42.3: Frequency and Period
- Page ID
- 92294
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The angular frequency \(\omega\) described earlier is a measure of how fast the oscillator oscillates; specifically, it measures how many radians of its motion the oscillator moves through each second, where one complete cycle of motion is \(2 \pi\) radians. A related quantity is the frequency \(f\), which describes how many complete cycles of motion the oscillator moves through per second. The two frequencies are related by
\[\omega=2 \pi f .\]
You can think of \(\omega\) and \(f\) as really being the same thing, but measured in different units. The angular frequency \(\omega\) is measured in units of radians per second ( \(\mathrm{rad} / \mathrm{s}\) ); the frequency \(f\) is measured in units of hertz \((\mathrm{Hz})\), where \(1 \mathrm{~Hz}=1 / \mathrm{sec}\).
The reciprocal of the frequency is the period \(T\), and is the time required to complete one cycle of the motion:
\[T=\frac{1}{f}=\frac{2 \pi}{\omega}\]
The period is measured in units of seconds. As shown in the plot of \(x(t)\) (Fig. 39.1.1), the period \(T\) is the time between peaks in the motion.