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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?

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  • $\begingroup$ Does it help if you think of the conversion factor from angular displacement to arc length as having units of $\mathrm{\frac{m}{rad}}$? For example: $$C = 2\pi\,\mathrm{rad}\cdot R\,\mathrm{\frac{m}{rad}} = 2\pi R\,\mathrm{m} $$ $\endgroup$ Aug 14, 2018 at 1:01
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/252288/2451 , physics.stackexchange.com/q/33542/2451 and links therein. $\endgroup$
    – Qmechanic
    Aug 14, 2018 at 8:00
  • $\begingroup$ In addition to the links Qmechanic has provided, see this relatively recent paper Dimensionless Units in the SI: "Here we consider dimensionless units as defined in the SI, e.g. angular units like radians or steradians and counting units like radioactive decays or molecules. We show that an incoherence may arise when different units of this type are replaced by a single dimensionless unit, the unit "one", and suggest how to properly include such units into the SI in order to remove the incoherence." $\endgroup$
    – Hal Hollis
    Aug 14, 2018 at 12:46

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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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The speed of a point on the circumference depends on its distance from the centre of the circle.

Let's assume the circle has a radius of $r$ m. Then, its circumference is $2{\pi}r$ m. If it rotates at 1 rpm, a point on the circumference will travel the whole circumference, or $2{\pi}r$ m, in 1 second. During that second it will also travel $360^{\circ}$, or $2\pi$ radians.

Angles are dimensionless and hence angle*radius/second corresponds to metre/second.

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