The Dimensionality of Torque
Just clearing up a little thing that’s always bugged me...
“Torque” is rotational force: the measure of how hard you’re twisting something. We don’t measure it directly; we can only take a definite measurement of it by gauging the amount of linear force that it exerts at a given distance from the axis of rotation. Because of how levers work, this force is high if you’re close to the axis and low if you’re a long ways off. The linear force times the length of the lever-arm from the center equals the torque. So we measure it in foot-pounds or newton-meters. If you double the length, you halve the force, and the product is the same, so the particular force and length numbers don’t matter — only the combined value does.
But wait — force times length equals work. Is torque in some sense the same thing as energy? Hell no. Torque is static; you aren’t doing work until you make it move. If you exert force at the end of the lever through some distance, thereby rotating something while exerting that torque, that’s work. If you continue to do so over more distance, that’s more work. Again, doubling the lever length halves the force at the end, but doubles the distance it has to travel for a given angle of turn, meaning that the particular lengths don’t matter — the work is the same for a given angle of turn.
Work equals torque times amount of rotation. A valid SI-ish unit of torque is not newton-meters, but newton-meters per radian. Or joules per radian, if you wish.
So why don’t people mention the radians? Because radians are defined as arc-length over radius: a distance divided by a distance. This cancels out to a pure number, a dimensionless ratio. Or so it is traditionally argued.
But we’ve clearly lost something there. The rotation is a very real physical thing, and its presence or absence makes all the difference, as we’ve seen, when relating torque to work. For that matter, just trying to solve for how far a projectile will fly, given its speed and angle, falls apart dimensionally if you say the angle is just a number. I think it’s time to acknowledge that when we divide a curved distance by a straight distance, what’s left is the curvature, and this should be treated as an independent dimension in its own right.
I notice there are contexts where angles of rotation don’t get neglected. For instance, the unit of magnetomotive force is the “ampere-turn”. In everyday terms, the strength of an electromagnet is determined by the amount of electric current multiplied by the number of times that the current winds around a circular or spiral path. Perhaps the fact that they happened to count full turns this time instead of radians (which they could have, and perhaps should have) made it harder to ignore.
The two are equivalent as a dimension; to convert you just multiply or divide by 2π, which is the number of radians per complete 360° turn. Many who work in fields involving rotation or oscillation do this all the time. For instance, people who work in acoustics, or in radio-frequency analog electronics, often measure oscillation frequencies not in complete cycles per second (Hz), but in radians per second, just to work with equations which have fewer instances of 2π in them. These radian frequencies are typically designated with ω, the lowercase Greek letter omega.
The use of Hz is a slippery slope in itself. It’s handy because it abstracts the idea of frequency — it says “per second” without being specific about what is being counted. But if you look at specific cases, it’s clear that there are dimensional distinctions to make. The rotation of a motor in turns per second is not really commensurable with the oscillation of a pendulum in swings per second, or the rate of fire of a machine gun in bullets per second — let alone something like the allocated frequency range of a wifi channel. But the idea of frequency as an independent concept is highly useful nonetheless. And there are certainly situations where different processes can be linked together so they are synchronized, as in a grandfather clock, wherein the swings of the pendulum are linked to rotations of the hour hand, with a fixed ratio of 21,600 between their frequencies.
Another dimensional abuse I object to is the practice, common in the aerospace industry, of measuring “specific impulse” in seconds. But the people who do this actually understand perfectly well that it’s incorrect; they’re not pretending a dimension isn’t there, they just prefer to make part of it implicit and unstated. (It started as a workaround to avoid having to convert between feet and meters, when NASA was trying to work with ex-Nazi rocket engineers.) When specific impulse is stated in seconds, it’s understood that you need to multiply that by the Earth’s gravity to arrive at the true figure, which is measured in meters per second. If you want to state it in terms which make sense in the original context, it’s momentum over mass, or Newton-seconds per kilogram... but since a Newton is kilogram-meters per second squared, that cancels down to meters per second.
(As an aside, we really ought to have a decent name for the unit of momentum, which is a kilogram-meter per second. It’s one of the most basic and essential of the derived units, and we have to refer to it by cumbersome compound names. We should, like, name it after Galileo... nope, woops, the “gal” is already taken, for a unit of acceleration which is no longer in active use because it’s based on centimeters. Aha! Let’s Name it the “bur”, after Jean Buridan.)
Traditional analysis regards mass, distance, and time as the only fundamental dimensions; since then we’ve added quantities such as electric charge, and (when in a generous mood) various arcane quantum properties for which the universe seems to have conservation laws, such as quark type and “color”, and whatever it is that is shared by electrons and electron-neutrinos, but distinguishes them from muons and muon-neutrinos. But these “material” values, as we might call them, don’t cover everything we need.
Some dimensional analysis enthusiasts have been saying that angles may not be the only neglected metric. A minority even argue that inertial mass, gravitational mass, and “substantial” mass (whatever that is) are at least two and maybe three subtly distinct dimensions which just happen to always coincide in value. (And indeed, the close relationship between energy, gravity, and inertia is still a mystery, despite apparent confirmation of the Higgs boson.) But even without that, we all too easily forget that length is not a single dimension when dealing with vectors in space — each axis is separate, just as you’d think from hearing the word “dimension” in the first place.
Donald Siano has shown that if you get sophisticated enough with vector dimensions, it can express the concept of a measure of angle or rotation as a derived quantity. He even shows that mathematical functions that we think of as pure numbers are incommensurable: for instance, sine and cosine are dimensionally distinct, and it makes no sense to add a cosine value to a sine, or (in further extensions by others) a logarithm to anything not logarithmic. Such mathematical expressions cannot correspond to anything physical, and indeed, you should not normally ever see them when doing even the most abstract math. So if we go for the full potato on using his “orientational” analysis, we don’t need a separate dimension just for angles. But that’s quite cumbersome for everyday use — if we don’t want to be constantly doing 4×4 matrix arithmetic to keep our dimensions straight, we can just treat angularity as a dimension.
On the opposite side from those who want to nitpick subtly distinct properties of mass, some argue that the universe’s fundamental constants mean that quantities such as time and distance and mass and charge really are all commensurable, and at bottom there’s only one fundamental dimension (probably best expressed in our terms as energy), and all natural laws are dimensionless. If so, so what, I say — that approach makes dimensional analysis less useful rather than more so. And if you think about vectors, space and time are still incommensurable, especially since in relativistic geometry, lengths on a timelike axis are imaginary numbers.
Update, 2023 — Check out XKCD 2748: