- Position $s$ is in units of ‘metres’ $\mathrm m$,
- speed $v$ is in ‘metres-per-second’ $\mathrm{m/s}$, and
- acceleration $a$ is in ‘metres-per-second-squared’ $\mathrm{m/s^2}$.

Now, what are the units of the equivalent *angular* properties?

- ‘Speed’ is ‘some distance per time’ and thus has ‘distance-units per time-units’.
**Angular speed**is also how far we move over time, but here the ‘how far’ is not*distance*. Rather, it is*angle*. We are dealing with ‘angle-units per time-units’, in SI units: ‘radians-per-second’.^{1}- Same for
**angular acceleration**with units of ‘radians-per-second-squared’.

But, previously we’ve invented the ‘radian’ unit to mean ‘no unit’!^{2} So, when we say ‘radians-per-second-squared’, we actually just mean ‘per-second-squared’. We would try to avoid writing, say, $2\;\mathrm{rad/s^2}$ or something like that, but instead simply $2\;\mathrm{/s^2}$.

This can look a bit confusing with the dash $/$ just “hanging” there. Why don’t we instead invent a **negative exponent** to mean ‘per’:

$$2\;\mathrm{/s^2}=2\;\mathrm{s^{-2}}$$

Translational vs angular units | |||||
---|---|---|---|---|---|

Linear property | Unit | Angular property | Unit |
||

Position | $$s$$ | $$\mathrm{m}$$ | Angular position | $$\theta$$ | $$-$$ |

Speed | $$v$$ | $\mathrm{m/s}$ ^{4} | Angular speed | $$\omega$$ | $$\mathrm{s^{-1}}$$ |

Acceleration | $$a$$ | $\mathrm{m/s^2}$ ^{3} | Angular acceleration | $$\alpha$$ | $$\mathrm{s^{-2}}$$ |

We know exponents to mean ‘multiplied with itself a number of times’. The added negative sign on the exponent now simply flips it, so we are ‘*dividing* with it a number of times’, or it is ‘multiplied with itself a number of times *in the denominator*’. A general **rule of algebra**:^{3}

$$\frac 1{a^b}=a^{-b}$$