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I had a neat realization during a conversation with my colleague Cameron today, so I thought I'd share. Feel free to tune this out if you're not a math/physics type.
Cameron was pondering the various "units" for angles: the common ones are degrees, radians, and rotations (a.k.a. revolutions). Physics students learn that many common equations (such as v = r ω) only work when angles are in radians: they given wrong answers if you use degrees or rotations. It's a standard story: radians are defined in terms of a ratio of lengths (arc length/radius) so they are automatically dimensionless (since a ratio like "meters/meters" cancels out).
But in what sense is "rotations" not dimensionless? After all, you're just counting things, so what you end up with ought to be a pure number (which has no units by, definition). For that matter, "degrees" is just counting things too (albeit more finely spaced things). Given that multiplying by a pure number can't possibly change the units of your answer, it's hard to understand on a conceptual level why angles measured in degrees or rotations shouldn't work in the equations. (The equations themselves are entirely clear about what works and what doesn't, mind you! It's just understanding them that's subtle.)
The resolution that I've come to is that "degrees" and "rotations" are not units in the sense that "seconds" or "meters" are units. They can't be, as noted. Instead, "degrees" and "rotations" have the same status as SI prefixes like "kilo" and "micro". After all, "kilo" effectively translates as "times 1000", a pure number, but you wouldn't expect your equation to give the right result if you removed a "kilo" that was supposed to be there! So I would assert that one can think of "degrees" as an SI "prefix" meaning "times 2π/360" and "rotations" as an SI prefix meaning "times 2&pi". Explicitly plugging in these definitions then just gives your result in terms of the base unit, the most natural one: radians.
It may not actually be all that helpful in practice (my advice is still "just use radians"), but I think this gives a cute new perspective on what these angular "units" actually mean.
Update: Another useful note: The choice of "base unit" occurs when we define the angle in radians: θ = s/r. That choice then carries through the rest of the equations that we use for angular motion. If we had instead defined the angle as θ = s/(2πr), the "base unit" would have been rotations, and that factor of 2π would show up in every other rotational equation. Radians are clearly the simplest choice, but there isn't anything privileged about it. (Just imagine the ugliness of equations throughout rotational motion if we decided to choose our fundamental definition of angle to be θ = 360s/(2πr)!)
Cameron was pondering the various "units" for angles: the common ones are degrees, radians, and rotations (a.k.a. revolutions). Physics students learn that many common equations (such as v = r ω) only work when angles are in radians: they given wrong answers if you use degrees or rotations. It's a standard story: radians are defined in terms of a ratio of lengths (arc length/radius) so they are automatically dimensionless (since a ratio like "meters/meters" cancels out).
But in what sense is "rotations" not dimensionless? After all, you're just counting things, so what you end up with ought to be a pure number (which has no units by, definition). For that matter, "degrees" is just counting things too (albeit more finely spaced things). Given that multiplying by a pure number can't possibly change the units of your answer, it's hard to understand on a conceptual level why angles measured in degrees or rotations shouldn't work in the equations. (The equations themselves are entirely clear about what works and what doesn't, mind you! It's just understanding them that's subtle.)
The resolution that I've come to is that "degrees" and "rotations" are not units in the sense that "seconds" or "meters" are units. They can't be, as noted. Instead, "degrees" and "rotations" have the same status as SI prefixes like "kilo" and "micro". After all, "kilo" effectively translates as "times 1000", a pure number, but you wouldn't expect your equation to give the right result if you removed a "kilo" that was supposed to be there! So I would assert that one can think of "degrees" as an SI "prefix" meaning "times 2π/360" and "rotations" as an SI prefix meaning "times 2&pi". Explicitly plugging in these definitions then just gives your result in terms of the base unit, the most natural one: radians.
It may not actually be all that helpful in practice (my advice is still "just use radians"), but I think this gives a cute new perspective on what these angular "units" actually mean.
Update: Another useful note: The choice of "base unit" occurs when we define the angle in radians: θ = s/r. That choice then carries through the rest of the equations that we use for angular motion. If we had instead defined the angle as θ = s/(2πr), the "base unit" would have been rotations, and that factor of 2π would show up in every other rotational equation. Radians are clearly the simplest choice, but there isn't anything privileged about it. (Just imagine the ugliness of equations throughout rotational motion if we decided to choose our fundamental definition of angle to be θ = 360s/(2πr)!)
no subject
I consider units to be a lot like data types in computer science: They restrict operations to help keep your computations sensible, and the types often really mean something. Units behave in certain understandable ways when you add them, multiply them, do calculus on them, etc.
I disagree on rad^2 being unitless (or more unitless than rad), though: It's the fundamental "unit" for solid angle. Also, rotational inertia is a tensor and not a scalar, so rotational energy really represents two steps of simplification from the tensor to a scalar.
If you ignore the whole tensor/vector/scalar property, then it behaves as you say: I'm just worried that that's not really a useful way of thinking about units.
This also has me thinking about natural units and the fine structure constant, which probably relates to this somehow.
no subject
Yeah, the more I think of it, the more I suspect that I'm just imagining keeping track of tensor rank in my units. Or at least, doing so would make this "rad^2=1" idea unnecessary. Good point about solid angle, too. (I knew I'd need to think some more before I knew whether the idea was even sensible. :) Ah well.)