Dimensional Analysis: a worksheet
Tuesday, October 11, 2016 · 18 min read
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In school, many of us learn about dimensional analysis as a way to convert between various units. There is, however, far more to this humble technique. In this paper, I would like to present a smörgåsbord of dimensional analysis pearls that I have found over the past few months.
Introduction
Many people argue that the metric system is better than the imperial system. This argument, however, is usually predicated on the relationship between various units—powers of ten are more easy to manipulate because we are taught arithmetic in base ten; additionally, the uniformity makes it easier to remember how to convert from centimeters to meters than from inches to furlongs. If we only consider units in absolutes, however, then is a meter really ‘better’ than a yard?
The theme of this paper is that there is no obvious quantity to use as a fundamental unit of length, mass or even time. In fact, this poses problems in certain domains where we cannot appeal to convention to establish units. For example: on the Voyager 1 probe, NASA placed a golden record designed to be found by extraterrestial life. Inscribed on this record, NASA provided Saganesque instructions for playback. The time taken for one rotation of the record is speciﬁed in terms of the period associated with the fundamental transition of the hydrogen atom [10].
“The fundamental transition of the hydrogen atom” is certainly more reasonable to explain to intelligent life than, say, a second (which would force the extraterrestial life forms to somehow measure the time Earth takes to complete its orbit). However, this choice still seems a little arbitrary^{∗}.
Why aren’t there obvious fundamental units? We can answer this question with a philosophical observation: nobody knows the absolute size of anything. For all we know, the universe could be really small, and us smaller still. All is not lost, however. Clearly, some properties of the world remain constant regardless of how we choose to measure them. I cannot become richer by measuring my income in cents rather than dollars. Percy Bridgman, a physicist who studied the properties of materials under extremely high pressure, stated this property eloquently in his 1931 treatise [3]:
[T]he ratio of the numbers measuring any two concrete examples of a secondary quantity shall be independent of the size of the fundamental units used in making the required primary measurements.
This statement, known as Bridgman’s Principle of absolute signiﬁcance of relative magnitude, is really just an expression of humility: nature is indifferent to our choice of units [11, 8]. That is, the period associated with the fundamental transition of the hydrogen atom will always be $2.23\times 1{0}^{17}$ times the period taken by Sol 3 to orbit the sun, regardless of which system of units are used to measure the two quantities. Units are arbitrary; ratios are invariant.
Exercises
 1.

 (a)
 According to Bridgman’s Principle, the ratio of a circle’s circumference to its diameter should be constant regardless of the units used to measure it (this constant, of course, is $\pi $). Is the ratio of a circle’s area to its radius independent of the units used? Does this violate Bridgman’s Principle?
 (b)
 By citing Bridgman’s Principle, explain why (for the purposes of this paper) Kelvins are a unit, but degrees Celsius are not.
 2.
 In Europe, fuel efficiency is measured in liters per kilometer. Note that both liters and kilometers can be expressed in terms of centimeters. Simplify ‘liters per kilometer’, and give a physical interpretation of the resulting units [7].
 3.
 The Pythagorean theorem that states that the square of the length of the longest
side of a right triangle is the sum of squares of the two shorter sides. Draw a right
trangle, and draw a line from the right angle, perpendicular to the longest side
(known as the hypotenuse). You now have three triangles: two small ones
contained in one large one.
 (a)
 Show that all three triangles are similar, that is, they have the same angles.
 (b)
 Convince yourself that the area of a triangle can be expressed as a function of one of the acute angles and the length of the hypotenuse.
 (c)
 Consider the relationship between the areas of the three triangles, then use dimensional analysis to derive the Pythagorean Theorem [14]. Why is this not a valid proof?
 4.

 (a)
 The derivative of a variable with respect to another is the rate of change of the former with respect to the latter. For example, the derivative of the position of an object with respect to time is its velocity. What are the units of position, time, and velocity? Arguing by analogy, construct rules for dimensional analysis of derivatives.
 (b)
 Integration is the opposite process of derivation: for example, the integral of the velocity of an object with respect to time is its position. Arguing by analogy, construct rules for dimensional analysis of integrals.
 (c)
 According to the Fundamental Theorem of Calculus, another way to
think about integration is that it corresponds to a measurement of the
area under a curve (i.e. between the curve and the $x$
axis). For example, with the equation $y=1x$,
the integral of $y$
with respect to $x$
from $0$
to $1$
is $0.5$—verify
this by drawing a picture.
Recall the formula for measuring the area of a rectangle. Assuming that the $x$ and $y$ axes represent values with units, do the units of area in this system correspond to the units of integration you found in the previous question? To you, does this feel like compelling evidence that the Fundamental Theorem of Calculus is true?
 (d)
 Consider the functions $f:\mathbb{B}\to \u2102$ and $g:\mathbb{A}\to \mathbb{B}$ where $\mathbb{A}$, $\mathbb{B}$, and $\u2102$ are all units. What are the units of ${f}^{\prime}$ and ${g}^{\prime}$? (Here, ${f}^{\prime}$ means the derivative of $f$ with respect to $x$.) Now, let $h\left(x\right)=f\left(g\left(x\right)\right)$. Convince yourself that $h:\mathbb{A}\to \u2102$. Then, use dimensional analysis to derive an equation for ${h}^{\prime}\left(x\right)$ in terms of $f$, $g$, and $x$. This expression is called the chain rule^{†}.
 5.
 In 1999, the Mars Climate Orbiter passed Mars on a trajectory that was too close to
the planet, causing it to pass through the upper atmosphere and disintegrate. The cause
of the MCO failure was determined [1] to be a software error: a missioncritical
piece of software produced output in nonSI units, in poundseconds rather than
Newtonseconds^{‡}.
It seems obvious that representing a number in a different set of units leads to a different answer. An interesting question to ask, however, is: are there meaningful calculations whose results do not change when evaluated with different units?
 (a)
 Suppose we were trying to discover the period $T$ of a pendulum. It seems like the important parameters we would need to know are the pendulum’s length $l$, and mass $m$. We also know that on Earth, the acceleration due to gravity is $g$. Using dimensional analysis, discover a formula $f$ for $T$ in terms of $l$ and $g$. Check this formula by running an experiment with a washer and a piece of string. You should discover that in reality, the period is approximately $T=6.28\times f$^{§}. Has dimensional analysis failed?
 (b)
 The idea that you can discover dimensionless constants like $6.28$ above by using dimensional analysis to create formulas, and then running experiments to solve for the constant, is formalized by Buckingham’s $\Pi $ Theorem^{¶} [4]. It states that with $n$ physical variables, $r$ of which are independent, you can derive $nr$ fundamental dimensionless quantities. Explain what is meant by ‘independent’. Using what you know about solving systems of linear equations, explain why the theorem makes sense intuitively.
 (c)
 If you plan on testing a $1:10$ scale model of an aircraft wing in a wind tunnel, it may not be immediately obvious to you whether you also need to correspondingly scale factors such as wind speed, air pressure, or temperature. Explain how to use Buckingham’s $\Pi $ Theorem to figure out which factors must be scaled, and in which direction, in order for the scale model to be an accurate representation of the real version^{∥}.
Surprisingly, Buckingham’s $\Pi $ Theorem can be used to deduce the fine structure constant. This constant is approximately $1\u2215137.036$, and has no units (like $\pi $). It can be computed using a formula involving the charge of an electron, the speed of light, Planck’s constant, and the capability of vacuum to permit electric field lines. Surprisingly, the calculation works regardless of which units are used. Richard Feynman called it [6] “one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man.”
 6.
 The first explosion of an atomic bomb was at the Trinity test in New Mexico,
1945. G. I. Taylor, a fluid mechanician at Cambridge University, asked his
colleagues at Los Alamos what the energy of the blast was—Los Alamos declared
that it was classified information. Taylor then resorted to dimensional
analysis [13]. His estimate (20kt of TNT) was remarkably close to the highly
classified value (22kt of TNT). This caused a great deal of embarrassment and as a
result Taylor was “mildly admonished by the US Army for publishing his
deductions” [2].
To make his estimate, Taylor used two photographs of the explosion, which he anecdotally took from the cover of LIFE magazine. Taylor asked, how does the radius of the blast grow with time? The other relevant factors are energy and the density of the surrounding medium (i.e. air). Find images of the Trinity blast that have timestamps (in milliseconds!) and a scale in meters^{∗∗}. Using the Buckingham $\Pi $ Theorem, estimate the amount of energy released by Trinity. Even getting this value right to within an order of magnitude is quite impressive.
 7.
 In the year 1215, the Magna Carta declared [12] that:
…there is to be one measure of wine throughout our kingdom, and one measure of ale, and one measure of corn, namely the quarter of London, and one breadth of dyed, russet and haberget cloths, that is, two ells within the borders; and let weights be dealt with as with measures.
One of the reasons for this clause was that due to the proliferation of too many systems of units, merchants could cheat their customers into paying more money for less goods.
Consider the following fictitious system, consisting of the units wizard and elf , with the conversion factor
$$\frac{2\phantom{\rule{0.3em}{0ex}}{\text{wizard}}^{2}}{\text{elf}}$$A merchant announces that for convenience, he will introduce the unit hobbit. He provides the following conversion factors to his customers:
$$\frac{3\phantom{\rule{0.3em}{0ex}}\text{hobbit}\phantom{\rule{0.3em}{0ex}}\text{elf}}{\text{wizard}},\frac{4\phantom{\rule{0.3em}{0ex}}{\text{wizard}}^{3}\phantom{\rule{0.3em}{0ex}}\text{hobbit}}{\text{elf}}$$You want to trade your eight $8\phantom{\rule{0.3em}{0ex}}{\text{wizard}}^{4}$ for some ${\text{elf}}^{2}$ in exchange. You perform the following calculation to determine the conversion:
$$\frac{8\phantom{\rule{0.3em}{0ex}}{\text{wizard}}^{4}}{1}\times \frac{\text{elf}}{4\phantom{\rule{0.3em}{0ex}}{\text{wizard}}^{3}\phantom{\rule{0.3em}{0ex}}\text{hobbit}}\times \frac{3\phantom{\rule{0.3em}{0ex}}\text{hobbit}\phantom{\rule{0.3em}{0ex}}\text{elf}}{\text{wizard}}=6\phantom{\rule{0.3em}{0ex}}{\text{elf}}^{2}$$The merchant disagrees, presenting his own calculation:
$$\frac{8\phantom{\rule{0.3em}{0ex}}{\text{wizard}}^{4}}{1}\times \frac{\text{elf}}{2\phantom{\rule{0.3em}{0ex}}{\text{wizard}}^{2}}\times \frac{\text{elf}}{2\phantom{\rule{0.3em}{0ex}}{\text{wizard}}^{2}}=2\phantom{\rule{0.3em}{0ex}}{\text{elf}}^{2}$$ (a)
 Has dimensional analysis failed? If not, explain what has gone wrong.
 (b)
 Using the premise $6\phantom{\rule{0.3em}{0ex}}{\text{elf}}^{2}=2\phantom{\rule{0.3em}{0ex}}{\text{elf}}^{2}$, prove that $0=1$.
 (c)
 Using what you know about solving linear equations, as well as Buckingham’s $\Pi $ Theorem, propose a method by which a set of conversion factors can be ‘audited’ to ensure that no such contradictions can arise.
 (d)
 Use your method to prove that the SI system is consistent. This should be reassuring.
 8.
 Ammonium nitrite is known to be highly unstable in its pure form.
I propose that it decomposes according to the following equation:
$\text{NH}\text{}\text{4}\text{NO}\text{}\text{2}$$\to $$\text{NO}\text{}\text{3}$$+$$\text{H}\text{}\text{2}\text{O}$.
 (a)
 Using techniques from this paper, give a purely mathematical argument that this reaction does not occur in nature. Is it surprising that you can predict the nonexistence of a reaction without knowing anything about the compounds involved?
 (b)
 Dilute acetic acid, $\text{CH}\text{}\text{3}\text{COOH}$, is the main ingredient in vinegar. Your friend is convinced that he has found a catalyst that lets him synthesize acetic acid from water, $\text{H}\text{}\text{2}\text{O}$, and methane, $\text{CH}\text{}\text{4}$, without producing any byproducts. You do not believe him. He insists that it must be possible because ‘all the atoms are there’. What is wrong with his argument?
 (c)
 Derive and explain a process by which you can balance any (possible) chemical equation without any ‘guess and check’. If you know how, write a program to do it.
References
[1] Mars climate orbiter mishap investigation board phase I report, 1999.
[2] Batchelor, G. The Life and Legacy of G. I. Taylor. 1996.
[3] Bridgman, P. Dimensional Analysis. 1931.
[4] Buckingham, E. On physically similar systems; illustrations of the use of dimensional equations. Phys. Rev. 4 (Oct 1914), 345–376.
[6] Feynman, R. QED: The Strange Theory of Light and Matter. 1985.
[8] Pienaar, J. A meditation on physical units, 2016.
[9] Rayleigh, J. W. S. On the question of the stability of the ﬂow of ﬂuids. Philosophical Magazine 4 (1892), 59–70.
[10] Sagan, C. Murmurs of Earth. 1978.
[11] Sonin, A. A. The Physical Basis of Dimensional Analysis. 2001.
[12] Summerson, H. The 1215 magna carta: Clause 35.
[13] Taylor, G. I. The formation of a blast wave by a very intense explosion. II. the atomic explosion of 1945. Proceedings of the Royal Society of London 201, 1065 (1950), 175–186.
[14] Torczynski, J. R. Dimensional analysis and calculus identities. The American Mathematical Monthly 95, 8 (1988), 746–754.
^{∗}There have in fact been efforts to standardize units based on fundamental physical constants like the speed of light in vacuum: such units are called ‘natural units’.
^{†}This problem was suggested by my friend David.
^{‡}To prevent such software errors, programming languages such as [5] ‘remember’ the units associated with a number and enforce that certain operations only happen on commensurable values. This generalizes to nonunitlike data types as well (for example, preventing you from dividing a number by an image, which makes no sense).
^{§}You might have noticed that the number 6.28 is roughly $2\times \pi $. Using some basic physics, we can explain why $\pi $ is involved.
^{¶}The idea had actually been around for almost 50 years before Buckingham published his 1914 paper about it. In particular, Rayleigh’s use of dimensional analysis to calculate the Reynolds number—an important constant used to study the motion of fluid in a pipe—was published in 1892 [9] and is now a classic textbook example. Buckingham’s use of the $\Pi $ symbol in this paper is what gives the theorem its name.
^{∥}Such a model is said to have similitude with the real version.
^{∗∗}Hunting for pictures is part of the fun, but if you get stuck, here are two hints:
 (a)
 https://commons.wikimedia.org/wiki/Category:Trinity_testand
 (b)
 http://blog.nuclearsecrecy.com/wpcontent/uploads/2012/03/TRNN11.jpg
All these images are in the public domain since they were taken as part of a federal government program.