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Math 314H Projectlet 1: Dimensional Analysis
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Due Date: Monday, February 17, 2003 
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Points: 15 
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\series bold 
About Projectlets:
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 Think of a projectlet as more than an exercise but less than a project.
 These are to be carried out by the smallest teams: individuals.
 Start by reading the assignment on your own.
 As with any homework assignment, it is OK to collaborate with others to
 the extent that you work jointly at solving a problem.
 It is 
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not
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 OK to copy someone else's work, and it should be clearly understood that
 each person is to write up his/her own work separately.
 
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Projectlets should be typed up in some document preparation system such
 as WordPerfect, MSWord, Latex, etc., or 
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very
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 neatly handwritten.
 Like any good writing writing exercise, projectlets should have, as Aristotle
 advises, a beginning (introduction), middle (main body of work) and end
 (conclusion).
 In a projectlet, these parts can be as small as a paragraph.
 Remember that part of your grade will be based on the quality of your written
 work.
 The paper you turn in should be a mix of equations, formulas and prose.
 You should write your answers in such a way that it can be read and understood
 by anyone who knows the material for this course.
 Your write-up should be self-contained.
 Do not write it up in the succinct style of a homework exercise for your
 instructor.
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Background: 
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Units are often neglected in math courses, but are very important for understand
ing a problem.
 Physicists are keenly aware of this; unit checks are commonly used for
 confirming that an equation is correctly formed.
 Such equations will usually involve physical quantities, such as mass,
 time, length, velocity, energy, etc.
 Physical quantities consist of a pure number with units attached to it.
 For instance, we might measure velocity as 
\begin_inset Formula $v=20$
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 feet per second.
 
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A physical law will involve physical quantities.
 It's reasonable to think that a physical law should be independent of units.
 Shouldn't Newton's law of gravity remain true whether we measure length
 in terms of meters or feet? We can abstract this idea of units a bit: both
 feet and meters are units of length.
 Think of length as a 
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dimension
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 of a quantity, and physical quantities as having dimensions attached to
 them, rather than a specific unit.
 Likewise, force has dimensions attached to it.
 Given a physical quantity 
\begin_inset Formula $q$
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, we can try to describe dimensions in terms of other, more fundamental
 dimensions.
 For example, if in some physical problem, 
\begin_inset Formula $f$
\end_inset 

 represents a force, then we can say the dimensions of 
\begin_inset Formula $f$
\end_inset 

 are 
\begin_inset Formula \[
\left[f\right]=\frac{\textrm{mass}\cdot \textrm{length}}{\textrm{time}^{2}}=MLT^{-2}\]

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where 
\begin_inset Formula $M,L,T$
\end_inset 

 are the fundamental dimensions of mass, length and time, respectively.
 There is a kind of calculus of dimensions here.
 For example, if in this same physical problem an object moves a distance
 
\begin_inset Formula $d$
\end_inset 

, then we could calculate the dimensions of 
\begin_inset Formula $fd$
\end_inset 

 (work) as
\begin_inset Formula \[
\left[Fd\right]=\left[F\right]\left[d\right]=MLT^{-2}\cdot L=ML^{2}T^{-2}.\]

\end_inset 

If this physical problem involves a mass 
\begin_inset Formula $m$
\end_inset 

, distance 
\begin_inset Formula $d$
\end_inset 

, force 
\begin_inset Formula $f$
\end_inset 

 and time 
\begin_inset Formula $t$
\end_inset 

, we see that 
\begin_inset Formula \[
\left[\frac{Fd}{m\left(d/t\right)^{2}}\right]=\frac{\left[F\right]\left[d\right]}{\left[m\right]\left[d\right]^{2}/\left[t\right]^{2}}=\frac{ML^{2}T^{-2}}{ML^{2}T^{-2}}=1.\]

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Thus, we would say that the quantity 
\begin_inset Formula $Fd/(m\left(d/t\right)^{2})$
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 is a 
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dimensionless
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 quantity for this problem.
 Are there any other dimensionless variables and how does one find them?
 What we do is add up all the exponents of each fundamental unit that occurs
 and set each sum equal to zero.
 To this end, it is convenient to set up a so-called 
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dimension matrix 
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of the form
\begin_inset Formula \[
\]

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One might wonder why this idea is worth anything at all.
 
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