angular velocity
Angular velocity tells you how fast something is spinning — the rotational twin of ordinary velocity. A vinyl record turning on a turntable, a fan blade whirring, the Earth completing one turn every day: each has an angular velocity that says how many radians of angle it sweeps out each second. Where linear velocity asks how many metres per second, angular velocity asks how many radians per second.
Precisely, angular velocity is the rate of change of the rotation angle. Its average value is omega = delta-theta / delta-t, and its instantaneous value is the derivative dtheta/dt; the units are radians per second (rad/s). It connects beautifully to how fast a point on the object moves: a point a distance r from the axis has a tangential speed v = r omega, so points farther out move faster even though the whole object shares one omega. For steady spinning, omega ties to the period T (time for one turn) and frequency f by omega = 2 pi / T = 2 pi f.
Angular velocity is really a vector: its magnitude is the spin rate and its direction points along the rotation axis, given by the right-hand rule (curl your right fingers the way it turns, your thumb points along omega). This is why a spinning wheel or gyroscope behaves so stubbornly — its angular velocity has a definite direction in space.
A record spins at 33 1/3 revolutions per minute. Each revolution is 2 pi rad, so omega = 33.33 x 2 pi / 60 = 3.49 rad/s; a groove 0.15 m from the centre moves at v = r omega = 0.15 x 3.49 = 0.52 m/s.
One shared omega, but the outer edge moves faster than the inner because v = r omega grows with r.
Do not confuse rad/s with rpm (revolutions per minute); convert with 1 rev = 2 pi rad and 1 min = 60 s before plugging into physics formulas.