Period and Frequency

13.2.1 Period and Frequency

When you push a child on a swing, the swing goes back and forth, returning to where it started after a fixed interval. That fixed interval is the period. The number of such back-and-forth trips in one second is the frequency. These two quantities are the heartbeat of any oscillatory motion — they tell you how fast the system repeats itself.

The Period ($T$)

The period of a periodic motion is the smallest time after which the motion repeats itself exactly. If you start a stopwatch at some instant, the period is the time you must wait until every detail of the motion — position, velocity, acceleration — is identical to what it was at the start.

Here $\omega$ is the angular frequency (in radians per second). The period is measured in seconds (s). For a simple pendulum, $T$ depends on the length of the string and the acceleration due to gravity; for a mass on a spring, it depends on the mass and the spring constant. But in every case, $T$ is the time for one complete cycle.

The Frequency ($\nu$ or $f$)

The frequency of a periodic motion is the number of complete oscillations performed per unit time. If the period is $T$ seconds, then in one second the system completes $1/T$ cycles. So

The SI unit of frequency is the hertz (Hz), named after Heinrich Hertz. One hertz means one cycle per second. In many older textbooks you will also see the unit "cycles per second" (cps), but hertz is the modern standard.

The Angular Frequency ($\omega$)

Angular frequency connects the period and frequency to the circular-motion analogy that underlies simple harmonic motion. If a particle moves uniformly around a circle of radius $A$ with constant speed, the angle it sweeps out per second is $\omega$. One full revolution is $2\pi$ radians, so

This is the bridge between the time-domain description (period, frequency) and the phase description (angular frequency). In the equation of SHM, $x(t) = A \cos(\omega t + \phi)$, the argument $\omega t$ advances by $2\pi$ radians over one period, which is exactly what makes the cosine repeat.

The Three-Way Relationship

The three quantities are not independent — knowing any one gives you the other two. The complete set of relations is:

Worked Example: A Simple Pendulum

Consider a simple pendulum of length $1.0$ m at a place where $g = 9.8$ m/s². The period of a simple pendulum (for small amplitudes) is

$T = 2\pi \sqrt{\frac{l}{g}}$ …