B Radian unit -- Why neglected in dimensional analysis?

1. Dec 26, 2017

Mr Genius

Why radians are usually neglected in dimensional analysis?

2. Dec 26, 2017

Staff: Mentor

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.

3. Dec 26, 2017

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.

4. Dec 26, 2017

sophiecentaur

Because radians are dimensionless - they are a ratio.

5. Dec 26, 2017

Staff: Mentor

6. Dec 26, 2017

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.

7. Dec 26, 2017

sophiecentaur

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

8. Dec 27, 2017

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.

9. Dec 27, 2017

Staff: Mentor

Did you read the Insights ariticle?

10. Dec 27, 2017

I like Serena

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

11. Dec 27, 2017

Staff: Mentor

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.

12. Dec 27, 2017

I like Serena

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.)

13. Dec 27, 2017

Staff: Mentor

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

14. Dec 27, 2017

I like Serena

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.

15. Dec 27, 2017

Staff: Mentor

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.

16. Dec 27, 2017

I like Serena

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?

17. Dec 27, 2017

Staff: Mentor

18. Dec 27, 2017

I like Serena

Hold on, it says:
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?

19. Dec 27, 2017

Staff: Mentor

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.

20. Dec 27, 2017

I like Serena

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.

21. Dec 27, 2017

Staff: Mentor

I gotta 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.

22. Dec 27, 2017

I like Serena

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.

23. Dec 27, 2017

Staff: Mentor

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.

24. Dec 27, 2017

I like Serena

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.

Last edited: Dec 27, 2017
25. Dec 27, 2017

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.