I'm having a hard time trying to understand the units between angular velocity and basic velocity of a circle. For angular velocity the units are Radian(s) per second(s) or degree(s) per second(s). The speed or velocity of the circles circumference is the angular velocity times the radius, but the units for this is meter(s) per second(s). So where did the radian go? It counts as a unit for angular velocity but why it doesn't count for the speed?
Answer
Actually the radian does not have units as it is defined as the ratio arclength/radius. Since the arclength is a distance and thus has units of meters, and the radius is also in meters, the ratio is dimensionless.
In particular for angular opening $\theta$ the arclength is $r\theta$ for a circle of radium $r$, and going around the circle in full once give a ratio $2\pi r/r=2\pi$ rad, where the arclength is the full circumference in this case.
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