# OSCILLATION

## CONCEPT

When a particle experiences repeated movement about a position of stable equilibrium, or balance, it is said to be in harmonic motion, and if this motion is repeated at regular intervals, it is called periodic motion. Oscillation is a type of harmonic motion, typically periodic, in one or more dimensions. Among the examples of oscillation in the physical world are the motion of a spring, a pendulum, or even the steady back-and-forth movement of a child on a swing.

## HOW IT WORKS

### STABLE AND UNSTABLE EQUILIBRIUM

When a state of equilibrium exists, the vector sum of the forces on an object is equal to zero. There are three varieties of equilibrium: stable, unstable, and neutral. Neutral equilibrium, discussed in the essay on Statics and Equilibrium elsewhere in this book, does not play a significant role in oscillation; on the other hand, stable and unstable equilibrium do.

In the example of a playground swing, when the swing is simply hanging downward—either empty or occupied—it is in a position of stable equilibrium. The vector sums are balanced, because the swing hangs downward with a force (its weight) equal to the force of the bars on the swing set that hold it up. If it were disturbed from this position, as, for instance, by someone pushing the swing, it would tend to return to its original position.

If, on the other hand, the swing were raised to a certain height—if, say, a child were swinging and an adult caught the child at the point of maximum displacement—this would be an example of unstable equilibrium. The swing is in equilibrium because the forces on it are balanced: it is being held upward with a force equal to its weight. Yet, this equilibrium is unstable, because a disturbance (for instance, if the adult lets go of the swing) will cause it to move. Since the swing tends to oscillate, it will move back and forth across the position of stable equilibrium before finally coming to a rest in the stable position.

### PROPERTIES OF OSCILLATION

There are two basic models of oscillation to consider, and these can be related to the motion of two well-known everyday objects: a spring and a swing. As noted below, objects not commonly considered "springs," such as rubber bands, display spring-like behavior; likewise one could substitute "pendulum" for swing. In any case, it is easy enough to envision the motion of these two varieties of oscillation: a spring generally oscillates along a straight line, whereas a swing describes an arc.

Either case involves properties common to all objects experiencing oscillation. There is always a position of stable equilibrium, and there is always a cycle of oscillation. In a single cycle, the oscillating particle moves from a certain point in a certain direction, then reverses direction and returns to the original point. The amount of time it takes to complete one cycle is called a period, and the number of cycles that take place during one second is the frequency of the oscillation. Frequency is measured in Hertz (Hz), with 1 Hz—the term is both singular and plural—equal to one cycle per second.

THE BOUNCE PROVIDED BY A TRAMPOLINE IS DUE TO ELASTIC POTENTIAL ENERGY. (Photograph by
Kevin Fleming/Corbis
. Reproduced by permission.)

It is easiest to think of a cycle as the movement from a position of stable equilibrium to one of maximum displacement, or the furthest possible point from stable equilibrium. Because stable equilibrium is directly in the middle of a cycle, there are two points of maximum displacement. For a swing or pendulum, maximum displacement occurs when the object is at its highest point on either side of the stable equilibrium position. For example, maximum displacement in a spring occurs when the spring reaches the furthest point of being either stretched or compressed.

The amplitude of a cycle is the maximum displacement of particles during a single period of oscillation, and the greater the amplitude, the greater the energy of the oscillation. When an object reaches maximum displacement, it reverses direction, and, therefore, it comes to a stop for an instant of time. Thus, the speed of movement is slowest at that position, and fastest as it passes back through the position of stable equilibrium. An increase in amplitude brings with it an increase in speed, but this does not lead to a change in the period: the greater the amplitude, the further the oscillating object has to move, and, therefore, it takes just as long to complete a cycle.

### RESTORING FORCE

Imagine a spring hanging vertically from a ceiling, one end attached to the ceiling for support and the other free to hang. It would thus be in a position of stable equilibrium: the spring hangs downward with a force equal to its weight, and the ceiling pulls it upward with an equal and opposite force. Suppose, now, that the spring is pulled downward.

A spring is highly elastic, meaning that it can experience a temporary stress and still rebound to its original position; by contrast, some objects (for instance, a piece of clay) respond to deformation with plastic behavior, permanently assuming the shape into which they were deformed. The force that directs the spring back to a position of stable equilibrium—the force, in other words, which must be overcome when the spring is pulled downward—is called a restoring force.

The more the spring is stretched, the greater the amount of restoring force that must be overcome. The same is true if the spring is compressed: once again, the spring is removed from a position of equilibrium, and, once again, the restoring force tends to pull it outward to its "natural" position. Here, the example is a spring, but restoring force can be understood just as easily in terms of a swing: once again, it is the force that tends to return the swing to a position of stable equilibrium. There is, however, one significant difference: the restoring force on a swing is gravity, whereas, in the spring, it is related to the properties of the spring itself.

### ELASTIC POTENTIAL ENERGY

For any solid that has not exceeded the elastic limit—the maximum stress it can endure without experiencing permanent deformation—there is a proportional relationship between force and the distance it can be stretched. This is expressed in the formula F = ks, where s is the distance and k is a constant related to the size and composition of the material in question.

The amount of force required to stretch the spring is the same as the force that acts to bring it back to equilibrium—that is, the restoring force. Using the value of force, thus derived, it is possible, by a series of steps, to establish a formula for elastic potential energy. The latter, sometimes called strain potential energy, is the potential energy that a spring or a spring-like object possesses by virtue of its deformation from the state of equilibrium. It is equal to ½ks2.

#### POTENTIAL AND KINETIC ENERGY.

Potential energy, as its name suggests, involves the potential of something to move across a given interval of space—for example, when a sled is perched at the top of a hill. As it begins moving through that interval, the object will gain kinetic energy. Hence, the elastic potential energy of the spring, when the spring is held at a position of the greatest possible displacement from equilibrium, is at a maximum. Once it is released, and the restoring force begins to move it toward the equilibrium position, potential energy drops and kinetic energy increases. But the spring will not just return to equilibrium and stop: its kinetic energy will cause it to keep going.

In the case of the "swing" model of oscillation, elastic potential energy is not a factor. (Unless, of course, the swing itself were suspended on some sort of spring, in which case the object will oscillate in two directions at once.) Nonetheless, all systems of motion involve potential and kinetic energy. When the swing is at a position of maximum displacement, its

A BUNGEE JUMPER HELPS ILLUSTRATE A REALWORLD EXAMPLE OF OSCILLATION. (
Eye Ubiquitous/Corbis
. Reproduced by permission.)
potential energy is at a maximum as well. Then, as it moves toward the position of stable equilibrium, it loses potential energy and gains kinetic energy. Upon passing through the stable equilibrium position, kinetic energy again decreases, while potential energy increases. The sum of the two forms of energy is always the same, but the greater the amplitude, the greater the value of this sum.

## REAL-LIFE APPLICATIONS

### SPRINGS AND DAMPING

Elastic potential energy relates primarily to springs, but springs are a major part of everyday life. They can be found in everything from the shock-absorber assembly of a motor vehicle to the supports of a trampoline fabric, and in both cases, springs blunt the force of impact.

If one were to jump on a piece of trampoline fabric stretched across an ordinary table—one with no springs—the experience would not be much fun, because there would be little bounce. On the other hand, the elastic potential energy of the trampoline's springs ensures that anyone of normal weight who jumps on the trampoline is liable to bounce some distance into the air. As a person's body comes down onto the trampoline fabric, this stretches the fabric (itself highly elastic) and, hence, the springs. Pulled from a position of equilibrium, the springs acquire elastic potential energy, and this energy makes possible the upward bounce.

As a car goes over a bump, the spring in its shock-absorber assembly is compressed, but the elastic potential energy of the spring immediately forces it back to a position of equilibrium, thus ensuring that the bump is not felt throughout the entire vehicle. However, springs alone would make for a bouncy ride; hence, a modern vehicle also has shock absorbers. The shock absorber, a cylinder in which a piston pushes down on a quantity of oil, acts as a damper—that is, an inhibitor of the springs' oscillation.

#### SIMPLE HARMONIC MOTION AND DAMPING.

Simple harmonic motion occurs when a particle or object moves back and forth within a stable equilibrium position under the influence of a restoring force proportional to its displacement. In an ideal situation, where friction played no part, an object would continue to oscillate indefinitely.

Of course, objects in the real world do not experience perpetual oscillation; instead, most oscillating particles are subject to damping, or the dissipation of energy, primarily as a result of friction. In the earlier illustration of the spring suspended from a ceiling, if the string is pulled to a position of maximum displacement and then released, it will, of course, behave dramatically at first. Over time, however, its movements will become slower and slower, because of the damping effect of frictional forces.

#### HOW DAMPING WORKS.

When the spring is first released, most likely it will fly upward with so much kinetic energy that it will, quite literally, bounce off the ceiling. But with each transit within the position of equilibrium, the friction produced by contact between the metal spring and the air, and by contact between molecules within the spring itself, will gradually reduce the energy that gives it movement. In time, it will come to a stop.

If the damping effect is small, the amplitude will gradually decrease, as the object continues to oscillate, until eventually oscillation ceases. On the other hand, the object may be "overdamped," such that it completes only a few cycles before ceasing to oscillate altogether. In the spring illustration, overdamping would occur if one were to grab the spring on a downward cycle, then slowly let it go, such that it no longer bounced.

There is a type of damping less forceful than overdamping, but not so gradual as the slow dissipation of energy due to frictional forces alone. This is called critical damping. In a critically damped oscillator, the oscillating material is made to return to equilibrium as quickly as possible without oscillating. An example of a critically damped oscillator is the shock-absorber assembly described earlier.

Even without its shock absorbers, the springs in a car would be subject to some degree of damping that would eventually bring a halt to their oscillation; but because this damping is of a very gradual nature, their tendency is to continue oscillating more or less evenly. Over time, of course, the friction in the springs would wear down their energy and bring an end to their oscillation, but by then, the car would most likely have hit another bump. Therefore, it makes sense to apply critical damping to the oscillation of the springs by using shock absorbers.

### BUNGEE CORDS AND RUBBER BANDS

Many objects in daily life oscillate in a spring-like way, yet people do not commonly associate them with springs. For example, a rubber band, which behaves very much like a spring, possesses high elastic potential energy. It will oscillate when stretched from a position of stable equilibrium.

Rubber is composed of long, thin molecules called polymers, which are arranged side by side. The chemical bonds between the atoms in a polymer are flexible and tend to rotate, producing kinks and loops along the length of the molecule. The super-elastic polymers in rubber are called elastomers, and when a piece of rubber is pulled, the kinks and loops in the elastomers straighten.

The structure of rubber gives it a high degree of elastic potential energy, and in order to stretch rubber to maximum displacement, there is a powerful restoring force that must be overcome. This can be illustrated if a rubber band is attached to a ceiling, like the spring in the earlier example, and allowed to hang downward. If it is pulled down and released, it will behave much as the spring did.

The oscillation of a rubber band will be even more appreciable if a weight is attached to the "free" end—that is, the end hanging downward. This is equivalent, on a small scale, to a bungee jumper attached to a cord. The type of cord used for bungee jumping is highly elastic; otherwise, the sport would be even more dangerous than it already is. Because of the cord's elasticity, when the bungee jumper "reaches the end of his rope," he bounces back up. At a certain point, he begins to fall again, then bounces back up, and so on, oscillating until he reaches the point of stable equilibrium.

### THE PENDULUM

As noted earlier, a pendulum operates in much the same way as a swing; the difference between them is primarily one of purpose. A swing exists to give pleasure to a child, or a certain bittersweet pleasure to an adult reliving a childhood experience. A pendulum, on the other hand, is not for play; it performs the function of providing a reading, or measurement.

One type of pendulum is a metronome, which registers the tempo or speed of music. Housed in a hollow box shaped like a pyramid, a metronome consists of a pendulum attached to a sliding weight, with a fixed weight attached to the bottom end of the pendulum. It includes a number scale indicating the number of oscillations per minute, and by moving the upper weight, one can change the beat to be indicated.

#### ZHANG HENG'S SEISMO-SCOPE.

Metronomes were developed in the early nineteenth century, but, by then, the concept of a pendulum was already old. In the second century A.D., Chinese mathematician and astronomer Zhang Heng (78-139) used a pendulum to develop the world's first seismoscope, an instrument for measuring motion on Earth's surface as a result of earthquakes.

Zhang Heng's seismoscope, which he unveiled in 132 A.D., consisted of a cylinder surrounded by bronze dragons with frogs (also made of bronze) beneath. When the earth shook, a ball would drop from a dragon's mouth into that of a frog, making a noise. The number of balls released, and the direction in which they fell, indicated the magnitude and location of the seismic disruption.

#### CLOCKS, SCIENTIFIC INSTRUMENTS, AND "FAX MACHINE".

In 718 A.D., during a period of intellectual flowering that attended the early T'ang Dynasty (618-907), a Buddhist monk named I-hsing and a military engineer named Liang Ling-tsan built an astronomical clock using a pendulum. Many clocks today—for example, the stately and imposing "grandfather clock" found in some homes—like-wise, use a pendulum to mark time.

Physicists of the early modern era used pendula (the plural of pendulum) for a number of interesting purposes, including calculations regarding gravitational force. Experiments with pendula by Galileo Galilei (1564-1642) led to the creation of the mechanical pendulum clock—the grandfather clock, that is—by distinguished Dutch physicist and astronomer Christiaan Huygens (1629-1695).

In the nineteenth century, A Scottish inventor named Alexander Bain (1810-1877) even used a pendulum to create the first "fax machine." Using matching pendulum transmitters and receivers that sent and received electrical impulses, he created a crude device that, at the time, seemed to have little practical purpose. In fact, Bain's "fax machine," invented in 1840, was more than a century ahead of its time.

#### THE FOUCAULT PENDULUM.

By far the most important experiments with pendula during the nineteenth century, however, were those of the French physicist Jean Bernard Leon Foucault (1819-1868). Swinging a heavy iron ball from a wire more than 200 ft (61 m) in length, he was able to demonstrate that Earth rotates on its axis.

Foucault conducted his famous demonstration in the Panthéon, a large domed building in Paris named after the ancient Pantheon of Rome. He arranged to have sand placed on the floor of the Panthéon, and placed a pin on the bottom of the iron ball, so that it would mark the sand as the pendulum moved. A pendulum in oscillation maintains its orientation, yet the Foucault pendulum (as it came to be called) seemed to be shifting continually toward the right, as indicated by the marks in the sand.

The confusion related to reference point: since Earth's rotation is not something that can be perceived with the senses, it was natural to assume that the pendulum itself was changing orientation—or rather, that only the pendulum was moving. In fact, the path of Foucault's pendulum did not vary nearly as much as it seemed. Earth itself was moving beneath the pendulum, providing an additional force which caused the pendulum's plane of oscillation to rotate.

Brynie, Faith Hickman. Six-Minute Science Experiments. Illustrated by Kim Whittingham. New York: Sterling Publishing Company, 1996.

Ehrlich, Robert. Turning the World Inside Out, and 174 Other Simple Physics Demonstrations. Princeton, N.J.: Princeton University Press, 1990.

"Foucault Pendulum" Smithsonian Institution FAQs (Website). <http://www.si.edu/resource/faq/nmah/pendulum.html> (April 23, 2001).

Kruszelnicki, Karl S. The Foucault Pendulum (Web site). <http://www.abc.net.au/surf/pendulum/pendulum.html> (April 23, 2001).

Schaefer, Lola M. Back and Forth. Edited by Gail Saunders-Smith; P. W. Hammer, consultant. Mankato, MN: Pebble Books, 2000.

Shirley, Jean. Galileo. Illustrated by Raymond Renard. St. Louis: McGraw-Hill, 1967.

Suplee, Curt. Everyday Science Explained. Washington, D.C.: National Geographic Society, 1996.

Topp, Patricia. This Strange Quantum World and You. Nevada City, CA: Blue Dolphin, 1999.

Zubrowski, Bernie. Making Waves: Finding Out About Rhythmic Motion. Illustrated by Roy Doty. New York: Morrow Junior Books, 1994.

## KEY TERMS

### AMPLITUDE:

The maximum displacement of particles from their normal position during a single period of oscillation.

### CYCLE:

One full repetition of oscillation. In a single cycle, the oscillating particle moves from a certain point in a certain direction, then switches direction and moves back to the original point. Typically, this is from the position of stable equilibrium to maximum displacement and back again to the stable equilibrium position.

### DAMPING:

The dissipation of energy during oscillation, which prevents an object from continuing in simple harmonic motion and will eventually force it to stop oscillating altogether. Damping is usually caused by friction.

### ELASTIC POTENTIAL ENERGY:

The potential energy that a spring or a spring-like object possesses by virtue of its deformation from the state of equilibrium. Sometimes called strain potential energy, it is equal to ½KS2, WHERE S is the distance stretched and k is a figure related to the size and composition of the material in question.

### EQUILIBRIUM:

A state in which the vector sum for all lines of force on an object is equal to zero.

### FREQUENCY:

For a particle experiencing oscillation, frequency is the number of cycles that take place during one second. Frequency is measured in Hertz.

### FRICTION:

The force that resists motion when the surface of one object comes into contact with the surface of another.

### HARMONIC MOTION:

The repeated movement of a particle within a position of equilibrium, or balance.

### HERTZ:

A unit for measuring frequency. The number of Hertz is the number of cycles per second.

### KINETIC ENERGY:

The energy that an object possesses due to its motion, as with a sled, when sliding down a hill. This is contrasted with potential energy.

### MAXIMUM DISPLACEMENT:

For an object in oscillation, maximum displacement is the furthest point from stable equilibrium. Since stable equilibrium is in the middle of a cycle, there are two points of maximum displacement. For a swing or pendulum, this occurs when the object is at its highest point on either side of the stable equilibrium position. Maximum displacement in a spring occurs when the spring is either stretched or compressed as far as it will go.

### OSCILLATION:

A type of harmonic motion, typically periodic, in one or more dimensions.

### PERIOD:

The amount of time required for one cycle in oscillating motion—for instance, from a position of maximum displacement to one of stable equilibrium, and, once again, to maximum displacement.

### PERIODIC MOTION:

Motion that is repeated at regular intervals. These intervals are known as periods.

### POTENTIAL ENERGY:

The energy that an object possesses due to its position, as for instance, with a sled at the top of a hill. This is contrasted with kinetic energy.

### RESTORING FORCE:

A force that directs an object back to a position of stable equilibrium. An example is the resistance of a spring, when it is extended.

### SIMPLE HARMONIC MOTION:

Harmonic motion, in which a particle moves back and forth about a stable equilibrium position under the influence of a restoring force proportional to its displacement. Simple harmonic motion is, in fact, an ideal situation; most types of oscillation are subject to some form of damping.

### STABLE EQUILIBRIUM:

A type of equilibrium in which, if an object were disturbed, it would tend to return to its original position. For an object in oscillation, stable equilibrium is in the middle of a cycle, between two points of maximum displacement.

### VECTOR:

A quantity that possesses both magnitude and direction.

### VECTOR SUM:

A calculation that yields the net result of all the vectors applied in a particular situation. Because direction is involved, it is necessary when calculating the vector sum of forces on an object (as, for instance, when determining whether or not it is in a state of equilibrium), to assign a positive value to forces in one direction, and a negative value to forces in the opposite direction. If the object is in equilibrium, these forces will cancel one another out.