natural (angular) frequency

Pluck a wine glass, a guitar string, or a child's swing and each rings at its own favourite rate — a pitch it 'wants' to vibrate at. That preferred rate is the natural frequency. It is built into the object by its stiffness and its mass, not by how hard you hit it: a tighter or lighter string sings higher, a looser or heavier one sings lower.

For the undamped spring-mass system m x'' + k x = 0, the natural angular frequency is omega_0 = sqrt(k/m), measured in radians per second. Stiffer spring (bigger k) means faster ringing; heavier mass (bigger m) means slower. From omega_0 you get the everyday frequency f = omega_0 / (2 pi) in cycles per second (hertz) and the period T = 2 pi / omega_0, the time for one full swing. The word 'angular' just means we count in radians (one full cycle is 2 pi) rather than in whole cycles.

The natural frequency is the single most important number of a vibrating system, because resonance happens when an outside push arrives near omega_0 — that is when small repeated nudges can build enormous swings. Engineers compute omega_0 for bridges, aircraft wings, and engine mounts precisely so they can keep everyday driving forces away from it. When damping is present the system actually rings slightly slower than omega_0, at omega_d = omega_0 sqrt(1 - zeta^2), but omega_0 remains the reference point everything is quoted against.