Why does radians disappear in equation $\tau = I\alpha$

Question

Since in terms of units, $\tau = Nm$, $I=kgm^2$ and $\alpha=\frac{radians}{s^2}$:

$I\alpha = kgm^2\cdot \frac{radians}{s^2} = \frac{kgm}{s^2}m\cdot radians = Nm\cdot radians$

But... $\tau$ does not have radians? Where does it disappear?

Answer 1

Torque has units of $Nm$. Meaning it has dimensions $[M][L]^2[T]^{-2}$. Radians on the other hand are completely dimensionless. When you look at any physical quantity which has radians and want to determine its dimensions, you treat that quantity as having none. For example, frequency $f$ has units $s^{-1}$ but angular frequency $2\pi f$ will have units $rad \ s^{-1}$. But both these quantities have the same dimensions, $[T]^{-1}$.