An object moving in a circle at constant speed isn’t moving at “constant velocity” at all — its direction keeps changing. To describe that motion cleanly we measure angles in radians and track how fast that angle is sweeping round: the angular speed.
📘 What you need to know
A radian is the angle where the arc length equals the radius; a full circle is 2π rad (= 360°).
Linear (tangential) speed:v = rω — same ω, but a bigger radius means a bigger linear speed.
In uniform circular motion, speed is constant but velocity keeps changing direction — so the object is still accelerating.
Why radians?
Degrees work, but radians link an angle directly to a length, which makes circular-motion formulas much simpler. One radian is defined as the angle at the centre of a circle subtended by an arc equal in length to the radius:
Definition of the radian
Δθ (rad) = s / r
A full circle has a circumference of 2πr, so going all the way round is an angle of 2πr / r = 2π radians — the same 360° you already know.
The radian is defined geometrically: when the arc length equals the radius, the angle is exactly 1 rad.
Degrees ⇄ radians: multiply degrees by π/180 to get radians; multiply radians by 180/π to get degrees. 360° = 2π rad, 180° = π rad, 90° = π/2 rad.
Angular displacement and angular speed
For an object moving round a circle, the angular displacement Δθ is the angle it has swept through, in radians. The angular speedω is how fast that angle is changing:
Angular speedω = Δθ / Δt = 2πf = 2π / T
where T is the time period (time for one full revolution) and f is the frequency. Notice angular speed doesn’t depend on the radius at all — a point near the centre and a point near the edge of a spinning disc both sweep through the same angle in the same time.
From angular speed to linear speed
The linear (tangential) speed — how fast the object is actually moving through space — does depend on the radius. A point further from the centre has further to travel to complete the same angle, so it must move faster:
Linear speedv = rω
Both points sweep the same angle in the same time (same ω), but the outer point covers a longer arc — a higher linear speed.
Quick reference: Δθ = s/r • ω = Δθ/Δt = 2πf = 2π/T • v = rω. Angular speed is the same everywhere on a rotating object; linear speed grows with radius.
Worked examples
WE 1
Converting an angle to degrees
Convert an angular displacement of π/3 rad into degrees.
A bird flies in a horizontal circle of radius 650 m with an angular speed of 5.25 rad s⁻¹. Find (a) its linear speed and (b) the frequency of one full circle.
(a) v = rωv = 650 × 5.25v ≈ 3410 m s⁻¹(b) ω = 2πf → f = ω / 2πf = 5.25 ÷ 2πf ≈ 0.836 Hz
WE 3
Period of a turning wheel
A wheel completes one full rotation every 0.40 s. Calculate its angular speed.
Use ω = 2π / Tω = 2π ÷ 0.40ω ≈ 15.7 rad s⁻¹
💡 Top tips
Always work in radians for circular motion formulas — set your calculator to RAD before using sin/cos in this context too.
T is the time for one full revolution, not an arbitrary time interval.
ω is the same for every point on a rigid rotating object, no matter how far from the centre — only v changes with r.
Remember the constant-speed paradox: uniform circular motion has constant speed but changing velocity, so it’s still accelerating.
⚠ Common mistakes
Mixing degrees and radians in the same calculation — convert first.
Using v = rω with the wrong radius — make sure it’s the actual radius of that point’s circular path.
Thinking constant speed means constant velocity in a circle — direction is always changing.
Confusing T (period) and f (frequency) — they’re reciprocals: f = 1/T.
Because the velocity direction keeps changing in a circle, there must be an acceleration — and therefore a force — pointing towards the centre. Up next: Centripetal Force, where we find out exactly what supplies that pull.
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