AT THE RIGHT AGE A child can climb onto a playground swing and learn through mimicry and experiment how to start it moving and how to "pump" it. Yet explaining the scientific principles behind swinging is hardly child's play. How does one initiate the pendulum motion other than by simply shoving off from the ground or having someone push? And then how does one build the amplitude (the angular extent of the swinging) when either standing or sitting in the swing?

Although the pastime of swinging is ancient, the relevant mechanics got little attention until 1968, when Peter L. Tea, Jr., and Harold Falk of the City College of the City University of New York published a study. They limited their examination to the case of someone standing in a swing that is already in motion. As you may have learned as a child, you can increase the amplitude by standing and squatting in synchrony with the swing's motion, standing when you pass through the lowest point and squatting at the two highest points. If the swing has rigid supports instead of ropes or chains, and if you are persistent, you may be able to swing higher than the swing's support bar. In fact, if the attachments of the supports to the bar allow it, you could even end up moving in full circles; some circus performers do it.

Why does the procedure of standing and squatting build the swing's amplitude? Tea and Falk pointed out that several factors are involved, but what is paramount is the matter of energy and just how you work to increase it. Picture yourself on a rope-supported swing; disregard any loss of energy to friction and air drag [left]. Suppose you are squatting when you reach the highest point of a swing to the rear. Your energy just then is entirely potential energy, which depends on the height of your center of mass from some reference level. Let the lowest point of the arc through which your center of mass swings be the reference level; assume that your center of mass is two meters above it.

As you descend, the energy is gradually transformed into kinetic energy and you gain speed. When you reach the lowest point, your energy is entirely kinetic energy and you are moving at peak speed. As you begin to ascend on the arc, the transformation is reversed: you slow down and then stop momentarily at the top of the arc.

How high you go depends on what you have done during the swing. If you have continued to squat, the upward motion is a mirror image of the downward motion, and your center of mass ends up just as high as when you began the forward swing, that is, at a height of two meters. If instead you stand when you are at the lowest point, you swing higher.

There are two reasons for the greater height. One of them is that after you stand your center of mass is higher as you begin the upward swing. Suppose standing raises your center of mass by .5 meter. Assume that standing does not change your kinetic energy (an assumption I shall further examine below); The amount of kinetic energy you have at the lowest point determines how high your center of mass goes during the rest of the forward swing. Since you have descended by two meters, you must have enough kinetic energy for your center of mass to swing back up by two meters. With the extra .5 meter added by the standing, your center of mass reaches a height of 2.5 meters at the top of the path. As you approach the top you rotate the swing platform around your center of mass, so that it too goes higher than before.

The amplitude is larger, then, but if you continue to stand, you will just swing back and forth to heights of 2.5 meters. If

you want to pump the swing to greater heights, you must squat each time you reach the end of an arc so that you can stand again when you pass through the lowest point. Of course, by squatting you lower your center of mass somewhat and sacrifice some of the height at the end of the arc. Yet you do not lose as much height as you gained by standing. The reason is that when you squat, the swing is at an angle to the vertical; you shift your center of mass along a slanted line rather than straight down. As you swing higher and thereby increase the angle, the height you lose with each squat decreases. If you can manage to swing up to an amplitude of 90 degrees, squatting does not lower your center of mass at all.

The other reason for the increase in amplitude when you stand at the lowest point is subtler. By standing you move your center of mass toward the axis about which you rotate, and that actually increases your kinetic energy rather than leaving it unchanged, as I assumed above. The increase may seem strange: normally when you lift something-including yourself when you stand up-you do not increase the kinetic energy of what you lift. Instead your work in the lift merely adds to the potential energy, because you have increased the height of what you lift.

The lift during swinging is different, because you are moving in circular motion. To see the point, consider a different and more conventional demonstration. When an ice skater spins on point with outstretched arms and then pulls her arms inward, she spins faster. This common textbook example displays a powerful but often nonintuitive principle of physics: the conservation of angular momentum. That form of momentum is the product of the rate of rotation and a factor called the moment of inertia, which depends on the quantity of mass and its distribution with respect to the axis of rotation. For the ice skater the moment of inertia is initially large because of the outstretched arms; it is smaller when the arms are pulled closer to the axis of rotation.

The angular momentum of an object can change only when a force acts on it in a certain way. If the force is not radial (if a mental extension of the force does not pass through the axis of rotation), the force creates a torque that changes the angular momentum. If the force is radial, though, it does not create a torque and the angular momentum does not change. When the ice skater pulls in her arms, the forces are radial and her angular momentum is unchanged by the action. The decrease in her moment of inertia is therefore matched by an increase in her rate of rotation.

She also increases her kineffc energy, which depends on her moment of inertia and the square of the Mte of rotation. Although the moment of inert}a is decreased, the change in the kinetic energy is dominated by the increased rate of rotation. The increase in energy comes from the work she does in pulling in her arms: she works, in effect, against centrifugal force. (Such a force is usually only a fictional device, because in most examples of rotation the notion that a force tugs outward on a rotating object is one of convenience rather than reality. Still, one can state mathematically that the ice skater does work against a centrifugal force, and the work increases her kinetic energy.)

The same principles apply when you stand up in a swing as it passes through its lowest point. By standing you decrease your moment of inertia, because you shift your center of mass toward the axis about which you rotate-the support bar of the swing. The forces you exercise are upward and radial to the rotation, and therefore they do not produce torques. Since your angular momentum cannot change while you stand, the decrease in your moment of inertia must be matched by an increase in your speed around the bar. As a result your kinetic energy increases too. The added energy comes from the work you do against centrifugal force. No such considerations apply to a squat at the top of the arc, because there the swing has slowed to a stop and no centrifugal force acts on you.

To see how your work against centrifugal force changes the height to which you swing, assume that the length of the ropes is five meters. You squat at a height of two meters, swing down, stand up and swing to the top of the arc. Your center of mass is then at a height of 2.97 meters; the additional .47 meter comes from your work against centrifugal force. After you squat your center of mass is at a height of 2.52 meters, which is somewhat higher than the previous result.

I do not mean to imply that your angular momentum remains constant throughout your swing, only that the act of standing does not change it. Except when you are at the lowest point in the arc, a mental extension of the force of gravity on you (your weight) does not pass through the axis of rotation, and the torque it creates continuously alters your angular momentum. When you swing upward the torque counters your rotation, decreasing your angular momentum until you have none when you reach the top of the arc. Then, as you descend, the torque reestablishes your angular momentum until it is largest when you reach the lowest point. One way to describe the advantage of standing at the lowest point is in terms of the angular momentum. Since it is largest there, the decrease in your moment of inertia when you stand up results in a larger increase in your speed than if you stood at some other point.

If you have been on a swing lately, you may have noticed that as you keep swinging the standing up becomes more difficult and the increase in amplitude with each pass through the arc becomes larger. At an early stage, standing is easy because the centrifugal force on you is small owing to your slow speed through the lowest point. When you raise your center of mass, you therefore do little work against the force and increase your kinetic energy only modestly. Also (because the angle of swing is small) when you squat, the decrease in the height of your center of mass is almost as as the increase in its height when you stand. Since the work is small and the loss of height is large, there is only a slight increase in amplitude with each pass through the arc.

Later, when the swing is going better, the centrifugal force is larger because of your faster speed through the lowest point. You might then have to struggle to stand, but in doing so you work more and increase your kinetic energy more than before. And when you squat, you lower your center of mass less. As a result the increase in amplitude with each pass through the arc is larger.

Pumping by standing up and squatting makes a swing go higher, but how do you get the swinging started without pushing off from the ground or having someone help you? One way was explored in 1970 by Bryan F. Gore of Central Washington State College. Imagine that you are standing on a stationary swing suspended by lightweight ropes. If you suddenly lean back and pull hard on the ropes, you can move them off the vertical toward you [see Figure 3]. The forces exerted on your hands by the ropes are then partially in the forward direction, and so you begin to accelerate forward. Since the ropes are tied to a fixed bar and to the platform that bears your weight, your pull on them is effectively directed against the bar and your weight.

The forces also yield a torque that rotates you about your center of mass, with the top half of your body tending to rotate forward and the bottom half (along with the platform) tending to rotate backward. The rotation is superimposed on your general forward motion and that of the swing; unchecked, it would move the top half of your body in front of the platform. As you reach the highest point in this first, short arc, you might stop the rotation about your center of mass by resisting it with your arms until you become aligned with the ropes. Instead you can actually shove the ropes forward, so that their distortion creates forces on you that are partially toward the rear [see Figure 3]. The torque from the ropes then eliminates your initial rotation and may send you rotating about your center of mass in the opposite direction.

Gore showed that after you first get the swing moving you can pump it by timing your pull and push on the ropes. The procedure works whether you are standing or sitting. When you swing forward, pull the ropes toward you so that their distortion generates forces on you in the forward direction. At the top of the arc push on the ropes to stop your rotation about your center of mass and to launch yourself toward the rear. As you swing backward, continue to push on the ropes. Near the top rear of the arc pull on the ropes to stop the rotation about your center of mass and generate a force in the forward direction. The work you do against the ropes feeds energy into the swing.

In 1972 John T. McMullan of the New University of Ulster demonstrated that a swing can be started from rest even if it has rigid supports rather than the flexible ropes in Gore's model The procedure is to stand on the swing with hands on the supports and arms bent, and then to fall backward until your arms are stretched out, stopping your fall. During the fall you and the swing function as a double pendulum: you rotate about the platform while the platform begins to rotate about the support bar. Once you stop your fall and your arms are rigid, you and the swing function together as a single pendulum. The energy of the motion comes from your initial fall, as potential energy is converted into kinetic energy. After you start the swing you can pump it by standing and squatting.

In 1976 the subject of stand-and-squat pumping was reexamined by Stephen M. Curry of the University of Texas at Dallas. He discovered that the rate at which energy is fed into the swinging has several surprising features. Suppose you are about to swing through an arc after having squatted. Suppose also that E_o is your energy just then, h is the height through which you shift your center of mass when you stand and L is the length of the ropes. When you again squat at the opposite end of the arc, you have an energy of about E_o(1 + 3h/L). After a complete oscillation (forward and backward) you have an energy of about E_o(1 + 3h/L)2, and after n complete oscillations you have an energy of about E_o(1 + 3h/L)²ⁿ. Provided h is much smaller than L, the energy then can be restated as approximately E_oexp(6nh/L), which indicates that the energy grows exponentially with the number of oscillations.

If the amplitude of the swinging happens to be small, the time T required for a complete oscillation is a constant that depends only on the acceleration of gravity and the length of the pendulum that you and the swing constitute. Suppose you somehow begin the swinging with an energy of E_o and pump for a time t The number of oscillations you have undergone is t/T, and your energy is then E_oexp[6(t/T)(h/L)], which shows that the energy also grows exponentially with time.

Curry noted two curious results of his derivation. First, the rate at which energy is pumped into the swing does not depend on your mass or weight. Although a tall person has an advantage over a short one because h has a larger value, there is no advantage in weight. The second result is curiouser. The calculation indicates that if the initial energy is zero, your energy after many pumps in which you stand and squat must still be zero, which certainly seems reasonable. But can the initial energy ever be truly zero? No, it cannot, because your swinging takes place in an environment of air molecules in thermal motion, which beat against you incessantly and provide some initial energy. From that supply you can build your swinging, but it takes standing and squatting for about four minutes before you see the results.

Curry carried the argument even further. Suppose you could somehow survive in an environment at a temperature of absolute zero, so that the air molecules could not help you to begin the swinging. Would that eliminate the possibility of pumping up a swing merely by standing and squatting? No, it would not, because one of the tenets of quantum mechanics disallows zero energy. Even at a temperature of absolute zero, you and the swing must still have a certain (albeit minuscule) energy, and from it you can amplify your swinging. In this case it would take about six minutes before you would see any result. The thought of pumping a swing at a temperature of absolute zero may be bizarre, but the pumping is allowed in theory.

In 1984 Juan R Sanmartin of the Polytechnic University of Madrid observed that the mechanics of pumping a swing by standing up and squatting can be related to the dramatic pendulum motion of a censer that has been swung during rituals in the Cathedral of Santiago de Compostela in northwestern Spain for the past 700 years. The censer, which weighs as much as a slender man, hangs by a stout rope from two wood rollers about 20 meters above it. The rope wraps around the rollers and runs down to a crew of men who control it at floor level.

After the censer is pushed to begin swinging, the men pump the pendulum by timing their pull on their end of the rope. When the censer passes through the lowest point of its arc, they pull hard to shorten the length of the pendulum by almost three meters. When the censer reaches its highest point of travel, the men relax their pull and allow the length of the pendulum to increase to its original value.

The pumping they accomplish is similar to the pumping contributed by standing and squatting on a playground swing. The men add energy to the swinging when they shorten the length of the pendulum and pull against the centrifugal force on the censer. After 17 pulls and about 80 seconds, the censer swings up by almost 90 degrees, rising to within a meter of the cathedral's vaulting. Its rapid passage through the lower part of the arc greatly fans the burning coals and incense it carries. It is quite a sight to see.

When a pendulum such as a swing is pumped twice during each complete oscillation, the pumping is said to be parametric. In 1969 A. E. Siegman of Stanford University described how parametric pumping of a swing can be demonstrated with a simple model. A pendulum bob is suspended by a long cord from a point near the rim of a small wheel that can be rotated in the vertical plane by a motor. Just below the wheel the cord passes between a pair of edges that serve as pivots.

The distance from the bob to a pivot point determines the frequency at which the pendulum would swing naturally if you gave the bob a small push If you adjust the motor so that the frequency at which the wheel turns is twice the natural frequency of the pendulum, the pendulum will start up from rest without any push from you. Although at first the bob merely wiggles, its energy is amplified parametrically whenever the wheel pulls upward on the cord-that is, twice during each complete oscillation of the pendulum.

Bibliography

PUMPING ON A SWING. Peter L. Tea, Jr., and Harold Falk in American Journal of Physics, Vol. 36, No. 12, pages 1165-1166; December, 1968.

HOW CHILDREN SWING. Stephen M. Curry in American Journal of Physics, Vol. 44, No. 10, pages 924-926; October, 1976.

O. BOTAFUMEIRO: PARAMETRIC PUMPING IN THE MIDDLE AGES. Juan R Sanmartin in American Journal of Physics, Vol. 52, No. 10, pages 937-945; October, 1984.

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