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# Negative unit exponents

• 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 propertyUnitAngular propertyUnit
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:5

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