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Our physics book is actually full of mistakes, so; Is angular frequency the same as angular frequency when the given time is period?

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Angular frequency and angular velocity (or more precisely, angular speed) are synonyms in the context of 2D rotational kinematics. They both refer to the rate at which angular displacement increases. It doesn't matter whether or not a time period is given.

Outside of rotational kinematics, angular frequency $\omega$ can also refer to $2\pi$ times the number of cycles per unit time for an arbitrary oscillating system. This contrasts with (just plain) frequency, $f$, which is simply the number of cycles per unit time. $$\begin{align} \omega &= \frac{2\pi}{T} & f &= \frac{1}{T} \end{align}$$ Angular frequency in this sense is a useful quantity when you're trying to describe the state of the oscillating system using sinusoidal functions. You will wind up using expressions like $\sin(2\pi f t)$ which can be more concisely written $\sin(\omega t)$.

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See the Wikipedia article on Angular frequency.

  • Angular velocity is $\vec \omega=\frac{d\vec\theta}{dt}$. It's magnitude called angular speed is $ \omega=\frac{d\theta}{dt}$. These are the number of radians per second something rotates.

  • Angular frequency $\omega$ is the number of radians per second in some periodic motion when a full round ($2\pi$ radians) is set as exactly one cycle in this periodic motion.

The terms angular frequency and angular speed (not velocity) are the same thing, as the Wikipedia article tells, but are used in different case (something rotating - e.g. a wheel - and something in periodic motion - e.g. a signal).

You might have seen angular frequency tied to regular periodic frequency like this: $\omega=2\pi f$. See also @dmckee's comment below; the choice of units is restricted for the frequency terms, while you are free to use any angular units per time unit for angular speed.

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    $\begingroup$ It is useful to apply the requirement that angular frequency is the angular speed expressed in radians per time unit, and the cyclic frequency is the angular speed expressed in cycles per time unit. Angular speed in general can be expressed in any angular units (degrees, grad, etc) but the angular frequency and periodic frequency have specified angular units (either for circular motion or for the motion to the phasor associated with some harmonic behavior). $\endgroup$ – dmckee Dec 4 '16 at 18:44

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