usually in physics (at least in the SI) angles are regarded as dimensionless. Is it possible to give a dimension (and a unit) also to angles and still have a system of units of measure as coherent as the previous one? Is there something in the concept of dimensional analysis that prevents physicians from doing that? Please note that I'm not saying it is convenient! I'm just asking if it is possible.
Read below only if you find this question stupid.
The dimensionless argument is often justified saying that the definition of "angle" is given by a length divided by a length.
I think this is just false since the definition of an angle is "two rays sharing a common endpoint", it is not even in the definition of "measure of an angle" since it is the comparison of the angle I want to measure with my fixed unit angle, something that is not related at all with length. Apparently this is only due to the operation we want to do with angles: multiply them with the radius of a circle to get the length of an arc. Here we can face the problem from two different points of view:
- Forget about multiplying angles and length together. A bit drastic but we might be able to construct our physics anyway. Honestly I don't remember doing this multiplication of angles and radius much often, especially in important theorems.
- In the same way in which we jump magically from charge to force in Coulomb's law we can invent a constant that like Coulomb's constant is not dimensionless and allows me to go from angles to lengths.
Another thing I was considering was to measure the radius with the same unit we have for length associating to each angle the measure of the corresponding arc on the unit circle, but I soon realized that this leads to problem when we do conversions since the unit circle in one unit is different from the unit circle in another unit...
Still if we can fix a certain length we can use it to fix the length of the circle we use, is much complicated since is like introducing a measure for angles by firstly introducing a second measure on length but what's the problem? My question is not if this method is efficient, it is about coherence!
Apparently on physics.stackexchange the difference between "one good method" and "the only way of doing it" is not well understood since the answers I got were embarassing: https://physics.stackexchange.com/questions/226332/dimension-of-an-angle
I hope to find better answers here. At least I would find a better definitions of dimension.