What is the Period and Frequency Calculator?
This calculator converts between two fundamental properties of any repeating motion or wave: the period (T) and the frequency (f). The period is the time it takes to complete one full cycle, measured in seconds. The frequency is how many cycles happen each second, measured in hertz (Hz). They are exact reciprocals of one another, so knowing one immediately gives you the other.
How to use it
Pick whether you want to find the period or the frequency. If you choose Period (from frequency), enter the frequency in hertz and the tool returns the period in seconds. If you choose Frequency (from period), enter the period in seconds and it returns the frequency in hertz. The result panel also shows the angular frequency \(\omega = 2\pi f\) in radians per second, which is handy for oscillation and simple harmonic motion problems.
The formula explained
The core relationship is $$T = \frac{1}{f}$$ and equivalently $$f = \frac{1}{T}$$ Because they are reciprocals, doubling the frequency halves the period. The angular frequency adds a factor of \(2\pi\) to express the rate in radians: $$\omega = 2\pi f = \frac{2\pi}{T}$$
Worked example
A musical note A4 vibrates at 440 Hz. Its period is $$T = \frac{1}{440} \approx 0.002273 \text{ seconds}$$ or about 2.27 milliseconds. Conversely, if a pendulum has a period of 2 seconds, its frequency is $$f = \frac{1}{2} = 0.5 \text{ Hz}$$ and its angular frequency is $$\omega = 2\pi \times 0.5 \approx 3.1416 \text{ rad/s}$$
FAQ
What units does this use? Frequency is in hertz (cycles per second) and period is in seconds. For kilohertz multiply Hz by 1000; for milliseconds divide seconds by 1000.
Can frequency be zero? No. A frequency of zero would mean an infinite period (no oscillation), so the tool guards against division by zero.
What is angular frequency for? Angular frequency \(\omega\) appears in trigonometric descriptions of waves like \(x(t) = A\cdot\sin(\omega t)\), making the math of oscillations cleaner.
