Angular Frequency Calculator
Angular frequency measures how fast something rotates or oscillates, expressed in radians per second (rad/s) rather than the cycles-per-second of ordinary frequency (Hz). This calculator converts freely between all three ways of describing the same motion: angular frequency, frequency and period, using the standard relationship omega = 2 pi f = 2 pi / T. Choose which quantity you want to start from, then enter a frequency in hertz, a period in seconds, or an angular frequency in rad/s directly, or pick a preset such as 50 Hz mains power, 60 Hz mains power, 440 Hz concert pitch, or a 100 rad/s mechanical shaft. The calculator instantly returns the angular frequency, ordinary frequency and period together, plus a breakdown showing revolutions per minute, revolutions per second, and a check value confirming omega divided by 2 pi matches your frequency. A step-by-step working panel shows how each figure was derived from your input, so you can follow the maths rather than just reading a number. Use it to convert a known frequency or rotational speed into the angular frequency needed for formulas such as reactance in AC circuits, simple harmonic motion, or wave equations, or to work back from an angular frequency to a practical frequency or period. Results are exact, calculated directly from the standard formula, and you can adjust the decimal precision shown from two to six places.
1. Input Mode
2. Common Presets
Full Conversion
Step-by-Step Working
Worked Example: Default values (f = 1 Hz)
Given: Ordinary frequency f = 1 Hz (one full oscillation per second).
Step 1: Apply the formula: omega = 2 * pi * f = 2 * 3.14159265... * 1 = 6.2832 rad/s
Step 2: Find the period: T = 1 / f = 1 / 1 = 1.0000 s
Result: Angular frequency = 6.2832 rad/s, period = 1.0000 s, frequency = 1.0000 Hz.
This matches the calculator output above with the default inputs.
What Is Angular Frequency?
Angular frequency (symbol omega, lowercase Greek letter) describes how quickly an object rotates or oscillates, measured in radians per second (rad/s). It is the rate of change of the phase angle of a sinusoidal waveform or rotating object.
While ordinary frequency (f) counts complete cycles per second, angular frequency counts radians traversed per second. Because one complete cycle spans 2 pi radians (approximately 6.2832 radians), the two are linked by the simple relationship:
omega = 2 pi f
Angular frequency appears throughout physics and engineering: in describing alternating current circuits, analysing simple harmonic motion (springs, pendulums), characterising wave propagation, and specifying the speed of rotating machinery.
The Formula
Three equivalent expressions let you move between any pair of quantities:
| To find | Formula | Notes |
|---|---|---|
| Angular frequency (omega) | omega = 2 pi f | f in hertz, omega in rad/s |
| Angular frequency (omega) | omega = 2 pi / T | T is period in seconds |
| Ordinary frequency (f) | f = omega / (2 pi) | Result in hertz |
| Period (T) | T = 2 pi / omega = 1 / f | Result in seconds |
Common Values
| Application | Frequency (f) | Angular Frequency (omega) | Period (T) |
|---|---|---|---|
| NZ/Australia mains power | 50 Hz | 314.16 rad/s | 0.0200 s |
| US/Canada mains power | 60 Hz | 376.99 rad/s | 0.0167 s |
| Concert A (music tuning) | 440 Hz | 2,764.60 rad/s | 0.0023 s |
| One revolution per second | 1 Hz | 6.2832 rad/s | 1.0000 s |
| Clock second hand | 1/60 Hz | 0.1047 rad/s | 60 s |
| Earth's rotation | 1/86400 Hz | 0.0000727 rad/s | 86,400 s |
Applications in Physics and Engineering
Angular frequency is central to the mathematics of oscillating and rotating systems:
- Simple harmonic motion: A spring-mass system oscillates at omega = sqrt(k/m), where k is the spring constant and m is mass. A pendulum oscillates at omega = sqrt(g/L) for small angles.
- AC electrical circuits: Impedance of capacitors and inductors depends on omega: inductive reactance = omega * L, capacitive reactance = 1 / (omega * C). Resonant frequency of an LC circuit occurs when omega = 1 / sqrt(LC).
- Wave physics: The dispersion relation for waves often takes the form omega = v * k, where v is wave speed and k is the wave number (radians per metre).
- Rotating machinery: Motor and engine speeds are often given in rpm (revolutions per minute). Converting to omega enables direct calculation of centripetal acceleration, torque, and power.
- Signal processing: Angular frequency simplifies the Fourier transform and the equations for filters, making it the natural unit for frequency in electronics and control engineering.
Converting RPM to Angular Frequency
Rotating machinery speeds are commonly given in revolutions per minute (rpm). To convert to angular frequency:
omega (rad/s) = rpm * 2 pi / 60
For example, a motor running at 1,500 rpm has an angular frequency of 1,500 * 2 pi / 60 = 157.08 rad/s.
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Sources and method: Formula omega = 2 pi f from Serway and Jewett, Physics for Scientists and Engineers (standard university text). SI unit for angular frequency is rad/s (radian per second). International Bureau of Weights and Measures (BIPM), SI Brochure 9th edition. Halliday, Resnick, and Walker, Fundamentals of Physics.
This calculator computes angular frequency using the exact mathematical relationship omega = 2 pi f. Results are exact within floating-point precision. The formula is standard and does not vary by context, unit system, or country.