ONE DAY IN THE MIDDLE OF the last century the French physicist Leon Foucault inserted a slender rod in the chuck of a lathe and plucked the free end of the rod so that it vibrated like a reed. When he started the lathe, he observed that the turning rod continued to vibrate in the same plane. Later he tried much the same experiment with a vertical drill-press. From the chuck of the drill-press he suspended a short pendulum consisting of a length of piano wire and a spherical weight. He set the weight to swinging and started the drill-press. The pendulum also continued to vibrate in the same plane. For a time nothing much came of these observations, but in them was the seed of an experiment destined to settle a classic scientific controversy.

In the year 1543 Nicolaus Copernicus had sent a copy of his new book, On the Revolutions of the Celestial Orbs, to pope Paul III with a note containing a historic understatement: "I can easily conceive, most Holy Father," he wrote, 'that as soon as people learn that in this book I ascribe certain motions to the earth, they will cry out at once that I and my theory should be rejected." Cry out hey did, and some, including a few scientists, were still crying out in 1850, when Foucault was invited to stage a science exhibit as part of the Paris Exposition scheduled for the following year. Being not only a gifted physicist but also something of a showman, he selected as the site of his exhibit the church of Sainte Genevieve, also known as the Pantheon.

From the dome of the Pantheon he hung a pendulum consisting of 200 feet of piano wire and a 62-pound cannon ball. On the floor, immediately below he cannon ball, he sprinkled a layer of fine sand. A stylus fixed to the bottom of the ball made a trace in the sand, thus recording the movement of the pendulum. Great care was taken during construction to exclude all forces except those acting vertically to support the system Tests were even made to assure symmetry in the metallurgical structure of the wire. Finally the ball was pulled to one side and tied in place with a stout thread. When the system was still, the restraining thread was burned.

The pendulum made a true sweep, leaving a straight trace in the sand. In a few minutes the thin line had expanded into a pattern resembling the outline of a two-bladed propeller. The pattern grew in a clockwise direction, and at the end of an hour the line had turned 11 degrees and 18 minutes. This could be explained only on the basis that the earth had turned beneath the pendulum. Copernicus was vindicated.

It is easy to visualize what had happened if one imagines a pendulum erected at the North Pole. The pendulum is hung, perhaps from a beam supported by two columns, in line with the earth's axis. The supporting structure corresponds to Foucault's lathe or drillpress-or the dome of the Pantheon. So long as the pendulum is at rest the whole affair simply turns with the earth, making one complete revolution every 23 hours and 56 minutes (the sidereal day), The pendulum is now drawn out of plumb and carefully released. The direction of the swing persists, and the earth turns beneath the pendulum. To an observer on the earth the plane of vibration appears to rotate in a clockwise direction, because at the North Pole the earth turns counter-clockwise. The same effect would be observed at the South Pole, except that there the rotations would be reversed. If the swing of the pendulum appears to turn clockwise at the North Pole and counter-clockwise at the South Pole, what happens at the Equator? Substantially no deviation is observed. Here the entire system-the earth, the supporting structure and the pendulum-is transported almost linearly from west to east [see illustration at the left].

At the poles the rate at which the swing of the pendulum appears to rotate is 15 degrees per sidereal hour; at the Equator it is of course zero degrees per sidereal hour. The rate varies with latitude; the higher the latitude, the higher the rate This neglects certain fine deviations caused by the curvature of the earth's orbit around the sun and by other perturbations. But it holds for the gross, easily observed, motion.

With the help of a few simple geometrical concepts it is easy to see why a pendulum appears to rotate more slowly as it approaches the Equator. Imagine an arc along an intermediate parallel of latitude through which the earth has turned during a short interval, say an hour or so [AB in illustration in Figure 2]. The angle subtended by the arc at the earth's axis increases at the rate of 15 degrees per hour-one full turn of 360 degrees per sidereal day. Now assume a pair of tangents [AC and AB] to the earth's surface subtended by the arc which meet on a projection of the axis at a point in space above the pole. The angle between the tangents increases in size at the same rate with which the pendulum's plane of vibration appears to rotate. The reason it does so becomes clear if one assumes that the pendulum continues to vibrate in the plane of the first tangent as it is transported to the position of the second. At the pole the two angles are equal; both increase at the rate of 15 degrees per hour. At the Equator the angle subtended by the arc at the earth's axis continues to increase at the rate of 15 degrees per hour. But the pair of tangents subtended by the arc at the Equator meet the projected axis at infinity. The angle vanishes, and its rate of increase does the same. Therefore the pendulum's rate of rotation is zero. Foucault demonstrated that the apparent rotation of the pendulum varies with the trigonometric sine of the latitude at which it is installed. Its rate at points between the poles and the Equator is equal to 15 degrees per hour multiplied by the sine of the latitude.

Amateurs who set up pendulums at New Orleans will observe an apparent rotation of about 7 degrees and 30 minutes per hour. They must wait two days for a full revolution. Those in Manila must wait four days; those on Howland Island in the South Pacific, about 40 days!

About the best way to gain an understanding of the Foucault pendulum is to make one. Like many such enterprises, this seems simple until it is tried. Many amateurs who have felt the urge to set up the apparatus have abandoned the idea because they had no Pantheon in which to hang a 200-foot pendulum and no cannon ball for a bob. This is no problem. Pendulums 10 or 15 feet long can be made to work handsomely with bobs weighing as little as five pounds. The most vexing problems encountered in making a pendulum have to do not with its size but with starting the bob in a true swing, maintaining the trueness of the swing, and supplying energy to the bob.

Foucault's method of starting the bob is still the most elegant. Many starting devices have been tried: mechanical releases, magnetic releases, mechanisms which accelerate the bob from dead rest, and so on. It is generally agreed that burning a thread is the simplest and best of the lot.

Until recent years the problem of making the pendulum swing true resisted some of the world's best instrument makers. It seemed clear that any method of suspension must have radial symmetry such as one would expect of the suspension device depicted in Figure 3. To assure this Foucault and subsequent experimenters took great pains in procuring wire of uniform characteristics and in designing the fixture to which the wire was attached. Roger Hayward, the illustrator of this department, tells me that the wire for the Foucault pendulum in the Griffith Observatory in Los Angeles was specially drawn and tied to a long two-by-four beam for shipment from an eastern mill to the West Coast. The designers were afraid that coiling the wire would destroy its symmetry. The pivot to which the wire was attached at first consisted of a set of gimbals with two sets of knife-edges at right angles to each other. Despite these precautions the completed pendulum insisted on performing figure eights and ellipses. Hayward, who had designed other exhibits for the Observatory, suggested that the wire simply be held in rigid chuck. This invited a break at the junction of the wire and the chuck, which could cause the wire to lash into a crowd of spectators. To minimize this hazard a crossbar was clamped to the wire just above the ring-shaped driving magnet. Thus if the wire had broken, the crossbar would have been caught by the magnet ring. Clamping the wire in a chuck cured the difficulty. The wire has now been flexing for more than 20 years without any apparent ill effect.

At about the time the Griffith pendulum was installed a French physicist named M. F. Charron devised a method for maintaining the true swing even 11 when the forces acting on the pendulum are measurably asymmetrical. Charron set out with the objective of designing a vise which would grip the wire rigidly and would also provide a long radius through which the wire could flex. He used a ferrule which at its upper end fitted snugly around the wire and at its lower end flared away from the wire [see illustration Figure 4].

The diameter of the hole at the lower end precisely accommodated the swing of the pendulum. It was observed that, when the pendulum swung true, the wire simply made contact with the inner surface of the ferrule at the end of each beat. But when the pendulum performed ellipses or other configurations, the wire rubbed against the inner surface of the ferrule and the energy responsible for the lateral component of motion was dissipated through friction. Stephen Stoot, a Canadian amateur, has suggested a modification for small pendulums which accomplishes the same result. He fixes a carefully centered washer around the wire just far enough below the point of suspension so that the wire touches the washer at the end of each swing [Figure 5].

The third basic problem is how to supply the pendulum with a periodic push in the precise direction in which it needs to go. No completely satisfactory mechanical solution has been devised. A variety of electrical drives are in use, however. Most of these feature a ring-shaped electromagnet which acts on an armature carried by the wire near the point of suspension. Power is applied during the portion of the swing in which the wire is approaching the magnet, and is interrupted when the wire moves away.

The arrangement in use at the Griffith Observatory is typical. Current for the ring magnet is supplied through a relay. The action of the relay, in turn, is controlled by a photoelectric cell. A beam of light, folded by mirrors so that the beam crosses itself at a right angle near the wire, actuates the photoelectric cell [see illustration in Figure 6]. When the suspension passes through the center of the ring magnet, a vane on the wire breaks the beam. The relay then operates, and applies current to the magnet. After an appropriate interval, during which the wire approaches the magnet, a time-delay relay breaks the circuit automatically.

In 1953 R. Stuart Mackay of the University of California described in the American Journal of Physics a novel method of driving Foucault pendulums. His apparatus consists of a simple coil bridged by a condenser into which power is fed continuously, and over which the bob swings freely. The method takes advantage of phase-shift effects which are essential to the operation of ordinary doorbells and buzzers. It requires no contacts, light beams or other arrangements to interrupt the current

"If one energizes a coil with alternating current," writes Mackay, "and sets an iron pendulum bob swinging immediately above it, the current through the coil will increase or decrease depending upon whether the bob is moving toward or away from the coil. The magnetic property of the iron influences the magnetic field set up by the coil and therefore the coil's inductance. The inductance, in turn, influences the flow of current. The fluctuations in the amplitude of the alternating magnetic field do not occur instantly. The effect is such that the current, and consequently the attractive force of the coil's magnetic field, is greater when the bob approaches the center of the coil than when it recedes. If the circuit is essentially inductive, enough energy will be transferred to the bob to maintain the pendulum's swing.

"The effect can be made stronger by placing a capacitor in series with the coil so that the combination resonates slightly below the 60-cycle frequency of the power line. In effect, the system is then working on one side of the resonance curve, where a given change in inductance causes a pronounced change in current.

"Some Foucault pendulums, particularly those set up as public displays, feature bobs of bronze or other nonmagnetic materials. These may also be driven by the coil. One takes advantage of the fact that such bobs can act as the secondary winding of a transformer. The magnetic field of the stationary coil and that set up by current induced in the bob are in opposition. Hence the coil and the bob are mutually repelled. As the bob approaches the coil the How of current increases in both. But because of electrical lag in the circuits the current, which starts to rise with the approach of the bob, does not reach its peak until the bob has traveled somewhat beyond the center of the coil. Accordingly the bob is subjected to a greater net repulsive force when it is moving away from the coil than when it is approaching the coil. Energy is thus made available to drive the pendulum.

"The forces acting between the coil and a nonmagnetic bob are smaller than those- between the coil and a magnetic bob. Moreover, energy dissipated by currents induced in the bob causes greater damping in the system. The effect can be shown easily by substituting a short-circuited coil for the bob. The change of current in both coils, induced by changes in inductance, can be enhanced by resonating each circuit near the power-line frequency, as in the case of magnetic bobs. When the capacitor of the bob coil is made slightly too large for resonance, the circuit becomes a trille inductive. The force between the coils is then repulsive; sustained oscillations result. If the capacitor is too small, an attractive force is produced which will not sustain the motion. A circuit diagram of the two coils appears in the accompanying illustration [Figure 7]. It is possible, of course, to attach a driving coil tuned for repulsion to the bottom of a nonmagnetic bob.

"The first coil I tried was wound with No. 16 wire. It was eight inches in diameter, two inches thick and had a two-inch hole in the center. It was used simply because it chanced to be on hand. The coil resonated at 60 cycles when it was placed in series with a 25-microfarad paper condenser. On the application of 10 volts it drove a three-inch iron bob on the first try. Considerably more power was delivered when the magnetic field was altered by laying the coil on a six-inch circle of 1/16-inch sheet iron. Should one wish to enhance the effect further, the coil may be inserted into a cylindrical core of iron [Figure 8],

"Which of the two drive systems is preferable, magnetic or nonmagnetic? A performance analysis indicates little choice either way, though the magnetic bob is probably simpler to make. It is true that over a number of cycles any small perturbing force can produce a marked effect on the swing of a Foucault pendulum, usually resulting in a slightly elliptical orbit. The 'plane' of oscillation of a pendulum swinging in an elliptical path will precess in the direction of tracing the ellipse at a rate roughly proportional to its area. Thus, if one does not take care, a Foucault pendulum can appear to turn at the wrong rate, or even to indicate that the earth is turning backward. It is tempting to suppose that the nonmagnetic bob will tend to avoid the 'magnetic potential hill 'that it 'sees,' that is, the repulsive force will tend to deflect the bob to one side if the swing does not pass directly through the center of the magnetic force. In contrast, a magnetic bob might appear to favor the center of the 'potential valley' it 'sees,' even though its normal swing would not necessarily bisect the field of force. Thus in either case one might expect some induced perturbation. In practice, neither has much effect on crosswise amplitude if a fairly heavy bob is used. Interestingly enough, since the motions in the two directions responsible for the ellipse are 90 degrees out of phase, and since changes in the magnetic field are controlled by the major motion, there is a slight tendency to damp out the minor motion.

"The matter of perturbing effects warrants further discussion, particularly from the viewpoint of comparing these systems with the conventional ring-magnet drive. It might seem that driving the pendulum from the top by means of a ring magnet would result in minimum sensitivity to asymmetries, but this is not strictly true A pendulum is less sensitive to asymmetries at its top. But a greater driving force is also required there. It is the percentage of asymmetry which interests us.

"An air-core coil is, of course, simpler to construct, install and maintain than the ring-magnet drive. It must be said, however, that this method of drive does suffer somewhat in comparison with the ring-magnet system in that the whole field is not turned on and off in the course of providing useful drive. The steady useless component of the field is not necessarily symmetrical. Consequently asymmetry may be added to the system continuously. This means that in this system the magnet must be made more perfect than in a system wherein the field changes the required amount by going fully off. When desired, the magnetic field of the air coil can be trimmed for symmetry with small tabs of magnetic iron.

"For the purpose of most demonstrations, however, extreme precision is unnecessary. With reasonable care in alignment the angular velocity of the pendulum's apparent deviation should fall within 15 per cent of the anticipated value."

Edward Prenner, a physician of New York City, substitutes a flashlight for the pierced cardboard. "It is only necessary," he writes, "to hold an ordinary pen-sized flashlight against the lower lid of the closed eye, and wiggle the flashlight rapidly from side to side. This gives a beautiful silhouette of the retinal vascular tree. It will be noticed that it is not quite possible to see the center from which the vessels radiate. This is another demonstration of the celebrated blind spot at the entrance of the optic nerve."

"Much the same effect can be perceived by directing the light beam against the white of the eye," says Heinz Norden of New York. "It is possible," he writes, "to train a pencil of light through the sclera, the tough whitish outer covering of the eyeball, in such a way that it will be reflected from one part of the retina to another. The first point of incidence is then visualized with the utmost clarity at the second point.

"Incidentally, the ability and habituation of the eye to ignore minuscule obstructions and flaws in vision can be studied in ways other than those Hayward describes, with little or no instrumentation and fascinating results. It is a process that might be called 'seeing seeing.' Almost everyone is familiar with the clusters and strings of tiny O-shaped figures, resembling microscopic images of bacteria, that appear against a bright field of vision when the lids are narrowed and that are called mouches volantes. These have been explained as shadows of inclusions in the vitreous humor, or as white blood corpuscles traveling through the cornea.

"With training and stronger light, especially from the side, they are readily perceived-or so it seems to me-as only the most prominent of an apparently three-dimensional reticular structure that fills the entire visual field. When, in such visualization, the eye is made to follow a pencil held before it, this cell network appears to remain fixed with reference to the cornea. I am convinced that it is the rear surface of that tissue.

"Most people are aware of a dance of tiny white points when they gaze at a brightly illuminated field, such as the clear sky, a dance that reminds one amazingly of Brownian movement. No less a personage than Albert Einstein insisted to me that these were 'shadows of blood corpuscles in the retina.' I remain unconvinced. Could they not indeed be what they appear to be-shadows of the giant molecules suspended and buffeted about in the tear secretion which constantly bathes the eyeball?

"It is fascinating to speculate that subjective evidence of the molecular structure of matter and of the cellular structure of organic matter may have been literally in front of our eyes long before man thought of the microscope."

Bibliography

COUPLED ELECTRICAL-MECHANICAL OSCILLATORY SYSTEMS. R. Stuart Mackay in American Journal of Physics, Vol. 21, No. 7, pages 575-576; October, 1953.

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