angular frequency

Angular frequency is a way of measuring how fast an oscillation cycles, but counted in radians per second instead of in cycles per second. It comes from the deep link between oscillation and rotation: imagine a point going steadily around a circle whose shadow traces the oscillation. Angular frequency is how fast that imaginary point sweeps around.

Precisely, angular frequency omega equals 2 pi times the ordinary frequency, omega = 2 pi f, and equivalently omega = 2 pi / T where T is the period. Its unit is radians per second (rad/s). For a mass-spring system omega = sqrt(k/m), and for a simple pendulum omega = sqrt(g/L). It is the omega that appears inside the sinusoid x(t) = A cos(omega t + phi), telling you how quickly the angle inside the cosine grows.

Physicists prefer omega because it makes the mathematics clean: the rate of change of cos(omega t) is -omega sin(omega t), with the 2 pi already absorbed, so calculus and wave formulas come out tidy. The one thing to keep straight: omega is not the same number as the frequency f. A full cycle is 2 pi radians, so omega is always 2 pi times bigger than f.