Angular Frequency Calculator
Convert between frequency (Hz) and angular frequency (rad/s) via ω = 2π·f. Includes period T = 1/f, a live rotating phasor diagram, and reference values from DC through GHz.
Input
Result
Reference: f, ω, T at common frequencies
| Source | f (Hz) | ω (rad/s) | T |
|---|---|---|---|
| Tidal cycle | ~1.2 × 10⁻⁵ | ~7.3 × 10⁻⁵ | ~12.4 hr |
| Earth rotation | ~1.16 × 10⁻⁵ | ~7.27 × 10⁻⁵ | ~24 hr |
| Heart rate (60 bpm) | 1 | 6.283 | 1 s |
| Sub-audible vibration | 10 | 62.83 | 0.1 s |
| Mains hum (Europe) | 50 | 314.16 | 20 ms |
| Mains hum (US) | 60 | 376.99 | 16.67 ms |
| A4 (concert pitch) | 440 | 2,765.5 | 2.273 ms |
| 1 kHz audio reference | 1,000 | 6,283.2 | 1 ms |
| 10 kHz (treble) | 10,000 | 62,832 | 0.1 ms |
| 20 kHz (top of hearing) | 20,000 | 125,664 | 50 µs |
| AM radio (1 MHz) | 1 × 10⁶ | 6.283 × 10⁶ | 1 µs |
| FM radio (100 MHz) | 1 × 10⁸ | 6.283 × 10⁸ | 10 ns |
| WiFi (2.4 GHz) | 2.4 × 10⁹ | 1.508 × 10¹⁰ | ~417 ps |
| Visible light (~600 THz) | 6 × 10¹⁴ | 3.77 × 10¹⁵ | ~1.67 fs |
About Angular Frequency
Angular frequency ω (omega) measures how fast a phase angle is changing — in radians per second. It's what you get when you imagine a sinusoidal signal as a rotating vector (phasor) on a 2D plane: the rate of rotation is ω. For a signal that completes f full cycles per second, the rotation rate is ω = 2π · f — because one full cycle is 2π radians. Angular frequency appears throughout physics and engineering: in simple harmonic motion (mass-spring systems, pendulums), in AC circuit analysis (impedance Z = jωL for inductors), and in signal processing (the Fourier transform variable).
Why physicists prefer ω over f
Calculus is cleaner with ω. The derivative of sin(ωt) is ω · cos(ωt), while the derivative of sin(2π·f·t) is 2π·f · cos(2π·f·t) — same answer, less notation. Differential equations, Laplace transforms, and electromagnetic theory all use ω natively. In particular, Euler's formula e^(jωt) = cos(ωt) + j·sin(ωt) elegantly encodes both the cosine and sine components of a sinusoid into a single complex exponential — making Fourier analysis far more compact. The cost: ω feels less intuitive ("314 rad/s" vs "50 Hz") for everyday work.
Why engineers prefer f over ω
Hertz directly maps to countable events: cycles per second. Measurement instruments report Hz. Specifications (audio frequency response, radio bands, clock speeds) all use Hz. Engineers do most of the conversion to/from ω only when crossing into theoretical analysis — for example, computing the resonant frequency of an LC circuit: ω₀ = 1/√(LC), then f₀ = ω₀ / (2π). To explore how frequency relates to wavelength and speed, see the frequency to period converter for T = 1/f.
The phasor picture
A real sinusoidal signal A·cos(ωt + φ) is the projection of a rotating vector onto the real axis. The vector has length A (amplitude), starts at angle φ (initial phase), and rotates counterclockwise at ω rad/s. After time t, the angle is ωt + φ. The phasor diagram on this page shows exactly that — a unit-length arrow rotating at the current ω (slowed down for visibility at high frequencies). Phasor analysis is the foundation of AC power calculations, filter design, and impedance matching in RF engineering.
Period vs frequency vs angular frequency
The three quantities are reciprocals/multiples of each other:
T = 1/f = 2π/ω
At 50 Hz mains: T = 20 ms, ω = 314.16 rad/s. At a quartz crystal's 32,768 Hz: T = 30.52 µs, ω = 205,887 rad/s. At A4 concert pitch (440 Hz): ω ≈ 2,765.5 rad/s. Pick the unit that matches your domain — and use the octave frequency calculator when you need to step through musical intervals in Hz.
Frequently Asked Questions
Why exactly 2π and not some other constant?
Is angular frequency the same as angular velocity?
Why does the phasor slow down at high frequencies?
What does ω·t in a sine wave mean?
sin(ωt) projects this angle onto the imaginary axis to give the wave's amplitude at time t. After 1 second, the phasor has rotated by ω radians; after 1/f seconds (one period), by exactly 2π radians (one full cycle). Phase is the "where in the cycle" coordinate; angular frequency is its rate of change.Why use radians instead of degrees?
sin(x) is cos(x) only when x is in radians. In degrees you'd get cos(x) · π/180. Similarly the Taylor series for sin/cos is simple only in radians. Engineering communicates with degrees (intuitive); theory uses radians (clean math). This tool bridges the two — input/output in Hz or rad/s, with degrees only shown for the phasor angle indicator.