When writing a complicated unit string in which several units are being multiplied together, is there any preferred way to order them? For example thermal resistivity could be written in (m K)/W or (K m)/W. To me both look correct and equally preferable, but is that true?
Thanks!
From my reading and usage, there definitely does seem to be a preference (for instance, one would never write the units of momentum as m kg/s), but I’ve never seen any formal rules laid out for what it is. Nor have I seen any rules for how to group units into compound units (for instance, J m[sup]-3[/sup] s[sup]-1[/sup] could also be W m[sup]-3[/sup]: Which is preferred?)
The APS style guide doesn’t say anything about this question.
It just comes down to convention, much like the ordering of variables in expressions. PV=nRT could just as well be VP=RnT. Some of these conventions can be traced back to whomever first wrote the expressions down. Some are lost to time. Some units or expressions have a natural ordering to prevent ambiguities or to look cleaner typographically. And some have no clear ordering consensus at all (e.g., ft-lb vs. lb-ft for torque).
Commonly encountered unit groups act as a single calculational and linguistic entity, so it’s natural for their “spelling” to become frozen for clarity. It’s also natural for past spelling conventions to influence future related ones. But I’ve never seen any attempt at a formal study of unit ordering (either descriptive or prescriptive).
A related topic is the choice of which symbols should represent which quantities. There are strong conventions (somewhat field-dependent) about what is implied by the choice of a symbol, yet these are similarly not (usually) codified and are (usually) of historical origin. A lot of meaning is conveyed through symbol conventions, in contrast to the ordering of units which doesn’t usually contain implied information.
It’s been forever since I’ve been in physics, and while I’m sure you’re right that you’d never write a (formal) formula out in some backwards way, there’s no reason you can’t do it, especially if you’re in the midst of an equation and it helps to get things cancelling out (or whatever other operations need to be done to it).
OTOH, using your example, I can see a physics 101 teacher tossing kg/s x m as a unit, just to see how many students catch it as momentum (work in a properly placed s[sup]-1[/sup] and you’ll get Newtons). Especially after the section on converting units.
If this is something that a teacher, supervisor, peer, the public is going to see, write them “properly” (or just use the actual unit P, N etc), but after the first line (or if it’s just for yourself), rearrange it in whatever way makes it easiest to work the algebra. No reason to drop a unit or flip a sign on line 3 of a 60 line equation just because you were trying to maintain convention.
It might be just coincidence (1 in 12 chance of it), but I do note that those variables happen to be in alphabetical order on both sides of the equation. Of course, the physicist’s version (PV = NkT) is then non-alphabetical, but that’s influenced by the molar version, which I think was derived first.
It’s funny you mentioned torque, Pasta, because the SI unit of torque is the joule. Except it’s never actually called that: It’s always a newton-meter. Even though a newton-meter is a joule.
Oh, and in the OP’s example of thermal resistivity, I’d be inclined to put the kelvins last, just to avoid any possibility of confusion with “kilo”. Though that might even be worse, because now it could be misread as millikelvins.
I think this leads to a principle that units are not just bits of math that we manipulate algebraically (although they are that too), but a notation for human use.
A joule is indeed a newton-meter, but in the context of torque, spelling it out makes it more obvious what’s going on: a force of one newton applied at a distance of one meter.
For energy, though, that’s not always a useful description. Maybe a watt-second is better, or a coulomb-volt, or a kg-m[sup]2[/sup]/s[sup]2[/sup]. All equivalent of course, but each gives an indication of where it came from. And if we don’t care about the origin, we just call them joules. We use the notation that is most descriptive even if they are mathematically identical.
And by the same token, the unit of electric field is either a newton per coulomb, or a volt per meter, depending on the context (and almost never a kg m s[sup]-3[/sup] A[sup]-1[/sup]).
I have to admit, I didn’t know that Newton and Joule were SI units. I see they’re ‘derived’ units. So, my question is, does that bastardize that point of SI units (being all base units) or does it simply expand what we consider to be SI units since things like Newton and Joule (and the others) are still base units (as opposed to, say, kiloJoule or MicroNewton).
The only thing I can come up with is that derived units are combinations of single SI units. So you could have mass x meter x amp, but not mass x (.5)meter x amp.
I wonder how many people try to come up with some kind of derived SI unit just to get their name in the books
Also, I wonder how many SI purists there are that reject SI derivatives as SI units.
I wonder how much of a new term becoming an SI unit involves the term being useful and part of everyday parlance.
Nitpick: whoever
That’s because they decided an angle wasn’t a unit, which would be silly, except that most of the time it makes the math easier, and so people go with it. Torque versus energy is one of the exceptions.
OK, put it this way: Is hbar the quantum of action, or of angular momentum? Now multiply it by a frequency: Is that an energy, or a torque?
Torque and Energy are not the same concept, but come out as using the same units if you ignore angles. Torque should have units Joule per radian, not Joule, as it is torque being applied over some angle that leads to work being done.
I’m not exactly sure what you’re getting at. A “derived” unit is not a base unit because it’s not something you can define independently. You can define the kilogram, second and meter independently. But once you’ve done that, the natural unit of energy is (kg m s^-2). We could have called it anything we wanted, but we couldn’t have defined it as some other value. If we had, we’d have had to deal with a conversion factor every time we calculated energy.
I was under the (mistaken) understanding that SI Units only included base units for which we have some set reference for and aren’t derived from anything else. Somewhere, there’s a hunk of metal, it’s one kilogram, we’ve agreed upon that, all other masses (and therefore weights) are based on that. One second was based on a portion of day (now the time it takes for something to vibrate one cycle, or something like that) and the list goes on. Disagree, and we can go back to the standard and measure it to get our base back.
To simplify what I was saying: I’m not sure why units derived from those are SI units. It’s not really something I want to get into an argument about, I don’t really care that much about it one way or the other, it just seems to me there’s no point in having them in there. I assume there’s a reason, though.
Like I said, I just thought SI units were base units only
To be honest, it’s been so long since I’ve had to do anything involving SI units, if you asked me I would have said Gram instead of Kilogram (fairly, too, I think).
The point of having them in there is that SI is about standardization. Units of energy is something people have found to be convenient and still find convenient, so SI has one single defined unit of energy, and where there are alternative ways to refer to units and compound units, such as J and Nm, the conversion factor will always be 1.
It’s not much of a system if you only have units for a half-dozen different sorts of measurements. A proper system of units will include a unit for every sort of measurable quantity. Now, some of those units won’t have names of their own, but they’re still perfectly valid units, and part of the system. For instance, the SI unit of momentum is the kg m/s. It’s kind of odd that this this unit, as often-used as it is, doesn’t have a name, but it’s still a unit, and it’s still part of the system.
I guess that makes more sense. Like I said, this isn’t something I’ve given a lot of thought too. I changed my major from physics to math in 1999, IIRC.
One thing that I don’t really get is why SI has seven base units. To my way of thinking, there are only two.
The kilogram is an obvious one in that it’s based on a physical artifact. The second is also obvious since it’s just X cycles of a cesium atom.
The meter, though, is known exactly if you know the second; it’s just the distance light goes in ~1/300000000 s. And not even that, really; light just happens to be the most easily accessible thing that moves at c. You wouldn’t need that if you had a 4-dimensional ruler. And really you don’t need a separate unit in the first place; just measure things in light-seconds (even metric-haters can get behind the light-nanosecond).
Next, the amp. According to Wikipedia: “The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 × 10[sup]-7[/sup] newton per metre of length.” Again, everything you need is derived from units that you know (the newton is a kg-m/s[sup]2[/sup], of course).
How about the kelvin? Well, it is defined as 1/273.16 of the triple point of water. So definitely a base unit as defined, but doing it this way seems silly. Instead, they can just fix the Boltzmann constant, which relates joules with kelvin. Joules are already known from kg-m[sup]2[/sup]/s[sup]2[/sup], so the kelvin pops right out.
Next up, the mole. Like the kelvin, the only reason this has a unit of its own is because no one has bothered to fix Avogadro’s constant. Do that, and you don’t need it as a separate unit.
Finally, the candela. This is the least deserving of base unit status by far, since it’s based on human perception. The correct unit of light intensity is the watt. People can apply whatever perceptual model they want afterward; you shouldn’t build it into the system.
So as far as I’m concerned, there are only two base units: mass and length-duration. The remainder are either (semi-disguised) derived units or bogus.
I should also mention that the kg and second are not the only two possible choices for “true” base units; one could start with the coulomb instead, and from that derive the amp, and from that the kilogram. But there are still just two degrees of freedom here.
It seems that torque being a vector cross product between force and distance and not being equivalent to energy which is a scalar ought to be sufficient reason for distinctive units.
You can, in principle, extend this logic so that there are zero base units. Here’s how:
First, remember Newton’s law of universal gravitation: all masses will attract all other masses. Assuming you’ve already defined the second, and used the speed of light to define the meter, you define a kilogram to be that mass which will cause an acceleration of such-and-such amount in another mass a distance of 1 meter away. This is an incredibly hard experiment to do, since gravitational forces due to normal-sized masses are incredibly weak, but it can in principle be done. This would be an operational definition of the kilogram.
Second, note that the product of any quantum-mechanical particle’s momentum and its wavelength are equal to Planck’s constant, h. This has units of kg m^2/s, and will have a particular numerical value in the units we’ve chosen. If you go carefully through the logic above, you’ll find that doubling the length of the second would double the length of a meter and double the mass of a kilogram; thus, if we re-defined our units this way, the numerical value of h would go up by a factor of four. In particular, this means that we could re-scale all our units in such a way that Planck’s constant was equal to 1.
In this way, we could define a set of units that is based solely on the fundamental constants of the universe: Newton’s constant, the speed of light, and Planck’s constant. In fact, theoretical physicists do this all the time; the procedure is slightly different, but the motivation is the same. The resulting units are called Planck units, and they make things easy when you’re doing theoretical calculations because you don’t have to carry around factors of G or h or c all the time.
I agree with you that the mole and the candela are bogus units, though.