Angular frequency ω vs angular velocity
Same units (rad s⁻¹), different meanings. Angular velocity = rate of change of angular displacement of something actually rotating. Angular frequency ω = 2πf — rate the oscillation moves through phase; an SHM particle moves in a line, nothing rotates. Only for the circular-motion projection are they numerically the same thing.
What this actually means
Same unit, different job. Angular velocity is a real rotation rate: something physically sweeps out angle. Angular frequency is a bookkeeping rate: how fast an oscillation moves through its cycle, 2π per period, with nothing actually turning.
An SHM mass moves along a straight line, so it has no angular velocity at all, yet ω appears in every one of its equations because phase is measured as an angle.
The two coincide numerically only in the circular-motion projection picture, where the turntable's angular velocity equals the shadow's angular frequency.
Prove it — watch it be true
- The peg on the turntable really rotates: that rate is angular velocity
- Its shadow moves on a straight line: nothing rotates, yet it oscillates with ω
- Same number, two meanings — the projection is where they meet