Angular Frequency Calculator
Angular frequency (omega, symbol omega) is the rate at which an object rotates or oscillates, measured in radians per second. Enter a frequency, period, RPM, angular displacement over time, or a spring-mass setup and get omega instantly, plus the reverse conversion back to each input form. The step-by-step panel shows the exact arithmetic so you can follow every calculation.
Formula
Worked example
A spring with k = 100 N/m and a 0.25 kg mass: omega = sqrt(100 / 0.25) = sqrt(400) = 20 rad/s. That gives f = 20 / (2*pi) ~3.18 Hz and T ~0.314 s.
What is angular frequency?
Angular frequency, written as the Greek letter omega (omega), describes how quickly an object rotates or oscillates in terms of radians per second. Whereas ordinary frequency f counts full cycles per second (hertz), angular frequency wraps those cycles into the radian measure of a circle: since one full cycle sweeps 2*pi radians, omega = 2*pi*f. Angular frequency is the natural language of wave physics, electrical engineering, and mechanical vibration, because differential equations for oscillating systems factor cleanly around omega rather than f.
The five formulas this calculator uses
Depending on what you already know, any of five equivalent routes reaches omega. (1) From ordinary frequency: omega = 2*pi*f. (2) From the period T: omega = 2*pi/T. (3) From rotational speed in RPM: omega = 2*pi * (RPM/60), because RPM/60 converts to revolutions per second and then each revolution contributes 2*pi radians. (4) From raw angular displacement delta-theta in time delta-t: omega = delta-theta / delta-t - this is the most fundamental definition. (5) For a spring-mass system: omega = sqrt(k/m), where k is the spring constant in N/m and m is the oscillating mass in kg. This last formula, derived from Hooke's law and Newton's second law, gives the natural angular frequency at which the system will oscillate freely.
Angular frequency in electrical circuits and waves
In AC electrical circuits, angular frequency appears in the impedance of inductors (Z = omega*L) and capacitors (Z = 1/(omega*C)), so it directly controls how much each component resists alternating current. The resonant frequency of an LC circuit - the frequency at which inductive and capacitive reactances cancel - is omega = 1/sqrt(LC). In wave physics, angular frequency appears in the wave equation as the product omega*t in the argument of a sine or cosine: y = A*sin(omega*t + phi). The ratio of angular frequency to wave number (radians per metre) gives the phase velocity of the wave.
How angular frequency relates to angular velocity
Angular frequency and angular velocity both carry the unit rad/s, and the symbols often overlap in textbooks. For uniform circular motion or steady oscillation, the two are identical: omega is simultaneously the rate of angle change and the radian frequency of the repeating pattern. The distinction appears only in non-uniform rotation: angular velocity can vary over time, while "angular frequency" conventionally refers to the constant rate in simple harmonic motion or steady-state AC signals. The calculator here computes the steady-state value; for time-varying rotation, you would need calculus-based angular kinematics instead.
Common angular frequency reference values
| System | Frequency (Hz) | Angular frequency (rad/s) | Period (s) |
|---|---|---|---|
| Mains electricity (US/Canada) | 60 | 376.99 | 0.0167 |
| Mains electricity (Europe/Asia) | 50 | 314.16 | 0.0200 |
| Middle C (musical note) | 261.6 | 1643 | 0.00382 |
| Human heart at rest | 1.17 | 7.33 | 0.857 |
| Car engine at idle (~800 RPM) | 13.3 | 83.8 | 0.075 |
| Earth rotation (one day) | 0.0000116 | 0.0000727 | 86400 |
| Pendulum clock (1 s half-swing) | 0.5 | 3.14 | 2.0 |
| Typical loudspeaker woofer | 80 | 502.7 | 0.0125 |
Approximate angular frequencies for familiar physical systems. Values are illustrative; real systems vary.
Frequently asked questions
What is the unit of angular frequency?
Radians per second (rad/s) is the SI unit. Because a radian is dimensionless, rad/s is dimensionally equivalent to s^-1 (per second), but the "rad" label is kept to avoid confusion with ordinary frequency in hertz, which also uses s^-1 as its formal unit.
How is angular frequency different from frequency?
Ordinary frequency f (in Hz) counts cycles per second. Angular frequency omega (in rad/s) counts radians per second. The link is omega = 2*pi*f, because one cycle covers 2*pi radians. Angular frequency is preferred in physics and engineering because oscillation equations - such as y = A*sin(omega*t) - are simpler with omega than with 2*pi*f written out every time.
How do I convert RPM to rad/s?
Divide RPM by 60 to get revolutions per second, then multiply by 2*pi to get radians per second. The formula is omega = (RPM / 60) * 2*pi. For example, 1200 RPM gives (1200/60)*2*pi = 20*pi ~62.83 rad/s. This calculator does this conversion automatically when you select the RPM mode.
What is the natural angular frequency of a spring-mass system?
It is omega = sqrt(k/m), where k is the spring constant in N/m and m is the attached mass in kg. This result comes from applying Newton's second law to a mass on a spring: the restoring force F = -kx produces acceleration a = -k/m * x, which matches the harmonic oscillator equation with omega^2 = k/m. The system oscillates freely at this frequency in the absence of damping.
Can angular frequency be negative?
Mathematically, a negative omega corresponds to rotation in the opposite direction (clockwise vs. counterclockwise). In most engineering contexts, omega is taken as a positive magnitude and direction is handled separately. In complex phasor notation used in electrical engineering, negative angular frequency can appear as a mirror image of a positive-frequency sinusoid in the Fourier spectrum.
How do I find the period from angular frequency?
T = 2*pi / omega. Since omega = 2*pi/T by definition, solving for T just reverses the relationship. This calculator shows the derived period automatically in the results for any input mode.