# Why does Wikipedia state that angular velocity is equal to angular frequency? [duplicate]

Isn't Wikipedia wrong? At the Wikipedia article Angular frequency it says that angular frequency and angular velocity are equal. But how on earth are they equal? Angular velocity is changing all the time while angular frequency is a constant(at least if period is constant ). The only case they might be equal is in constant velocity.

Even in circular motion , suppose it's not moving in constant velocity. Why would they be equal? Angular velocity is changing every second. While angular frequency is just a constant. If you have such a system the angular velocity would be d0/dt which is different every second. While the angular frequency would be the repetitions per second multiplied by 2 pi.

Angular frequency is a one value every time. And velocity you can't even say it's value because it's dependant on time.

I don't really understand the statement in Wikipedia which says both are equal. Can you explain?

Edit: this is even more bizarre in harmonic non circular motions. Suppose you have a pendulum oscillating. Then the angular frequency would be a constant measure of an imaginary circle. While angular velocity would be defined as the derivative of the angle. The two are certainly not equal. One is a changing derivative and one is constant. But even in circular non constant motions it doesn't make sense as explained above.

• I am not sure angular frequency is a constant. If you look at a chirped laser pulse that you would obtain from non linear optical pulse generation system, you would see that the angular frequency is defined as a function of time – Ronan Tarik Drevon Jun 1 '18 at 23:38
• "Angular velocity is changing all the time while angular frequency is a constant." - ??? Consider an ideal record player which 'spins' the LP at a constant 33 1/3 rpm. The angular velocity of the LP is constant (as is the angular frequency). Now consider a non-ideal record player with non-zero wow and flutter - neither the angular velocity nor the angular frequency is constant. – Alfred Centauri Jun 1 '18 at 23:58
• Related question by the OP: Is angular frequency the same as angular velocity or are they different? – Alfred Centauri Jun 2 '18 at 0:01
• – Alfred Centauri Jun 2 '18 at 0:02
• "Angular velocity is changing all the time" In what system? And angular velocity of what? You've linked to a Wikipedia article on a general concept but your earlier question focused on a specific system. I've added an answer to physics.stackexchange.com/q/409588 to clarify what I mean by asking what is alleged to have an angular velocity. – dmckee --- ex-moderator kitten Jun 2 '18 at 1:04

Maybe you’re getting tripped up on what is meant by “angular velocity”. Fear not; it’s confusing!

Angular velocity is not, as you may perhaps be thinking, the instantaneous velocity of something revolving around. Rather, it is the rate of change of angle. In analogy to positional velocity, which is the time derivative of position ($dx/dt$), angular velocity is the time derivative of angle ($d\theta/dt$). Thus, angular velocity has the same units and magnitude as angular frequency. But like positional velocity, angular velocity is a vector (defined with the right hand rule like angular momentum), denoting the “direction” of $d\theta/dt$.

• I do not understand. Of course it's the rate of change of an angle. But what if the velocity isn't constant? The rate of change would be different every second. On the other hand if the period is constant the angular frequency would be constant. – bilanush Jun 2 '18 at 7:24
• @bilanush Yes, of course you can come up with idiosyncratic situations where angular velocity is modulated faster than the period such that the period remains constant. But then the instantaneous angular frequency would be modulated as well. Angular frequency is not simply the inverse period <-- that would be the average angular frequency, averaged over a period. – Gilbert Jun 2 '18 at 11:53
• That's what I am trying to understand. I understood that angular frequency is the average velocity in one period. Because it doesn't care about a certain point of time it just calculates the frequency per cycles multipled by 2 pi. So by definition it must be some sort of an average of a given full circle. So why does Wikipedia equate between the two? Do you mean that Wikipedia only meant that the average velocity in a cicle is equal to average frequency given by 2nd f? – bilanush Jun 2 '18 at 13:49
• @bilanush the average angular frequency can be related to inverse period, but the angular frequency itself is defined generally just as Wikipedia suggests: it’s the magnitude of the (possibly time-varying) angular velocity. – Gilbert Jun 2 '18 at 18:09
• Gilbert Ha now I start to understand something. So what you are saying is that angular frequency in definition isn't just defined as 2pi f, but rather has a broader general definition which is d0/dt and this IS INDEED instantaneous. But apart from that, there is definition of the average frequency or velocity and for this alone 2pi f would be the equivalent of it. It's stange that people write 2pi f= d0/dt. It doesn't equal. Only the average is equal. I hope I am right? – bilanush Jun 2 '18 at 22:24

I don't see it saying that they are equal; it states that angular velocity is a vector quantity, and that angular frequency is its magnitude.

"Angular frequency [...] is the magnitude of the vector quantity angular velocity."

perhaps you're referring to the phrase that follows,

"The term angular frequency vector is sometimes used as a synonym for the vector quantity angular velocity."

In which case the vector part of angular frequency vector is an important distinction to make, and why it can be used as synonym to angular velocity.

• source of quotes is the article linked in question, consulted on June 1, 2018 en.wikipedia.org/wiki/Angular_frequency – Alexandre Aubrey Jun 2 '18 at 0:12
• As far as I understand it . It sounds like the only difference is that one is a magnitude and one is a vector. I am not even seeing how one is the magnitude of the other? They may have completely different magnitudes. Suppose a circular motion in which the velocity is changing constantly but the period is always constant. In what sense can you claim one being the magnitude of the other? Angular frequency would be a number . And angular velocity would bexpect derivative of time. – bilanush Jun 2 '18 at 0:15
• Wikipedia here is being overly precise with the wording, imo... it's very typical for physicists to use the terms angular frequency/velocity to refer to the same thing – Burrito Jun 2 '18 at 4:39