Radian unit -- Why neglected in dimensional analysis?

[Mr Genius]

Why radians are usually neglected in dimensional analysis?

 

[mfb]

It doesn’t always disappear, e.g. if you use the radius of something to find a length along the circle. This becomes even more prominent if you use things like the small angle approximation.

 

[Baluncore]

Because angles can be treated as a ratio, as a proportion of a full circle.

The unit employed to specify an angle decides the magnitude and dimension of the coefficients you need to use when evaluating, for example, a sine or cosine of an angle by expanding a polynomial.

 

[sophiecentaur]

Why radians are usually neglected in dimensional analysis?

Because radians are dimensionless - they are a ratio.

 

[anorlunda]

By the way, we have a PF Insights article where the author proposes creating a unit for angles. Comments on the article argued against.

https://www.physicsforums.com/insights/can-angles-assigned-dimension/

 

[RobMac]

Need to add to this explanation: "Because radians are dimensionless - they are a ratio"...of a distance by another distance. Many ratios are not dimensionless.

The angle in radians of an arc of a circle is the distance of the arc divided by the radius of the circle. By similarity, this is independent of the radius of circle used.

 

[sophiecentaur]

Need to add to this explanation: "Because radians are dimensionless - they are a ratio"...of a distance by another distance. Many ratios are not dimensionless.

The angle in radians of an arc of a circle is the distance of the arc divided by the radius of the circle. By similarity, this is independent of the radius of circle used.

The "ratios" I think you are referring to are not ratios - they are Quotients.
Meters per Second is not a ratio.

 

[I like Serena]

My problem with 'ignoring' radians, is in angular frequency.
If we have an angular frequency of 1 s^-1, it could either mean 1 radian per second (rad/s), or 1 revolution per second (Hz).
Both are ratios, but there does not seem to be any way to distinguish them, other than actually specifying radians.
And ignoring the distinction leads to failure in dimensional analysis.

 

[anorlunda]

y problem with 'ignoring' radians

Did you read the Insights ariticle?

 

[I like Serena]

Did you read the Insights ariticle?

I thought I did.
In particular I've found:

3.5 Frequency and Angular Frequency
In wave expressions, frequency, f, is the number of cycles per unit time, while angular frequency, ω, is radians per unit time. ω=2πf. Clearly ω should have dimension ΘT⁻¹. The dimension for ff depends on whether the factor π is to be taken as an angle or as a dimensionless number performing a conversion of units. Taking f as having dimension T⁻¹ appears to be best.

Reference https://www.physicsforums.com/insights/can-angles-assigned-dimension/

All in all, I didn't find anything pertaining to my particular comment. Not in the article, nor in the comments on it.
The article does say that angular frequency is different from frequency, although I do consider that ambiguous. For starters, angular velocity seems to be a better term than angular frequency, but that's not something we can change. And it also doesn't help in the dimensional analysis.

Anyway, apparently I'm missing something. Can you clarify?

 

[anorlunda]

Anyway, apparently I'm missing something. Can you clarify?

IMO the thing you're missing is the arguments for and against in the article and in the comments on the article. They apply to both angles and angular velocity.

In AC electrical work, we typically use "angular frequency" $2\pi{f}$ 377 radians per second on a 60 Hertz system. Angular velocity would be an inappropriate term in that use.

 

[I like Serena]

IMO the thing you're missing is the arguments for and against in the article and in the comments on the article. They apply to both angles and angular velocity.

In AC electrical work, we typically use "angular frequency" $2\pi{f}$ 377 radians per second on a 60 Hertz system. Angular velocity would be an inappropriate term in that use.

Can you be more specific please?
As I see it, s^-1 is still very much ambiguous in dimensional analysis, regardless of how we name the quantities.
(I have to admit that naming the quantities is a separate topic, although in my language (Dutch), we actually call it angular velocity and not angular frequency. One of the things I had to learn in university is that it was different in English.)

 

[anorlunda]

As I see it, s-1 is still very much ambiguous in dimensional analysis

I don't understand your confusion. If s (seconds) is an unambiguous unit, why is $s^{-1}$ ambiguous?

 

[I like Serena]

I don't understand your confusion. If s (seconds) is an unambiguous unit, why is $s^{-1}$ ambiguous?

Because it's not clear whether it is rad/s, or revolutions/s or just s^-1 (or whatever you want to call it). The difference being a factor of $2\pi$ that we do not want to ignore.

 

[Nugatory]

Because it's not clear whether it is rad/s, or revolutions/s or just s^-1 (or whatever you want to call it). The difference being a factor of $2\pi$ that we do not want to ignore.

I don't know that there is any satisfactory resolution to this notational ambiguity, except to look at how the expression is being used in context. However, this is a problem when dealing with angular frequencies when the angle may or may not have bee expressed in radians, whereas this thread started out asking about the dimension of the radian itself.

 

[I like Serena]

I don't know that there is any satisfactory resolution to this notational ambiguity, except to look at how the expression is being used in context. However, this is a problem when dealing with angular frequencies when the angle may or may not have bee expressed in radians, whereas this thread started out asking about the dimension of the radian itself.

I'm still not clear why it would be bad to acknowledge $\text{rad}$ as an actual unit. It seems to me that is would solve the ambiguities in dimensional analysis, since as far as I can tell, saying it's a ratio does not rid us of the ambiguities. That's because it's ambiguous what the reference is for the ratio. Am I missing something?

 

[anorlunda]

This article https://en.wikipedia.org/wiki/General_Conference_on_Weights_and_Measures describes the organization to make your proposal to.

I don't get a vote on those standards bodies, but if I did, I would vote yes on your proposal.

 

[I like Serena]

This article https://en.wikipedia.org/wiki/General_Conference_on_Weights_and_Measures describes the organization to make your proposal to.

I don't get a vote on those standards bodies, but if I did, I would vote yes on your proposal.

Hold on, it says:

The SI supplementary units (radian and steradian) become derived units.
​

Doesn't that mean that they have been recognized as official units?
To be honest, I'm a bit confused about the whole mess. Are radians recognized as a unit by SI or not?

 

[anorlunda]

The International System of Units (SI) specifies a set of seven base units from which all other SI units of measurement are derived. These SI derived units are either dimensionless, or can be expressed as a product of one or more of the base units, possibly scaled by an appropriate power of exponentiation.

It's like Catch 22, It is a dimensionless derived unit.

Is there any value in discussing this? We have no special insight or influence on those committees.

 

[I like Serena]

It's like Catch 22, It is a dimensionless derived unit.

Is there any value in discussing this? We have no special insight or influence on those committees.

I believe there is still value in it. And as far as I can tell it's not mentioned explicitly in our Insight article.
Those committees do not say how to do dimensional analysis do they?
They just recognize that radians are a derived unit, whatever that means exactly.
Doesn't it make sense to track radians in every dimensional analysis?
The only reason to deviate that I can think of, is in a series expension of the sine, in which case we can probably still come up with some form that it's still a unit for dimensional analysis.

 

[Nugatory]

in a series expension of the sine

I got to say... That seems like a really good reason to me.

in which case we can probably still come up with some form that it's still a unit for dimensional analysis.

... and that sounds like a cure that's worse than the disease to me. What dimensionful combination of base units could be an improvement over the trivial (that is, dimensionless) combination that the standards body has settled on?

Conversely, if it's not a combination of base units, then it would have to be a base unit in its own right, which creates a different set of problems.

 

[I like Serena]

I got to say... That seems like a really good reason to me.
... and that sounds like a cure that's worse than the disease to me. What dimensionful combination of base units could be an improvement over the trivial (that is, dimensionless) combination that the standards body has settled on?

Conversely, if it's not a combination of base units, then it would have to be a base unit in its own right, which creates a different set of problems.

Okay, so let's take a closer look at the sine expansion, which I didn't do yet:
$$\sin x = \sum \frac {d}{dx^k} \sin0 \cdot \frac{x^k}{k!}$$
That is, we're dividing by $\operatorname{rad}^k$, and we're multiplying by $\operatorname{rad}^k$, yielding a dimensionless quantity.
So in retrospect, even the expansion of the sine behaves perfectly reasonable with respect to radians.

In other words, radians is a good unit for dimensional analysis.

 

[Nugatory]

What are we going to do with:
$$\sin x = \sum (-1)^k\frac{x^{2k+1}}{(2k+1)!}$$

And you still haven't said what combination of base units might be appropriate as an alternative to the trivial dimensionless one.

 

[I like Serena]

What are we going to do with:
$$\sin x = \sum (-1)^k\frac{x^{2k+1}}{(2k+1)!}$$

And you still haven't said what combination of base units might be appropriate as an alternative to the trivial dimensionless one.

That $(-1)^k$, is actually $(-1)^k \text{ rad}^{-2k-1}$, which cancels (as far as the unit is concerned) with $x^{2k+1}$.
We're just leaving out that we're making use of how radians are defined.
And I'm just saying that if something is an angle and if we express it in radians, that we should mark it with the unit radians (or alternatively with degrees, gradians, grons, or revolutions).
And if we use for instance gradians, we need to modify that -1 as well.

 

[Baluncore]

It does not matter if length is measured in metres or furlongs, the dimension will always be Length.
Likewise seconds, hours and seasons will always have the dimension of Time.
Speed can be mph, kph or metres/sec, whatever the units used, speed will have the dimension Length / Time.
It is pretty obvious that units carry dimensions, but that dimensions are not specific units.

 

[fresh_42]

Revolutions are counted be higher values than $2 \pi$.
Radians is a derived (from a circle with radius one meter) SI unit, which doesn't mean it cannot be a dimensionless number, because it is the ratio of two lengths, and $\frac{m}{m}=1$. It simply means, we are allowed to write $1 \operatorname{rad}$, in contrast to e.g. $1 \operatorname{Oz}$. It makes mathematically perfect sense to consider it as a pure number, as in

What are we going to do with:
$$\sin x = \sum (-1)^k\frac{x^{2k+1}}{(2k+1)!}$$

However, it can also make sense to treat it formally as a unit, e.g. when checking a physical calculation where the angles don't cancel out or in a case like this:

It doesn’t always disappear, e.g. if you use the radius of something to find a length along the circle. This becomes even more prominent if you use things like the small angle approximation.

In the end it remains a ratio of two quantities of the same dimension, length. And as such $1 \operatorname{rad} = 1$.

 

[I like Serena]

It does not matter if length is measured in metres or furlongs, the dimension will always be Length.
Likewise seconds, hours and seasons will always have the dimension of Time.
Speed can be mph, kph or metres/sec, whatever the units used, speed will have the dimension Length / Time.
It is pretty obvious that units carry dimensions, but that dimensions are not specific units.

My point exactly.
We can define angles with respect to either radians, degrees, gradians, grons, or revolutions.
Still, we have to make sure to identify that unit of angle, because otherwise our calculations come to naught.

 

[anorlunda]

We're running out of new things to say; this thread is closed. PM a mentor if you want it opened to say something new.