Table of Contents
What Is Angular Frequency?
Angular frequency (omega) is a measure of how fast something oscillates or rotates, expressed in radians per second (rad/s). It equals 2 pi times the ordinary frequency in Hertz. While frequency counts complete cycles per second, angular frequency measures the rate of change of the phase angle.
Angular frequency appears throughout physics: in simple harmonic motion (pendulums, springs), electromagnetic waves, AC circuits, quantum mechanics (Schrodinger equation), and signal processing. It simplifies many formulas by eliminating factors of 2 pi that would otherwise appear repeatedly.
Angular Frequency Formulas
Where f is frequency in Hz, T is period in seconds, k is spring constant, m is mass, g is gravitational acceleration, and L is pendulum length.
Common Angular Frequencies
| System | Frequency | ω (rad/s) |
|---|---|---|
| AC mains (60 Hz) | 60 Hz | 376.99 |
| AC mains (50 Hz) | 50 Hz | 314.16 |
| Middle C (music) | 261.6 Hz | 1,643.4 |
| 1 second pendulum | 0.5 Hz | 3.14 |
| Earth rotation | 1.16e-5 Hz | 7.27e-5 |
FAQ
What is the difference between frequency and angular frequency?
Frequency f counts complete cycles per second (measured in Hertz). Angular frequency omega counts radians per second, where one complete cycle is 2 pi radians. They are related by omega = 2 pi f. Angular frequency is preferred in physics equations because it eliminates 2 pi factors.
Why is omega used instead of f?
Many physical equations become simpler with omega. For example, the energy of a photon is E = hbar*omega (instead of E = hf). The phase of an oscillation is phi = omega*t (instead of phi = 2*pi*f*t). Using omega reduces clutter in equations involving trigonometric functions.
What is resonance frequency?
Every oscillating system has a natural angular frequency at which it vibrates most easily. When driven at this frequency, the amplitude of oscillation reaches a maximum. This is resonance, which can be constructive (tuning a radio) or destructive (bridge collapse from wind-induced vibrations).