Unit of angular velocity in oscillatory motion

Question

Suppose an angular oscillatory motion. In the function below, $\alpha$ and $\alpha_o$ are angles measured in radian, $\omega$ is circular frequency ($2\pi/T$) measured in [radian/s] and $t$ is time.

$$ \alpha = \alpha_o\sin(\omega t) $$

Angular velocity is obtained by taking derivative with respect to time:

$$ \frac{d\alpha}{dt} = \alpha_o\omega\cos(\omega t) $$

I get confused with the unit of angular velocity. To me it looks like:

$$ \frac{d\alpha}{dt} = \alpha_o[rad]\omega[rad/s]\cos(\omega t)[1] =[rad^2/s] $$

Why I cannot get $rad/s$ for angular velocity?

Answer 1

Radian is defined in such a way that it has no unit. So we cannot mathematically differentiate between a numerical constant and a unit. And this leads to many confusions. Using degree as unit of angle will make this more clear.

Let $\alpha,\alpha_o,\omega,t$ be numerical value

$$ \alpha^\circ = \alpha_o^\circ\sin\left(\left(\frac{\omega^\circ}{s}\right)\left(\frac{\pi}{180^\circ}\right)\left(t\space s\right)\right) $$

Angular velocity is obtained by taking derivative with respect to time:

$$ \left[\frac{d\alpha}{dt}\left(\frac{^\circ}{s}\right)\right] = \left[\alpha_o^\circ\left(\frac{\omega^\circ}{s}\right)\left(\frac{\pi}{180^\circ}\right)s\cos\left(\left(\frac{\omega^\circ}{s}\right)\left(\frac{\pi}{180^\circ}\right)\left(t\space s\right)\right)\right]$$

$$\implies \left(\frac{^\circ}{s}\right)=\left(\frac{^\circ}{s}\right)$$

Answer 2

It is radians squared. The problem is you are using two different radians. One is the physical angle $\alpha$, which oscillates with amplitude $\alpha_0$. Fine...you could measure that in degrees, say if it were a latitude or longitude. That might make the point clearer, or at least less abstract.

The other angle is the temporal phase angle of the oscillatory function:

$$\phi(t) = \omega t $$

So, these two angles live in different spaces, and their radians aren't the same thing. Of course, you can also just not use radians, and they are dimensionless (with good reason: space radians and time radians don't mix well, as you discovered).