I know there are duplicates. But the answers seem to disagree and also I have more specific questions related to this title.
First of all, most questions on this site which ask this question have at least two answers of which one is at variance with the other.
Some say they are the same thing, just that one is scalar 'magnitude ' and one is a vector with 'direction'. On the other hand you have some answers saying they are completely different things apart from the case of complete cyclical motions of constant velocity.
Don't these answers contradict each other? You have these types of answers posted on a few of such questions.
Wikipedia too, in 'angular frequency ' states that it's just the same as angular velocity. How is that possible? Isn't it wrong? At least in oscillating systems which don't have complete circular motion with constant velocity. The angular velocity is just the derivative of theta while angular frequency is omega.
I am very confused. When I am asked about an angular velocity of an oscillator, does that mean omega? Or does that mean derivative of theta?
The two don't seem to be identical at all. For omega you get a constant which is dependent on some given quantities. While the derivative of theta turns out to be completely different and is dependent on t=time and has a cos or sin functions for it's own function.
To sum it up. There is a lot of confusion here , and seemingly contradictionary material as well. Can someone please make it clear?-
Best Answer
In a simple harmonic oscillator, or any vibrating system, the angular frequency is a scaling of the frequency by $2\pi$, e.g. $\sin(2\pi ft)$, $\omega = 2\pi f$.
This ensures that the sine function will complete one cycle in one period $T = 1/f$.
Angular velocity is a definition related to angular motion. For an object rotation about an axis $\omega = (\theta_f - \theta_i)/(t_f - t_i)$.
An object rotating with a constant angular speed (or velocity) we have $\theta(t) = \omega t + \omega_0$. The Cartesian coordinates of the point are related to the angular variable by $x = r\cos(\omega t + \omega_0)$ etc. So you see that the individual coordinates of the "rotating" object appear to move as simple harmonic oscillators.
From a purely physical point of view I would say these two quantities are manifestly different because the basic definitions are different. Once does not need "rotation" to define a periodic motion. However they can be related. Keep in mind that the function $\sin(2\pi ft)$ can also describe an oscillating voltage or other quantity that has not physical relation to rotation or vibration.
From a mathematical point of view one may see them as the same. This equivalence can be seen from the complex representation of $\sin(x)$ as $\operatorname{Im}(e^{ix})$. In the complex plane $e^{i2\pi ft}$ can be seen as "rotating" about the origin of the complex plane. Sometimes when two quantities have the same units it hints at a deep connection, sometimes not.
My personal perspective is that they are different quantities for the reasons described above, but there are interesting and useful connections between the two depending on use and mathematical representation.
I hope that lightens your confusion.