So far every kind of motion you’ve met has travelled in a straight line. Now we turn a corner — literally. When something goes round in a circle, its ordinary (linear) speed only tells half the story, because its direction is changing the whole time. To describe circular motion properly we need a new idea: how fast the object is sweeping round the centre. That’s angular velocity, and it comes with its own natural unit for measuring angles — the radian.
📘 What you need to know
An object in uniform circular motion moves at constant speed but constantly changing velocity, because its direction keeps changing
A radian is the angle at the centre of a circle where the arc length equals the radius; a full circle is 2π radians
Angular displacementθ = arc length ÷ radius = s/r
Angular velocityω is the rate of change of angular displacement, measured in rad s−1
Linear and angular speed are linked by v = rω
For one full turn: ω = 2π/T = 2πf, where T is the period and f the frequency
Why circular motion needs a new idea
Imagine an ice puck sliding across a perfectly smooth rink. With no resultant force, Newton’s first law says it keeps going in a straight line at constant velocity — forever. Now tie a string from the puck to a fixed pin in the ice. As the puck moves, the string pulls it towards the pin, and instead of shooting off in a straight line it curves round into a circle.
The velocity (red) always points along the tangent — at right angles to the radius. Its size stays the same, but its direction changes at every point. That’s why circular motion needs its own way to measure “how fast it goes round”.
The string’s pull always acts at 90° to the puck’s motion, so it never speeds the puck up or slows it down — it only bends its path. The size of the velocity stays constant while its direction continuously changes. We call this uniform circular motion: uniform speed, but not uniform velocity.
This trips people up every year, so let’s be clear: “constant speed” and “constant velocity” are not the same thing. Speed is just a number; velocity is a number with a direction. Going round a roundabout at a steady 30 km/h, your speed is constant but your velocity is changing every instant, because you’re always turning. And a changing velocity means you’re accelerating — even though the speedometer never moves.
Measuring angles in radians
Degrees are fine for everyday life, but for circular motion physicists use radians instead. The reason is simple: radians link the angle directly to the distance travelled around the circle, which makes every equation cleaner.
Here’s the definition. Take a circle and lay the radius along the curved edge (the arc). The angle you sweep out at the centre while the arc length equals one radius is one radian.
One radian is the angle at the centre when the arc length equals the radius. Because a full circle’s arc (its circumference) is 2πr, a whole turn is 2π radians — about 6.28.
Since the circumference of a circle is 2πr and each radian “uses up” one radius of arc, a full circle contains 2π radians. That gives you the conversions:
A few worth knowing by heart: a full turn 360° = 2π rad, a half turn 180° = π rad, and a right angle 90° = π/2 rad.
Angular displacement
Instead of measuring how far around the circle an object has travelled in metres, in circular motion we measure the angle it has swept out at the centre. This is the angular displacementθ, and it follows straight from the radian definition:
Angular displacementθ = s ÷ r
where s is the arc length (the distance travelled around the circle, in metres) and r is the radius (m). Because you’re dividing a length by a length, θ comes out as a pure number — that’s why the radian has no units of its own.
WE 1
The tip of a fan blade is 0.30 m from the centre. As the blade turns, the tip sweeps through an arc of 0.45 m. Find the angular displacement in radians, and convert it to degrees.
Step 1 — use θ = s ÷ rθ = 0.45 ÷ 0.30θ = 1.5 radStep 2 — convert to degrees (× 180/π)1.5 × 180 ÷ π = 85.9°Just under a right-and-a-half; radians are bigger “steps” than degrees, so the number is smaller.
Angular velocity
Now we can define the star of the show. Angular velocityω (the Greek letter omega) is how fast the angular displacement is changing — how quickly the object sweeps round the centre:
Angular velocityω = Δθ ÷ Δt
Its unit is radians per second (rad s−1). One full turn is 2π radians and takes one period T, so for anything going round and round:
Angular velocity, period and frequencyω = 2π ÷ T = 2πf
where T is the time for one complete revolution (s) and f is the frequency — the number of revolutions per second (Hz). These are the same T and f you’ll meet again in waves and simple harmonic motion.
A quick way to keep ω straight in your head: it’s just “angle per second” the same way ordinary speed is “distance per second”. Swap metres for radians and you’ve basically got the same idea, only measured around a circle instead of along a line.
Linking angular and linear speed
The object still has an ordinary linear speed v as it moves along the circle. How does that connect to ω? Start from θ = s/r, so the arc length s = rθ. Divide both sides by time and the “angle per second” on the right becomes ω:
Linear and angular speedv = rω
This little equation carries a big idea. For a fixed angular velocity, the linear speed grows with radius: points further out have to cover more distance in the same time, so they move faster. Everyone on a spinning disc shares the same ω, but the rim races along while the centre barely moves.
P and Q sit on the same disc, so they share the same angular velocity ω. But Q is further out, so its linear speed v = rω is larger — shown by the longer red arrow.
Putting the pieces together, the linear speed can also be written in terms of the period, which is often the handiest form:
Linear speed from periodv = 2πr ÷ T
🛠️ Working with angular velocity
Check your angle units. Anything going into ω or a circular-motion equation must be in radians, not degrees.
Spot what you’re given — a period T, a frequency f, or an angular velocity ω? They’re all linked by ω = 2π/T = 2πf.
Need a linear speed? Reach for v = rω (or v = 2πr/T).
State the units clearly: rad s−1 for ω, m s−1 for v, Hz for f.
WE 2
A child sits 1.8 m from the centre of a playground roundabout that completes one full turn every 4.0 s. Find (a) the angular velocity and (b) the child’s linear speed.
Part (a) — angular velocity from the period
ω = 2π ÷ T
ω = 2π ÷ 4.0ω = 1.6 rad s⁻¹Part (b) — linear speed
v = rω
v = 1.8 × 1.571 = 2.83v = 2.8 m s⁻¹Check: v = 2πr/T = 2π(1.8)/4.0 = 2.8 m s⁻¹. Same answer.
WE 3
A disc spins at a steady angular velocity of 8.0 rad s−1. Point P is 2.0 cm from the centre and point Q is 5.5 cm from the centre. Find the linear speed of each point.
Step 1 — both points share the same ω
use v = rω with r in metres
Step 2 — point P (r = 0.020 m)v = 0.020 × 8.0 = 0.16 m s⁻¹Step 3 — point Q (r = 0.055 m)v = 0.055 × 8.0 = 0.44 m s⁻¹vP = 0.16 m s⁻¹, vQ = 0.44 m s⁻¹Same ω, but Q moves 2.75× faster — exactly the ratio of the radii.
💡 Top tips
Radians, always. Put your calculator in radian mode for circular motion, and convert any angle given in degrees before using it.
Convert cm to metres before using v = rω — a radius in centimetres is a classic slip.
T is one full revolution. Don’t confuse the period with the time for a fraction of a turn.
Same disc, same ω. Every point on a rigid rotating object shares one angular velocity, even though their linear speeds differ.
Quick recap: Circular motion needs radians (arc = radius = 1 rad; a full turn = 2π). Angular velocity ω = Δθ/Δt measures how fast you sweep round the centre, and links to linear speed by v = rω. For a full turn, ω = 2π/T = 2πf.
⚠ Common mistakes
Leaving the calculator in degree mode when the physics needs radians
Thinking constant speed means constant velocity — in a circle the direction (and so the velocity) always changes
Forgetting to convert a radius from cm to m before applying v = rω
Mixing up period and frequency — remember f = 1/T
Assuming two points on the same spinning object have the same linear speed — they share ω, not v
We’ve said the object is always accelerating because its direction keeps changing — but we haven’t yet worked out how big that acceleration is or which way it points. That’s next: Centripetal Acceleration, the acceleration that’s forever aimed at the centre of the circle, and the force that has to be there to cause it.
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