What's the Difference Between Period, Frequency, and Angular Frequency?

Period, frequency, and angular frequency all describe how quickly a repeating motion occurs, but they use different measurements. For IB Physics SL and HL, their relationships are $f=\frac{1}{T}$ and $\omega=2\pi f=\frac{2\pi}{T}$.

How Are They Different?

These quantities connect cycle counting to phase. Frequency describes how many cycles occur in a time interval, while period describes the duration of each cycle. Angular frequency is useful because simple harmonic motion is represented by sine or cosine functions, whose arguments are angles in radians. Thus, $\omega t$ represents the phase accumulated after time $t$.

One complete cycle corresponds to an angular change of $2\pi$ radians. Therefore, an oscillator completing $f$ cycles each second passes through $2\pi f$ radians of phase each second, giving $\omega=2\pi f$.

For example, suppose a mass completes one oscillation every $0.50\text{ s}$:

$$f=\frac{1}{T}=\frac{1}{0.50}=2.0\text{ Hz}$$ $$\omega=2\pi f=2\pi(2.0)=4\pi\approx12.6\text{ rad s}^{-1}$$

The motion therefore has a period of $0.50\text{ s}$, a frequency of $2.0\text{ Hz}$, and an angular frequency of $12.6\text{ rad s}^{-1}$.

The same conversions apply beyond SHM, including waves and circular motion. For example, a rotating object with $T=4.0\text{ s}$ has $f=0.25\text{ Hz}$ and $\omega=1.57\text{ rad s}^{-1}$. In SHM, angular frequency describes the rate of phase change rather than the angular velocity of a rotating body.

A common misconception is that frequency and angular frequency are interchangeable. They are not: $f$ counts cycles per second, whereas $\omega$ measures radians of phase per second, so they differ by a factor of $2\pi$.

Exam Technique

In topic C.1 simple harmonic motion, identify the quantity and unit before substituting. Use $\omega$, not $f$, in $a=-\omega^2x$. For a Calculate question, show the conversion, substitution, and final value with the appropriate unit.