The difference between angular frequency and frequency?

[bolzano95]

I looked up and read the definitions in several different books, but still don't get it. Is someone willing to explain it to me on a really simple level?

 

[lekh2003]

Angular frequency is a rotation rate. This is represented by the value, ω. Angular frequency can have the units radians per second. Frequency on the other hand might refer the simple harmonic motion or any object with a repeating motion. Frequency is generally in the units Hertz and can be rpm (which would be converted from angular frequency).

Edited: Information which I made an error with which jtbell included in Post 3 correctly.

 

[jtbell]

I looked up and read the definitions in several different books, but still don't get it.

Can you give us an example of something specific that confuses you?

Either one can be used to describe periodic or rotational motion. Frequency (usually with the symbol $f$ or $\nu$) is cycles (or revolutions) per second. Angular frequency (usually with the symbol $\omega$) is radians per second. 1 cycle (or revolution) equals $2\pi$ radians, so $\omega = 2 \pi f$.

 

[sophiecentaur]

on a really simple level?

There really isn't a 'really simple level' answer because the only reason for using Angular frequency (Radians per second) is to make some already-not-simple calculations easier. We could get along perfectly well without radians when discussing and calculating oscillations and rotations; it's just that the equations would be littered with the symbol "π". Mathematicians have to deal with long enough strings of symbols in formulae that it's always worth while avoiding unnecessary baggage which can hide important patterns.
A better answer to your question would have to depend on where you happen to be with your Maths knowledge. I think pretty well everyone has asked the "why ω" question at some stage but it really is a very useful concept.

 

[bolzano95]

I have manly problems with wave phase kx+wt. w here represents angular frequency, but I’m confused, because when I looked at it, I thought that’s angular velocity, but in textbook it was written angular frequency. I suppose I’m a little bit confused about notation.
But nevertheless, do I understand correctly that angular velocity= angular frequency? Why different names? When do we talk about angular velocity and when do we talk about angular frequency?

 

[sophiecentaur]

They are the same thing. "Radians per second" is obviously a 'rate' expression and that can either relate to the speed at which an angle (measured in radians) changes or, when it's inside a trig function, it relates to the fraction of a complete turn per second - which is a 2π X frequency. I can't think of an occasion where there's a clash of meanings and I think it's as simple as that.

 

[PeroK]

I have manly problems with wave phase kx+wt. w here represents angular frequency, but I’m confused, because when I looked at it, I thought that’s angular velocity, but in textbook it was written angular frequency. I suppose I’m a little bit confused about notation.
But nevertheless, do I understand correctly that angular velocity= angular frequency? Why different names? When do we talk about angular velocity and when do we talk about angular frequency?

The two are very different. Angular velocity (about a given point) is a vector quantity defined by:

$\vec{\omega} = \frac{\vec{r} \times \vec{v}}{r^2}$

Where $\vec{r}$ is the displacement from the given point.

Note that in the special case of two-dimensional motion (in the x-y plane, say) the angular velocity is always in the z direction. In this case, we can simplify the concept of angular velocity to a "signed scalar" $\omega$, which is taken to be positive for "clockwise" motion and negative for anti-clockwise motion:

$\omega = \pm \frac{|\vec{r} \times \vec{v}|}{r^2}$

And, in the special case of circular motion, this reduces further to:

$\omega = \frac{d\theta}{dt} = \pm \frac{v}{r}$

Angular frequency, however, is a general term for the frequency with which an oscillation occurs. For example, a simple pendulum oscillates according to:

$\theta = A\cos(\sqrt{\frac{g}{l}}t) = A\cos(\omega t)$

Here $\omega = \sqrt{\frac{g}{l}}$ is the angular frequency of the oscillation (which is constant) and is not the same as the angular velocity with which the pendulum is swinging at any point (which varies with time). The two are related by:

$\omega' = \frac{d \theta}{dt} = -A\omega \sin(\omega t)$

Where $\omega'$ is the angular velocity of the pendulum's circular motion and $\omega$ is the angular frequency of the pendulum's simple-harmonic oscillation.

In the special case of circular motion: $x = R\cos(\omega t), \ y = R\sin(\omega t)$, we have:

$\theta = \omega t$ and $\frac{d \theta}{dt} = \omega$

So, the same parameter $\omega$ represents the angular frequency of the oscillation of the $x$ and $y$ coordinates and the angular velocity of the circular motion itself. In this special case, the two can be seen as the same.

 

[sophiecentaur]

Angular velocity (about a given point) is a vector quantity

i was arm waving a bit in my post. of course you can't put a vector inside a trig function. Proper use of symbols helps - as always.

 

[jtbell]

When do we talk about angular velocity and when do we talk about angular frequency?

I say "angular frequency" when I'm talking about oscillating motion (e.g. mass moving back and forth on a spring). I say "angular velocity" when I'm talking about an object rotating (e.g. a spinning top) or moving in an orbital path around some point (e.g. Earth around the sun).

 

[sophiecentaur]

"Angular speed" even.

 

[slow]

Hi. Maybe I should start a new thread for my question. In that case I beg to warn me.

Flat electromagnetic wave in vacuum, without polarization of any kind. In that case, no field has a helical configuration, or something similar. An angular frequency also appears in the temporal part of the argument of the wave function, as if there were something that rotates in the propagation. We have all learned to state mathematically what is observed in a wave to obtain the function according to the phenomenon. That is not what worries me. What worries me is the mental need to put every term of a physical equation in correspondence with a real phenomenon. And I see the need to put an angular frequency in correspondence with something physical that rotates.

Is my need justifiable or is it simply an undue attachment to physical models?

 

[olivermsun]

Not sure if I understand your question, but if a motion is "periodic" then sure it makes sense to map the motions (and its equations) to position on a circle, aka angular position or phase. Then you make use of lots of convenient properties that are already well understood.

 

[sophiecentaur]

Hi. Maybe I should start a new thread for my question. In that case I beg to warn me.

Flat electromagnetic wave in vacuum, without polarization of any kind. In that case, no field has a helical configuration, or something similar. An angular frequency also appears in the temporal part of the argument of the wave function, as if there were something that rotates in the propagation. We have all learned to state mathematically what is observed in a wave to obtain the function according to the phenomenon. That is not what worries me. What worries me is the mental need to put every term of a physical equation in correspondence with a real phenomenon. And I see the need to put an angular frequency in correspondence with something physical that rotates.

Is my need justifiable or is it simply an undue attachment to physical models?

No problem with a 'Physical Model" here because there is no QM involved.
Unpolarised light can either be regarded as the sum of a number of individual polarised waves, all at slightly different frequencies. This is what you get from light from the atoms in a conventional light bulb or discharge tube. The net E field at any time and place will have just one (vector) value in a random plane. Alternatively, you can look on it as a single wave in which the E field orientation is constantly changing. This involves a finite bandwidth for the wave because it involves some variations of phase. They both boil down to the same thing, of course.
Circular or elliptical polarisation involves the E field rotating by 360 degrees each cycle. As with plane polarisation, it is more 'ordered' than unpolarised waves.

 

[slow]

Hi. Thank you for your answers. I will try to specify my concern better. Is there in that flat wave without polarization, traveling in a vacuum, something physical that is rotating while the wave advances at the speed of propagation? We know that there is no mass in the wave (that which was old times called mass at rest). Then I can not think of mass turning. I can not think of turning charge either, because a longitudinal component is required to have a non-zero charge density, and that is forbidden in vacuum. So I can not find a way to think of something physical that rotates while the wave propagates and that is the physical correlate of the angular frequency, so that only the rotating vector of an abstract diagram is the only thing that is associated with $\omega$.

 

[sophiecentaur]

Then I can not think of mass turning. I can not think of turning charge either,

What do you already know about EM waves? That statement leads me to believe that you have not learned much yet about the nature of EM waves. There really is no quick Arm Waving description of EM waves but one thing you have to realize from the start is that there is nothing Mechanical and that EM waves have no mass or charge. You have to know the basics before polarisation can mean anything to you at all. There are so many possible hits on Google if you search for "Electromagnetic Waves" and, later "circular polarisation. I suggest you get reading before you start asking questions.

 

[slow]

sophiecentaur, I have reread notes # 14 and # 15 that we have written. I have the impression that we have aimed the same.

 

[jtbell]

An angular frequency also appears in the temporal part of the argument of the wave function, as if there were something that rotates in the propagation.

The same thing happens in the description of any simple harmonic motion, e.g. of a mass bobbing up and down as it hangs on a spring. There is no rotation, but we nevertheless use a trigonometric function of a "phase angle", e.g. $y = y_0 \cos (\omega t + \phi_0)$. As olivermsun noted, it happens that we can make a correspondence between the motion of the mass and an object moving in a circular path at constant speed.

 

[slow]

Thank you very much to all the people who have helped to dispel my doubts.

 

[Khashishi]

Angular frequency is frequency times $2\pi$ radians. That's all.

The term angular frequency suggests something that is rotating, but it is common in wave mechanics to use the term angular frequency even in cases where there is no physical rotation.