Period, Frequency and Angular Frequency

Period T is the time for one full cycle, measured in seconds. Frequency f is how many cycles happen per second, measured in hertz, and it is just the reciprocal of period: f = 1/T. Angular frequency omega measures the same repetition, but in radians per second instead of cycles per second. Since one full cycle corresponds to 2 π radians, omega = 2 π f, and combining with f = 1/T gives T = 2 π / omega. These three numbers all describe the identical motion; they are just different units for the same clock.

Physicists lean on omega because the equations of simple harmonic motion are built from sines and cosines, and those functions take angles, not cycle counts. Writing x(t) = A cos(omega t) makes omega the natural rate variable: it drops straight out of solving the differential equation d^2x/dt² = -omega² x, where omega² equals k/m for a spring or g/L for a pendulum. Using frequency in hertz there would force awkward factors of 2 π into every derivative and every energy calculation.

This sets up everything else in oscillations. Once you know omega for a system, you know its period, its frequency, and the argument inside every position, velocity, and acceleration equation. It also connects simple harmonic motion to uniform circular motion: if an object moves in a circle at constant angular speed omega, the shadow of that object cast onto a line through the circle's center moves back and forth exactly like a mass on a spring with that same omega. That picture is why the word angular survives even though nothing is literally spinning in a mass-spring system.