Convert between angular frequency (ω), frequency (f), and period (T) using ω = 2πf. Includes RPM, deg/s, and electromagnetic wavelength outputs.
Angular frequency (ω) measures how fast something oscillates or rotates in radians per second. It is related to ordinary frequency (f) by ω = 2πf and to the period (T) by ω = 2π/T. While frequency counts cycles per second, angular frequency counts radians per second, making it the natural unit for circular and sinusoidal motion.
This calculator converts freely between angular frequency, frequency, and period. Enter any one value and instantly get the other two, along with derived quantities like RPM, degrees per second, and — for electromagnetic waves — the corresponding wavelength and photon energy. Preset buttons cover common scenarios from heartbeats to visible light.
Angular frequency appears throughout physics: in simple harmonic motion (x = A cos(ωt)), in AC circuits (V = V₀ sin(ωt)), in wave equations, and in quantum mechanics (E = ℏω). This calculator makes converting between the various representations quick and error-free. Check the example with realistic values before reporting.
Converting between frequency, angular frequency, and period is straightforward but tedious — especially when you also need RPM, degrees per second, or electromagnetic quantities. This calculator does all conversions instantly and provides a reference table spanning 15 orders of magnitude in frequency.
The photon energy output is a bonus for optics and quantum mechanics problems, connecting oscillation frequency to energy in both joules and electron-volts.
ω = 2πf = 2π/T. f = ω/(2π) = 1/T. T = 1/f = 2π/ω. RPM = f × 60. For EM waves: λ = c/f, E = hf.
Result: 376.99 rad/s
US mains AC at 60 Hz has angular frequency ω = 2π × 60 = 376.99 rad/s, with a period of 16.67 ms.
Angular frequency is a unifying concept across many branches of physics. In classical mechanics, it describes the oscillation rate of springs, pendulums, and rotating bodies. In electrodynamics, it characterizes the oscillation of electromagnetic fields. In quantum mechanics, it connects energy to oscillation through E = ℏω, one of the most profound relationships in physics.
Electrical engineers work with angular frequency daily when analyzing AC circuits. The impedance of capacitors (1/jωC) and inductors (jωL) depends directly on ω, making it the natural variable for frequency-domain analysis. In control theory, transfer functions are expressed in terms of jω, and Bode plots use ω as the horizontal axis.
The electromagnetic spectrum spans a remarkable range of frequencies: from radio waves at a few Hz to gamma rays at over 10²⁰ Hz. Each region has unique properties and applications. Radio and microwave frequencies are used for communication, infrared for thermal imaging, visible light for vision, ultraviolet for sterilization, X-rays for medical imaging, and gamma rays for cancer treatment.
Frequency (f) counts full cycles per second and is measured in Hz. Angular frequency (ω) counts radians per second and equals 2πf. One full cycle = 2π radians, so ω is always 2π times larger than f.
Angular frequency simplifies many physics equations. In SHM, x = A cos(ωt) is cleaner than x = A cos(2πft). In quantum mechanics, E = ℏω uses the reduced Planck constant naturally.
Multiply by 2π. For example, 60 Hz × 2π ≈ 376.99 rad/s.
Visible light ranges from about 2.7 × 10¹⁵ rad/s (red) to 4.7 × 10¹⁵ rad/s (violet), corresponding to frequencies of 430–750 THz.
ω (rad/s) = RPM × 2π/60. So 3600 RPM = 3600 × 2π/60 = 376.99 rad/s (same as 60 Hz).
ℏ = h/(2π) ≈ 1.055 × 10⁻³⁴ J⋅s. It appears naturally with angular frequency: E = ℏω = hf.