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: $$C = 2\pi\,\mathrm{rad}\cdot R\,\mathrm{\frac{m}{rad}} = 2\pi R\,\mathrm{m} $$ $\endgroup$– Alfred CentauriCommented Aug 14, 2018 at 1:01
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$\begingroup$ Possible duplicates: physics.stackexchange.com/q/252288/2451 , physics.stackexchange.com/q/33542/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Aug 14, 2018 at 8:00
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$\begingroup$ 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 HollisCommented Aug 14, 2018 at 12:46
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2 Answers
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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.
answered Aug 14, 2018 at 2:05
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$\begingroup$ Only if you believe arc length and radius length are the same thing which they are not. Radian is a perfectly legitimate unit, equal to [arc-meter/meter], for example, where [arc-meter] is here intentionally made a distinct unit. Before you cry foul, note that length and time also have the same units in some natural units in physics and speed is then unitless. Bottom line: whether something is a unit in a system depends on whether it is considered conceptually different enough from something else, which is not absolute. $\endgroup$– NimrodCommented Oct 18 at 6:20
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$\begingroup$ @Nimrod There’s nothing to believe here. Are you suggesting there is a system of units where radian is not unitless? $\endgroup$ Commented Oct 18 at 11:22
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$\begingroup$ Yes, that is what I suggested. It's a mere convention that we decide to use the same unit for arc length as for radius. We can make a different choice. NIST even has a paper on this, see: arxiv.org/abs/1409.2794. $\endgroup$– NimrodCommented Oct 18 at 20:02
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$\begingroup$ Sorry I don’t follow. There is no confusion as to using the same unit for radius and arclength, and the radian is defined as the ratio of these and thus dimensionless by definition. There are other definitions of angles but not of radians. As your link points out, computing the cosine of a dimensionful quantity is meaningless. $\endgroup$ Commented Oct 18 at 20:51
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$\begingroup$ Nope, the NIST paper marks radius of curvature with units of m/rad, a different unit than for arc length. Handling function arguments is an entirely different issue. The paper explains very clearly what to do -- $\cos(y)$ where $y$ is really $y$ rad is shorthand for $cos(y\textrm{ rad} \times 1/\textrm{rad})$. $\endgroup$– NimrodCommented Oct 18 at 21:21
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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.
