CE5

Harmonic motion is exhibited by a weight vibrating on the end of a spring. In A, the displacement of the object from its equilibrium position and the acceleration of the object are at a maximum; the velocity of the object is zero. In B, the displacement and acceleration are zero, while the velocity is at a maximum. In C, the displacement and acceleration are again at a maximum, and the velocity is again zero.

harmonic motion, regular vibration in which the acceleration of the vibrating object is directly proportional to the displacement of the object from its equilibrium position but oppositely directed. A single object vibrating in this manner is said to exhibit simple harmonic motion (SHM). More complex harmonic motion can be analyzed as combinations of two or more simple harmonic motions. Examples of objects whose motion approximates SHM are a pendulum swinging in a small arc, a mass bouncing at the end of a stretched spring, and air molecules vibrating back and forth as a sound wave passes. Simple harmonic motion is a periodic motion; that is, it repeats itself at regular intervals. The time required for one complete vibration of the object is the period of the motion. The inverse of the period is the frequency, which is the number of vibrations per unit of time. The maximum displacement of the object from its central position of equilibrium is the amplitude of the motion. At maximum displacement the velocity of the object is zero; the velocity is greatest when the object passes through its equilibrium position. These terms are commonly used to describe any periodic phenomenon, e.g., wave motion and the rotation or revolution of an astronomical body. For any real harmonic motion, various forces act to reduce the amplitude with each vibration, i.e., to damp the motion. If these forces are small compared to the restoring force arising from the original displacement, then the object will vibrate a number of times with successively smaller amplitudes until the motion gradually dies out; this is known as damped harmonic motion. For a certain value of the damping forces, the object returns to its original position in a minimum amount of time and comes to rest at that position; such motion is termed critically damped. If the damping forces are large compared to the restoring force, the object returns slowly to its original position without vibrating at all; the system is said to be overdamped.

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2023, Columbia University Press. All rights reserved.

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