Everything you learned about motion along a line has a twin for things that turn: wheels, fans, planets, a door. The turning versions use angles instead of distances, and the angle unit that makes everything work is the radian, which you met with arc length.
Why it works
A radian measures an angle by the arc it cuts: $\theta = \dfrac{s}{r}$, arc length over radius. A full circle has arc $2\pi r$, so a full turn is $\dfrac{2\pi r}{r} = 2\pi$ rad. Because $\theta$ is a ratio of two lengths, the radian has no real unit, and that is exactly why formulas such as $s = r\theta$ and $v = r\omega$ work with no conversion factor, but only when the angle is in radians.
Key idea
Angular position $\theta$ in radians ($1\ \text{rev} = 2\pi\ \text{rad} = 360^\circ$). Angular velocity $\omega = \dfrac{\Delta\theta}{\Delta t}$ in rad/s. Angular acceleration $\alpha = \dfrac{\Delta\omega}{\Delta t}$ in rad/s². Counterclockwise is usually positive, and $\alpha$ against the turning means slowing down.
| Along a line | Turning | Unit |
|---|---|---|
| position $x$ | angle $\theta$ | rad |
| velocity $v = \Delta x / \Delta t$ | angular velocity $\omega = \Delta\theta / \Delta t$ | rad/s |
| acceleration $a = \Delta v / \Delta t$ | angular acceleration $\alpha = \Delta\omega / \Delta t$ | rad/s² |
Worked examples
Worked example: Revolutions per minute to rad/s
A record turns at $45$ rpm (revolutions per minute). Find $\omega$ in rad/s.
- Each revolution is $2\pi$ rad, and each minute is $60\ \text{s}$.
- $$\omega = 45\ \frac{\text{rev}}{\text{min}} \times \frac{2\pi\ \text{rad}}{1\ \text{rev}} \times \frac{1\ \text{min}}{60\ \text{s}} = 4.71\ \text{rad/s}.$$
Answer: $4.71\ \text{rad/s}$
Worked example: Slowing down
A ceiling fan at $180$ rpm is switched off and stops in $12\ \text{s}$. Find its average angular acceleration, taking its direction of turning as positive.
- $\omega_0 = 180 \times \dfrac{2\pi}{60} = 18.8\ \text{rad/s}$ and $\omega = 0$.
- $\alpha = \dfrac{0 - 18.8}{12}\ \text{rad/s}^2 = -1.57\ \text{rad/s}^2$.
- Negative: the angular acceleration points against the turning, so the fan slows.
Answer: $-1.57\ \text{rad/s}^2$
Worked example: Arc length
A point $0.30\ \text{m}$ from a wheel's axle turns through $120^\circ$. How far does it travel?
- Convert first: $120^\circ \times \dfrac{\pi}{180^\circ} = 2.09\ \text{rad}$.
- $s = r\theta = (0.30\ \text{m})(2.09) = 0.63\ \text{m}$.
Answer: $0.63\ \text{m}$
Common mistake
Mixing degrees, revolutions and radians in one calculation. Putting $120$ (degrees) into $s = r\theta$ gives an arc of $36\ \text{m}$ for a $0.30\ \text{m}$ wheel, far more than its whole circumference ($1.9\ \text{m}$). Convert every angle to radians before using $s = r\theta$ or $v = r\omega$, and check that an arc for less than one turn is less than $2\pi r$.