The behavior of any top is due mainly to the effect of gravity. Every atom of the top's mass is pulled downward by gravity, but the net pull is more readily envisioned as being through the center of mass, which lies somewhere inside the top, usually at its geometric center. The weight of the top can be represented by a vector pointing downward from the center of mass. It seems logical that because of this pull the top would be less likely to remain standing than to topple over (as you would if you were leaning away from the vertical).

The difference is that the top is spinning. As a result the downward pull of gravity gives rise to the surprising rotation of the top about the vertical. This reaction is not easy to visualize because you are more familiar with nonspinning objects. Normally a force on an object causes an acceleration in the direction of the force. When a spin is involved, the force may result in a motion perpendicular to the direction of the force. Such an unfamiliar motion is part of the fascination of tops.

The spinning top has angular momentum. Where linear momentum is the product of the mass and the velocity of an object, angular momentum is the product of the mass distribution (the moment of inertia) and the angular velocity. The top has angular momentum because it is spinning around its long body axis and because it has a mass distributed around that axis. The angular momentum is a vector that lies along the body axis, the axis around which the top is symmetrical.

The only way to change the angular momentum of an object is by means of a torque. A torque is scientifically defined (not too differently from the common usage) as the product of (1) the force on an object and (2) a lever arm that extends from a pivot point to a line drawn through the force perpendicular to the lever arm. The pivot point for a top is obviously at the pointed end touching the floor or a tabletop. The lever arm extends horizontally from that point to a vertical line drawn through the center of mass, which is where gravity is considered to be pulling on the top.

Gravity provides not only a downward force but also a torque to change the angular momentum of a top. The torque does so in a simple way: it redirects the angular momentum, rotating the vector about the vertical axis. Since the vector must continue to be along the body axis, that axis also rotates about the vertical in the motion called precession. It maps out a cone centered on the vertical. (For the present I shall treat the point of the top as being fixed where it touches the floor.)

If the top is spinning counterclockwise as it is viewed from overhead, the precession around the vertical is also counterclockwise. If the top is spinning clockwise, the precession is clockwise. If the top were not spinning, the torque effect of gravity would cause it to fall to the floor, which it actually does during the last stage of its spin because of the effect of friction on the point on which the top is spinning.

Nearly all tops fall somewhat as they start to spin. The reason is an energy requirement. If the top is to precess as a result of the torque due to gravity, it must have kinetic energy. Occasionally it gains energy from the launching. More often the energy must come from the initial fall, during which the decrease in the potential energy of the top is transformed into kinetic energy.

This simple story of the top fails the first time you examine the motion of a real top. The top does not just spin and precess; its body axis nods in what is called nutation. As it nutates the angle between the body axis and the vertical varies between two values determined by the mass distribution and kinetic energy of the top and its initial angle with the vertical. It is possible to illustrate the types of nutation by tracing the position of the upper end of the body axis on a sphere centered on the point where thc top touches the floor. The angle between the body axis and the vertical is limited by two circles drawn around the vertical. The circles represent the range through which the top can lean away from the vertical during nutation.

In one type of nutation the body axis weaves between the two limiting circles harmonically, touching each circle tangentially. The precession of the axis around the vertical is always in one direction, either clockwise or counterclockwise as seen from above. In the illustration in Figure 4 the rotation is counterclockwise.

In the second type of nutation the body axis loops between the two limiting circles but still touches each one tangentially. The direction of travel of the body axis periodically changes between clockwise and counterclockwise. In spite of this reversal the precession has an average value that is in one direction or the other. In the illustration the average precession is counterclockwise.

The third type of nutation traces cusps on the imaginary sphere. The path of the body axis meets one of the limiting circles tangentially as it does in the other types. The path meets the other limiting circle perpendicularly. The precession is consistently in one direction, but the rate of precession varies from maximum at the lower limiting circle to zero at the upper limiting circle.

The type of nutation that occurs (if any) depends on the initial conditions of the top's spinning. Since there are many perturbations in starting a top, you do not always have full control over those conditions. Suppose the launching imparts to the top a precession velocity that is in the same direction as the precession velocity gravity provides. Then the top precesses with harmonic nutation. Regardless of where the top is during its cycle of nutation, either its initial precession velocity or the velocity provided by gravity guarantees its continued precession in one direction around the vertical.

If the initial precession velocity is opposite to the precession resulting from gravity, the nutation is looped. During the lower part of the loop the gravitational precession drives the top around the vertical in the preferred direction. During the higher part of the loop, where the body axis reaches its upper limiting circle, the precession due to gravity is exhausted and only the initial precession is left. Until the top can drop and be driven around the vertical again by gravity it precesses in the opposite direction.

The third type of nutation is often seen when a spinning top is initially held at an angle to the vertical and then released. It has no initial precession and so it just falls to the lower limiting circle The fall is necessary because the resulting decrease in the potential energy of the top provides energy for the precession brought about by the torque exerted by gravity. When continued nutation brings the body axis back to its initial angle with the vertical, the top regains its initial potential energy. Precession then stops momentarily, since no energy is left for it. Afterward the top again falls to the lower limiting circle and precession continues until the top again is raised by nutation.

The limiting circles originate from three severe restrictions on spin, precession and nutation. The total energy (kinetic and potential) of the top must remain constant. (I shall discuss the effects of friction below.) The angular momentum along the body axis must also remain constant in amount, although its direction can change, because there is no torque along the axis to change it. Finally, the angular momentum along the vertical must remain constant for the same reason. (Once precession begins, the calculation of these two angular momentums becomes harder because the angular velocity then includes both the spin of the top and the precession velocity.) Because of these three strict rules the body axis is limited to a certain range of angles with the vertical. If the body axis dropped below the lower limiting circle or rose above the upper one, it would do so only by disobeying the rules. (Tops do rise above the upper circle but only because of friction.)

Most mathematical models of tops are devised with the simplifying assumption that the kinetic energy of the spin is much greater than the change in potential energy as the top weaves in nutation. Such a top is said to be "fast." With this assumption several features of the motion can be related to the rate at which the top spins. A lower rate increases the nutation weave but decreases the rate of nutation. The average rate of precession also depends on the spin, being faster with a slower spin. You can easily see these relations when you spin a top on the door. As friction gradually robs the top of spin the precession rate increases and nutation becomes slower and more pronounced. Finally, just before the end, the top swings sluggishly up and down as it precesses around the vertical faster than ever.

The precession rate and the extent and frequency of nutation also depend on the mass and shape of the top. In general a greater mass increases each of these components of the motion. (Of course, a more massive top is more difficult to spin at a given rate than a less massive one.)

If the top is spinning quite rapidly, the small amount of nutation that should be present might be eliminated by the friction on the point of the top. Then the top would appear to precess uniformly around the vertical without nutation. This uniformity, however, is due only to the intervention of the friction.

Truly uniform precession is possible if the top has coincident limiting circles. If it does, it must be released in such a way that its angle with the vertical corresponds to the angle of the limiting circles. Such a top does not nutate, since it can neither rise above the limiting circles nor fall below them. Since it does not fall, however, it has no way to convert some of its potential energy into energy for precession. Your starting technique or the initial impact of the top on the floor would have to provide the impetus to start precession.

There are actually two possible types of precession. Only the slower one is related to the gravitational pull on the top. The faster one is rarely seen in a top but can turn up under the proper initial conditions. The classical explanation of the faster precession involves a top (usually considered as an ellipsoid in textbook discussions) that is free to roll around on a horizontal plane while it spins. No forces or torques act on this imaginary top, not even gravity. In this situation t he top cannot slide into the plane or break contact with it.

Once the top is spinning with its body axis at some angle to the vertical it begins to precess around the vertical in a way that maintains the contact between it and the horizontal plane. The plane is usually called the invariant plane; the path of the top's contact point on the invariant plane is called the herpolhode, and the path of the top's contact point o n the top itself is called the polhode. As one writer put it, the motion leads to a jabberwockian statement: The polhode rolls without slipping on the herpolhode lying in the invariant plane. The precession in this idealized case is the fast precession a real top might have.

My colleague James A. Lock has pointed out that fast precession can be seen in a spinning football and a softdrink bottle thrown into the air. A quarterback throws a football with a spin to keep the ball stable during flight, taking advantage of its streamlined shape. In a long pass the football often has a noticeable wobble, which may be partly due to air drag. Even without drag the ball will wobble if it is thrown to develop fast precession.

The precession is easier to see if a softdrink bottle is tossed upward with spin. (The bottle should be empty; sloshing fluid would interfere with the spin.) If you toss the bottle with its body axis vertical, you may not be able to see any precession. With a less careful toss the spinning bottle will precess around the vertical in the fast-precession mode. The motion is not caused by gravity because the bottle is in flight. During the precession the bottom of the bottle traces out the herpolhode on an imaginary invariant plane that is perpendicular to the vertical.

Fast precession is rare with a real top because it is so much faster than the precession that depends on gravity. Much more energy is needed to send the top ., :into fast precession. Unless the top is somehow given that energy during the initial stages (for example, if a fortuitous launching imparts the energy and the required initial motion), it will settle instead into slow precession.

With some care you can set a top spinning so that its body axis is vertical. (The technique of launching a top with two lengths of tape, which I shall describe below, is useful here.) If the spin rate is higher than the critical value, which varies according to the type of top, the top stays upright. At a spin rate lower than the critical value the top nutates between the vertical and some limiting circle corresponding to a larger angle to the vertical.

Both kinds of behavior can be seen in a real top with a spin rate that is initially above the critical value and a body axis that is initially vertical. The spin has two stages: first a sedate vertical spinning and then a frantic nutation. The transformation is brought about by the friction at the point where the top touches the floor. Although the initial spin rate is high enough for the vertical spinning, friction eventually pushes the spin rate below the critical value and the top begins to lean and to nutate and finally to topple

Can a top that is initially tilted ever nutate until it is vertical? In principle it cannot because it will always lack the energy to become precisely vertical. It can come close, however, if the upper limiting circle is quite small. Then the top can nutate upward in such a way that it passes close to the vertical before it nutates downward again. This situation will probably not arise often.

A keen observer of tops would not agree with much of what I have said. Many tops do rise to a vertical spin (such a top is called a sleeper), and so my statements are not entirely correct. A top rises not through a process of nutation but through a subtle interplay of the stem of the top and the friction between the point of the top (the end of the stem on which the top spins) and the floor. The stem of any real top is not the infinitesimal point I have been assuming so far. A better model would be a hemispherical stem.

Such a stem would tend to rotate on the floor because of the top's spin. Meanwhile the top is also trying to precess. Since I am now considering a more realistic stem instead of a stationary point of contact with the floor, I must visualize the precession as moving the top around its center of mass, which remains stationary. The contact point on the floor moves, circling below the center of mass, while the upper section of the top circles above the center of mass.

As a result the hemispherical stem is attempting to roll over the floor in two ways, one due to the spin of the top around its body axis and the other due to the precession driving the stem over the floor. The first way is much faster than the second, and so the stem slips. As it slips the friction between it and the floor points in a direction opposite to the direction of the slip, as is shown in the bottom illustration in Figure 6. It is this friction that rotates the body axis of the top to make the spin vertical. To calculate the torque I treat the pivot point as the center of mass. The lever arm is the distance between that point and the place where the stem touches the floor. The torque raising the top is equal to the multiplication of the lever arm and the friction.

Of course, the friction on the stem is also stealing energy from the top. The trick in designing a good sleeper is to shape the stem so that the top becomes vertical before it loses too much energy. Hence a good sleeper has a stem with a small radius of curvature. Tops with stems that have a more gradual curvature take longer to rise and may not reach the vertical before the spin declines below the critical value needed to sustain a vertical position. A whip top, which is lashed with a whip to add energy to the spin, has a stem with a very small radius of curvature; it rises to be a sleeper within seconds of being lashed.

Most of the commercially available tops are shaped like a pear. Some are spun by the fingertips. Others are wound with string and thrown to the floor. Some handcrafted tops are mounted in a wood holder until the string wrapped around their spindle is unwrapped with a brisk pull; then they fall spinning to the floor.

Although all these types are usually found in toyshops, they by no means exhaust the variety of tops that can be (or once could be) found around the world. A beautiful description of the many kinds of top that have been made can be found in the book by D. W. Gould cited in the bibliography for this issue. Some tops have multiple faces for the purpose of gambling and telling fortunes. Some are flat disks pierced by a thin rod. Others are flat disks with a central bump on which they spin. (Several of them can be deployed at one time from a single holder.) The most remarkable feature of tops (apart from their rotational mechanics) is how they were developed in such a variety of shapes by people working almost independently around the world.

Dubois recently wrote me about the tops he and his students make from common items such as buttons, needles and machine screws. Often these unlikely objects need only a little fixing before they can serve as tops. A point may have to be ground or a slot may be required in a spindle so that a cord can be inserted. Sometimes several pieces are assembled into a top with glue. (Dubois says epoxy is best.)

Since friction plays a role in the behavior of tops, Dubois experiments on different surfaces. Paper, thin plastic sheets (such as food wrap) and bed sheets work well with some tops but not with others. Dubois has found that a linoleum floor is the only surface on which all his tops work adequately, although most of them work better on some other surface. You can keep track of where the point of a top travels by spinning it on carbon paper, soot-covered paper, ink-covered glass or some other surface on which the point can leave a mark.

Some of Dubois's tops can be spun by a quick snap of the fingers. He holds the upper stem of the top between his thumb and second finger, keeping both the point and his palm downward. Then he snaps those fingers, causing the top to spin outward on the table or the floor. If there is no upper stem, he holds the stem with his palm upward. Then when he releases the top with a snap, the point is properly downward. Another launching procedure is to hold the stem between the forefinger of the left hand and the thumb of the right hand; when the two digits are pulled quickly in opposite directions, the top is spun.

Fast spins can be achieved with some tops by wrapping a cord or a strip of cloth around the stem or some other cylindrical part of the top. A typewriter ribbon works well. (The ink should of course be removed first.) You can slip one end through a notch in the stem and then wrap the ribbon around the stem several times. When you throw the top outward across the floor while holding the end of the ribbon, the top is made to spin rapidly.

This technique does not work well with small or fragile tops. With them Dubois prefers a double wrapping of ribbon around the stem. Both strips are started in the same way in the notch. As the top is turned in the hand the double layer of ribbon is wrapped around it six to 12 turns. Then instead of throwing the top to unwrap the ribbon, pull the two strips horizontally in opposite directions so that the top is twirled. With double layer wraps of up to 14 revolutions Dubois can make an upholstery tack spin at about 15,000 revolutions per minute. With a finger snap the speed is about 8,000 r.p.m.

If the top has a smooth stem, as an upholstery tack does, the ribbon is difficult to wrap because it slips easily. Dubois coats the surface with a layer of epoxy to provide more friction. The added friction also means that when the end of the ribbon is reached during the launching, the top is deflected by a final tug rather than slipping away freely.

Some of Dubois's tops sing as they spin, either because they scratch the surface on which they are spinning or because they generate turbulence and vibration in the air. The spin of a singing top must exceed 3,300 r.p.m. One of Dubois's faster tops sings two octaves above A (440 hertz). If the top is spinning on a membrane of some kind, the membrane acts as a sounding board to enhance the audibility of the sound. Do not stretch the membrane too tight or it may be punctured by the spinning top.

Spin and precession can be monitored with a repetitive-flash strobe light. The frequency of the light is varied in order to match either the spin or the precession. Some of the tiny tops Dubois has sent me turn at speeds of more than 100,000 r.p.m., which is possible because the top's moment of inertia around its body axis is small and the energy imparted at launching results in a high rotational speed. The commoner pear-shaped top has a much greater moment of inertia around its body axis, so that the launching energy results in a much lower rotational speed.

The most fascinating top I have ever encountered is the Tippe Top, which I described in this department for October, 1979. The top usually has a hemispherical bottom and a central stem. It is spun with the bottom downward, a logical position because that section is heavier than the stem. Soon after the top is released, however, it inverts so that it spins on the stem. The heavier hemispherical end is lifted, seemingly in defiance of gravity.

The cause of the odd inversion is friction with the surface on which the hemisphere is turning. Spinning the Tippe Top on a soot-covered pane of glass will give a picture of the path of the top. Such a picture, made and analyzed in 1960 by Frank F. Johnson of Hasbrouck Heights, N.J., appears in the bottom illustration at the left. Johnson's analysis is cited in the bibliography for this issue.

What would a top do on a slightly inclined surface? Would its point of contact simply move around the way it does on a horizontal surface? Ledo Stefanini of the University of Bologna recently published a detailed answer to this question. Like other people studying the role of friction in the behavior Of a top, he considered the stem to be hemispherical.

If the top is spinning without precession, it will move horizontally across the inclined plane in a straight line. If it is precessing, however, four things are possible, depending on the slope of the inclined plane and how much the top tilts from the vertical. One possible path for the stem is a spiral along a line down the plane. The other patterns resemble the nutation patterns of an ideal top with a sharp stem that does not move across the floor. In some cases the stem traces out loops on the plane as it moves primarily in a horizontal direction. Or it may map out cusps. For another set of angles the stem first climbs the inclined plane and then descends while also moving horizontally across it.

You might like to try various combinations of angles on an inclined plane. Stefanini published a tracing he made from a top weaving across carbon paper he had tacked onto an inclined plane You can make your own tracings to catalogue the possible motions.

Bibliography

MECHANICS OF THE GYROSCOPE: THE DYNAMICS OF ROTATION. Richard F. Deimel. The Macmillan Company, 1929.

THE TIPPY TOP. Frank F. Johnson in American Journal of Physics, Vol. 28, No. 4, pages 406-407; April, 1960.

THE TOP: UNIVERSAL TOY, ENDURING PASTIME. D. W. Gould. Clarkson N. Potter, Inc., 1973.

A REALISTIC SOLUTION OF THE SYMMETRIC TOP PROBLEM. T. R. Kane and D. A. Levinson in Journal of Applied Mechanics, Vol. 45, No. 4, pages 903-909; December, 1978.

BEHAVIOR OF A REAL TOP. Ledo Stefanini in American Journal of Physics, Vol. 47, No. 4, pages 346-350; April, 1979.

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