Angular Frequency Calculator
Find angular frequency ω (radians per second) from an oscillation period, ordinary frequency (Hz), the parameters of a spring–mass system (ω = √(k/m)), or a simple pendulum (ω ≈ √(g/L)).
Input method
s
ω = 2π / T = 2π·f — radians swept per second
- 1
Full circle
6.28319One oscillation sweeps 2π radians. - 2
Angular frequency ω
2π ÷ 2 = 3.1416
How does this calculator work?
Angular frequency ω (rad/s) = 2π/T = 2π·f. For a spring–mass oscillator ω = √(k/m); for a simple pendulum ω ≈ √(g/L) where g = 9.807 m/s². ω and ordinary frequency f are related by ω = 2π·f, and period T = 2π/ω.
Formula
How this is calculated
Angular frequency ω is the rate at which the phase of an oscillation advances, measured in radians per second. It is related to ordinary frequency f (Hz) by ω = 2π·f and to period T by ω = 2π / T. Because one full oscillation sweeps 2π radians, ω and f carry the same physical information with a factor of 2π between them.
For a simple harmonic oscillator (SHM), ω is determined by the restoring-force constant. For a spring–mass system: ω = √(k / m), where k (N/m) is the spring constant and m (kg) is the attached mass. A stiffer spring or a lighter mass produces a higher ω. For a simple pendulum (point mass on a massless string, small oscillation angles under about 15°): ω ≈ √(g / L), where g ≈ 9.807 m/s² and L is the length from pivot to centre of mass. The pendulum formula is an approximation that becomes exact only at zero amplitude.
The calculator uses standard gravity g = 9.80665 m/s² for the pendulum. Both system formulas assume ideal conditions — a massless spring/string, no damping, and small angles for the pendulum. Real systems add damping and non-linearity that shift the resonant frequency slightly.
Frequently asked questions
Ordinary frequency f (Hz) counts full oscillations per second; angular frequency ω (rad/s) measures radians per second. They differ by a factor of 2π: ω = 2π·f. Angular frequency is more convenient in wave equations and SHM mathematics because phase advances linearly with ω·t.
Newton's second law for a mass on a spring gives m·a = −k·x, which is the SHM equation ẍ = −ω²·x with ω² = k/m. Taking the square root gives ω = √(k/m). A stiffer spring (larger k) or lighter mass (smaller m) both increase ω and reduce the period.
No — ω ≈ √(g/L) is the small-angle approximation, accurate to within 1% for swings below about 14° from vertical. For larger amplitudes the true angular frequency is lower (the period is longer), requiring elliptic integrals for the exact result.
TG we-Calculate Editorial Team. (2026). Angular Frequency Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/angular-frequency-calculator
TG we-Calculate Editorial Team. "Angular Frequency Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/angular-frequency-calculator.
TG we-Calculate Editorial Team, "Angular Frequency Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/angular-frequency-calculator
@misc{wecalculate_angular_frequency_calculator, title = {Angular Frequency Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/angular-frequency-calculator}}, year = {2026}, note = {TG we-Calculate} }
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