Dimensional Analysis and Units
Dimensional Analysis and Units/SI episodes were first published in two parts in August 2018, the first is here and the second is here.
This is an episode on two subjects. One of them sounds amazing but is actually much more mundane, and one of them sounds mundane but is actually pretty amazing. But, in time-honoured Physical Attraction fashion, I’m going to start some way off the beginning and then find my way to the starting line.
In Paris, city of light, city of love, city of clichés, you can visit all kinds of places. You can visit the Eiffel Tower, which commemorates the anniversary of the French Revolution. Robespierre and many of the others involved in that conflict had a dream that, when they overthrew the Ancien Regime — that creaking power-structure that had dominated and oppressed France for so long — that they would bring in a new age of rationality and intellectual enlightenment.
But like so much in the French revolution, the good sentiment that “reason will prevail” devolved into madness and chaos. If you’re being charitable to the revolutionaries, you can say that they were ahead of their time — that they rejected the dogma of the Catholic church, but in 1793, struggled to find an alternative worldview to put in its place. If you’re being more reasonable, you can say that this devotion to rationality — like the revolution itself — struggled under the weight of its own contradictions. Who gets to define what is and isn’t irrational? Who sets the standards?
So it was that you end up with the almost oxymoronic and violently atheistic “Cult of Reason”, with people encouraged to worship and venerate ideals such as “truth”, “virtue”, and “reason” in place of the old deities. This degenerated into Robespierre’s “Cult of the Supreme Being”, which tried to restore god without the power of the Catholic Church. The Cult of the Supreme Being, along with Robespierre and Saint-Just’s tendency to centralise control in a tiny undemocratic committee for “public safety”, and execute a hell of a lot of people during the Great Terror, lead in part to the downfall of this first iteration of the French Revolution. A few years and political convulsions later, this would lead to the rise of Napoleon, and he left his mark on the city in the form of an imperial arch in the old Roman style to celebrate his military victories: the Arc du Triomphe.
Yet there is another legacy of the French Revolution’s push for rationality that you can find in Paris. If you go to the Musée des Arts et Métiers, you can see all kinds of things. A supercomputer from 1985, which now has less raw processing power than a couple of mobile phones. A hilarious pair of binoculars from the 17th century which got around the tricky “focusing on faraway objects” thing by being so long that no object is truly far away. Then there’s the Avion III, an early attempt at an aircraft shaped like a bat and powered by steam.
“The first flight was attempted on 14 October and most sources agree ended almost immediately in a crash without ever leaving the ground. Later in his life, the inventor claimed that there had been a flight of 100 m (328 ft) on this day, and said he had two witnesses to confirm it. Whatever actually happened, the French military was unimpressed with the demonstration and cancelled any further funding.”
But also, in this museum, there’s a lump of metal. Okay, to be more precise, a bar of a special alloy, 90% platinum and 10% iridium, that is almost exactly one metre long.
The “almost exactly” part is a fairly new invention, because — up until 1960 — it was exactly a metre. Not to the nearest millimetre, or the nearest micrometre. This bar was exactly one metre long, because this bar defined what the length of a metre was. In fact, from 1799 right the way up until 1960, the answer to the question “What is a metre?” was pointing at some metal bar and saying “It’s the length of this bar.”
It might seem a little arbitrary — but in a lot of ways, units are just something that we all agree upon. They are yardsticks for describing the world. The idea of a day we can all sort of comprehend: after all, it defines when the earth rotates on its axis. It’s the cycle by which things get light and dark again. Once a day, you will probably feel an overwhelming urge to sleep, and so your conscious experience of the world is divided into these neat little units — ignoring for a minute the fact that, if you measure days by light or by sleeps, the length of the day will vary pretty wildly. The Egyptians had twenty-four hours to the day, although funnily enough they varied, keeping twelve for both day and night, so that in the winter, daytime hours were shorter than nighttime hours.
But what is an hour — what is a minute — what is a second? These are really just convenient subdivisions, chosen because they’re easy to divide, and because they correspond to our experience. Various ancient civilizations have come up with similar lengths of time, which maybe implies that the speed of clocks is somewhat related to the clock speed of the processor in our own heads. The Babylonian second was three and a half of our seconds. It’s about the length of a thought. It’s about the length of a heartbeat, our own, internal, pulsing, irregular clocks. Those strange timekeepers of mortality, the heartbeat; dependent, in a very non-linear way, on physical and emotional intensity: beating faster when you’re at your most alive. As if to remind you that, regardless of our attempts to label it and put it in little, constant boxes marked “rational” and “scientific”, the human perception of time is slippery and far more dependent on our emotions and psychology than we’d like to think.
Things were much worse in the days before we arbitrarily started labelling and pinning down definitions about what a second was, and what a good unit of length was. Previous definitions had been pretty ridiculous — and if you think about it, a failure to agree on a unit of length universally is a pretty big deal. If I promise you 200m of cloth and I give you 200 of my metres, but it’s only 190 of your metres, you’re going to feel ripped off. And don’t even get me started on the foot. Henry VIII reportedly tried to define a foot in terms of his own height, while in the 16th century, the procedure for determining the length of a foot was literally to grab 16 random people from your Church on Sunday, line them all up foot to foot, and then divide that length by 16. I wish I was kidding. So the French revolutionaries might seem a little obsessive with their incredibly-carefully-measured bar, but they had a point.
You could repeat the calculations of physics measuring lengths in units based on your own height, and temperatures based on your own body heat. (That might sound stupid, but let me remind American listeners from our thermodynamics episode that this is part of what the Fahrenheit scale is actually based on.) Physicists regularly do work in units that are convenient for the particular situation — for example, we talked in particle physics episodes about using the “electron volt” as a unit of energy, and particle physicists will often talk about masses in terms of energies as well. You’ll regularly hear someone say that the “mass of the electron is around half an MeV”. The unit size is convenient, because saying it’s 9.11 x 10^-31 kg is cumbersome and difficult to remember, and it makes most of your numbers unreasonably small.
[Talk more about the French revolutionary history, introduction of other units of time and measurement, failed attempt to decimalize time]
So the initial attempt by those radical French revolutionaries to institute the metric system and some kind of universal adherence to truth and logic didn’t catch on. They had the right idea — standard measurements of length, time, and mass in the metre, second, and kilogram — but took things a little far. There was a chaotic rollout of a sort of decimalized calendar; alongside famously declaring the first year of the revolution to be “year zero” of a new calendar, and renaming all of the months after the weather they associated with that month, there was even an attempt to introduce decimalised time. They kept 12 months, but they were 30 days each and divided into 3 weeks of 10 days; the day was divided into 10 hours of 100 minutes of 100 seconds.
So after the Committee of Public Safety had their political downfall, and Robespierre himself actually managed to botch his own suicide and end up with the far more poetic fate of being guillotined after having guillotined so many other people… the French authorities later quietly discarded a lot of the nonsense that had been suggested and quietly kept some of the stuff that made a little more sense. The metre was defined by a metal bar; the kilogram, which was based on the mass of a certain volume of water, was a small block of platinum manufactured specifically to define the kilogram. This would remain the definition of the kilogram and the metre for over a century.
They use platinum because it doesn’t corrode. The issue, of course, is that — well — no length is really constant. Changes in temperature cause metals to expand and contract, for example. The scientists quickly realised this, and so — if you ever want to retrieve and precisely measure that 1kg, say — you have to do it at a certain temperature and pressure. They use platinum because it’s not reactive and doesn’t expand very much when temperatures change. But, eventually, defining the length of a metre by a stick of metal in some lab somewhere wasn’t constant enough. For a while, they defined it using something that everyone could hope to agree on — the emission lines from krypton gas. You’ll remember that when atoms get “excited” — energy is transferred to their electrons — they can “de-excite” and lose that energy by emitting photons. It’s this that causes those beautiful glowing clouds of gas in outer space that you might have seen pretty pictures of; the gas gets heated up and depending on what it’s made of, the atoms de-excite with different wavelengths, different colours. Every atom of the same isotope of an element is the same, and, under the same conditions, it will emit radiation of the same wavelength. So they picked the orange line of krypton because it’s nice and visible. For a while, it was defined as some multiple of this wavelength.
But this suffered in a similar way to other standard measurements, because, after all, what is a measurement? Any good measurement has some uncertainty. You don’t know precisely the value. If you measure your height with a tape measure, it might be to the nearest centimetre, or even millimetre, but below that — uncertainty. And really, it’s the same with any measurement, even if it’s only billionths of metres that you’re unsure about.
The problem was that as we kept pushing at that uncertainty limit, suddenly, it became clear that what we were using wasn’t ideal. We discovered that the bars of metal were changing too much; we worked out that the line for krypton was slightly asymmetrical so you’d have to pick a point to measure it for consistency, and so on. Eventually, because time could be measured more accurately than light, they defined the metre with the current definition that it has — which is in terms of the speed of light in vacuum, which is as far as we know one of the few fundamental constants built into the structure of our universe — and the second.
So, at long last, the metre is officially:
The metre is the length of the path travelled by light in vacuum during a time interval of 1 ⁄ 299 792 458 of a second.
About time, then. Time was at first the hardest to measure. People tried to make things nice and consistent by measuring things in terms of fractions of a year, or of a day — but these things aren’t as solid and immutable as you’d like to think. We’re constantly putting in leap days and leap seconds to correct for our years getting out of sync with the rotation of the Earth; and, what’s more, because of the tidal drag on the oceans, the Earth’s rate of rotation is gradually slowing down. The moon, through its drag on the oceans, is gradually shifting the length of a day to be closer to that of a lunar month — it’ll keep going until they’re both 47 days long. But before you have to worry about your Monday lasting for weeks, we’ll all be dead: we only gain about 2.3 milliseconds on the day every century. You might think, after listening to the TEOTWAWKI specials where we detailed all of the various threats to human civilization over the next century, that days getting a few milliseconds longer is a pretty tiny consideration to worry about. But I think it’s both important and cute that someone, somewhere, in a dusty room somewhere, is keeping track of this kind of thing for us.
https://en.wikipedia.org/wiki/History_of_the_metre
https://www.reddit.com/r/Metric/comments/48mvsw/can_one_visit_the_bipm_innear_paris/
https://physics.stackexchange.com/questions/122830/why-is-the-second-the-si-base-unit-for-time
https://sizes.com/time/time_atomic.htm
https://en.wikipedia.org/wiki/SI_base_unit
The first clocks relied on things like sundials, burning candles of a certain length, or mechanical processes that require constant winding and slow down over time — so none of these was particularly ideal either.
Anyway: it was soon realized that atomic decays offer the most accurate possible measurements of time, and it’s using atomic clocks that the most accurate measurements can be determined for time units which then get converted into a second. Ideally the clock is as close to absolute zero as possible when this decay is measured, and it’s currently determined by a transition in what’s called the hyperfine energy levels of caesium. (By the way, this is our physics based chat-up line: “Baby, I’d better include both the spin-orbit coupling and the electron-nuclear-spin interaction in a consideration of your energy levels, because you’re not just fine — you’re hyper-fine”. Yes, I know. One day I’ll explain all of that, promise.)
Out of the three first units to be defined in the international system, it’s actually only mass — only the kilogram — that is still defined by a lump of metal in a vault somewhere. Which is insane, I know. It’s called the International Prototype Kilogram and is made out of 90% platinum, 10% iridium. What’s even crazier is the security precautions around this thing. It’s in a locked vault, kept in filtered air at a constant temperature and pressure. It is taken out and carefully remeasured every forty years to ensure that the value of the kilogram is consistent. It requires three separate keys, presumably wielded by stern and serious-looking Agents of Science who will only reveal their name, rank and serial number if questioned. There are SIX identical sister copies, in case… um… in case we lose five of them. All of this feels a little bit like they’re making up for the embarrassment of having to deal with the fact that our definition of a kilogram comes from a lump of metal. But when you’re trying to be this precise, like — weighing the thing and getting 1kg is not enough. You need to be sure that the gravitational field is the same every time you weigh it. It’s that sensitive.
I am going to spare you the literally pages and pages of documents produced by… what I can only describe as some of the nerdiest nerds ever to have nerded, who are lobbying for their own pet measurement system to take over from this lump of Parisian metal. Suffice it to say that people have demonstrated that the lump is gradually changing in mass by millionths of a gram, mostly due to things accumulating on the surface which can’t be cleaned away. They will rant about how it is an embarrassment to science, and should be burned at the stake. You can see VERY LENGTHY DETAILED EXPLANATIONS of why we should be using, say, magnetic forces, or the masses of a certain number of atoms, or so on to define the kilogram. One day, we probably will, and the IPK will be consigned to a museum somewhere and eventually forgotten about as some quirk of history. But for the moment, it has its time in the sun. All masses, all forces are defined through me, says the kilogram. Fear me. I am not just a lump of metal. I am a lump of metal that weighs exactly 1kg.
[Describe the system international, the base units that are used, and the modern definitions of all of these. Now need to add on other than MLT units.]
So we have these units — the metre, kilogram, and second, that are all SI base units. And we can derive other units from them, which are called the SI derived units. So, for example: we know that force is mass times acceleration. So we can define force as mass times length divided by time-squared. That’s how 1 newton of force is defined; it’s the force needed to cause 1kg of mass to accelerate by one metre per second, per second. Similarly, energy is force multiplied by distance, so a joule of energy is the energy you get by applying a force of 1N over a distance of 1m. You can derive things like the pascal, for pressure, in a similar way. Since they can be derived from the other units, they aren’t “base units”.
Of course, as we learned more about electromagnetism, it was clear that there were more things to be measured than just mass, weight, and time. There was this thing called electrical charge, for a start, that clearly needed some units. The trouble is that measuring electrical charge turned out to be exceedingly difficult. You might think it would be easy — you just measure forces — but measuring static charges, charges that aren’t moving, is far less easy than measuring currents. The unit of current, the Ampere, was relatively easy to measure — it’s actually a little strange, as it’s currently defined as “the amount of current that needs to flow in parallel wires, separated by a metre, to produce 2 x 10^-7 N of force.” You can see that defining it in terms of this experimental setup hints at the way that these things were originally measured. So in SI, charge is actually a derived unit: 1 Coulomb is the amount of charge that flows past a point when a 1 Amp current flows for one second. From the ampere, then, you can get charge, electric field, magnetic field — stuff like that.
This is odd in some ways, because nature does in fact have a fundamental unit of charge — the charge on the electron. We don’t see any subatomic particles that aren’t multiples of the charge on the electron — even quarks, which can never be separated from the nucleus, are still only exactly 1/3 or 2/3s the charge on the electron. Nature has a well-defined unit of charge, then. But it’s tiny.
However, the Coulomb is a very big unit of charge. Lightning strikes, which seem about as dramatic an electrical event as we can imagine, only result in transferring 15 Coulombs of charge. And you need around ten million billion electrons to get up to a Coulomb of charge. But I suppose one or other of the units has to be either tiny, or ridiculously large. Similarly, it means that the magnetic field unit, the Tesla — calm down Elon Musk fans — is a little large for our usual scales. The Earth’s magnetic field has to be measured in microTesla, your fridge’s magnetic field in miniTesla, and the strongest steady magnetic field humans have ever produced is around 45T. On the other hand, 16T was enough to levitate a frog — literally due to diamagnetism in the water of the frog. The same calculations indicate that 50T should be enough to levitate a human, but it’s not that simple: the only reason the frog levitates is that there’s a huge magnetic field *gradient*; this is what determines the force.
So you’d need to create a magnetic field that’s 50T at your feet and 0T at your head, which is an insanely difficult engineering problem. But, amazingly — at least if the frogs are anything to go by — you might not be harmed that much by this magnetic field gradient, as long as it’s sufficiently smooth that different parts of your body aren’t being pulled apart. I would say “don’t try this at home”, but, if you have access to a 50T magnet, you probably also have access to liquid nitrogen and all kinds of cool stuff like that, so you can try whatever you want.
It doesn’t end with the Ampere, though. Base units also include the mol, which is really just a number that defines the amount of a substance, the Kelvin absolute scale of temperature, which we discussed pretty thoroughly in the thermodynamics episodes, and the candela. Candela is a little weird: it’s defined as the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540⋅1012540⋅1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. Which, in non-gobbledygook, is essentially “how bright things appear.” That’s why this particular frequency is chosen — it’s the frequency of light that humans are most sensitive to. The candela, as the name suggests, is kinda similar to the luminous intensity of a single candle.
You’re probably thinking — okay, but couldn’t we derive something similar in terms of, say, energy and time? Something like — how much energy from the source emerges from a particular angle at a particular time, that kind of thing? Well, yes, we could — that’s the luminosity, and that’s really what physicists use — but this sense of “intensity” is intrinsically linked to the human eye and its own sensitivity. So the candela is a little wishy-washy compared to the others, but it’s important for industry to have a definition of light in terms of “how bright something is”. After all — if you just had luminosity, then things really depend on the frequency of the radiation concerned. You could get a lightbulb and a radio emitter, or an x-ray emitter, that had the same luminosity — but you wouldn’t even be able to see the other two. That’s why we’re stuck with the candela.
You can see that, around the edges of this system, there are some pretty concerning inconsistencies. The candela is useful, but basing something on human eyesight and how bright candles are seems a little old-fashioned. The kelvin scale of temperature gets a lot of flack, too, with some people pointing out that you can derive it from energy: since, after all, the temperature is really just a measurement of the thermal energy of a body. From another perspective, temperature is the Lagrange multiplier that emerges from the statistical mechanics approach to maximise entropy. The point here is that, although you CAN technically define a temperature in terms of an energy, the temperature quantity is useful enough that it gets its own units.
NEXT EPISODE, WE’LL TALK ABOUT WHY UNITS ARE IMPORTANT, AND WHAT DIMENSIONAL ANALYSIS IS!
Thanks for listening to this episode of Physical Attraction…
DIMENSIONAL ANALYSIS
The units we’ve been stuck with, then, are a little strange. But really, it’s not just the candela that obviously depends on humans for a real definition. A metre makes sense as a unit of distance for a human: it’s comparable to human scales. Stretch out your arms: about a metre. Your height: about a metre. Certainly not 10m or 10cm. Similarly, kilograms for mass: a kilogram is approximately the weight of something you can hold in your hands. But if you compare this to the fundamental building blocks of the universe — the most common things — grams and kilograms are pretty arbitrary. Stars are closer to 10³⁰kg, subatomic particles are closer to 10^-30kg; in this intermediate range, the human domain, the domain of plants and people and things, we define our terms, and set our yardstick.
There are proposals that will be enacted in a couple of years to change the definitions of these quantities, to be more like the metre. The metre is now determined in terms of a fundamental physical constant — one of the numbers, the rules, the laws, if you like, that’s hard-coded into the Universe (as far as we can tell.) The speed of light in a vacuum is the same everywhere, and we define a metre as some fraction of that speed of light. That speed has an exact value, and, if we get any better at measuring it, this is what changes the length of a metre. Doing things this way makes everything a little more neat and tidy, so it’s been proposed to determine other SI units as multiples of fundamental physical constants, or combinations of them. Basically, this will mean that the kilogram lump of metal will become obsolete. As early as 2018 — so possibly already when you hear this — that kilogram lump will be consigned to the dustbin of history — although not an actual dustbin because it’s 90% platinum — and replaced by some fraction of planck’s constant and the speed of light and so on. Thus another link between the old world and the brave new one will be severed — but, instead, the ways we describe the world will be linked more intimately to the laws of physics, and the presumably heavily-armed swat team that guard the kilogram and defend it with their lives can stand down and take up flower-arranging instead.
At this stage, you’re probably thinking: it seems like people have spent way too long thinking about incredibly precise definitions for all of these units. In some ways, that’s correct, but if you’d ever spent ages getting your maths wrong because you faffed around with too many conversion factors, you’d understand completely. There’s a reason we don’t use miles and so on: you’d have to come up with a very consistent definition for these units, you’d have to derive new units for force and so on, and you’d have to deal with a whole bunch of unpleasant conversion factors. 1 Newton acting for 1 m gives you 1 Joule and you can carry around fewer numbers, which you invariably forget and which mess up your calculation, leaving you scratching your head for hours trying to catch where you went wrong.
But units are important for another reason — a method so simple and elegant and yet with such amazing results in physics that it almost does live up to its incredibly cool sounding name. I’m talking, of course, about dimensional analysis.
Dimensional analysis is really just a super fancy name for checking that your units match. If I tell you that I add one apple to one banana and end up with two oranges, you’ll probably be sceptical that my fruit recognition licence is really as legit as I claimed. But the reason you know that’s obviously wrong is dimensional analysis; you can’t add an apple to a banana and get a couple of oranges.
Similarly, in physics, imagine you’ve come to the end of some incredibly long calculation. Maybe you’re trying to derive the amount of time it will take for you to get over Firefly being cancelled. But you realize straight away that it will depend on all kinds of different factors: how many new shows are getting produced a year? How good are the shows? How quickly does your emotional intensity decay over time? That kind of thing. So you throw it all into a super complicated equation, solve that, and you get a formula for t that depends on all of the parameters you threw in.
But it’s late, you had to work across seventeen pages of algebra, and maybe — eyes filling with tears as you realize that Joss Whedon is moving on, so why can’t I — you don’t know if the formula is correct. One test is to see if the units of that formula are actually in terms of time, at all. If they’re not — if they’re in units of sadness instead — you’ve made a mistake somewhere.
This gives you an invaluable chance to check your work for consistency, and more than once I’ve had to throw out an answer for giving times instead of masses or similar problems like this. But the power of dimensional analysis is far greater than that. If used carefully, it actually has some amazing predictive powers about nature and science.
In 1950, just a few years after the first nuclear explosion had occurred at the Trinity test in Los Alamos, a mathematician called G.I. Taylor published a fairly unusual mathematical paper. It was unusual because it contained a piece of highly classified information that hadn’t been leaked to the public: the yield, or explosive force, of that first nuclear test.
This had been kept secret by the US military and was considered to be a highly sensitive piece of information. How had Taylor got his hands on it?
The answer was dimensional analysis. Although the yield of the bomb had been kept secret, photos and video of the mushroom cloud had been published in 1947 to a fascinated and rather terrified audience. They watched it and saw what Oppenheimer saw, that we discussed in our TEOTWAWKI specials: the slowly dawning realisation, as the cloud slowly spread, that humans had unlocked incredible power.
But GI Taylor saw something else: he realised that he could use this to estimate the yield of the nuclear bomb. And you can do it, more or less, with a simple case of a little intuition and some dimensional analysis.
So imagine ourselves in the future. We’ve got a nice neat formula that links together the energy of the initial blast with information about the fireball. What’s in that formula?
Well, the fireball is expanding, so we’d better include its radius, R. Since it’s expanding, time, t, since the explosion, is also going to be important. The energy of the bomb’s explosion will probably determine how quickly the fireball expands, right? Chuck that in. And, finally — the density of the air surrounding the bomb. This isn’t immediately apparent, until you think — okay, what if we exploded the bomb in incredibly dense treacle, rather than air? Clearly this impacts how quickly the shock wave and the fireball can spread.
So we have identified the physical parameters relevant to the problem of the expanding fireball. We’ve got the energy, the time, the radius, and the density. We also know that there’s probably going to be some dimensionless constant of proportionality involved. This is really something of a guess, but things rarely scale with each other perfectly. For example, the volume of a sphere scales with the cube of the radius, but it’s not equal to the cube of the radius — that kind of thing. That constant might turn out to be one, but you need to throw it in there.
Then we can actually use dimensional analysis to work out how all the quantities relate to each other. Let’s say that the formula we’re looking for lets us write the radius of the bomb blast in terms of the constant, the time, the energy, and the density.
We know that, if this is true, the dimensions have to match. So we can ask ourselves: how can you multiply together a time, an energy, and a density to give you a radius?
It turns out — and you can do the maths yourself — that there’s only one way to do this so that the units match. This formula must be true:
So this is pretty incredible. You can plot the radius to the power of 5 against time squared, and the gradient of that is the Energy of the bomb, times a constant, divided by the density of the surrounding medium. Plug in the density of the air, and you have some constant times the (highly classified!) energy of the bomb.
Whenever you come up with a formula in physics, the best thing to do is to sanity check it. That means you think “okay, what do I expect to happen, based on my knowledge of the laws of physics and… common sense in general?”
So. R⁵ = Et² / rho. We can see straight away that the radius increases as time increases, which is good, because we want our fireball to grow. We can also see that the rate of growth depends on two things — the energy, and the density of the surrounding air. If the energy of the blast is bigger, the fireball grows more quickly. Makes sense: bigger boom! If the density of the surroundings is higher, the bomb blast grows more slowly. Okay, that also makes sense: shock waves might spread less quickly through treacle than air. Does this mean that the formula is necessarily correct? No, it doesn’t mean it’s correct. But it doesn’t mean that it’s obviously wrong. It fits well with our common sense. Similarly, if you discovered that your Firefly happiness formula predicted that you’d get sadder over time, rather than recovering… you’d need to explain it. Maybe it’s the Young Sheldon effect.
All that’s left is the constant. Working out that constant was the really clever bit for Taylor — it turns out to be very close to 1, and you can work it out by considering how shock waves that AREN’T due to nuclear blasts behave. That was how he managed to, with fairly simple maths, derive a value for the classified explosive energy of that first nuclear bomb.
You can formalise this method of dimensional analysis to figure stuff out using something called Buckingham-Pi theorem, an incredibly useful result. What it does is tell you how to construct dimensionless quantities from the parameters of interest, that do have dimensions. You can then use that to construct simple rules about how your physical system will behave. And, amazingly often, when you go through and solve the actual calculation — you find out that the dimensional analysis estimate was bang on the money.
This is really important for a couple of reasons. First off, you don’t need to know all that much for dimensional analysis: you just need to know which parameters of the system you’re interested in. You don’t need to know the underlying equations that govern the physics. You could use dimensional analysis to calculate, for example, fluid flow in a pipe. And you wouldn’t have to solve the Navier-Stokes equations, the equations that tell you how fluids behave and are notoriously often difficult to solve. You wouldn’t even need to know that equation.
What’s more, if you’re running an experiment, it can tell you about how what you’re trying to measure might depend on other parameters in the system. This is so, so important, because the best way to get to the right answer is often to have some idea of the answer that you’re trying to get to in the first place. In other words, if you can find a formula that looks good and makes physical sense when you sanity check it, and it agrees with the evidence and measurements, you can save yourself a whole lot of work trying to work out the underlying equations and solving them.
You can even get super profound results from just considering dimensional analysis, if you’re smart to begin with. Let’s say that I have one of the physicist’s favourite problems — a block attached to a spring. You move the mass a little bit, and it starts to oscillate, bouncing up and down. Is there a way to calculate the time period — how long it takes for a full cycle of oscillations, so that the mass returns to where it was?
Similar story. What does it depend on? We’re just estimating, so let’s ignore friction and air resistance. Well, you might expect the block mass to matter, and the stiffness of the spring, too. You might also initially think that the gravitational field is important, too. After all, surely if the weight of the spring is bigger, that has an impact?
But when you go through and do the dimensional analysis, you find there’s no way to sensibly get time out of the spring constant, which tells you how much force you need to extend the spring, gravity, and the mass. You know that the spring constant is important, because you’ve tested a couple of different springs, so assume that gravity can be discarded. (The reason for this turns out to be that the mass is important because it determines how strong the spring has to be to snap back the block in a certain amount of time. Basically, the WEIGHT of the block — including gravity — just determines the equilibrium position of the spring. The mass is only important because it tells you how the spring forces can accelerate the block.)
It turns out the dimensionless combination gives T = constant * sqrt(m/k). Let’s do a sanity check: a stronger spring takes less time to oscillate (it has a shorter time period — okay, makes sense, because the spring is responding with a bigger force.) If the mass is bigger, the oscillations take longer — there’s more inertia in the system, it’s harder to change the motion of the mass. This is exactly what you get from Newton’s Second Law, but we didn’t need to use Newton’s Second Law — just units and a bit of logic.
This process is incredibly invaluable in physics — both for sanity-checking your answers, and for estimating how quantities might increase with each other to get simple formulae to solve and estimate the problems. Often a more careful analysis leads to very similar results from the quick, back of the envelope dimensional analysis. And in things like fluid mechanics, it becomes even more useful: but that’s for another day.
I hope I’ve given you some insight in the last couple of episodes into the incredibly geeky but very important people who keep track of our units, and how you can learn to stop worrying and dimensionally analyse the bomb. Next time you tell someone you can’t compare apples to oranges, you can tell them that you worked it out with the mathematical method of dimensional analysis, and watch their heads explode. Then you can work out how the radius depends on time. It’s that useful.
I’m going to finish with some non-SI units that have been defined over the years. Physicists are often very nerdy, many of us love terrible jokes, and many of us have a weird obsession with units, so it seems appropriate.
[Jokes — NON SI UNITS / new SI base units to finish off, the Scaramucci]
So those of you who follow US politics will know that Anthony Scaramucci briefly served as the White House Director of Communications — at least, until he was fired for giving an obscene and probably drug-addled interview with a magazine, remember that? Well, his position was so short that he defined a new unit of political time. The Scaramucci is 11 days long. Similarly, from slightly longer ago, people got so sick of columnist Thomas Friedman saying that “the next six months will be crucial in determining the fate of the Iraq War” that they defined six months as one Friedman.
Sometimes these funny units are useful. In some cases, we have units, but people don’t always understand what they mean. What if I told you that you’d been exposed to a microsievert of radiation? Should you be worried, or is this fine?
Well, bananas are naturally slightly radioactive. They are high in Potassium, as John Darnielle failed to note in his banana song, so every time you eat one, it’s slightly radioactive. But this probably doesn’t bother you too much. So when I say that one microsievert is 10 bannana equivalent doses, or BEDs, you probably wouldn’t feel too nervous. The average dose of radiation for people around the Three Mile Island accident was 80 microsieverts — it might worry you, put like that, but 80 bananas doesn’t sound so bad. It takes 4–5 Sieverts to kill a person.
Similarly, micromorts measure the risks from day-to-day activities. A micromort is equivalent to a one in a million chance of dying — so, driving 370km in your car boosts your death risk by a micromort. Scuba-diving is 5 micromorts.
Douglas Adams defined the “Sheppy” as 7/8 of a mile. He said it was the “closest distance at which sheep remain picturesque” and, yeah, if you’ve ever got close up to a sheep, you’ll get what he means. If anyone can define the closest distance at which I remain attractive, please send your answers on a postcard.
Andy Warhol famously said that “in the future, everyone will be famous for fifteen minutes.” So, by now, you’ve probably been listening to this episode for two Warhols.
A unit of distance that space people like is the light year — a phenomenally huge distance that is simply how far light travels in a single year. The nearest star is around four light years away. But you can get a much smaller unit by considering how facial hair grows — a unit of distance called the beard-second. Apparently it’s about 5 nanometres — so an atom is about 0.2 beard-seconds across.
Because Carl Sagan, the famous astronomer and science communicator, had a catchphrase — “billions and billions” (often useful when you’re talking about outer space in terms of human-sized units!) a “sagan” is a large quantity of anything. Similarly, the physicist Paul Dirac — who made immense contributions to theoretical physics and quantum mechanics — was also famously very quiet and seldom spoke. In some ways, he’s portrayed by history as this archetype of the quiet, reserved, immensely intelligent physicist. He was so reluctant to indulge in idle chatter that his friends defined a unit of information flow: the Dirac. It’s one word per hour.
When I was an undergrad, I used to turn in incredibly lengthy problem sheets. To give you a sense of perspective, you’d usually get 1–2 problem sheets a week. The longest one I ever turned in was over a hundred pages of A4. So other people defined their “Hornigold factor”, which was the number of times longer my problem set was than theirs. I guess it measures conciseness. I could certainly define an empirical law that the number of shows it actually takes to cover a topic is between 2 and 3 times as many as the number I think it will take.
I’ve thought of another one: how about a personal unit of time that measures how long, on average, you can go without checking your phone, or social media? We can call them distraconds.
Can you guys come up with any more humorous or useful units? I’ll read out good suggestions in a future episode!
Anyway: I hope this nerdy foray into physical units and dimensional analysis has been fun. Until next length/speed!
Thanks for listening
PLUGS
