SUPPOSE YOU CONNECT TWO identical pendulums with a spring, hold one of them and pull the other one sideways to stretch the spring. Then release them both. How will the system behave?

Intuition suggests that the interconnecting spring will force the initially stationary pendulum into motion until both pendulums swing chaotically, each with roughly the same amount of energy. Surprisingly, the pendulums do not end up in this state. Instead the energy is periodically exchanged between them in such a way that sometimes one of them stops and at other times the second one stops. What accounts for these phenomena?

In a typical example of an oscillating system that displays periodic transfers of energy two identical pendulums hang from a horizontal rod that in turn is' suspended from a rigid support by two lengths of string. When one of the pendulums is set swinging parallel to the plane of the rod, it begins to pass its energy to the other pendulum. After the transfer is complete the energy is transferred back. Sometimes one of the pendulums is stationary; at other times the second one is stationary.

In another example the pendulums are suspended from a string and swing 'perpendicular to the segment of it that extends between them. In a related example the pendulums are hung from a small-bore pipe supported by a string running through its interior. The pendulums swing perpendicular to the plane of the pipe. In both versions the energy is periodically exchanged between the pendulums as their swinging waxes and wanes.

In a quite different example, called the resonant-spring pendulum, an object is fastened to the bottom of a vertical spring; the top of the spring is attached to a rigid support. If you pull the object down and then release it, the spring oscillates vertically for a while the swinging diminishes and the vertical oscillations reappear. The assembly behaves the same way if you start with the pendulum motion.

The fourth example is called the Wilberforce pendulum after L. R. Wilberforce, an English physicist who studied it in 1894. A small object with extended arms is attached to the bottom of a vertical spring. You pull the object down and release it, which makes the spring oscillate vertically. Soon the oscillations die out as the object begins to swing back and forth about a vertical axis through its center. When the spring oscillations disappear, all the energy of the system is in the rotation. Then the energy is transferred back to the spring oscillations As they build up, the rotation disappears and so the cycle begins again.

To explain these systems I start with the two pendulums connected by a spring. Although in practice this system is more cumbersome to set up than the others, the energy transfer may be easier to visualize. I shall ignore such matters as friction and air drag.

Suppose you hold pendulum A while you pull pendulum B to the right and then release them both. As B begins to swing left and right, it pushes and pulls on the spring, which then pushes and pulls on A. As A begins to swing, the spring generally opposes the motion of B and promotes the motion of A. Thus it is responsible for the transfer of energy from B to A. Once the energy is fully drained from B the situation is reversed.

One can examine the motion of the pendulums in terms of their normal modes: two ways in which they can swing. If they swing in either of the ways, no energy is exchanged and each pendulum's amplitude (the extent of its swing) is constant.

The normal modes arise from symmetry. Suppose pendulums A and B are both pulled the same distance to without changing the length of the spring. Indeed, they swing just as they would without the spring. Since the spring transfers no energy, this synchronous swinging is one of the normal modes.

The other normal mode can be created by pulling the pendulums equally far to opposite sides. When they are released simultaneously, their motions are mirror images. In this normal mode the spring is stretched and compressed by the pendulums. Therefore it does influence their motion. The force it exerts on A is symmetric to the force it exerts on B. For example, when it pushes A to the left, it pushes B just as much to the right. Since the symmetry in the forces precludes a transfer of energy between the pendulums, their amplitudes remain constant.

When a pendulum is free to swing by itself, its swinging frequency is proportional to the square root of the ratio F of the acceleration of gravity to the length of the pendulum. In the first normal mode each pendulum swings at that frequency. In the second normal mode the pendulums swing at a higher frequency because of the driving force from the spring.

For example, after the pendulums swing toward each other, compressing the spring, they return to the vertical not only because of gravity but also because of a push by the spring. When the pendulums swing away from each other, the spring exerts a pull. The push-and-pull action drives the swinging at a high frequency.

Normal modes are important because any motion of the system can be described as a combination of them. Suppose you actuate the system by displacing and then releasing only pendulum B. Thereafter the movement of each pendulum is expressible as a multiplication of two sinusoidal variations. One of them is rapid, having a frequency that is the average of the frequencies for the normal modes. Whenever a pendulum is active, it swings at this frequency.

The other sinusoidal variation modifies the amplitude of the swinging. The frequency of the modification is equal to the difference in the frequencies of the normal modes, and so it is lower than the frequency at which the pendulums swing. The result is a gradual variation in the swinging amplitude of each pendulum. At maximum F amplitude the pendulum has all the energy of the system. At zero amplitude the pendulum is at rest. Since the modifications of amplitude are exactly out of step, one pendulum peaks in amplitude as the second comes to rest. For other ways of activating the system the pendulums may swing differently, but they will still display beats in which their amplitudes vary.

You may have heard audible beats if you have ever listened to two pure tones that differ slightly in frequency. The sound you hear is not either of the two waves actually reaching your ears but rather a sound with a frequency t that is the average of the frequencies of the two waves. Its amplitude, which is related to the intensity of the sound you hear, varies at a rate that is the difference between the two frequencies. Hence the sound you hear waxes and wanes as the two waves beat against each other.

The system in which two pendulums are coupled by a rod has been analyzed by Joseph Priest and James Poth of Miami University in Ohio. The system came to their attention during a production of My Fair Lady, when an actor accidentally bumped a suspended stage panel. The panel hung from one end of a horizontal rod; an identical panel hung from the other end. The rod was suspended by two cables.

When the first panel began to swing, parallel to the rod (in the vertical plane passing through the rod), its energy was gradually transferred to the other panel. Thereafter the energy passed periodically from one panel to the other, creating a noticeable distraction from the play. Priest and Poth studied a similar system in which two pendulums are supported by a horizontal rod. The string of each pendulum runs from the rigid support of the system to the rod, once around the rod and then to the pendulum bob.

The motion of the pendulums is coupled by means of the rod. When one pendulum is made to swing parallel to the rod, it moves the rod, which then moves the other pendulum. These pendulums, like the spring-coupled ones, have two normal modes in which they do not exchange energy. Any motion of the system that is not a normal mode is expressible in terms of the normal modes. In such cases the system displays beats.

As before, the normal modes arise from symmetry. In the first normal mode the pendulums swing exactly in step. Since in this mode the rod swings too, the effective length of the pendulum must be measured from the pendulum bob all the way to the rigid support from which the rod is suspended, because the full length takes part in the swinging. The frequency associated with this mode is low because of the long effective length of the pendulums.

In the second normal mode the pendulums move in opposite directions and the rod is motionless. The effective length of the pendulum is now measured from the bob only to the rod, because only that much of the string takes part in the swinging. The frequency of this normal mode is high because of the short pendulums.

The coupling between the pendulums involves only motion parallel to the rod. If one pendulum swings perpendicular to the rod (that is, perpendicular to the vertical plane passing through the rod), that motion is retained because the string winds and unwinds around the rod without moving it much. If a pendulum swings both parallel and perpendicular to the rod, the perpendicular swing is maintained and the parallel swing generates beats.

Priest and Poth demonstrated this motion with small scale models and also with a larger model. In the larger model two playground swings hung from a rod suspended from a ceiling. A student sat in each swing. After one of the students was pushed and began to swing, the system moved like the pendulums in the small models.

The related system in which pendulums hang from a string and swing perpendicular to its length has been studied by Michael J. Moloney of the Rose-Hulman Institute of Technology in Terre Haute, Ind. The motion of the pendulums is coupled because each swinging pendulum pulls the interconnecting string, which then influences the motion of the other pendulum. Each pendulum also twists the string Since this action complicates the mathematical analysis, Moloney reduces its importance by placing the pendulums at least 10 centimeters apart.

In one normal mode the pendulums swing in step. The effective length of the pendulum is then the vertical distance from the bob to the rigid support because that full length participates in the swinging. In the other normal mode the pendulums swing exactly out of step, which decreases their effective length and results in a higher frequency. The system displays beats when the pendulums swing in some pattern other than a normal mode.

The resonant-spring pendulum is related to the systems I have discussed so far, but it lacks normal modes and is harder to analyze mathematically. Nevertheless, you can easily set up such a system and can analyze its behavior in approximate terms.

If you pull the mass downward, stretching the spring, and then release it, the spring's vertical oscillations are soon replaced by a pendulum motion. Since once you release the spring the energy of the system remains constant, the energy of the pendulum motion must come from the energy in the oscillations of the spring. When the transfer is complete, it reverses, so that the spring oscillations reappear and the pendulum motion dies out.

This exchange of energy is optimal if the mass attached to the spring stretches the spring to about 4/3 of its initial length. The purpose of the increase in length is to decrease the pendulum's frequency in order to establish a relation between the frequencies associated with pure spring oscillations and those associated with pure pendulum motion. The spring oscillations must be twice the frequency of the pendulum motion. Then the instability that develops in the system is large enough to drive the mass into pendulum motion.

Martin G. Olsson of the University of Wisconsin at Madison was one of the first to study the spring-pendulum system. To increase the spring length by the proper amount one must test various weights hung from the spring. Not all springs are suitable for the demonstration. J. G. Lipham and V. L. Pollak of the University of North Carolina at Charlotte have pointed out that in practice one must consider the mass of the spring itself in addition to the mass hung from it.

They found that to create the proper relation of frequencies the spring must be stiff. Lipham and Pollak made a pendulum with a spring they bought from the Central Scientific Company (11222 Melrose Avenue, Franklin Park, III. 60131-1364). It is part of a "Hooke's Law Apparatus" (catalogue number 73 960). The mass of the spring is 3.4 grams, the unstretched length five centimeters. A force of about .41 newton stretches it by one centimeter.

To make a weight for the spring Lipham and Pollak melted lead in a 50milliliter Pyrex beaker, allowed it to cool and then got it out by breaking the glass. A cotter pin or a length of wire threaded through a hole punched in the lead served to attach the weight to the spring. Lipham and Pollak tested the weight on the spring and then trimmed it until they found the optimum mass, which turned out to be 73 grams. (If you trim too much lead, add slotted weights from a common laboratory balance to the block to "tune" the apparatus.)

Lipham and Pollak also devised a way to use weak springs in the demonstration. (The task is to decrease the frequency of the pendulum motion without altering the frequency of the spring oscillations.) They lengthened the pendulum by adding a string between the spring and the support from which it was suspended.

If the spring could be released with no horizontal displacement, it would oscillate only vertically; the pendulum motion would not develop. In practice, however, the release always includes some horizontal displacement of the mass hanging on the spring. The ensuing instability in the oscillations builds up the pendulum motion.

When the spring oscillations dominate, as they do initially, the excitation of the pendulum motion is said to be an example of parametric resonance. The term indicates that the pendulum motion depends on a quantity (the length of the pendulum) that is itself oscillating. The result is a transfer of energy to the pendulum motion. (Parametric resonance also accounts for one's ability to pump a playground swing to a greater amplitude. The periodic shift of the swinger's legs and torso alters the effective length of the pendulum—the swing—so that energy is fed into the pendulum motion.)

When the pendulum motion dominates, energy is fed back to the spring oscillations because of a simpler resonance: the spring is periodically pulled at a frequency matching its oscillation frequency. Each time the mass swings to its maximum displacement during the pendulum motion it pulls on the spring. Since for each swing of the pendulum the spring is pulled, twice, the pulling frequency (twice that of the pendulum motion) matches the frequency of the spring oscillations. The oscillations begin to build up. Energy drains from the pendulum motion until it is almost entirely in the oscillations of the spring.

Olsson observed that sometimes the mass on the end of the spring moved in a relatively stable path as the pendulum motion built up and died down. In some cases the path was in the shape of a U. In other cases it was an inverted U. In still other cases the path was not well defined.

Recently H. M. Lai of the Chinese University of Hong Kong reexamined the spring pendulum, extending and correcting the prior mathematical analysis. He found that the stable paths are Lissajous figures (named after Jules Antoine Lissajous, a French physicist of the 19th century) that depend on several parameters of the apparatus and in particular on' the relative phases of the spring and pendulum motions.

The path can vary in several ways as pendulum motion builds up and decays [see illustration below]. It may begin in the shape of a U that distorts into a bow-tie figure. As the pendulum motion nears its maximum the path returns to the shape of a U. During the decay of the motion the path distorts into a bow-tie shape is before, but now with the reverse motion. Similar variations of the path take place, if the mass initially follows a path in the shape of an inverted U.

The Wilberforce pendulum has seldom been discussed since the original publication. It is another example of a system that can oscillate in normal modes and can display beats, when it is not in a normal mode. The normal modes involve both the vertical motion of the object on the spring and its rotation about a vertical axis.

The coupling between the vertical motion and the rotational motion is achieved by the coiling of the spring. If you stretch a spring, its end rotates too. In one of the normal modes of oscillation the end of the spring makes a screwing motion in the same direction as the coiling of the spring. In the other normal mode the screwing motion is in the opposite direction.

For the pendulum to perform best the frequency of pure rotational oscillations must match the frequency of pure spring oscillations. One achieves the match by fitting the object hung from the pendulum with extended arms that make it possible to change the frequency of the rotational oscillations. Two symmetrically placed bolts, each with a nut, extending out from the vertical axis of the pendulum will serve. Moving the nuts outward decreases the frequency of the rotations oscillations and moving them inward increases it. Adjust them until)he frequencies of the oscillations match.

You can put the pendulum in one of its normal modes of oscillation, if you I pull down on it while you turn the end through the proper angle. When you release the pendulum, it oscillates vertically and rotates back and forth with unvarying amplitudes, that is, without any transfer of energy between the vertical and rotational oscillations. If you pull the pendulum down without rotating the end, the two normal modes beat when you release it. At times the pendulum oscillates vertically with no rotation and at other times it rotates back and forth with no vertical oscillations.

Much more can, be learned by building and studying oscillating systems that display normal modes or parametric resonance. Can you find additional examples of either type? Since the Wilberforce pendulum seems to have been neglected over the years, you may also want to analyze it in greater detail.

Bibliography

VIBRATIONS AND WAVES. A. P. French. W. W. Norton & Co., Inc., 1971.

TEACHING PHYSICS WITH COUPLED PENDULUMS. Joseph Priest and James Poth in The Physics Teacher, Vol. 19, No. 2, pages 80-85; February, 1982.

ON THE RECURRENT PHENOMENON OF A RESONANT SPRING PENDULUM. H. M. Lai in American Journal of Physics. Vol. 52, No. 3, pages 213-223; March 1984.

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