Make a list of everyday examples of oscillatory (back-and-forth) motion you have experienced. What commonalities do you notice between them? How common do you think oscillatory motion is in the real world?
An oscillation typically occurs when an object is displaced away from a stable equilibrium such that a net force acts to return the object to this resting position. The forces that lead to oscillatory motion are typically called restoring forces. A restoring force always acts opposite an object’s displacement from equilibrium. Hooke’s Law, in which the force is directly proportional to displacement, is the canonical example restoring force.
Several quantities are key to understanding oscillatory motion. The first, the period, is familiar from uniform circular motion, and corresponds to the total amount of time for the oscillation to progress through one full cycle. The second, the frequency, corresponds to the rate at which the oscillation proceeds. Lastly, the amplitude describes the maximum deviation from equilibrium.
The oscillation frequency is the number of cycles completed during a given time interval, and is thus related to the period by
\begin{equation*}
f = \frac{2\pi}{T}
\end{equation*}
Frequency is measured in cycles per second, or Hertz (Hz). The angular frequency is instead the number of radians completed during a given time interval, and is given by
\begin{equation*}
\omega = 2\pi f = \frac{2\pi}{T}
\end{equation*}
Angular frequency is measured in radians per second.
The amplitude of an oscillatory function is the function’s maximum deviation from equilibrium. It can be represented generically with \(A\) or with a more evocative name for a particular function, such as \(x_{\text{max}}\) for position, which can help make the units of the amplitude more obvious.
The most fundamental oscillatory motion, simple harmonic motion, can be described by a sine or cosine function. More complicated oscillatory motion can be described either as a sum of sinusoidal functions or as a sinusoidal function multiplied by other functions such as exponential functions.
Pendulum Figure by Ideophagous published by Wikimedia Commons, the free media repository under Creative Commons Attribution-Share Alike 4.0 International.
