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The unit for angular acceleration $\alpha$ is:

$$\mathrm{rad/s^2}$$

The unit for torque is $\mathrm{Nm}$:

$$\mathrm{kg\ m^2/s^2}$$

And their relationship with Inertia is:

$$I = \tau/\alpha$$

So shouldn't the unit for for Inertia be:

$$\mathrm{kg\ m^2/rad}$$

yet everywhere I read says it is simply $\mathrm{kg\ m^2}$ instead. How does the $\mathrm{rad}$ unit fall off?

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See also Simple Harmonic Motion - What are the units for $\omega_0$? and https://en.wikipedia.org/wiki/Joule#Confusion_with_newton-metre

Here's a somewhat shorter explanation reflecting my own (possibly incorrect) intuition:

Radians aren't "real" units; they're just a trick to keep track of which quantities involve angles and which don't, since it's usually a mistake to get those mixed up. However, it's occasionally valid to mix those two types of quantities, and then we drop the radians. Torque is one such place.

It's probably possible to be fully rigorous about this and make radians an actual unit, but I've never seen it done.

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  • $\begingroup$ +1 "It's -- possible to be fully rigorous about this and make radians an actual unit" - Actually the right starting point is to name the Dimension, which is Angle, and then we can allow a variety of units, two of which can be chosen as bases - the cycle, and the radian. We also have, in dimensional analysis terms that a steradian is Angle^2, though the scaling terms are spherical not linear (4pi vs (2pi) squared) $\endgroup$
    – Philip Oakley
    Commented Apr 27, 2018 at 22:48

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