Simple harmonic motion

Simple harmonic motion is a concept that is chiefly relevant in physics. It refers to an object moving repetitively, back and forth, in a medium, such that its displacement (i.e., movement away from the equilibrium position) in one direction is equal to its displacement in the other direction. Essentially, the term describes the movement of a pendulum swinging back and forth, so that it swings the same distance to the left as it swings to the right. The same motion can be found when a weight is suspended from a spring. When the weight is at rest, the system is stable and does not move, but if the weight is pushed or pulled downward, it begins to exhibit simple harmonic motion—it travels upward from its point of equilibrium the same distance that it was pulled or pushed downward.

Background

The use of the term “harmonic” stems from the fact that the vibrations that musical instruments make in order to produce sound are of the same character. When musicians strike gongs, tap drums, or blow on horns, they are actually using the instrument to create vibrations that travel through the air and are interpreted by listeners’ ears as sounds. These vibrations take the form of sound waves, which cause particles in the medium (typically, the air) to vibrate back and forth in a wave pattern. Thus, because the motion can be described mathematically in the same way as musical vibrations, the physical phenomenon has been dubbed simple harmonic motion.

Overview

One of the qualities of simple harmonic motion as physicists describe it is that a restoring force acts toward the equilibrium position and is proportional to the displacement from that position. The oscillation can carry on indefinitely only in a vacuum. In the ordinary world, the oscillation will gradually decrease because of friction, as the oscillating object would encounter resistance from molecules of oxygen, nitrogen, carbon dioxide, and other trace materials in the air. The resistance will cause the object to slow down slightly as it travels to and from the equilibrium position, so that with each cycle of movement, the object travels a shorter distance in both directions. The continuation of this behavior eventually brings the movement of the object to a halt, again resting motionlessly.

The defining characteristic of simple harmonic motion is that the restoring force is proportional to the displacement from the equilibrium position, although properties such as the period may depend on the characteristics of the system. Differences in simple harmonic motion may result from both the properties of the system and the influence of external forces. To use the example of friction, the effect of air resistance depends on factors such as size, shape, speed, and mass, which can affect how quickly oscillations are damped.

The gradual slowing down seen with simple harmonic motion under conditions where friction is present is known as “damping.” The process involves some of the energy in the simple harmonic motion system being removed from that system and transferred elsewhere, bit by bit. An example of this can be seen in an automobile’s suspension, which uses springs and shock absorbers to help reduce the jolting effect of driving over uneven surfaces. Driving over a bump compresses the spring, and the spring then begins to push back in the opposite direction, following the pattern of simple harmonic motion. Shock absorbers are then used to siphon energy out of the system—otherwise the back-and-forth motion of the spring would continue much longer than most passengers would find comfortable. Another example of damping is when one pushes a child on a swing. The child is first pulled backward, then swings forward when released almost as far as the child had been pulled back. During each subsequent cycle, the motion is dampened as energy is transferred from the system through friction.

The study of simple harmonic motion has been of tremendous importance in a wide range of fields. Principles of simple harmonic motion are also used in devices such as accelerometers, resonators, and vibration-control systems. One of the most famous is the Foucault pendulum, a relatively heavy pendulum observed by nineteenth-century scientist Jean Bernard Léon Foucault. Foucault used the motion of the pendulum to demonstrate that the Earth rotates on its axis. He did this by setting up a pendulum over a floor covered with a smoothed layer of sand. He attached a pointer to the bottom of the pendulum, so that the pointer would trace a line in the sand as the pendulum swung. Because the pendulum's plane of oscillation tends to remain fixed, observers expected to see the pointer on the pendulum trace the same line in the sand over and over. Instead, they saw a series of lines, each one slightly to the right of the previous one. Because the pendulum’s plane of oscillation tends to remain fixed while the Earth rotates beneath it, the only other conclusion was that the Earth was moving underneath the pendulum. Foucault pendulums are displayed in many museums and educational institutions as demonstrations of Earth's rotation.

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