Radians and Angular Velocity

The radian — the angle subtended at the centre of a circle by an arc equal in length to the radius.

Because the circumference is 2πr, a complete revolution is 2π radians:

2π rad = 360°

so 1 rad ≈ 57.3°. Converting is a matter of multiplying by π/180 or 180/π, and a calculator left in the wrong mode is a silent and very common source of wrong answers.

Arc length —

s = rθ

with θ in radians. This simple relationship is the reason radians are used at all: in degrees the same formula would carry an awkward conversion factor.

Angular velocity (ω) — the rate at which angle is swept out:

ω = Δθ / Δt

measured in rad s⁻¹. For one complete revolution in period T:

ω = 2π / T = 2πf

Linking angular and linear speed — a point at radius r travels an arc s = rθ, so dividing by time gives

v = ωr

Example: a record turning at one revolution every 0.50 s has ω = 2π ÷ 0.50 = 12.6 rad s⁻¹. A point 0.15 m from the centre moves at v = 12.6 × 0.15 = 1.9 m s⁻¹, while a point at the edge moves faster — same ω, different v.

The distinction that matters — every point on a rotating rigid body has the same angular velocity but a different linear speed, increasing with distance from the axis. A child at the rim of a roundabout and one near the middle complete each turn together, yet the one at the rim is moving much faster.

Frequency and period — f = 1/T as always, so a higher frequency means a larger ω. Questions often give revolutions per minute, which must be converted to revolutions per second before use.