Frequency and Period: T = 1 / f

Quick Answer

Frequency and period are two views of the same oscillation, and they are reciprocals: T = 1 / f. Frequency (f) counts how many cycles happen each second, measured in hertz. Period (T) is the time one cycle takes, measured in seconds. For example, a 2 Hz oscillation repeats twice a second, so its period of oscillation is 1 / 2 = 0.5 seconds. Open the simulator, drag the frequency slider or tap a beat, and watch the period and angular frequency update live.

Frequency vs Period

Frequency and period describe the same repeating motion from opposite ends. Frequency answers "how many cycles per second," and period answers "how many seconds per cycle." Because one is cycles-per-second and the other is seconds-per-cycle, multiplying them always gives one: f times T = 1. That is why the relationship between frequency and period is a reciprocal, not a proportion. Raise the frequency and the period shrinks; lower it and the period grows. They move in opposite directions, locked together by T = 1 / f.

How It Works

The canvas shows an oscillator repeating its cycle at the frequency you set. A readout converts that frequency to period directly through T = 1 / f, so you never do the arithmetic by hand. To convert frequency to period yourself, divide one by the frequency in hertz. The tap-the-beat input works the other way: it times the gap between your taps, reads that as a period, and reports the matching frequency, which is how a metronome tempo in cycles per second is found from real timing.

What Is Angular Frequency?

Angular frequency, written omega, measures the same oscillation in radians per second instead of cycles per second. One full cycle is 2 pi radians, so omega = 2 pi f. To find angular frequency from frequency, multiply the hertz value by 2 pi. For example, a 1 Hz signal has an angular frequency of about 6.28 rad/s. Frequency vs angular frequency is simply a change of units: ordinary frequency counts whole cycles, while angular frequency counts the radians swept out, which is the form that appears in the equations for simple harmonic motion.

Examples

For example, European mains runs at 50 Hz, so the period is 1 / 50 = 0.02 seconds, or 20 milliseconds per cycle, which is why a 50 Hz hum feels continuous. For example, a resting heart rate of 60 beats per minute is 1 Hz, so the period is 1 second between beats; double the rate to 120 bpm (2 Hz) and the period halves to 0.5 seconds. For example, tap the beat at a steady tempo and the tool reports both the frequency in hertz and the period in seconds, turning a musical tempo into physics units.

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