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The relation between the "regular" frequency $f$ and the angular frequency $\omega$ ($\omega = 2\pi f$) is clear to me. However, every time I see "rotations per second" I really get confused as to what is meant there. How do I know whether the author means frequency or angular frequency, if "rotations per minute" can refer to both the frequency of rotations and the angular frequency of rotations as they both have the exact same unit $1 \over s$?

Before you mark this question as a duplicate, I've seen many questions regarding the differences between $f$ and $\omega $ but that's NOT what I'm asking here. I just want to know how to distinguish them when I come across a problem involving rotations and waves.

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“Rotations per second” always means regular frequency.

“Radians per second” always means angular frequency.

A “rotation” always means a full rotation, which is through $2\pi$ radians.

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    $\begingroup$ I would say frequency can be used for more than just rotations though. $\endgroup$ – Aaron Stevens Feb 1 at 22:03
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Angular frequency corresponds to the rate at which an angle is changing, so you will most likely find it as part of the argument of trig functions/complex exponentials. It is usually understood like "radians per time"

"Regular" frequency is usually anything else, typically like a "number of events per time". This could be talking about number of rotations, cycles, or anything really. As long as it has a will defined period between the same events.

In either case, the terminology is not as essential as knowing what you are denoting as $f$ or $\omega$ really means in the system you are looking at. They are usually interchangeable anyway through simple conversions.

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Frequency and angular frequency don't have the same units. Angular frequency has the units $rad/s$, unlike frequency that has $s^{-1}$.

In general, $\omega$ is the angular speed - the rate change of angle (as in a circular motion).

Frequency ($f$) is $1/T$ or the number of periodic oscillations or revolutions during a given time period.

Angular speed or angular frequency, relates the same idea to angles - how much angle is covered over a time period.

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