Physics – Why Are Units of Angular Acceleration the Same as Angular Velocity Squared? Exploring Physics Units

Question

According to this answer, the units for angular velocity squared are $\mathrm{rad}/s^2$. The units for angular acceleration are also $\mathrm{rad}/s^2$. Why is this the case?

Answer 1

They have the same units of $\mathrm s^{-2}$ only if you don't use $\mathrm{rad}$ unit for bookkeeping (which you can indeed avoid because radians are technically dimensionless, similarly to turns and other auxiliary units). But if you do try to distinguish angles from dimensionless numbers, then they are not the same: the unit of angular acceleration is $\mathrm{rad}/\mathrm s^2$, and that of square of angular velocity is $\mathrm{rad}^2/\mathrm s^2$.

If radian is dimensionless, why do we not "simplify" $\mathrm{rad}^2$ to $\mathrm{rad}$? For the same reason as why we introduced $\mathrm{rad}$ in the first place: it's not required to use this symbol, but it does help us remember that we have an angle somewhere. Similarly, if we square it, we must use $\mathrm{rad}^2$ because now we have a square of that angle. But as the unit is dimensionless, it's technically not necessary. It's simply for convenience. We could invent a bunch of other dimensionless units to aid us in bookkeeping, but once we've done it, we must keep their correct powers, otherwise these units are simply useless.