If I square a value units of radians, is the result in units of radians squared or is it still radians?

Question

I am writing a paper on circular motion. A function given is $$T=Msω^2L$$ The units for $ω$ are $\text{rad}/s$. What are the units for $ω^2$? Are they $\text{rad}^2/s^2$ or $\text{rad}/s^2$? If they are the latter, why does $ω^2$ have the same units as those for angular acceleration?

Answer 1

Radians are dimensionless, since they are defined as the ratio of arclength to radius when you look at what an angle cuts out of a circle. That ratio is length/length. So if $\omega$ has units radians/second the dimensional analysis treats that as $1/s$. Then $\omega^2$ has units $1/s^2$.