Considering that angular velocity is:
$w = \frac{2\pi}{t} \frac{[rad]}{[s]}$;
and that linear velocity is:
$ v = w \frac{[rad]}{[s]} \times r[m]$;
I would expect linear velocity to be expressed as $v = x\frac{[rad]\times[m]}{[s]}$;
but, as far as I know, it's always expressed in meters per second.
My question is: Where do the radians disappear to?
$\endgroup$ 62 Answers
$\begingroup$Well, radian is a dimensionless physical quantity. So, mentioning it while mentioning the unit of any physical quantity is not a big deal. Just like, for the angular velocity or angular frequency $\omega$, you have the unit $rad/s$ or rather $s^{-1}$, since the unit in both cases is $\mathrm{Hz}$ (hertz).
$\endgroup$ 7 $\begingroup$You can consider that $r$ is not strictly the radius, but the conversion factor between the angular speed and the linear speed, in $\frac{[m]}{[rad]}$.
Now imagine that the angles are counted in degrees. You would need to use
$w = \frac{360}{t} \frac{[°]}{[s]}$;
$ v = w \frac{[°]}{[s]} \times r'\frac{[m]}{[°]}$
where
$r'\frac{[m]}{[°]}=\frac{2\pi}{360}r\frac{[m]}{[rad]}$.
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