$$\require{cancel}$$

# 3.12 Periodic Processes

An orbit is a kind of motion that repeats regularly. The Earth or any other planet passes any particular point in its orbit after a fixed interval of time — called the period of the orbit. Orbits are just one example of a broad class of physical phenomena, called periodic processes. Waves and vibrations are other examples. We can use an ordered deck of cards as an analogy for a periodic sequence. Imagine a deck of cards where the suits are separated and the cards within a suit are arranged in numerical order. The cards of any particular value are arranged periodically. For example, the Jacks would be at positions 10, 13, 26, and 39 in the deck. The spacing is uniform, in a cycle or "period" of 13 cards. The same period applies for any other value of card. Waves and vibrations are all around us.

Polar view of the orbit of Mercury. Click here for original source URL.

Orbits are repeatable motions in time and space: the Earth passes through any particular point in its orbit every 365.25 days and it follows the same path through space every year. Every 89 days Mercury reaches its maximum elongation or angular separation from the Sun. Every 75 years Halley’s comet returns from the depths of space to brighten our skies. These cyclic motions are repeatable and predictable. Spinning motions are also cyclic; every 24 hours the Sun rises due to the Earth’s rotation.

Orbits are an example of a wide range of physical phenomena called periodic processes. Let’s look at some other examples. Oscillations or vibrations occur when an object moves backward and forward or from side to side in a regular way. A pendulum is a good example — a weight freely suspended by a string will swing from side to side repeatedly. The time it takes to finish a cycle of motion depends only on the length of the string. Galileo discovered this interesting fact while he was a young student in Pisa. During a service at the cathedral he noticed that a swinging altar lamp had a period that did not depend on how wildly or gently it was swinging. This periodic motion of a pendulum became an essential part of clocks for hundreds of years.

A needle on a tuning fork carved these figures on a glass plate covered with carbon black. As the plate is moved from left to right, the sinewave-shaped swinging motion appears. Click here for original source URL.

Mechanical objects can vibrate in a regular way. If you hold a wooden meter rule (or a thin metal ruler) so that it is projecting off the edge of a table and flick the end, you will set up a vibration or a rapid periodic motion. Usually this motion is a blur, but if the ruler is long enough, you can see that several cycles of motion occur every second. Mechanical vibrations can be very rapid. Most musicians using a tuning fork that vibrates 440 times per second! Now we can see the connection to yet another periodic process: waves. The rapid vibration of the arms of the tuning fork sets up a periodic variation in the density of air that we perceive as a sound wave. A vibration of 440 times per second corresponds to A below middle C on the musical scale.

A number of apparently diverse phenomena — orbits, rotations, oscillations, vibrations, and waves — share common features. They all follow a cycle; that is, a motion or behavior that repeats. The time it takes to complete a cycle is the period, which is usually labeled T. The number of cycles per second is the frequency, labeled f. These two quantities are simply related

f = 1 / T

The unit of frequency is the Hertz, after the German physicist Heinrich Hertz. In the case of orbits we use a closely related quantity called the angular frequency, denoted by the Greek letter omega (ω). Each orbit, an object sweeps through an angle of 360° or 2π radians. So angular frequency is the number of radians moved through per second

An example of periodic motion: Pendulum cylinder rotation. Click here for original source URL.

w = 2ωf = 2ω / T

Progressing towards smaller physical systems, the period decreases and the frequency increases. Smaller objects rotate (or oscillate or vibrate) faster! This behavior is obvious within the solar system, where planets with smaller orbits have shorter periods. You can see it for yourself in the example of the ruler vibrating off the edge of a table. As the ruler vibrates, slide it so that more or less is projecting off the table and watch how the frequency of vibration changes. The same behavior is also familiar from the world of vibrations and sounds. Short piano wires have more rapid vibrations than long piano wires, and they lead to higher frequency (or higher pitched) sounds. Also, the highest frequency sounds come from the smallest brass or woodwind instruments — think of a trumpet compared to a tuba or a flute compared to a bassoon.

There is another striking similarity between the different periodic processes. Their motions in one dimension are described by a sinusoidal variation. The motion or displacement from the center position is given by

X = Xmaxcos(ωt) = Xmaxcos(2ωft)

The maximum amount of motion is given by the quantity Xmax. Positive and negative positions just correspond to motions to the left or right of the central or resting position so the total deflection is 2Xmax. For the case of the Earth-Sun orbit, X is the apparent separation of the Earth from the Sun when the orbit is viewed in the ecliptic plane. For the case of the rotating Earth, X is the apparent separation of any point on the surface from the rotation axis, when the rotation is viewed in the plane of the equator. In all the other examples, X is the displacement of the object from the central or resting position. For sound waves, X is the variation in air density as the wave passes any point. It is remarkable that such a wide range of phenomena can be described by the same simple equation.