shows the main physical characteristics of a wobbling object. The object is in contact with the supporting surface at a single point. An axis of symmetry extending along the length of the object is usually at an angle to the vertical. As the wobbler spins around this axis the axis turns around the vertical. Since the center of mass is stationary as the object rotates, the bottom of the wobbler roll in a circle around the vertical axis pass precession, the movement usually seen when a toy top has been set spinning.

The force that generates precession is developed at the point where the wobbler touches the surface on which it is turning. Although each part of the wobbler is pulled downward by gravity, the combined weight can be assumed to act through the center of mass. If the wobbler were not also spinning around its axis of symmetry, the pull on the center of mass would cause the object to fall over.

A force usually makes an object accelerate in the direction of the force, but a force acting on something that is spinning creates a torque that makes the object turn. The torque on a wobbler causes precession. If the rate of rotation is too low or too high for the object to roll without slipping, friction quickly increases the rate of spin until the slipping stops.

What the torque is actually doing is redirecting the angular momentum of the wobbler. This quantity is the product of the object's distribution of mass (its moment of inertia) and its rate of spin. The angular momentum can also be described as a vector that lies in an imaginary vertical plane containing the axis of symmetry around which the wobbler is spinning. Rotating about the vertical axis that passes through the center of mass, the vector maps out an imaginary cone. Hence the axis of symmetry also rotates about the vertical. Lacking gravity and the torque it generates, an object could be made to spin without precessing. A wobbler then would not wobble.

A wobbler's rate of precession depends in part on the angle between the axis of symmetry and the vertical. When Whitehead and Curzon analyzed this relation a surprise emerged. In general a wobbler will precess steadily at any angle of inclination between approximately zero (with the axis of symmetry nearly vertical) and 90 degrees (with the axis nearly horizontal). The surprise was that the wobbler will not precess steadily in a certain zone that is determined by its shape. The forbidden zone is easy to determine if the object rolls around on a circular rim, as coins and bottles do, but is hard to determine if the shape is more irregular.

Whitehead and Curzon worked with a cylinder in their study of the requirements for steady precession. The range of inclination angles at which a cylindrical object cannot precess steadily depends on the shape of the cylinder. In particular the ratio of half of the cylinder's length to its radius is crucial. A coin is a cylinder in which the ratio is small, with beer cans and most other cylinders the ratio is large.

The results of the analysis are shown in the graph in Figure 3. The ratios of half length to radius are on the horizontal axis and the inclination angles are on the vertical one. For example, the inclination angle of a coin is small when the coin is nearly flat and large when it is on or near its edge.

The cylinder precesses steadily when its axis of symmetry is outside the forbidden zone. That zone is marked by the angles 8₁ and 8₂. The angle is 0₁ when the cylinder's center of mass is directly above the support point, in which case gravity provides no torque for the precession. At 8₂ the angular-momentum vector is vertical and torque again does not cause precession.

Ordinarily these angles differ. They are the same only for a cylinder with a half-length-to-radius ratio equal to (3^1/2)/2 (half of the square root of 3). A cylinder with a smaller ratio precesses more like a coin and would be recorded on the left side of the graph. Long cylinders such as cans precess in a way that is reflected on the right side.

A coin spun on a surface begins to wobble in a steady precession when the conditions are as indicated in section a of the graph. As it gradually loses energy to friction its angle of inclination decreases. Initially the rate of precession decreases too, but when the angle of inclination approaches zero, the precession rate begins to rise. You can detect the change with your ears as well as your eyes. As the angle of inclination decreases, the clatter of the coin first drops in frequency (reflecting the initial decrease in the rate of precession) and then rises.

If you are looking down on the coin, you can also monitor the spin rate. At first it is too high for you to be able to see the face of the coin clearly. As the coin approaches its final horizontal position the spin rate diminishes and the face of the coin becomes clearer. The increasing clatter from the increasing precession rate contrasts sharply with the decreasing spin rate.

How can the precession rate of a wobbling coin increase while the coin is losing energy to friction? The coin has three types of energy. Two are rotational kinetic energies associated with the spin around the axis of symmetry and the precession around the vertical. The third type is gravitational potential energy. Every part of the coin except the point in contact with the table has potential energy, but the situation is easier to understand if the entire amount is associated with the center of mass.

The coin has to tilt increasingly as it wobbles because friction with the surface gradually removes the energy of the spin around the axis of symmetry. With a smaller rate of spin the axis of symmetry must move closer to the vertical. As a result the center of mass drops closer to the surface and the potential energy of the coin decreases. The loss of energy (both kinetic and potential) forces the coin to precess faster, somewhat like a rubber ball that bounces more frequently as it loses energy.

In theory you could also make a coin spin in the conditions represented by section b of the graph, but it would be difficult. The coin begins to spin while it is nearly vertical. As it loses energy to friction its angle of inclination increase to 90 degrees. In principle the coin would come to rest balanced on its edge. This position is unlikely in practice because the coin is too sensitive to small perturbations. It would probably fall over and lie on the surface.

A cylinder with a ratio of half length to radius of (3^1/2)/2 shows the greatest range of inclination angles at which it can

precess smoothly. Longer cylinders such as bottles wobble in two different ways, which are represented by regions c and d on the graph. A cylinder set wobbling when it is nearly upright performs as shown in region c. You could start a bottle in this way by putting one hand on each side of it while it is inclined almost at the balancing angle of 8₁ and rapidly moving your hands in opposite directions.

The bottle rolls on its rim. It gradually speeds up, and the angle of inclination decreases as the bottle approaches the vertical. Since the bottle is tall, it can be affected by nonuniformities in the rim and the table. The wobble is usually unstable.

A cylinder wobbling under the conditions represented in region d begins with its axis of symmetry almost horizontal. The angle of inclination may be just slightly larger than 0₂. As with a coin the angle of inclination cannot be exactly 8₂ because the precession and spin would be infinitely fast. In losing energy the cylinder begins to lie down and its rate of precession decreases steadily. When it lies flat, it still has in principle some minimum rate of precession, but in fact it is quickly stopped by friction with the surface.

You can start a small cylinder in this type of motion with a snap of a finger. The cylinder precesses smoothly and gradually descends to the surface, where it rolls about more irregularly. Its clatter steadily decreases in frequency.

Whitehead realized early in his work that the wobble of a cylinder could be prolonged with jets of air. He rigged an apparatus in which jets of air were directed tangentially at a cylinder of aluminum eight centimeters long and three centimeters in diameter. Launched by hand at a large angle of inclination, the cylinder wobbled in a way represented by region d of the graph. Precession frequencies of up to 100 cycles per second could be achieved by adjusting the jets. Although the motion was stable, the cylinder had a tendency to drift across the table. Whitehead therefore set out to build a better apparatus.

The result is a container with a Plexiglas base 15 centimeters in diameter. Whitehead machined the base so that it is concave, its radius of curvature is 50 centimeters. The top surface is polished to eliminate rough features that would interfere with a stable wobble. A layer of rubber an inch thick is glued to the surface to provide better resistance to the air. The concave surface keeps the wobbler from straying from the center.

Air jets from a standard laboratory supply of air are directed into the container near the base. They maintain the energy of the wobbler, overcoming its losses to friction. The air goes out of the container through the top, which is a flat piece of Plexiglas with holes in it.

The main problem faced by Whitehead was how to freeze both the precession and the spin of the wobbler. Usually the two proceed at different rates. The solution was to deploy a wobbler in which the spin rate is an integral multiple of the precession rate. If a stroboscope is then set to flash at the precession rate, both the precession and the spin are frozen.

Whitehead chose as a wobbler a carefully machined cylinder of aluminum with a half-length-to-radius ratio of 2.71. When such a cylinder wobbles at an angle of inclination of 64 degrees, its spin rate should be twice as high as its precession rate. That angle is close to 82 for the cylinder. By adjusting the rate of flow in the air jets he could achieve fine control of the angle of inclination and thereby of the precession rate. The stroboscope was operated at a frequency of 50 hertz.

Whitehead had to remachine the cylinder a bit to achieve the desired ratio of 2 in its precession and spin rates. A precision of about . I millimeter was necessary. When the cylinder was right, he set it in motion by hand to wobble under the conditions of region d. It quickly developed a stable wobble near the desired angle of inclination. Whitehead adjusted the flow of air to maintain the angle. Under these conditions the wobble could be maintained for three or four days. A cylinder of this kind could be kept wobbling longer if the concave base of the apparatus were made of a material harder than Plexiglas. To keep wobbling smooth on Plexiglas calls for an occasional repolishing of the surface.

Olsson has studied the wobble of a coin, analyzing the motion with methods that could also be applied to a child's top. The two objects are actually quite similar. When Olsson sets a coin spinning on its edge, the spin is at first stable against perturbations from the table and from gravity. Friction gradually slows the spin, which soon drops below a critical value determined by the coin's mass, radius and moment of inertia. Then the coin begins to wobble.

Olsson also demonstrates wobbling in a large-scale way, spinning an aluminum disk that is an inch thick and about as big as a manhole cover. It wobbles with quite a noise, and toward the end, when the disk is almost flat, the racket is impressive. In a classroom where the seats are on a sloping floor the students can both watch the spin rate and hear the precession rate. The contrast between the decrease of the one and the increase of the other is mesmerizing.

In October, 1979, I described the rattle-back, a toy that resists spinning in one direction but spins easily in the other.

Many readers have asked where such a toy can be bought. It is now being sold by Toltoy, Inc. (5439 Schultz Drive, Sylvania, Ohio 43560), under the name of Space Pet.

Last December I examined the phenomenon of bubbles in beer and other carbonated drinks. Many people have written to me amplifying my explanation of why bubbles starting at the bottom of a glass of beer grow larger as they rise. I attributed the expansion to reduced hydrostatic pressure. My critics point out that the bubbles become larger than they would if reduced hydrostatic pressure were the only thing affecting them and that the bubbles must also be collecting dissolved carbon dioxide gas as they rise.

I erred too in stating that gas bubbles originate only on the sides of the glass where pockets of air or carbon dioxide gas already adhere to crevices. Such pockets do serve as nuclei for the formation of bubbles, but another important source is pointed out by Thomas D. Kunkle of the Los Alamos National Laboratory. The liquid contains microscopically small regions of gas left over from the cavitation that resulted when the beer was poured into the glass. I would have thought that the molecules of gas in such regions would quickly diffuse into the liquid, where they would dissolve. According to Kunkle, however, the regions are stabilized against diffusion by surface-active molecules that bond together to form a "skin" one molecule thick. These regions of gas then act as nucleating agents for the formation of bubbles as more gas comes out of solution.

Rinehart S. Potts of Glassboro, N.J., asked why one should avoid cleaning beer glasses with a detergent. It matters only if you want lots of bubbles in the beer after the cavitation from pouring the drink has subsided. A detergent is likely to coat the surface of the glass, and so the crevices there are less able to serve as nucleating agents.

Paul C. Condit of San Anselmo, Calif., and Richard W. Hill of East Lansing, Mich., both told me about the practice of salting beer that has been poured into a glass. Some people salt beer because they like the taste, others because the salt generates more bubbles. The adsorbed pockets of air on each grain of salt serve as nucleating sites.

A "yard" container for beer or ale is a distinctively shaped structure that is about a yard long, with a round bottom and a long, narrow neck. The drinker must know how to drink out of it or he will be drenched by a sudden gush of fluid. I said the trick is to tap the neck lightly so that bubbles will flow up the tilted container to replace liquid that is gently flowing down to the drinker. Martin Reid of Surrey in England describes a better technique, which works even if the fluid is uncarbonated (as some ales are in England). He rotates the neck of the yard glass as he drinks. In this way fluid travels along the sides of the neck as well as the bottom, and the drinker achieves more control over the flow.

H. D. Westphal of Puchheim in West Germany wrote to me about a way to remove the excess head of foam that can build up in a poured beer. Weissbier (or weizenbier), which is brewed from wheat, has yeast added to it so that the final stage of brewing takes place in the bottle. The beer contains much more carbon dioxide than most other beers do, and so a big head develops when weissbier is poured. The traditional glass for the beer is tall, curved along the sides and flat on the inside of the bottom. It is rinsed in cold water, a small amount of which is left on the bottom. Then the bottle is put in the glass with the neck close to the bottom. As beer emerges and foam is generated the bottle is pulled upward at the same rate and the foam is sucked into it. The bottle is then set aside so that the beer that materializes as the bubbles burst can be drunk later.

Two letters reminded me of stories I have heard for years but have not been able to confirm. James L. Ealy, Jr., of Pottstown, Pa., said that when an unopened bottle or can of beer has been inadvertently shaken, the carbon dioxide that has been released in it can be redissolved by tapping lightly on the container with a knife or some other object. Then one can open the container without releasing a gush of foam and liquid. I cannot explain how vibrations of the container and its contents could drive the gas back into solution.

Alvaro de Paiva Abreu of Rio de Janeiro reminded me that a carbonated drink in a container that has been opened can be kept from going flat for a while by putting the handle of a metal spoon in the neck. If this maneuver does in fact delay the loss of gas, how does it do so? I would welcome explanations of both of these tricks.

Bibliography

COIN SPINNING ON A TABLE. M. G. Olsson in American Journal of Physics, Vol. 40, Pages 1543-1545; October, 1972.

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