The first time I encountered dimensional analysis, I didn’t recognize it for the magic wand that it is; it seemed to be just a form of mathematical hygiene — useful for avoiding mistakes, but not much else. To get a sense of what I mean by hygiene, consider a question that came up in my own household (arising from the incident described in Endnote 1): how do we convert 6 feet per second to units of miles per hour? We know that there are 60 × 60 = 3600 seconds in an hour and 5280 feet in a mile, but suppose we aren’t sure what to do with those numbers to get our answer; we suspect we should multiply 6 by 3600 / 5280 or by 5280 / 3600, but we’re not sure which. To keep ourselves on the right path, we use something called the factor-label method: we attach units to those numbers and we multiply three fractions that carry those units along in their numerators and denominators, with the three fractions representing the assertions “The speed is 6 feet per second”, “3600 seconds equals 1 hour”, and “1 mile equals 5280 feet”:
This gives an answer with units of miles per hour, after we’ve cancelled sec with sec and ft with ft. If we had tried to combine the fractions the wrong way, multiplying where we should have divided or vice versa, the appearance of unfamiliar units in the answer (like square-foot-hours per square mile per second) would have hopefully alerted us to our mistake.
But dimensional analysis can do much more for us. In many cases, it gives us a way to figure out the answer to a physics problem without actually applying the laws of physics. Bertrand Russell once wrote: “The method of ‘postulating’ what we want has many advantages; they are the same as the advantages of theft over honest toil.” In many ways, dimensional analysis gives us the same bargain: it offers us a way to cheat the universe at its own game.
YOU’RE ON THE MOON, MIKE
The video The Math Behind Michael Jordan’s Legendary Hang Time by Andy Peterson and Zack Patterson discusses the kinds of feats basketball phenomenon Michael Jordan would be able to perform elsewhere in our solar system. Here “hang time” refers to the time that MJ is airborne. Peterson and Patterson go through lots of work, some of it behind the scenes. Starting from the equation of motion h = h0 + v0 t + (1/2) g t2 (which they include in the video), they derive equations (not included) for the hang time thang and the peak jump height hpeak as a function of the strength of gravity g found on other worlds (based on the assumption that, on all those worlds, MJ pushes off with an initial velocity of v0 = 4.51 meters per second). For instance, they figure out that MJ’s hang time on the Moon would be almost six seconds, as opposed to just under a second on Earth.
I’m guessing that Andy and Zack derived the formula thang = 2 v0 / g from the equation of motion as described in Endnote 5. But they need not have worked so hard! Dimensional analysis teaches us that there has to be a formula for thang of the form C v0 / g for some constant C, and once we know that, it’s easy to deduce MJ’s hang time on the Moon from his hang time on Earth: since the Moon’s gravity is about six times weaker, MJ’s hang time on the moon has to be about six times longer, or about 6 seconds.
In the same way, dimensional analysis teaches us that there has to be a formula for hpeak of the form D v02 / g for some constant D, so that peak height, like hang time, is inversely proportional to the strength of gravity; if MJ can jump to a height of about 1 meter on Earth, he could jump to a height of 6 meters on the Moon.
THE MATH BEHIND THE MAGIC
Before I explain how Andy and Zach could have saved themselves some algebra and didn’t even need to write down Michael Jordan’s equation of motion, let’s do a little physics hygiene and check that the formula for thang that I gave above makes sense in terms of units. The velocity v0 is expressed in meters per second, while g (the uniform acceleration caused by being in a world’s gravitational field) is expressed in meters per second per second: (m/sec)/sec, i.e., m/(sec × sec) = m/sec2. Let’s leave out the numbers and just do fraction arithmetic with the units:
We get the answer we were hoping for, but there’s something overly specific about this calculation; we’ve used meters to measure height and seconds to measure time, but if we’d used feet to measure height and minutes to measure time, our answer would still have come out in units of time:
The less parochial way to describe what’s going on is to phrase things in terms of dimensions rather than units. We say that the dimension of v0 is L/T (L = Length, T = Time) and the dimension of g is L/T2, so that the dimension of v0 / g is
Now our magic is going to get a bit metaphysical. Describing the specific state of the universe involves quantities that have dimensions like Length or Time, but the fundamental laws governing the universe can all be expressed in terms of relations between derived quantities that are dimensionless. Maybe physicists can explain why this is so, but for now, we’ll take it as an axiom, tied in with the fact that the universe has no preferred scale for quantities like Length or Time. (See Endnote 4 for some caveats about this over-broad statement.) thang / (v0 / g) is a dimensionless quantity; it “has units of 1” (that is, all the dimensions cancel out). There are other dimensionless quantities that can be expressed in terms of thang, g, and v0, but they’re all powers of this one (see Endnote 3), or sums of such powers, or things like that.
The upshot is that if the laws of physics can be applied to deduce some consistent numerical relationship between these three quantities, it must be of the form
thang / (v0 / g) = constant.
And indeed, a reckoning that takes into account Newton’s laws of ballistics (see Endnote 5) tells us that thang / (v0 / g) is 2. But the point is, even without solving Newton’s equations, dimensional analysis tells us that if there is a way to solve for thang in terms of v0 and g, then thang / (v0 / g) must be some constant C. That is, there must be a formula of the shape thang = C v0 / g. And that’s enough to tell us that when the strength of gravity goes down by a factor of 6, the hang time goes up by a factor of 6.
Something similar happens for the physics-story involving the quantities v0, g, and hpeak (the jumper’s peak height). You can check that the dimension of v02 / g is (L/T)2 / (L/T2) = L, so the derived quantity hpeak / (v02 / g) is dimensionless. Since hpeak / (v02 / g) is the only dimensionless derived quantity around, it must be some mathematical constant. That is, there must be a formula of the shape hpeak = D v02 / g, telling us that when the strength of gravity goes down by a factor of 6, the peak jump height goes up by a factor of 6.
Our formulas also tell us something of terrestrial import that isn’t quite so obvious: if here on Earth MJ could increase his initial velocity v0 by a factor of just 1 percent, his hang time would go up by 1 percent and his peak height would go up by slightly more than 2 percent (since 1.01 squared is 1.0201).
ROASTING A DODO
I was initiated into the magic side of dimensional analysis in Tom MacMahon’s biomechanics course during my senior year as an undergraduate, and you can learn about it from the book “On Size and Life” that MacMahon wrote with John Bonner. (Or, for a quick overview, see https://en.wikipedia.org/wiki/Dimensional_analysis.) Bonner and MacMahon call dimensional analysis “Physics Made Easy”, but as they point out, this is true only up to a point. They include all the requisite caveats (around page 120 of their book), but since I’m not trying to be scholarly, I won’t. Instead, I’ll apply the method to the problem of roasting a dodo.
Dodos are still extinct (see Waldrop), so you can’t actually cook one, but the magic of dimensional analysis gives us an easy way to figure out how you’d cook it, assuming that the magic of genetic engineering ever lets us bring it back to life.
Just as our Michael-on-the-moon scenario involved a quadratic equation, cooking involves an equation, but a much more complicated one: a PDE (partial differential equation) that models how heat varies both spatially and over time. An honest solution would depend on details about the shape of the bird, and since there’s no simple formula for that shape, there’d be no simple formula for the cooking time; you’d have to solve the PDE using numerical methods. An empirical approach would be to just cook a few birds and use trial and error to see how much time is required. (If you want, you might take the somewhat whimsical point of view that the bird is serving as an analog computer for solving the PDE.)
But here’s the thing: because the PDE would be a consequence of underlying laws of physics, which are (by our axioms) dimensionally consistent, the PDE would have to be dimensionally consistent, and the solution to it would also have to be dimensionally consistent. What’s more, the physical constants and parameters going into the mathematical model are few: specifically, they are the size of the bird (s) and three characteristics of bird-flesh: thermal conductivity (k), mass-density (ρ), and specific heat (C). Here are the dimensions of these quantities, in terms of length (L), time (T), mass (M) and temperature (Θ):
How can we get a dimensionally-consistent formula of the form tcooking = constant sa kb ρc Cd? The only way to choose the exponents a, b, c, and d so that sa kb ρc Cd has dimensions of time is to take a=2, b=−1, c=1, and d=1 (see Endnote 6).
Dimensional analysis has told us that cooking time should equal to a constant times C ρ s2 / k. Meanwhile, weight W grows like s3, so that s2 grows like W2/3; our analysis thus predicts that cooking time should grow like the 2/3 power of weight. So if we model a dodo as a very large turkey, then knowing that an unstuffed 24-pound turkey needs about 5 hours at 325° F, we can estimate that the cooking time for an unstuffed 48-pound dodo, twice as heavy as the turkey, would be longer by a factor of 22/3, giving us an approximate roasting time of 8 hours.
Alternatively, if we model a dodo as an extremely large chicken, then knowing that an unstuffed 6-pound chicken needs about 2 hours at 350° F, we can estimate that the cooking time for a 48-pound dodo, eight times as heavy as the chicken, would be longer by a factor of 82/3= 4, giving us an approximate roasting time of (once again) 8 hours.
BIKING ON MARS
If Tom MacMahon, who used his knowledge of biomechanics to redesign Harvard’s indoor track in the 1980s, were alive today, he’d be eager to tackle new projects. Perhaps Elon Musk would be asking him to design athletic facilities for a Martian colony. Tom would certainly be the person for the job. His application of dimensional analysis to the tilt of a bicycle turning a corner gives us a way to figure out, even before we put people and bicycles on Mars, how far bicycles will lean when bicyclists turn corners in Martian gravity.
Say Musk wants the colony to have a circular bicycle track that’ll let cyclists get a good work-out, biking on level ground at speeds up to 30 miles per hour (an easier speed to maintain on Mars than on Earth). But this track will be only 200 feet across (big domes are expensive on Mars), so Musk is worried that traveling in such tight circles, the Martian cyclists will find their bikes tilting inward to an uncomfortable degree, perhaps even to the point that their tires will slip outward. Can MacMahon help?
MacMahon knows that the physics of bicycles is complicated (for instance, explaining why bicycles are as stable as they are is a controversial subject of contemporary research), but he also knows how to cheat. A basic principle of engineering, called the principle of dynamic similarity, is that you can confidently model a system by a prototype if all the dimensionless quantities that can be expressed in terms of the system’s parameters have the exact same values in the prototype. In the case of a bicyclist traveling on a circular track, the relevant parameters are the velocity of the bike, the radius of the track, and the strength of gravity. The only dimensionless way to combine them is v2 / g r (check: (L/T)2 / (L/T2) (L) = 1). Since gravity on Mars is 2.6 times weaker than gravity on Earth, we can model the Martian bike-track without leaving Earth as long as we increase the radius by the same factor. And if the cyclist on Earth goes slower by some factor, we just have to make the track smaller by the square of the factor. MacMahon can build a track on Earth that’s about 34 feet across and have a student ride around it at 20 mph; whatever the tilt of the terrestrial bicycle is, dimensional analysis tells us that the tilt of the Martian bicycle on the smaller track will be the same.
At this point MacMahon would switch over to analyzing the issue of skidding, once again making use of dimensional analysis to determine what sort of Earthly analogues can be used to predict what will happen on Mars. I’m guessing that he’d find a way to rescale the size of the track and the speed of the bicycle to get dynamic similarity to prevail. But I’m neither an engineer nor a physicist, so I’ll have to abandon this fantasy, or leave others more expert than me to continue it.
THE POWER OF DIMENSIONAL ANALYSIS
We perform measurements on the world and record our answers as real numbers. Using real numbers to encode magnitudes is a fantastically useful device that’s one of the keys to the success of the scientific enterprise. One number system can be applied to so many diverse domains of science! But this flexibility comes at a cost. When we replace a magnitude by a number, we’re erasing important information about the meaning of the number. What dimensional analysis tries to do is bring some of this meaning back. When we write down an expression like
those numerators and denominators aren’t just numbers; they’re dimensioned numbers, or (as George Hart more properly calls them in his book “Multidimensional Analysis”), dimensioned scalars.
Hart says on page 6, “the point of view taken here is that dimensions are so important that they are brought into the number system rather than being left as some kind of annex to our calculations.” This is not just a philosophical issue. It’s an intensely practical one. Many expensive mistakes in large engineering projects occurred because of conversion errors, that is, the failure to take units into account properly. People and their computer delegates start trafficking in numbers, forgetting that the numbers arrive in our world with units attached. These units give the numbers their context and their meaning. Sever the link between a number and its meaning, and you risk subverting that meaning (and wasting millions of dollars).
Of course, since dimensional analysis ignores the distinction between different units that have the same dimension (feet and meters, say), computer systems and engineering-team protocols that merely kept track of dimensions wouldn’t have prevented the costly unit-conversion mistakes I’ve mentioned. But keeping track of dimensions is a first step on the path towards keeping track of units, and more broadly, keeping track of what a number means, whether it’s a team or a computer that’s working with that number.
Hart writes in his book that the process “of divining complex nonobvious facts about the world from apparently a priori arguments … affects some people as having a deep, almost mystical power.” You can count me as one of those people. If you too find this kind of divination fun, here’s one last example of the magic of dimensional analysis that you can try for yourself: What happens to the period of a pendulum when you double the mass of the pendulum (keeping the length fixed) or when you double the length (keeping the mass fixed) or when you double the strength of gravity? You don’t need to know any physics, beyond knowing the dimensions of the relevant physical quantities. See Endnote 9 for the answer.
Thanks to Sandi Gubin, Brian Hayes, Michael Kleber, Christian Lawson-Perfect, Keith Lynch, Henri Picciotto, Shecky Riemann and Glen Whitney.
Next month: The Genius Box.
1. My daughter was riding her hoverboard, spinning in the tightest and fastest circle she could, and she asked me “How fast am I going?” My answer, “About one revolution per second”, didn’t satisfy her; she wanted to know her speed in miles per hour. I’m ashamed to admit it now, but I pedantically told my daughter that her question didn’t make sense because rotational speed can’t be measured in miles per hour. This isn’t the right answer to give a nine-year-old, even a bright and mathematically knowledgeable one. What I should have said is that different parts of her hoverboard and her body were moving at different speeds; I should have given her an intuitive sense of why that was the case; and I should have proceeded to work with her to compute the approximate speed of her feet. Those feet were traveling in a circle of radius r = 1 foot, and π is about 3, so her feet were traveling a distance of 2 π r = 6 feet every second. I should have talked to her about how to convert 6 feet per second into miles per hour. (That’s where the problem with which I began this essay came from.) By the time I realized the right way to respond to my daughter’s question, she’d lost interest. I did eventually tell her the result of the calculation — 4 mph — and she confirmed that this was about what the manufacturer listed as the hoverboard’s maximum speed.
2. Can anyone tell me what movie has a deathbed scene that ends with the line “You’re on the moon, Mike” right after the Mike in question (a woman) dies?
3. Consider a product of the form thanga v0b gc, where a, b, and c are undetermined exponents. For what values of a,b,c is this product dimensionless? thang has units of T (Time), v0 has units of L/T (Length per Time), and g has units of L/T2 (Length per Time-squared), so thanga v0b gc has units of Ta (L/T)b (L/T2)c = Lb+c Ta−b−2c. Setting b+c=0 and a−b−2c=0, we find that we must have a=c and b=−c. That is, the triple (a,b,c) must be some multiple of the fundamental solution (1,−1,1). Putting it differently, if thanga v0b gc is dimensionless, it must be a power of thang1 v0−1 g1, or thang g / v0.
4. Actually, the universe as we understand it does have a preferred length scale, namely the Planck length, and the laws of physics that we use cease to work when we try to apply them to phenomena that are too small (or too large). A more careful way to state my point is that for phenomena on roughly the same scale as Michael Jordan (the domain of classical physics), there doesn’t seem to be a preferred length scale or time scale. So physical law, as embodied in relationships between magnitudes, can be expressed in a scale-independent way in which all the fundamental relationships are numerical relationships between dimensionless quantities.
5. It would probably have been better for the video-makers to write the equation of motion as h = h0 + v0 t − (1/2) g t2 (note the minus sign), since gravity is accelerating MJ downward, not upward, and since g is usually taken to be positive, not negative. To derive the formula for hang-time from h = h0 + v0 t − (1/2) g t2, we need to find the time at which the jumper’s height h returns to its initial value h0. That is, we need to find the non-zero value of t that satisfies h0 = h0 + v0 t − (1/2) g t2, or equivalently, (1/2) g t2 = v0 t. Dividing both sides by t, we get (1/2) g t = v0, and solving for t, we get thang = 2 v0 / g.
To derive the jump-height, we set h0 = 0 for convenience and use the fact that a downward-curved parabola that crosses the horizontal axis at t = 0 and t = 2 v0 / g will reach its highest point exactly halfway in between, at t = v0 / g; halfway through his jump, MJ will have height hpeak = v0 t − (1/2) g t2 = v0 (v0 / g) − (1/2) g (v0 / g)2 = v02 / g − (1/2) v02 / g = (1/2) v02 / g.
6. If your high-school algebra skills are sharp, finding the magic combination of exponents is a slightly tedious but not onerous task. The units of sa kb ρc Cd are (L)a (M L T−3 Θ−1)b (M L−3)c (L2 T−2 Θ−1)d = La+b−3c+2d T−3b−2d Mb+c Θ−b−d, which equals T (also known as L0 T1 M0 Θ0) if and only if a,b,c,d satisfy a+b−3c+2d=0, −3b−2d=1, b+c=0, and −b−d=0. This is a system of four linear equations in four unknowns. The ones to focus on first are the second and fourth equations, which constitute a system of two equations in two unknowns with the unique solution b=−1, d=1. With these values we can use the equation b+c=0 to find c=1 and then use the equation a+b−3c+2d=0 to find a=2.
I got this nice example of dimensional analysis from pages 75−78 of “On Size and Life”.
7. You shouldn’t take my 8-hour estimate too seriously. For one thing, cooking tables like the one at the Huffington Post don’t show cooking times that grow like the 2/3 power of weight; the actual exponent seems to be a bit smaller, presumably because big birds aren’t just scaled-up replicas of small birds of the same species. (See last month’s discussion of elastic similarity.) For another thing, most mathematical models only work well for phenomena that lie in a certain scale-range, and extrapolation beyond the range in which we’ve done experiments is always a chancy game. In our case, we’re extrapolating from chickens and turkeys to birds that are over twice as big and that had a very different lifestyle, so from the start we should be skeptical. I’d welcome input from those who know about the physics of cooking. For instance, are thermal conductivity, mass-density, and specific heat fairly similar for different kinds of meat, or at least for different kinds of fowl?
8. The angle of tilt of a bicycle is dimensionless since an angle measured in radians is just the ratio between an arc-length and a radius, both of which have dimensions of L, and since an angle measured in degrees has the same dimensions as an angle measured in radians.
9. Once again using the L,T,M dimensional system, we find that the mass m of the arm has dimensions of M, the length ℓ of the arm has units of L, and the strength of gravity g has units of L/T2. The only dimensionally consistent relationship is that the period is proportional to sqrt(ℓ/g). So, doubling m has no effect on the period, doubling ℓ increases the period by a factor of sqrt(2), and doubling g decreases the period by a factor of sqrt(2). It should be mentioned that this assumes that the angle of deflection θ remains small; the period doesn’t depend strongly on θ as long as θ is close to zero, but when θ gets far from zero, there is an important dependence. Also, it’s worth mentioning that when θ is small, the proportionality factor is an interesting mathematical constant, namely, 2π. That is, when θ is small, the period of a pendulum is given to high accuracy by the formula 2π sqrt(ℓ/g). Turning this around: you can estimate pi with a pendulum! See
When θ is large, the principle of dynamic similarity still applies: for a fixed θ, the period of the pendulum is still proportional to sqrt(ℓ/g). It’s just that the proportionality constant isn’t that close to 2π.
10. The title of this essay is a bit of a steal from one of my heroes, the writer Oliver Sacks, and more specifically from the title of his book “An Anthropologist on Mars”. I never had the privilege of meeting him, but since I’ve already imaginatively resurrected Tom MacMahon, I’ll do the same for Dr. Sacks — and while I’m at it I’ll put him on Mars and give him a bicycle. (He’d prefer a motorcycle, but Musk doesn’t want the pollutants in his dome.) I’ll end this essay with the image of the revived and rejuvenated neurologist speeding around the Martian track on a bicycle, reveling in the mixture of old and new sensations, even as he carefully eavesdrops on his own cerebellum calibrating itself to the gravity of a new world.
John Tyler Bonner and Thomas MacMahon, On Size and Life.
John S. Denker, Dimensional Analysis: https://www.av8n.com/physics/dimensional-analysis.htm
John S. Denker, Introduction to Scaling Laws: https://www.av8n.com/physics/scaling.htm
George Hart, “Multidimensional Analysis: Algebras and Systems for Science and Engineering.”
Howard Waldrop, “The Ugly Chickens”; in the collection “Howard Who?”. Available at
Neat post — I love your examples. I myself wrote a similar post a couple years ago, with some dimensional analysis tricks that you might enjoy (including a fascinating historical example of how simple dimensional analysis foiled the US Army’s secrecy…). http://hardmath123.github.io/dimensional-analysis.html
I recently also learned of another example (not in the above post) from Ian Stewart, who describes how you can calculate how many times a slice of bread will flip before it hits the ground. Based on an approximation of how tall creatures can reasonably evolve to be given some value of g, he goes to show that bread is likely to fall butter-side-down on all worlds!
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This can be presented to high school math classes! How to predict the radius of an atomic explosion, as a function of time, with nothing more than a little high school algebra and elementary physics? And then accurately know the power of the bomb which caused it?
In 1940, five years before there were any atomic bombs, an English physicist was commissioned to do just that – a formula for the future, so to speak. He predicted that the radius R would be nothing more than proportional to some unknown powers of the energy of the bomb (E), the density of the air (ρ) and the time (t).
R = K (Ea) x (ρb) x tc
In the mks system,
m = K (kg m2 sec-2)a x (kg m-3)b x secc
Equating coefficients, a + b = 0, 2a – 3b =1, and -2a + c = 0, so a = 1/5, b = -1/5 and c = 2/5
So R = K (E/ρ)1/5 x t2/5
It was discovered, in an actual blast after the war, that K ≈ 1.0327… and could be taken as 1.
So for a given E and rho, log R = Some Constant + 2/5 log t
In 1946, an atomic fireball was measured every thousandth of a second. When log R was plotted vs log t, a straight line was determined, with slope equal to 2/5.
These results ( R vs t) were made public and Taylor, the physicist, announced that the energy of the bomb must have been 16.8 kilotons. He was subsequently interrogated under the Official Secrets Act, since the energy of the bomb was classified. “But the energy is simply equal to the fifth power of the radius times the air density divided by the square of the time. You had me figure that out back in 1940.”
This anecdote is from the book “In Praise of Simple Physics” by Paul Nahin: http://press.princeton.edu/titles/10690.html
Lawrence S. Braden
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Intersting indeed! A small typo in image https://mathenchant.files.wordpress.com/2018/02/032-inset5.png : specific heat has dimension L^2, not L^-2.
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Thanks for bringing this mistake to my attention! I’ll fix this if and when I publish the piece someplace.