# Difference between angular velocity and angular frequency

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## Main Question or Discussion Point

I have seen so many questions and confusion about the difference between angular velocity/speed and angular frequency. Usually, answers were always given in the context of uniform circular motion (angular speed) and simple harmonic oscillation (angular frequency), but this is what causes the confusion (I think).

Let $\omega_v$ be the angular velocity and $\omega_f$ be the angular frequency.

Let's consider a nonuniform circular motion (not constant tangential speed) and recall that the relationship between tangential speed $v$ and angular velocity $\omega_v$ is given by $v=R\omega_v$ where $R$ is the radius of the circular shape made by the rotation. Now, since $v$ changes throughout the rotation $\omega_v$ also changes, for example a rock tied to a string being swirled vertically therefore its angular speed is lowest at the top and highest at the bottom. The angular frequency then measures the "speed" with which the rotation takes to cover a full cycle (let's assume in this case it is $2\pi$) for a given period $T$, say $T=2$ s, so that $\omega_f = \pi$ rad/s which is a constant throughout the rotation. Throughout the rotation, $\omega_v$ could be $2$ rad/s, $1.5$ rad/s, $4$ rad/s, $5$ rad/s, or even $\pi$ rad/s which is just a coincidence, but this shows how different $\omega_v$ from $\omega_f$. It is when there is uniform circular motion that $\omega_v = \omega_f$.

"So even though the angular speed might change from this value to that value in a cycle, it will always take the same "speed" (angular frequency) for it to cover a cycle."

I think a clearer definition so as to make the two distinct would be,

Definition. The angular velocity is the instantaneous rate of change of the angular displacement where the angular speed is just the magnitude of angular velocity.
Definition. Angular frequency is the speed with which the rotation covers the whole cycle for a given period.

Note: Angular frequency is a constant of a given oscillation or rotation whereas angular speed can vary (nonuniform circular motion).

Can anyone comment on my thoughts on this?

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mfb
Mentor
The angular frequency then measures the "speed" with which the rotation takes to cover a full cycle
That would be something like the average angular frequency over some time interval.
If you have angular acceleration, angular frequency is not a very useful concept.
Definition. Angular frequency is the speed with which the rotation covers the whole cycle for a given period.
Where do you start with a rotation? In general there is no special point of "phase 0".

That would be something like the average angular frequency over some time interval.
If you have angular acceleration, angular frequency is not a very useful concept.Where do you start with a rotation? In general there is no special point of "phase 0".
I'm also thinking about the angular frequency definition that I provided to be like an average but that is just based on the interpretation of the formula $\omega_f = \frac{2\pi}{T}$ assuming one cycle is $2\pi$. What then would be a better definition that would CLEARLY distinguish it from angular speed.

There is no special point indeed but to cover one cycle is what I mean.

mfb
Mentor
Just use it as angular speed with an additional factor of 2pi. What is more useful depends on the situation.

Just use it as angular speed with an additional factor of 2pi. What is more useful depends on the situation.
But that is the point of the confusion, mixing the two of them. Also, what do you mean by adding a factor $2\pi$?

mfb
Mentor
But that is the point of the confusion, mixing the two of them.
Then don't mix them, just use one.
Also, what do you mean by adding a factor $2\pi$?
Oh wait, ignore that part, the angular frequency has that factor relative to the regular frequency already.

In higher dimensions, angular velocity has a direction while angular frequency is the magnitude of the angular velocity. In two dimensions you only have a sign as possible difference.

Then don't mix them, just use one.Oh wait, ignore that part, the angular frequency has that factor relative to the regular frequency already.

In higher dimensions, angular velocity has a direction while angular frequency is the magnitude of the angular velocity. In two dimensions you only have a sign as possible difference.
But that is what I want to make clear in this thread. So you're saying the magnitude of the angular velocity is angular frequency hence IS angular speed? Why is it this is the first time I'm hearing this.

How do you explain a varying angular speed? Since by definition angular frequency is a constant $\frac{2\pi}{T}$.

mfb
Mentor
That formula applies only if the angular velocity is constant.

That formula applies only if the angular velocity is constant.
Angular speed you mean, then why is that this is a confusion if the only difference is just the words being used, books or other sources could just simply say angular velocity is the instantaneous rate of change of the angular displacement and angular frequency aka angular speed is the magnitude PERIOD. There should be nothing to be confused of if you put it that way.

An example would be Walter Lewin's "Million dollar" video

Why bother with the long winded explanation if it's just as simple as being a vector (angular velocity) or a scalar (angular frequency/speed).

Angular speed you mean, then why is that this is a confusion if the only difference is just the words being used, books or other sources could just simply say angular velocity is the instantaneous rate of change of the angular displacement and angular frequency aka angular speed is the magnitude PERIOD. There should be nothing to be confused of if you put it that way.
Copying text from the wikipedia links, because I am not sure you've read it:
In physics, angular frequency ω (also referred to by the terms angular speed, radial frequency, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit time (e.g., in rotation) or the rate of change of the phase of a sinusoidal waveform (e.g., in oscillations and waves), or as the rate of change of the argument of the sine function.

Angular frequency (or angular speed) is the magnitude of the vector quantity angular velocity. The term angular frequency vector {\displaystyle {\vec {\omega }}} is sometimes used as a synonym for the vector quantity angular velocity.​

mfb wrote it already in his post #6.

How do you explain a varying angular speed? Since by definition angular frequency is a constant $\frac{2\pi}{T}$.
Angular frequency is constant only in case of uniform circular motion. Then, also the angular speed is constant. But in general, the circular motion can be accelerated, and in such case the frequency is not constant, neither the angular speed.

Angular frequency is constant only in case of uniform circular motion. Then, also the angular speed is constant. But in general, the circular motion can be accelerated, and in such case the frequency is not constant, neither the angular speed.
I understand what you're saying but talking as if the two were different and each implies the other is confusing because as you've said, they are exactly the same thing just different names. So we could just drop the angular speed convention and use angular frequency.

I understand what you're saying but talking as if the two were different and each implies the other is confusing because as you've said, they are exactly the same thing just different names. So we could just drop the angular speed convention and use angular frequency.
Well, the world is not perfect, and the naming is not always as strict as it could be. In case of physical quantities, one of the reasons why we have different names could be that the different names were used in different domains, eg. engineering vs. physics (as a science). An example could be torque vs. moment of force...

We just have to deal with that and keep it in mind...

Well, the world is not perfect, and the naming is not always as strict as it could be. In case of physical quantities, one of the reasons why we have different names could be that the different names were used in different domains, eg. engineering vs. physics (as a science). An example could be torque vs. moment of force...

We just have to deal with that and keep it in mind...
How do you resolve the pendulum equation,

$\theta(t) = \theta_0 \sin(\omega t)$

where its first derivative is the angular speed $\theta'(t)$ which obviously varies with time and $\omega$ (angular frequency) is a constant,

$\theta'(t) = \theta_0 \omega \cos(\omega t)$

At the beginning you considered only the circular motion, pendulum is different case...
Let's consider a nonuniform circular motion
For pendulum, I don't think it make sense to talk about angular frequency. Pendulum has the constant period $T$, so the frequency of oscillations is $f = 1 / T$ . Of course, you can define and use $\omega = 2 \pi f = \frac{2 \pi}{T}$ , but you shouldn't call it angular frequency. It doesn't represent rate of change of the angular displacement of the body.

At the beginning you considered only the circular motion, pendulum is different case...

For pendulum, I don't think it make sense to talk about angular frequency. Pendulum has the constant period $T$, so the frequency of oscillations is $f = 1 / T$ . Of course, you can define and use $\omega = 2 \pi f = \frac{2 \pi}{T}$ , but you shouldn't call it angular frequency. It doesn't represent rate of change of the angular displacement of the body.
Then what is the physical meaning of $\frac{2\pi}{T}$ in the case of the pendulum?

I would say, the physical meaning of $\omega = \frac{2 \pi}{T}$ is the same as meaning of $f = \frac{1}{T}$. It is a frequency. Multiplying by a constant factor of $2 \pi$ doesn't change the physical meaning.

• BvU
BvU
Homework Helper
2019 Award
At the risk of messing up a clear thread: $\ \sin \omega t \$ is the real part of $\ e^{i\omega t} \$ and in the complex plane that $\omega t$ is an angle.

Personally, I'd say that's more of a mathematical motivation to call it an angular speed than a physical one, but we definitely can't do without math, so we gladly adopt a lot of the lingo.

robphy
Homework Helper
Gold Member
At the risk of messing up a clear thread: $\ \sin \omega t \$ is the real part of $\ e^{i\omega t} \$ and in the complex plane that $\omega t$ is an angle.
The $\mbox{Real part}\ \Re(z)=\displaystyle\frac{z+\bar z}{2}$.
So, since $e^{i\theta} =\cos\theta + i\sin\theta$
and $\overline{e^{i\theta}}=e^{-i\theta} =\cos\theta - i\sin\theta$,
we have
$\mbox{Real part}\ \Re(e^{i\theta})=\cos\theta$.

• BvU
robphy
Homework Helper
Gold Member
How do you resolve the pendulum equation,

$\theta(t) = \theta_0 \sin(\omega t)$

where its first derivative is the angular speed $\theta'(t)$ which obviously varies with time and $\omega$ (angular frequency) is a constant,

$\theta'(t) = \theta_0 \omega \cos(\omega t)$
Since $\theta$ is the angular-position,
I would call $\theta'(t)$ the angular-velocity (in analogy to $v=dx/dt$)
(generally it's a vector, as mentioned above).
I would call its magnitude $|\theta'(t)|$ the angular-speed.

In $\sin(\omega t)$, I would call $\omega$ the phase-frequency
in order to avoid "angular-frequency", which is a potentially confusing term in this context
since there are two different angles here:
• the angle $\theta$ in space (describing the configuration of the pendulum in ordinary space),
• the angle $\frac{2\pi t}{T}$ in phase (describing the configuration of a clock-hand describing the advance of time).

• lomidrevo