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                <a href="/"><span class="left-word">Comfortably</span>&nbsp;<span class="right-word">Numbered</span></a>
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        <article id="postcontent" class="centered">
            <section>
                <h1>Dimensional Analysis: a worksheet</h1>
                <center><em><p>An article with absolute significance</p>
</em></center>
                <h4>Tuesday, October 11, 2016 &middot; 18 min read</h4>
<p><em><strong>Editor's note:</strong> This post was automatically converted
from a LaTeX document. While some attempts have been made to preserve the
original layout, it is impossible to faithfully reproduce the original
formatting. If you would like a copy of the PDF version, please contact
me.</em>

<br/>

     <!--l. 32--><p class="indent" >      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.
</p>
   <h3 class="likesectionHead"><a 
 id="x1-1000"></a>Introduction</h3>
<!--l. 41--><p class="noindent" >Many people argue that the metric system is better than the imperial system. This
argument, however, is usually predicated on the relationship <em>between</em> 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&#x2019; than a yard?

</p><!--l. 49--><p class="indent" >   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 <em>Voyager 1</em> probe,
NASA placed a golden record designed to be found by extraterrestial life. Inscribed on
this record, NASA provided Sagan-esque 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 <span class="cite"> [<a 
href="#Xvoyager">10</a>]</span>.
</p><!--l. 58--><p class="indent" >   “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<span class="footnote-mark"><a 
href="#fn1x0" id="fn1x0-bk"><sup class="textsuperscript">∗</sup></a></span><a 
 id="x1-1001f1"></a>.
</p><!--l. 66--><p class="indent" >   Why aren&#x2019;t there obvious fundamental units? We can answer this question with a
philosophical observation: nobody knows the <em>absolute</em> 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 <span class="cite"> [<a 
href="#Xbridgman">3</a>]</span>:
     </p><div class="quote">
     <!--l. 74--><p class="noindent" >[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.</p></div>
<!--l. 76--><p class="noindent" >This statement, known as <em>Bridgman&#x2019;s Principle of absolute signiﬁcance of relative magnitude</em>, is
really just an expression of humility: nature is indifferent to our choice of units <span class="cite"> [<a 
href="#Xsonin">11</a>, <a 
href="#Xmeditation">8</a>]</span>.
That is, the period associated with the fundamental transition of the hydrogen atom will
always be <!--l. 80--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mn>2</mn><mo 
class="MathClass-punc">.</mo><mn>2</mn><mn>3</mn><mo 
class="MathClass-bin">×</mo><mn>1</mn><msup><mrow 
><mn>0</mn></mrow><mrow 
><mo 
class="MathClass-bin">−</mo><mn>1</mn><mn>7</mn></mrow></msup 
></math>
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.
</p><!--l. 85--><p class="noindent" >
</p>
   <h3 class="likesectionHead"><a 
 id="x1-2000"></a>Exercises</h3>
<!--l. 88--><p class="noindent" >

     </p><dl class="enumerate-enumitem"><dt class="enumerate-enumitem">
   1. </dt><dd 
class="enumerate-enumitem">
          <dl class="enumerate-enumitem"><dt class="enumerate-enumitem">
      (a) </dt><dd 
class="enumerate-enumitem">According to Bridgman&#x2019;s Principle, the ratio of a circle&#x2019;s circumference
          to its diameter should be constant regardless of the units used to measure
          it (this constant, of course, is <!--l. 92--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>π</mi></math>).
          Is the ratio of a circle&#x2019;s <em>area</em> to its radius independent of the units used?
          Does this violate Bridgman&#x2019;s Principle?
          </dd><dt class="enumerate-enumitem">
      (b) </dt><dd 
class="enumerate-enumitem">By citing Bridgman&#x2019;s Principle, explain why (for the purposes of this
          paper) Kelvins are a unit, but degrees Celsius are not.
          </dd></dl>
     </dd><dt class="enumerate-enumitem">
   2. </dt><dd 
class="enumerate-enumitem">In Europe, <strong>fuel efficiency</strong> is measured in liters per kilometer. Note that both
     liters and kilometers can be expressed in terms of centimeters. Simplify
     ‘liters per kilometer&#x2019;, and give a physical interpretation of the resulting
     units <span class="cite"> [<a 
href="#Xxkcd">7</a>]</span>.
     </dd><dt class="enumerate-enumitem">
   3. </dt><dd 
class="enumerate-enumitem">The <strong>Pythagorean theorem</strong> 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.
     <!--l. 113--><p class="indent" >
          </p><dl class="enumerate-enumitem"><dt class="enumerate-enumitem">
      (a) </dt><dd 
class="enumerate-enumitem">Show  that  all  three  triangles  are  similar,  that  is,  they  have  the  same
          angles.
          </dd><dt class="enumerate-enumitem">
      (b) </dt><dd 
class="enumerate-enumitem">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.
          </dd><dt class="enumerate-enumitem">
      (c) </dt><dd 
class="enumerate-enumitem">Consider the relationship between the areas of the three triangles, then
          <em>use dimensional analysis to derive the Pythagorean Theorem</em> <span class="cite"> [<a 
href="#Xpythagoras">14</a>]</span>.
          Why is this not a valid proof?
          </dd></dl>

     </dd><dt class="enumerate-enumitem">
   4. </dt><dd 
class="enumerate-enumitem">
          <dl class="enumerate-enumitem"><dt class="enumerate-enumitem">
      (a) </dt><dd 
class="enumerate-enumitem">The <em>derivative</em> 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.
          </dd><dt class="enumerate-enumitem">
      (b) </dt><dd 
class="enumerate-enumitem"><em>Integration</em>  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.
          </dd><dt class="enumerate-enumitem">
      (c) </dt><dd 
class="enumerate-enumitem">According to the <strong>Fundamental Theorem of Calculus</strong>, 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 <!--l. 142--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>x</mi></math>
          axis). For example, with the equation <!--l. 142--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>y</mi> <mo 
class="MathClass-rel">=</mo> <mn>1</mn><mo 
class="MathClass-bin">−</mo><mi 
>x</mi></math>,
          the integral of <!--l. 143--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>y</mi></math>
          with respect to <!--l. 143--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>x</mi></math>
          from <!--l. 143--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mn>0</mn></math>
          to <!--l. 143--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mn>1</mn></math>
          is <!--l. 143--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mn>0</mn><mo 
class="MathClass-punc">.</mo><mn>5</mn></math>—verify
          this by drawing a picture.
          <!--l. 146--><p class="noindent" >Recall the formula for measuring the area of a rectangle. Assuming that
          the <!--l. 147--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>x</mi></math>
          and <!--l. 147--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>y</mi></math>
          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?
          <em>To you, does this feel like compelling evidence that the Fundamental
          Theorem of Calculus is true?</em>
          </p></dd><dt class="enumerate-enumitem">
      (d) </dt><dd 
class="enumerate-enumitem">Consider the functions <!--l. 152--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>f</mi> <mo 
class="MathClass-punc">:</mo> <mi 
>𝔹</mi> <mo 
class="MathClass-rel">→</mo> <mi 
>ℂ</mi></math>
          and <!--l. 153--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>g</mi> <mo 
class="MathClass-punc">:</mo> <mi 
>𝔸</mi> <mo 
class="MathClass-rel">→</mo> <mi 
>𝔹</mi></math>
          where <!--l. 153--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>𝔸</mi></math>,
          <!--l. 154--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>𝔹</mi></math>,
          and <!--l. 154--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>ℂ</mi></math>
          are all units. What are the units of <!--l. 155--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><msup><mrow 
><mi 
>f</mi></mrow><mrow 
><mi 
>′</mi></mrow></msup 
></math>

          and <!--l. 155--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><msup><mrow 
><mi 
>g</mi></mrow><mrow 
><mi 
>′</mi></mrow></msup 
></math>?
          (Here, <!--l. 155--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><msup><mrow 
><mi 
>f</mi></mrow><mrow 
><mi 
>′</mi></mrow></msup 
></math>
          means the derivative of <!--l. 156--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>f</mi></math>
          with respect to <!--l. 156--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>x</mi></math>.)
          Now, let <!--l. 156--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>h</mi><mrow ><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow> <mo 
class="MathClass-rel">=</mo> <mi 
>f</mi><mrow ><mo 
class="MathClass-open">(</mo><mrow><mi 
>g</mi><mrow ><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></mrow><mo 
class="MathClass-close">)</mo></mrow></math>.
          Convince yourself that <!--l. 157--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>h</mi> <mo 
class="MathClass-punc">:</mo> <mi 
>𝔸</mi> <mo 
class="MathClass-rel">→</mo> <mi 
>ℂ</mi></math>.
          Then, use dimensional analysis to derive an equation for <!--l. 158--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><msup><mrow 
><mi 
>h</mi></mrow><mrow 
><mi 
>′</mi></mrow></msup 
><mrow ><mo 
class="MathClass-open">(</mo><mrow><mi 
>x</mi></mrow><mo 
class="MathClass-close">)</mo></mrow></math>
          in terms of <!--l. 158--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>f</mi></math>,
          <!--l. 158--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>g</mi></math>,
          and <!--l. 159--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>x</mi></math>.
          This expression is called the <strong>chain rule</strong><span class="footnote-mark"><a 
href="#fn2x0" id="fn2x0-bk"><sup class="textsuperscript">†</sup></a></span><a 
 id="x1-2014f2"></a>.
          </dd></dl>
     </dd><dt class="enumerate-enumitem">
   5. </dt><dd 
class="enumerate-enumitem">In 1999, the <strong>Mars Climate Orbiter</strong> 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 <span class="cite"> [<a 
href="#Xmars">1</a>]</span> to be a software error: a mission-critical
     piece of software produced output in non-SI units, in pound-seconds rather than
     Newton-seconds<span class="footnote-mark"><a 
href="#fn3x0" id="fn3x0-bk"><sup class="textsuperscript">‡</sup></a></span><a 
 id="x1-2016f3"></a>.
     <!--l. 176--><p class="noindent" >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?
     </p><!--l. 182--><p class="indent" >
          </p><dl class="enumerate-enumitem"><dt class="enumerate-enumitem">
      (a) </dt><dd 
class="enumerate-enumitem">Suppose we were trying to discover the period <!--l. 182--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>T</mi> </math>
          of a pendulum. It seems like the important parameters we would need to
          know are the pendulum&#x2019;s length <!--l. 184--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>l</mi></math>,
          and mass <!--l. 184--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>m</mi></math>.
          We also know that on Earth, the acceleration due to gravity is <!--l. 185--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>g</mi></math>.
          Using dimensional analysis, discover a formula <!--l. 186--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>f</mi></math>
          for <!--l. 186--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>T</mi> </math>
          in terms of <!--l. 186--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>l</mi></math>
          and <!--l. 186--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>g</mi></math>.
          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
          <!--l. 188--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>T</mi> <mo 
class="MathClass-rel">=</mo> <mn>6</mn><mo 
class="MathClass-punc">.</mo><mn>2</mn><mn>8</mn><mo 
class="MathClass-bin">×</mo><mi 
>f</mi></math><span class="footnote-mark"><a 
href="#fn4x0" id="fn4x0-bk"><sup class="textsuperscript">§</sup></a></span><a 
 id="x1-2018f4"></a>.
          Has dimensional analysis failed?

          </dd><dt class="enumerate-enumitem">
      (b) </dt><dd 
class="enumerate-enumitem">The idea that you can discover dimensionless constants like <!--l. 194--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mn>6</mn><mo 
class="MathClass-punc">.</mo><mn>2</mn><mn>8</mn></math>
          above by using dimensional analysis to create formulas, and then running
          experiments to solve for the constant, is formalized by <strong>Buckingham&#x2019;s
          <!--l. 196--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>Π</mi></math>
          Theorem</strong><span class="footnote-mark"><a 
href="#fn5x0" id="fn5x0-bk"><sup class="textsuperscript">¶</sup></a></span><a 
 id="x1-2020f5"></a> <span class="cite">
           [<a 
href="#Xbuckingham">4</a>]</span>. It states that with <!--l. 203--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>n</mi></math>
          physical variables, <!--l. 204--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>r</mi></math>
          of which are independent, you can derive <!--l. 204--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>n</mi><mo 
class="MathClass-bin">−</mo><mi 
>r</mi></math>
          fundamental dimensionless quantities. Explain what is meant by ‘independent&#x2019;.
          Using what you know about solving systems of linear equations, explain
          why the theorem makes sense intuitively.
          </dd><dt class="enumerate-enumitem">
      (c) </dt><dd 
class="enumerate-enumitem">If you plan on testing a <!--l. 209--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mn>1</mn> <mo 
class="MathClass-punc">:</mo> <mn>1</mn><mn>0</mn></math>
          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&#x2019;s <!--l. 212--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>Π</mi></math>
          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<span class="footnote-mark"><a 
href="#fn6x0" id="fn6x0-bk"><sup class="textsuperscript">∥</sup></a></span><a 
 id="x1-2022f6"></a>.
          </dd></dl>
     <!--l. 220--><p class="noindent" >Surprisingly, Buckingham&#x2019;s <!--l. 220--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>Π</mi></math>
     Theorem can be used to deduce the <strong>fine structure constant</strong>. This constant is approximately
     <!--l. 222--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mn>1</mn><mo 
class="MathClass-bin">∕</mo><mn>1</mn><mn>3</mn><mn>7</mn><mo 
class="MathClass-punc">.</mo><mn>0</mn><mn>3</mn><mn>6</mn></math>, and has no
     units (like <!--l. 222--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>π</mi></math>).
     It can be computed using a formula involving the charge of an electron, the speed
     of light, Planck&#x2019;s constant, and the capability of vacuum to permit electric field
     lines. Surprisingly, the calculation works <em>regardless of which units are used</em>.
     Richard Feynman called it <span class="cite"> [<a 
href="#Xfeynman">6</a>]</span> “one of the greatest damn mysteries
     of physics: a magic number that comes to us with no understanding by
     man.”
     </p></dd><dt class="enumerate-enumitem">
   6. </dt><dd 
class="enumerate-enumitem">The first explosion of an atomic bomb was at the <strong>Trinity test</strong> 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 <span class="cite"> [<a 
href="#Xtaylor">13</a>]</span>. 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” <span class="cite"> [<a 
href="#Xbatchelor">2</a>]</span>.
     <!--l. 240--><p class="noindent" >To make his estimate, Taylor used two photographs of the explosion, which he
     anecdotally took from the cover of <em>LIFE</em> 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<span class="footnote-mark"><a 
href="#fn7x0" id="fn7x0-bk"><sup class="textsuperscript">∗∗</sup></a></span><a 
 id="x1-2024f7"></a>. Using
     the Buckingham <!--l. 254--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>Π</mi></math>
     Theorem, <em>estimate the amount of energy released by Trinity</em>. Even getting this
     value right to within an order of magnitude is quite impressive.
     </p></dd><dt class="enumerate-enumitem">
   7. </dt><dd 
class="enumerate-enumitem">In the year 1215, the <strong>Magna Carta</strong> declared <span class="cite"> [<a 
href="#Xmagnacarta">12</a>]</span> that:
          <div class="quote">
          <!--l. 259--><p class="noindent" >…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.</p></div>
     <!--l. 263--><p class="noindent" >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.
     </p><!--l. 267--><p class="noindent" >Consider the following fictitious system, consisting of the units <em>wizard</em> and <em>elf </em>,
     with the conversion factor
</p>

<div class="math-display"><!--l. 269--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="block" ><mrow 
>
                              <mfrac><mrow 
><mn>2</mn><mspace width="0.3em" class="thinspace"/><msup><mrow 
><mstyle 
class="text"><mtext  >wizard</mtext></mstyle></mrow><mrow 
><mn>2</mn></mrow></msup 
></mrow>
     <mrow 
><mstyle 
class="text"><mtext  >elf</mtext></mstyle></mrow></mfrac>
</mrow></math></div>
     <!--l. 271--><p class="nopar" >
     </p><!--l. 273--><p class="noindent" >A merchant announces that for convenience, he will introduce the unit <em>hobbit</em>. He
     provides the following conversion factors to his customers:
</p>
<div class="math-display"><!--l. 276--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="block" ><mrow 
>
                       <mfrac><mrow 
><mn>3</mn><mspace width="0.3em" class="thinspace"/><mstyle 
class="text"><mtext  >hobbit</mtext></mstyle><mspace width="0.3em" class="thinspace"/><mstyle 
class="text"><mtext  >elf</mtext></mstyle></mrow>
   <mrow 
><mstyle 
class="text"><mtext  >wizard</mtext></mstyle></mrow></mfrac>  <mo 
class="MathClass-punc">,</mo> <mfrac><mrow 
><mn>4</mn><mspace width="0.3em" class="thinspace"/><msup><mrow 
><mstyle 
class="text"><mtext  >wizard</mtext></mstyle></mrow><mrow 
><mn>3</mn></mrow></msup 
><mspace width="0.3em" class="thinspace"/><mstyle 
class="text"><mtext  >hobbit</mtext></mstyle></mrow> 
          <mrow 
><mstyle 
class="text"><mtext  >elf</mtext></mstyle></mrow></mfrac>
</mrow></math></div>
     <!--l. 279--><p class="nopar" >
     </p><!--l. 281--><p class="noindent" >You want to trade your eight <!--l. 281--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mn>8</mn><mspace width="0.3em" class="thinspace"/><msup><mrow 
><mstyle 
class="text"><mtext  >wizard</mtext></mstyle></mrow><mrow 
><mn>4</mn></mrow></msup 
></math>
     for some <!--l. 281--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><msup><mrow 
><mstyle 
class="text"><mtext  >elf</mtext></mstyle></mrow><mrow 
><mn>2</mn></mrow></msup 
></math>
     in exchange. You perform the following calculation to determine the
     conversion:
</p>

<div class="math-display"><!--l. 284--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="block" ><mrow 
>
              <mfrac><mrow 
><mn>8</mn><mspace width="0.3em" class="thinspace"/><msup><mrow 
><mstyle 
class="text"><mtext  >wizard</mtext></mstyle></mrow><mrow 
><mn>4</mn></mrow></msup 
></mrow>
      <mrow 
><mn>1</mn></mrow></mfrac>    <mo 
class="MathClass-bin">×</mo>    <mfrac><mrow 
><mstyle 
class="text"><mtext  >elf</mtext></mstyle></mrow> 
<mrow 
><mn>4</mn><mspace width="0.3em" class="thinspace"/><msup><mrow 
><mstyle 
class="text"><mtext  >wizard</mtext></mstyle></mrow><mrow 
><mn>3</mn></mrow></msup 
><mspace width="0.3em" class="thinspace"/><mstyle 
class="text"><mtext  >hobbit</mtext></mstyle></mrow></mfrac> <mo 
class="MathClass-bin">×</mo><mfrac><mrow 
><mn>3</mn><mspace width="0.3em" class="thinspace"/><mstyle 
class="text"><mtext  >hobbit</mtext></mstyle><mspace width="0.3em" class="thinspace"/><mstyle 
class="text"><mtext  >elf</mtext></mstyle></mrow> 
   <mrow 
><mstyle 
class="text"><mtext  >wizard</mtext></mstyle></mrow></mfrac>    <mo 
class="MathClass-rel">=</mo> <mn>6</mn><mspace width="0.3em" class="thinspace"/><msup><mrow 
><mstyle 
class="text"><mtext  >elf</mtext></mstyle></mrow><mrow 
><mn>2</mn></mrow></msup 
>
</mrow></math></div>
     <!--l. 289--><p class="nopar" > The merchant disagrees, presenting his own calculation:
</p>
<div class="math-display"><!--l. 291--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="block" ><mrow 
>
                 <mfrac><mrow 
><mn>8</mn><mspace width="0.3em" class="thinspace"/><msup><mrow 
><mstyle 
class="text"><mtext  >wizard</mtext></mstyle></mrow><mrow 
><mn>4</mn></mrow></msup 
></mrow>
      <mrow 
><mn>1</mn></mrow></mfrac>    <mo 
class="MathClass-bin">×</mo>  <mfrac><mrow 
><mstyle 
class="text"><mtext  >elf</mtext></mstyle></mrow> 
<mrow 
><mn>2</mn><mspace width="0.3em" class="thinspace"/><msup><mrow 
><mstyle 
class="text"><mtext  >wizard</mtext></mstyle></mrow><mrow 
><mn>2</mn></mrow></msup 
></mrow></mfrac> <mo 
class="MathClass-bin">×</mo>  <mfrac><mrow 
><mstyle 
class="text"><mtext  >elf</mtext></mstyle></mrow> 
<mrow 
><mn>2</mn><mspace width="0.3em" class="thinspace"/><msup><mrow 
><mstyle 
class="text"><mtext  >wizard</mtext></mstyle></mrow><mrow 
><mn>2</mn></mrow></msup 
></mrow></mfrac> <mo 
class="MathClass-rel">=</mo> <mn>2</mn><mspace width="0.3em" class="thinspace"/><msup><mrow 
><mstyle 
class="text"><mtext  >elf</mtext></mstyle></mrow><mrow 
><mn>2</mn></mrow></msup 
>
</mrow></math></div>
     <!--l. 296--><p class="nopar" >
     </p><!--l. 299--><p class="indent" >
          </p><dl class="enumerate-enumitem"><dt class="enumerate-enumitem">
      (a) </dt><dd 
class="enumerate-enumitem">Has dimensional analysis failed? If not, explain what has gone wrong.
          </dd><dt class="enumerate-enumitem">
      (b) </dt><dd 
class="enumerate-enumitem">Using the premise <!--l. 302--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mn>6</mn><mspace width="0.3em" class="thinspace"/><msup><mrow 
><mstyle 
class="text"><mtext  >elf</mtext></mstyle></mrow><mrow 
><mn>2</mn></mrow></msup 
> <mo 
class="MathClass-rel">=</mo> <mn>2</mn><mspace width="0.3em" class="thinspace"/><msup><mrow 
><mstyle 
class="text"><mtext  >elf</mtext></mstyle></mrow><mrow 
><mn>2</mn></mrow></msup 
></math>,
          prove that <!--l. 303--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mn>0</mn> <mo 
class="MathClass-rel">=</mo> <mn>1</mn></math>.
          </dd><dt class="enumerate-enumitem">
      (c) </dt><dd 
class="enumerate-enumitem">Using what you know about solving linear equations, as well as Buckingham&#x2019;s
          <!--l. 306--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>Π</mi></math>
          Theorem, propose a method by which a set of conversion factors can

          be ‘audited&#x2019; to ensure that <em>no</em> such contradictions can arise.
          </dd><dt class="enumerate-enumitem">
      (d) </dt><dd 
class="enumerate-enumitem">Use your method to prove that the SI system is consistent. This should
          be reassuring.
          </dd></dl>
     </dd><dt class="enumerate-enumitem">
   8. </dt><dd 
class="enumerate-enumitem"><strong>Ammonium nitrite</strong> is known to be highly unstable in its pure form.
     I propose that it decomposes according to the following equation:
     <!--l. 319--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mstyle 
class="text"><mtext  >NH</mtext></mstyle>  <mstyle 
class="text"><mtext  ></mtext></mstyle> <mstyle 
class="text"><mtext  >4</mtext></mstyle><mstyle 
class="text"><mtext  >NO</mtext></mstyle>  <mstyle 
class="text"><mtext  ></mtext></mstyle> <mstyle 
class="text"><mtext  >2</mtext></mstyle></math><!--l. 319--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mo 
class="MathClass-rel">     →</mo></math><!--l. 319--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mstyle 
class="text"><mtext  >NO</mtext></mstyle>  <mstyle 
class="text"><mtext  ></mtext></mstyle> <mstyle 
class="text"><mtext  >3</mtext></mstyle></math><!--l. 319--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" > <mo 
class="MathClass-bin">+</mo></math><!--l. 319--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mstyle 
class="text"><mtext  >H</mtext></mstyle>  <mstyle 
class="text"><mtext  ></mtext></mstyle> <mstyle 
class="text"><mtext  >2</mtext></mstyle><mstyle 
class="text"><mtext  >O</mtext></mstyle></math>.
     <!--l. 322--><p class="indent" >
          </p><dl class="enumerate-enumitem"><dt class="enumerate-enumitem">
      (a) </dt><dd 
class="enumerate-enumitem">Using techniques from this paper, give a <em>purely mathematical</em> argument
          that this reaction does not occur in nature. Is it surprising that you can
          predict the non-existence of a reaction without knowing anything about
          the compounds involved?
          </dd><dt class="enumerate-enumitem">
      (b) </dt><dd 
class="enumerate-enumitem">Dilute <strong>acetic acid</strong>, <!--l. 327--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mstyle 
class="text"><mtext  >CH</mtext></mstyle>  <mstyle 
class="text"><mtext  ></mtext></mstyle> <mstyle 
class="text"><mtext  >3</mtext></mstyle><mstyle 
class="text"><mtext  >COOH</mtext></mstyle></math>,
          is  the  main  ingredient  in  vinegar.  Your  friend  is  convinced  that  he
          has found a catalyst that lets him synthesize acetic acid from water,
          <!--l. 329--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mstyle 
class="text"><mtext  >H</mtext></mstyle>  <mstyle 
class="text"><mtext  ></mtext></mstyle> <mstyle 
class="text"><mtext  >2</mtext></mstyle><mstyle 
class="text"><mtext  >O</mtext></mstyle></math>,
          and methane, <!--l. 330--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mstyle 
class="text"><mtext  >CH</mtext></mstyle>  <mstyle 
class="text"><mtext  ></mtext></mstyle> <mstyle 
class="text"><mtext  >4</mtext></mstyle></math>,
          without producing any byproducts. You do not believe him. He insists
          that it must be possible because ‘all the atoms are there&#x2019;. What is wrong
          with his argument?
          </dd><dt class="enumerate-enumitem">
      (c) </dt><dd 
class="enumerate-enumitem"><em>Derive and explain a process by which you can balance any (possible)
          chemical equation without any ‘guess and check&#x2019;.  If you know how,
          write a program to do it.</em>
          </dd></dl>
     </dd></dl>
<!--l. 1--><p class="noindent" >
</p>
   <h3 class="likesectionHead"><a 
 id="x1-3000"></a>References</h3>
<!--l. 1--><p class="noindent" >

     </p><div class="thebibliography">
     <p class="bibitem" ><span class="biblabel">
  [1]<span class="bibsp">   </span></span><a 
 id="Xmars"></a>Mars climate orbiter mishap investigation board phase I report, 1999.
     </p>
     <p class="bibitem" ><span class="biblabel">
  [2]<span class="bibsp">   </span></span><a 
 id="Xbatchelor"></a>Batchelor, G. The Life and Legacy of G. I. Taylor. 1996.
     </p>
     <p class="bibitem" ><span class="biblabel">
  [3]<span class="bibsp">   </span></span><a 
 id="Xbridgman"></a>Bridgman, P. Dimensional Analysis. 1931.
     </p>
     <p class="bibitem" ><span class="biblabel">
  [4]<span class="bibsp">   </span></span><a 
 id="Xbuckingham"></a>Buckingham, E. On physically similar systems; illustrations of the use
     of dimensional equations. Phys. Rev. 4 (Oct 1914), 345–376.
     </p>
     <p class="bibitem" ><span class="biblabel">
  [5]<span class="bibsp">   </span></span><a 
 id="Xfrink"></a>Eliasen, A. Frink.
     </p>
     <p class="bibitem" ><span class="biblabel">
  [6]<span class="bibsp">   </span></span><a 
 id="Xfeynman"></a>Feynman, R. QED: The Strange Theory of Light and Matter. 1985.
     </p>
     <p class="bibitem" ><span class="biblabel">
  [7]<span class="bibsp">   </span></span><a 
 id="Xxkcd"></a>Munroe, R. What If? 2014.
     </p>
     <p class="bibitem" ><span class="biblabel">
  [8]<span class="bibsp">   </span></span><a 
 id="Xmeditation"></a>Pienaar, J. A meditation on physical units, 2016.
     </p>
     <p class="bibitem" ><span class="biblabel">
  [9]<span class="bibsp">   </span></span><a 
 id="Xrayleigh"></a>Rayleigh, J. W. S. On the question of the stability of the ﬂow of ﬂuids.
     Philosophical Magazine 4 (1892), 59–70.
     </p>
     <p class="bibitem" ><span class="biblabel">
 [10]<span class="bibsp">   </span></span><a 
 id="Xvoyager"></a>Sagan, C. Murmurs of Earth. 1978.

     </p>
     <p class="bibitem" ><span class="biblabel">
 [11]<span class="bibsp">   </span></span><a 
 id="Xsonin"></a>Sonin, A. A. The Physical Basis of Dimensional Analysis. 2001.
     </p>
     <p class="bibitem" ><span class="biblabel">
 [12]<span class="bibsp">   </span></span><a 
 id="Xmagnacarta"></a>Summerson, H. The 1215 magna carta: Clause 35.
     </p>
     <p class="bibitem" ><span class="biblabel">
 [13]<span class="bibsp">   </span></span><a 
 id="Xtaylor"></a>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.
     </p>
     <p class="bibitem" ><span class="biblabel">
 [14]<span class="bibsp">   </span></span><a 
 id="Xpythagoras"></a>Torczynski, J. R.  Dimensional analysis and calculus identities.  The
     American Mathematical Monthly 95, 8 (1988), 746–754.
</p>
     </div>

   <div class="footnotes"><!--l. 64--><p class="indent" >      <span class="footnote-mark"><a 
href="#fn1x0-bk" id="fn1x0"><sup class="textsuperscript">∗</sup></a></span>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&#x2019;.</p>
<!--l. 160--><p class="noindent" ><span class="footnote-mark"><a 
href="#fn2x0-bk" id="fn2x0"><sup class="textsuperscript">†</sup></a></span>This problem was suggested by my friend David.</p>
<!--l. 174--><p class="noindent" ><span class="footnote-mark"><a 
href="#fn3x0-bk" id="fn3x0"><sup class="textsuperscript">‡</sup></a></span>To prevent such software errors, programming languages such as <span class="cite"> [<a 
href="#Xfrink">5</a>]</span> ‘remember&#x2019; the units associated
with a number and enforce that certain operations only happen on commensurable values. This generalizes
to non-unit-like data types as well (for example, preventing you from dividing a number by an image, which
makes no sense).</p>
<!--l. 191--><p class="noindent" ><span class="footnote-mark"><a 
href="#fn4x0-bk" id="fn4x0"><sup class="textsuperscript">§</sup></a></span>You might have noticed that the number 6.28 is roughly <!--l. 191--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mn>2</mn><mo 
class="MathClass-bin">×</mo><mi 
>π</mi></math>.
Using some basic physics, we can explain why <!--l. 191--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>π</mi></math>
is involved.</p>
<!--l. 203--><p class="noindent" ><span class="footnote-mark"><a 
href="#fn5x0-bk" id="fn5x0"><sup class="textsuperscript">¶</sup></a></span>The  idea  had  actually  been  around  for  almost  50  years  before  Buckingham  published  his
1914  paper  about  it.  In  particular,  Rayleigh&#x2019;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 <span class="cite"> [<a 
href="#Xrayleigh">9</a>]</span>  and  is  now  a  classic  textbook  example.  Buckingham&#x2019;s  use  of  the
<!--l. 203--><math 
 xmlns="http://www.w3.org/1998/Math/MathML"  
display="inline" ><mi 
>Π</mi></math>
symbol in this paper is what gives the theorem its name.</p>
<!--l. 216--><p class="noindent" ><span class="footnote-mark"><a 
href="#fn6x0-bk" id="fn6x0"><sup class="textsuperscript">∥</sup></a></span>Such a model is said to have <em>similitude</em> with the real version.</p>
<!--l. 254--><p class="noindent" ><span class="footnote-mark"><a 
href="#fn7x0-bk" id="fn7x0"><sup class="textsuperscript">∗∗</sup></a></span>Hunting for pictures is part of the fun, but if you get stuck, here are two hints:
                       </p><dl class="enumerate-enumitem"><dt class="enumerate-enumitem">
                (a)  </dt><dd 
class="enumerate-enumitem"><a 
href="https://commons.wikimedia.org/wiki/Category:Trinity_test" class="url" >https://commons.wikimedia.org/wiki/Category:Trinity_test</a>and
                       </dd><dt class="enumerate-enumitem">
                (b)  </dt><dd 
class="enumerate-enumitem"><a 
href="http://blog.nuclearsecrecy.com/wp-content/uploads/2012/03/TR-NN-11.jpg" class="url" >http://blog.nuclearsecrecy.com/wp-content/uploads/2012/03/TR-NN-11.jpg</a></dd></dl>
<!--l. 254--><p class="noindent" >All these images are in the public domain since they were taken as part of a federal government
program.</p>                                                                                                                                                 </div>

            </section>

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