Physics · Further Mechanics · A2 Level

Circular Motion & Angular Velocity

A point on a spinning CD, a satellite in orbit, and the tip of a clock's second hand are all doing the same kind of motion — moving at constant speed around a circle. That word "constant" hides something strange: even though speed never changes, velocity changes every instant. Untangling that apparent contradiction is where circular motion starts.

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Measuring angles in radians

Degrees are a somewhat arbitrary unit — 360 of them in a circle, for historical reasons connected to Babylonian astronomy. Circular motion is described more naturally using the radian, defined so that an angle of $1$ radian is subtended at the centre of a circle by an arc equal in length to the radius.

$\theta \text{ (in radians)} = \dfrac{\text{arc length}}{\text{radius}} = \dfrac{s}{r}$

Since the full circumference is $2\pi r$, a complete revolution corresponds to $\dfrac{2\pi r}{r}=2\pi$ radians — so $2\pi\,\text{rad}=360°$, and $1\,\text{rad}\approx57.3°$. The advantage of radians isn't just tidiness: equations linking angular and linear quantities (like $v=\omega r$, coming up shortly) only take their simple form in radians — degrees would require an awkward conversion factor every time.

A circle centred at O with a radius r drawn to point A, and an arc from A to point B whose length s is equal to the radius, showing the angle θ between OA and OB is exactly 1 radian. — By definition, 1 radian is the angle at the centre O when the arc AB is exactly as long as the radius r — that's the only fact the whole radian system is built from.

Angular velocity

Just as velocity describes how fast a linear position changes, angular velocity $\omega$ describes how fast an angle is swept out:

$\omega = \dfrac{\Delta\theta}{\Delta t}$

with units of radians per second (rad s⁻¹). For an object moving at constant speed around a circle, one full revolution ($2\pi$ radians) takes one period $T$, so:

$\omega = \dfrac{2\pi}{T} = 2\pi f$

where $f=1/T$ is the frequency of revolution in hertz. This pair of relationships — $\omega=2\pi/T=2\pi f$ — is used constantly throughout circular motion and reappears in oscillations, so it's worth being fluent switching between $\omega$, $T$ and $f$.

Derivationω = 2π/T = 2πf, from angular velocity's own definition

Start from the definition $\omega=\dfrac{\Delta\theta}{\Delta t}$.

For an object moving at constant speed around the circle, one complete revolution sweeps out $\Delta\theta=2\pi$ radians (a full turn), taking a time $\Delta t$ equal to one period, $T$.

The period $T$ is defined as the time for exactly one revolution, so it's the natural choice of $\Delta t$ to substitute here.

Substituting these values into the definition gives $\omega=\dfrac{2\pi}{T}$.

Frequency is defined as the number of revolutions per second, $f=\dfrac{1}{T}$, so $T=\dfrac{1}{f}$. Substituting into $\omega=2\pi/T$: $\omega=\dfrac{2\pi}{1/f}=2\pi f$.

This is why $\omega=2\pi/T=2\pi f$ isn't two separate formulas to memorise — both fall straight out of the same definition, just written in terms of $T$ or $f$.

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Linking angular and linear speed

An object moving around a circle of radius $r$ still has an ordinary speed $v$ — metres travelled per second along the path. That linear speed and the angular velocity are directly related.

Derivationv = ωr, from the radian's own definition

From the definition of the radian, arc length $s=r\theta$ (rearranging $\theta=s/r$).

Divide both sides by the time taken, $\Delta t$: $\dfrac{s}{\Delta t}=r\dfrac{\theta}{\Delta t}$.

The left-hand side, distance travelled over time, is the linear speed $v$. The right-hand side contains $\theta/\Delta t$, which is the angular velocity $\omega$.

So $v = \omega r$ — linear speed equals angular velocity multiplied by radius.

This makes intuitive sense: at fixed $\omega$, doubling the radius doubles the distance travelled per revolution, so it doubles the speed too.

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Why constant speed still means changing velocity

Velocity is a vector — it has both magnitude (speed) and direction. For an object travelling around a circle at constant speed, the magnitude never changes, but its direction changes constantly: velocity in circular motion is always tangential — along the tangent to the circle at the object's current position, perpendicular to the radius.

As the object moves around the circle, that tangent direction sweeps around too. A change in direction, even with fixed magnitude, is still a change in the velocity vector — and a changing velocity means a non-zero acceleration. This is the key idea the next lesson builds on: an object in uniform circular motion is constantly accelerating, even at constant speed, because its direction isn't constant.

A particle moving around a circular path shown at four positions, with a tangential velocity arrow drawn at each position perpendicular to the radius line at that point. — The velocity vector at each point on the circle is tangential — perpendicular to the radius — and its direction rotates continuously even though its length (the speed) stays fixed.

Speed vs velocity in circular motion — the exam-critical distinction

Uniform circular motion means constant speed, not constant velocity. Examiners exploit this: an object going around a circle at unchanging speed is still accelerating, because acceleration is the rate of change of velocity (a vector), and the direction component is changing throughout, even while the magnitude isn't.

Key takeaways

The radian is defined by $\theta = s/r$; a full revolution is $2\pi$ radians ($360°$).
Angular velocity $\omega = \dfrac{\Delta\theta}{\Delta t} = \dfrac{2\pi}{T} = 2\pi f$, measured in rad s⁻¹.
Linear speed and angular velocity are linked by $v=\omega r$.
Velocity in circular motion is always tangential — perpendicular to the radius at that point.
Uniform circular motion has constant speed but continuously changing direction, so the velocity vector is always changing — meaning the object is always accelerating, even though its speed never does.

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