THREE PROPERTIES OF THE SIMPLE pendulum have long fascinated experimenters. The properties are the pendulum's ability to keep time, to sense relative motion and to measure the force of gravity. Although the maser has now outmoded the pendulum as an accurate timekeeper, many experimenters continue to find work with pendulums diverting. Amateurs have recently solved some mechanical problems of long standing that were responsible for variations in the rate at which pendulums swing and have constructed pendulum clocks that keep time to within a thousandth of a second per week.

Other experimenters have been developing new versions of pendulums for sensing the earth's rotation on its axis, vibrations induced in the earth's surface by earthquakes and periodic disturbances in the local gravitational field.

One of these is Nils E. Lindenblad of Princeton, N.J., who took up the study of pendulums two years ago at the age of 70 after a career of 45 years as an electronics engineer. Lindenblad writes:

"My experiments with pendulums were undertaken primarily as a means of tapering off work habits that had to be changed as a result of retirement. I first thought of tinkering with seismographs of the kind used for detecting long earthquake waves, but I soon learned that most of the basic problems in the operation of these instruments had been solved electronically by the use of amplification and feedback. Further improvements, I was told, would doubtless entail refinement of circuits-just the kind of work I wanted to avoid.

"Scaling down my ambitions somewhat, I set out to develop a small mechanical pendulum of good quality and long period that could be used as a reference mass: a mass that tends to stand still when neighboring objects move. The pendulum I had in mind would be designed to swing six times per minute, a rate of vibration equivalent to a period of 20 seconds. Simple pendulums, such as those consisting of a weight suspended by a thin wire, vibrate at a rate that varies inversely with the square root of their length. One designed for a period of 20 seconds turns out to be about 330 feet long! My problem, then, was to find ways of reducing the length to the proportions of my basement workshop without introducing frictional or other mechanical losses that would seriously impair the performance of the apparatus.

"Pendulums of long period and small size are not new. A classical design, for example, is the compound pendulum. This apparatus consists of a rod that carries a bob on each end. When the rod is suspended at one end, it vibrates as a simple pendulum. As the pivot point is moved toward the middle of the rod the period increases. If the bobs are of equal mass, the period becomes infinite when the pivot point is shifted to the middle of the rod.

"A compound pendulum, however, cannot be used as a reference mass. When the apparatus is displaced, the resulting forces of inertia developed by the two bobs act in opposition. Hence the bobs move if the object to which the pendulum is attached moves, and no reference effect is achieved.

"Another classical apparatus of long period and small size is the horizontal pendulum, essentially a cantilever hinged at one end to a rigid support. When the axis of the supporting hinge lies in the horizontal plane, the cantilever swings as a simple pendulum of short period. The period increases as the axis departs from the horizontal. The mechanism then swings like a barn door that is more or less out of plumb. Pendulums of this type are commonly used as the reference mass in seismometers. The pendulum senses the horizontal component of earthquake waves even though components of the arc traversed by the bob lie in both the horizontal and the vertical planes. I thought it would be interesting to attempt the development of a reference-mass pendulum that swung only in the vertical plane.

"In examining the essential properties of pendulums I found it helpful to consider their geometry from two points of view. Customarily the period of vibration is calculated in terms of the length of the pendulum. To find the period in seconds divide the length (in centimeters) by the acceleration of gravity (980 centimeters per second per second), take the square root of the quotient and multiply it by twice pi (6.2832). In other words, the period increases as the length increases.

"It is just as valid, and on occasion more helpful, to think of the period as a function of the curvature of the arc through which the bob swings. For a swing of a given distance from the vertical the period of vibration increases as the curvature of the arc becomes flatter. As the curvature approaches a straight line the period increases without limit.

"This relation leads to the explanation of why the sensitivity of pendulums to external forces increases as the period increases. Work must be done to push a pendulum away from the vertical because the bob is then raised by an amount that depends on its excursion from the vertical as determined by the curvature of its path. Work thus done in elevating the bob is stored by the bob as potential energy that subsequently causes the pendulum to swing. If the arc through which the bob swings is relatively fiat, the bob rises only slightly when it is pushed from its rest position, and so less work is needed to start it swinging. In other words, the sensitivity of pendulums to external forces varies inversely with the curvature of the arc traversed by the bob.

"With these principles in mind I explored various linkages that have been developed for transforming circular motion into linear motion. For one experiment I selected a straight-line motion similar to one developed by James Watt, the inventor of the steam engine. The linkage, as modified, consists of a horizontal bar supported at one end by a hinged lever and suspended at the other end by a flat ribbon of spring steel [see Figure 1]. The ribbon in turn was hinged in the middle by means of a short length of spring stock turned at right angles to give the suspension lateral freedom. If the ribbon and the lever are equal in length, the midpoint on the horizontal bar will move in a straight line when pushed along its axis through a limited distance. A bob that has a hole through its center of gravity and is slid into position at the middle of the bar will move as though it were suspended by a wire of infinite length forming a pendulum of infinite period.

"If the position of the bob on the bar is shifted toward the suspension member and the system is set in motion along the axis of the bar, the bob will move through an arc that curves more or less upward depending on its distance from the midpoint. The bob will oscillate at periods ranging from seconds to minutes. When the bob is shifted in the other direction, from the middle of the bar toward the supporting lever, the system soon becomes unstable because the arc through which the bob moves then curves downward. Accordingly the bob falls and comes to rest at one or the other of its limits of excursion. In Figure 1 the positions of the bob that result in an infinite period, a short period and the unstable condition are designated respectively A, B and C. The proportions of the structure can be varied, and the performance of the apparatus can be predicted in advance of construction, by making a diagram similar to the one in the upper part of the illustration.

"A demonstration model of the pendulum was assembled on a framework consisting mostly of iron pipe and rod. Iron pipe of small diameter comes in a range of sizes that telescope snugly into each other. The pipe therefore lends itself nicely to making bearings, sliding adjustments and so on. The material is easy to cut and form with ordinary hand tools. I assemble it into rigid structures by brazing or silver-soldering the joints. My facilities are those of the typical amateur. All work is performed on a bench two feet wide and three feet long. My only power tool is a quarter-inch electric drill. The tool I value most, however, is a gas-air torch. I use it for silver soldering and for heating the pipe before making bends.

"The spherical bob of this pendulum and others to be described is made of lead, which can be obtained inexpensively from dealers in plumbing supplies. I made molds for casting the lead by coating plastic balls with copper. After applying the coating, I made a hole in each ball and heated the ball so that the plastic would melt. That left a copper shell into which I poured the molten lead. The copper exterior gives the bob an attractive appearance.

"The plastic balls can be bought in toy stores. To apply the copper plating attach an electrical connection to the surface of the plastic. The connection consists of a disk of thin brass about a quarter-inch in diameter to which a short copper wire has been soldered. The disks were cut from shim stock, which can be bought from dealers in automobile supplies. A disk is attached to the plastic by a film of quick-drying cement.

"The ball and the disk must be washed with a detergent to remove the film of grease that is invariably present; they must also be painted with a thin solution of Aquadag or an equivalent substance that conducts electric current. (Aquadag is a product of the Acheson Colloid Company, Port Huron, Mich. 48060.) When the conducting film has dried, the ball can be plated. (A conducting silver paint is made by E:. I. du Pont de Nemours & Co.)

"My plating solution consists of copper sulfate, sulfuric acid and distilled water in the proportions of 32 ounces of copper sulfate and eight ounces of sulfuric acid, by weight, to each gallon of water. Copper flashing, of the kind available from dealers in building supplies, can be used for the anode of the plating bath. The plating current should be limited to about .02 ampere per square inch of surface until a thin film of copper forms over the entire area to be plated. The current can then be increased to .3 ampere per square inch of surface. The direct-current supply must be capable of maintaining a potential of between one and four volts when it is under load. An automobile storage battery can be used as the supply if a suitable rheostat for regulating the current is included in the circuit. An instrument for measuring current is essential.

"Another mechanical linkage, known as Scott-Russell's straight-line motion, was modified to operate as a pendulum of any desired period. It is free to vibrate in any vertical plane. In principle it consists of two rods, one of which is hinged to the other in the form of a distorted T [see Figure 2]. The weight of the assembly is supported by the leg of the T through an appropriate hinge, such as a pivot or a spring suspension. The bob is carried by one end of the crossbar of the T.

"The motion of the opposite end of the crossbar is constrained; that end is free to move up and down but not sideways. The constraint was accomplished by attaching three radial guy wires to the end of the crossbar and anchoring them to the supporting framework. In the diagram the leg of the T is represented by the line od' and the crossbar by the line dd". The motion of the d" end of the crossbar is restricted to the line of". When the proportions of the rods are made as illustrated by the solid lines od', dd' and d'd", the bob will traverse the straight line bb' and the period of the pendulum will be infinite. If the length of the portion of the crossbar between d' and d" is made shorter but all other dimensions are retained as indicated at e'e", the curvature of the arc will increase inversely with the length of e'e". The system will then function as a conventional pendulum. When the length d'd" is increased, as at f'f", the path of the bob will bend downward as indicated by the arc of. The pendulum will then be unstable and the bob will fall from its center position to one of the limits of excursion.

"The pendulum can be made in other proportions. I always analyze a new structure graphically before attempting construction. First I draw the straight, broken line of", establish point a and through it construct the perpendicular bb'. I usually make the length oa less than 20 percent of the length oc. Then with the arbitrary radius oc I draw the arc c'cc". Finally I construct the arc through which the bob will swing. This arc fixes the period at which the pendulum will vibrate. It is drawn through a, from an appropriate center point on the line of". I next draw the pendulum arm proportioned so that the bob will traverse the arc just drawn through a. With an arbitrary point such as e' on the arc cc" as a center, and with ac as a radius, a short arc is drawn to intersect the arc previously drawn through a, thus locating point e. A straight line that projects from ee' intersects of" at e", thus establishing the length e"e', which yields the arc ae and establishes the desired period of vibration.

"Pendulums of this type can be made to swing in a single plane by constructing the member od' in the form of an inverted U. The legs are appropriately hinged to a supporting base.

"The remarkable sensitivity of long-period pendulums can be demonstrated by adjusting the rate of vibration to five or fewer swings per minute and setting the apparatus on a heavy workbench. Level the base carefully. Then the pressure of a finger on the bench top will set the bob swinging.

"A demonstration model of the pendulum just described was constructed to vibrate freely in all vertical planes. In this version of the basic mechanism the supporting lever takes the form of a circular cage, with radial cross braces attached at 120-degree intervals to the top and bottom rims [see Figure 3]. The bottom of the cage is flexibly supported at the center, where the cross braces meet, by a short steel wire that extends downward from a small tripod rigidly anchored to the base. The pendulum arm, dd", is similarly flexibly coupled to the radial cross members of the cage at the point where they join at the top. When the pendulum arm is proportioned so that the bob traverses an arc with a radius of 200 feet, the bob vibrates at a period of about eight seconds. It then duplicates within a space of less than two cubic feet the performance of the huge pendulum that was set up more than a century ago by the French physicist Jean Bernard Leon Foucault in Paris to demonstrate the rotation of the earth.

"Pendulums equipped with small bobs swing for a shorter time-given the same amount of inherent loss-than those that have large, heavy bobs because the small bobs store less energy. A good pendulum equipped with a four-pound bob will normally swing for about an hour when it is given an initial push of 10 degrees, if care has been taken to minimize frictional losses at the points of suspension. The duration of the swing could not be extended appreciably, however, even by reducing all frictional losses to zero, because a substantial amount of energy would still escape through the supporting structure. The swing of the bob causes the support to sway from side to side. This vibration is communicated to the bench, the floor and the ground. The loss through the support can be minimized only by anchoring the apparatus to a massive pier. Such piers are costly and cumbersome. A practical solution is to compensate for the loss by driving the pendulum electrically. Several effective driving mechanisms for pendulums of the Foucault type have been described in this department [June, 1958].

"While I was considering the minute quantity of energy required to sustain vibration in a well-made pendulum, it occurred to me that perhaps a pair of pendulums could be used to demonstrate local pulsations in the gravitational field. If the pendulums were closely spaced and tuned to the same period, gravitational attraction between the bobs should cause the pendulum at rest to start swinging when its companion vibrates. To test this assumption I erected two pendulums side by side [see Figure 4]. Each of the bobs was suspended by two nonmagnetic wires about five feet long. In the case of the master pendulum, which would be driven electrically, the wires were arranged in the form of a V so that the bob would swing in a fixed plane. A lead sphere four inches in diameter and weighing 14 pounds served as the bob. The suspension wires were attached to screw eyes in the ceiling.

"The driving mechanism consisted of a coil, which had several hundred turns of magnet wire and was attached to the suspension wires near the top, and a permanent magnet in the form of a cylinder that occupied the space in the center of the coil. One end of the magnet was anchored to the wall. Limit switches were actuated by the suspension wiles at each end of the swing. One switch, when closed, applied current to a relay equipped with two contact springs. One contact spring, which was connected in parallel with the limit switch, maintained current in the coil of the relay when the switch opened, thus keeping the relay locked down when the suspension wires swung away from the switch. The second contact spring applied current to the driving coil. The magnetic force between the field of the coil and that of the magnet drove the pendulum through one swing. At the end of the swing the second limit switch was operated by the approaching suspension wires. This switch interrupted the current in the coil of the relay, releasing the relay. This action removed power from the coil during the return swing.

"The second pendulum, which was designed to sense any gravitational effect, consisted of a copper rod 10 inches long suspended at each end by nonmagnetic wires attached at the top to a metal bracket. The bracket was anchored to the cinder-block wall. A three-quarter inch sphere of lead at the center of the copper rod served as the bob. The rod vibrated in the direction of its axis when the pendulum swung. I tuned the pendulums to resonance by adjusting the length of the master pendulum. The master pendulum was arranged to be in sharp resonance with the sensing pendulum only in the direction of the axis of the rod supporting the bob. A partition of sheet aluminum served as an electrostatic shield between the bobs, and the sensing pendulum was fully enclosed by a housing of transparent plastic to protect it from air currents.

"I assumed that the sensing pendulum would swing through only a small angle, and accordingly I set up an optical system for projecting an enlarged image of one of the suspension wires on a distant screen. The projector consisted of a lamp housing fitted with a condensing lens of 100-millimeter focal length (for concentrating light on the suspension wire) and an achromatic objective lens of 12.4-millimeter focal length for projecting a magnified image of the wire on the screen.

(The lenses and other parts of the optical system are available from the Edmund Scientific Co., Barrington, N.J. 08007.)

"When the driven pendulum was set in motion, the sensing pendulum gradually swung into step. Its movement was barely perceptible at the end of two minutes, but within 45 minutes the amplitude of the swing had built up to seven wire diameters, or about .05 inch. Should this response be ascribed to gravitational coupling between the bobs or to mechanical coupling between the supports? To test the latter possibility I suspended a second sensing pendulum on the opposite side of the master pendulum. This one was made a quarter of the length of the other two so that it would vibrate twice as fast. It was suspended by a single wire so that it would be free to vibrate in any plane. The bob and its suspension were nonmagnetic and weighed two pounds.

"I anticipated that if the second pendulum responded to the local gravitational effect, it would vibrate in a plane at right angles to the plane of the master pendulum. Accordingly I set up an optical system to detect vibration in this plane. The system differed from the first one only by the insertion of a 90-degree prism to bend the projected light in the direction of the screen.

"As in the first experiment, the swing of the short pendulum reached perceptible amplitude within two minutes and thereafter continued to increase for about 45 minutes. The fact that the short pendulum vibrates in a plane at 90 degrees to the plane of the master pendulum, and at half the period of the master pendulum, indicates that the response is indeed the result of gravitational coupling. By improving the instrumentation it should be possible to use the experiment for determining the universal constant of gravity. Mathematically the reduction of the data would not be easy. Newton's law, which states that the gravitational attraction between two bodies varies directly in proportion to their masses and inversely in proportion to the square of the distance between them, is valid only for masses so small and widely spaced that they can be regarded as mathematical points. With closely spaced pendulum bobs every point in one bob attracts every point in the other; moreover, the distance between the bobs varies as the bobs swing. The solution of such problems involves some difficult equations in the integral calculus. For this reason I have been content with the simple fun of making the demonstration."

Bibliography

MECHANISMS FOR THE GENERATION OF PLANE CURVES. I. I. Artobolevskii. The MacMillan Company, 1964.

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