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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)!)
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Units in general can be a complicated topic: I frequently have trouble explaining how torque and work have different units, despite both being Newtons times Meters. Are there other such quirks hiding deeper in physics?
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I should probably sit down sometime and figure out what's really going on with the units of torque and energy. It's a little bit of a fib to insist that torque and work have different units, of course, since simply multiplying a torque by an angle in radians (which as noted have no units) does give work after all. The best comment I can make offhand is that really, I think they do have the same units, but that having the same units is not a sufficient condition for two quantities to be fundamentally "the same thing" (though it is certainly a necessary one). It just so happens that the laws of physics never arrange themselves to add a torque to an energy, even though the units would match up fine.
Maybe I should be careful saying that, though: it's always possible that there's some subtle exception that I haven't thought of. (I'm choosing not to count my earlier "torque times angle" example as an exception.) After all, the "coincidence" that pressure has units of energy density gets used directly when the two are added together in Bernouli's theorem, and the connection between them is actually pretty deep.
Hmm. Crazy new conjecture (that I'd need to do some work to even be sure it's sensible): what if "radians" isn't truly unitless, but only has units in some "topological" sense. More specifically, what if rad^2 is truly unitless, but rad alone is not? This isn't entirely crazy: I could imagine some subtle connection to spin and to SU(2) as the double cover of SO(3). Torque (defined as dL/dt) has one power of radians, while torque*angle has two (and is indeed an energy). Rotational energy 1/2 I ω^2 obviously has units of rad^2.
You know, if I could actually prove there was something to this, I could probably get a paper out of it. Interesting thought.
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Stu:You know, if I could actually prove there was something to this, I could probably get a paper out of it. Interesting thought.
Grand unification theory, here we come! I always knew you'd get it some day ;-)
--Beth
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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.
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Same deal for rotational units. And why are radians a more natural unit for rotational measurements than revolutions? I would think (at the laymens' level) that 1 revolution = 1 is much more natural than 1 revolution = 6.28...
Note: not sure I really want an answer to that question - the real point is that all unit systems are arbitrary to varying degrees, er, amounts.
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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.)
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Obviously radians for circular angles: in addition to making physics equations "prettier", it simplifies things like series-expansions of the trig functions. And as John points out, solid angles have a "natural" set of units --- and it isn't radians! Likewise the quaternions have a "natural" scaling/representation, where the scalar part is equal to 1 (not pi or e or i or somesuch). And so forth.
mixing unit with dimensions
To convert from m to km, I multiply by 1 = (1 km)/(1000 m).
To convert from rad to degree, I multiply by (360 deg)(2pi rad). Ergo, deg<->kilometer, not kilo, unless you want to add the ad-hoc rule the degree is a short name for degreeradian.
If I stick a measurement is Coulombs into |F|=q_1q_2/|r|^2, I get the wrong answer. That doesn't mean the Coulombs is in the same status as kilo. Also, note that even thought Coulombs and statcoulombs are both units of the same physical quantity they in fact have different dimensions: [Coulombs] = [C], whereas [statcoulombs]=[L^(3/2)M^(1/2)T^(-1)], since SI has 4 fundamental units whereas cgs only has 3.
Different units can have the same dimensions. Different physical quantities can have the same units are/or dimensions. Phase-space volume and angular momentum are both multiples of h, that doesn't make them equivalent quantities.
Moral: units, dimensions, and physical quantities are all related yet distinct. Hence, merriment ensues.
--Itai
radians are a unit dangit!
First you assert that degrees and rotations aren't units because they are just counting things. Huh? What unit isn't? You are taking some concept defining a measurement to cut it up into and counting the number of units it now divides in. This is true of all units.
Now for radians physicists insist that radians have no units because A=rθ for θ measured in radians, and A and R have the same units. I disagree A=crθ where c is a universal constant with units of 1/[units of angle] for rotation c=2pi [1/degree] for radians c=1 [1/radian].
Now Torque has units of Joules/radian, not the same as energy and so not the same thing.
To bad we would have to fight decades (centuries?) of convention.
radians are dimensionless, but revolutions are not
I suggest treating Nemene's constant c in the same way that Itai reminds us that 1/(4 pi epsilon_0) is treated when switching from SI to cgs. I'll lay out my rudimentary understanding of switching from SI to cgs and then try to apply the same method to Nemene's constant c. I request that a theorist, who is more comfortable with cgs than I am, show me if I'm wrong.
In SI, Coulomb's law reads F = 1/(4 pi epsilon_0) q_1 q_2 /r^2, and 1/(4 pi epsilon_0) has a value of approximately 9e9 N m^2/C^2 or equivalently 9e18 dyne cm^2/C^2. If we want to avoid having to rewrite the constant 1/(4 pi epsilon_0) we can change the units of charge such that this constant is numerically equal to 1. I'll call this new unit of charge statC'. We simply define 1 statC' to be equal to approximately 3e-10 C. This makes the constant 1/(4 pi epsilon_0) equal to 1 dyne cm^2/statC'^2. Note that it still has units. This saves on computation, but you still have to remember the units of 1/(4 pi epsilon_0). So we go farther and redefine what we mean by charge to incorporate the units of 1/(4 pi epsilon_0) into the q's, keeping the quantity q_1 q_2/(4 pi epsilon_0) the same. Now the constant 1/(4 pi epsilon_0) is 1 with no units. This leaves us with the cgs system of measuring charge in units of statC (without the prime). Note that when measuring a charge, choosing units of statC or statC' will give numerically the same value, but with different dimensions. The dimensions of statC can be expressed in terms of other fundamental dimensions, whereas C or statC' could not be. I would argue that statC and C are not units of the same physical quantity. We have, after-all, redefined what we mean by charge.
Now, as Nemene states, an arclength is given by s = c r theta, where c is a constant equal to 2 pi (1/revolutions), which I'm trying to argue has dimensions of [1/angle]. I can define a new unit of angle rad' such that c is numerically equal to 1 but it will still have the same dimensions and units of (1/rad'). In order to be able to completely ignore the constant c, I need to redefine what I mean by angle and incorporate the dimensions of c into the angle. This gives us a different system with the unit of rad. In this new system, the constant c, is equal to 1 and is dimensionless. This also results in our new definition of angle, as measured in radians, being of different dimensions than our original definition of angle, as measured in revolutions. It just so happens that our new definition of angle, as measured in rad, is dimensionless. This new system is what I claim we typically work with, and when we say "You have to use radians." We mean, "If you use revolutions (or degrees), then you need to remember the constant c and its dimensions." In the same way if you use SI, you need to remember the constant 1/(4 pi epsilon_0).
This seems far too cute, so I assume that I'm forgetting something. Please enlighten me.
Nathan