During Uniform Circular Motion, the linear speed of the particle is defined as the radius times the angular speed. $$ v = r\omega $$ The units of linear speed is meters/second (m/s). But the units of $ r\omega $ is $ m\frac {rad}{s} $. How is this possible? And why does the unit "radian" cancel out?
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$\begingroup$ Thanks for the question. I teach students circular motion right now and I'll mention it for them too. If you were confused, they also can be puzzled :) $\endgroup$– Andrei Z.Commented Oct 24, 2021 at 9:15
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$\begingroup$ Related: physics.stackexchange.com/q/252288/123208 $\endgroup$– PM 2RingCommented Oct 24, 2021 at 9:24
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$\begingroup$ Thank you and Welcome!! $\endgroup$– VinayCommented Oct 24, 2021 at 11:15
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1 Answer
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Rad is a dimensionless unit. It is defined as such an angle that it intercepts an arc on a circle with the length equal to the radius of a circle. Angle in radians is the length of an arc over radius, therefore meter over meter: $\theta = L_{arc}/R$. They are dimensionless in nature and people write "rad" simply for convenience.
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$\begingroup$ I write radians or rad to indicate that I'm talking about an angle. $\endgroup$ Commented Oct 24, 2021 at 17:36