Insights Can Angles be Assigned a Dimension? - Comments

[Aufbauwerk 2045]

I think this is an important topic. What is more basic to physics than how we measure things?

In SI units the angle is not a base quantity. We state the angle in radians, which is a ratio of two lengths, namely the length of the subtended arc to the length of the radius, giving a dimensional ratio of L/L = 1. So we say the angle is dimensionless.

Some people have argued for making the angle a base quantity. See for example https://arxiv.org/ftp/arxiv/papers/1604/1604.02373.pdf.

It may be more convenient to keep angles dimensionless. Consider two similar triangles. Perhaps they are both 30-60-90 triangles but the hypotenuse of triangle #1 is twice the length of that of triangle #2. The corresponding angles are equal, but the corresponding sides are not. I suppose it's fair to ask why this is more convenient.

I think this question is related, at least subjectively, to time. The ancients came up with 360 degrees because it corresponds to a 360-day year in some ancient calendar. You can associate a point moving around on the circumference of a circle with the passage of time. We don't care how long that circumference is. We just want to know how many units of time have passed. We can associate units of time with degrees around a circle.

For example, consider our standard analog clock. It may be a wristwatch or Big Ben. In either case, we know that when the little hand is at a certain angle from straight up, it means 20 minutes past the hour.

In physics, in general, the study of periodic motion is an enormously important topic. Therefore, we want our system to be convenient for the mathematics of periodic motion.

I plan to read the above paper and think about this some more. It would also be a good time to review Bridgman's Dimensional Analysis.

 

[Stephen Tashi]

It's your topic. Maybe "why does dimensional analysis work?"

My question is "What do we mean when we say an particular convention of dimensional analysis 'works'?". For example, what criteria are we using to say that assigning angles a dimension "works" or "doesn't work" ?

 

[haruspex]

My question is "What do we mean when we say an particular convention of dimensional analysis 'works'?". For example, what criteria are we using to say that assigning angles a dimension "works" or "doesn't work" ?

My criterion is quite simple: can a dimension be assigned in a way which is consistent and has the power to detect blunders? I believe I have shown that the answer is yes.
But from your posts in the thread, I thought you were focused on the more fundamental question of why DA works at all. E.g., your observation regarding products versus sums.

 

[scottdave]

Interesting concept. One thing I noticed was: in cases where you multiply (or divide) by 2pi, the article suggests that pi has the dimension of Angle. Since there are 2pi radians in a full circle (or full cycle of oscillation), then shouldn't it be 2pi containing the dimension, rather than just pi?
I do like the fact that this way puts a direction in as a dimension, when quantities are vectors.

 

[scottdave]

Interesting concept. I'm not sure if I am ready to adopt it, but it gives some things to think about. You have certainly put a lot of thought into how to handle various situations. In your section 3.6 discussing Planck's Constant. You show hbar to have dimension of ML2T^−1Θ, but I think you should have ML2T^−1Θ^-1, to reflect π in the denominator, and to cancel out the ω.

 

[haruspex]

Interesting concept. I'm not sure if I am ready to adopt it, but it gives some things to think about. You have certainly put a lot of thought into how to handle various situations. In your section 3.6 discussing Planck's Constant. You show hbar to have dimension of ML2T^−1Θ, but I think you should have ML2T^−1Θ^-1, to reflect π in the denominator, and to cancel out the ω.

Θ²=1, so Θ and Θ^-1 are the same.