1.3: The Unit Factor Method
- Page ID
- 56771
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Sometimes you will need to convert one unit to another unit. The trick for doing this: multiply by one as many times as necessary. You can always multiply a number by 1 without changing that number. The secret is writing the number 1 in a particularly clever way. Here are some ways you can write the number 1:
\(\ \begin{gathered}
1=\left(\frac{60 \mathrm{~min}}{1 \mathrm{hr}}\right) \\
1=\left(\frac{2.54 \mathrm{~cm}}{1 \mathrm{in}}\right) \\
1=\left(\frac{1 \mathrm{M}_{\odot}}{2.0 \times 10^{30} \mathrm{~kg}}\right)
\end{gathered}\)
(The M⊙ in the last example is the standard symbol for the mass of the Sun.)
If you have an expression in one set of units and you need them in another set of units, you just multiply by one as many times as necessary. Cancel out units that appear anywhere on both the top and bottom in your huge product, and you will be left with a number and another set of units. A simple example: convert the length 2.500 yards into centimeters:
\(\ 2.500 \mathrm{yd}=(2.5 \mathrm{yd})\left(\frac{36 \mathrm{in}}{1 \mathrm{yd}}\right)\left(\frac{2.54 \mathrm{~cm}}{1 \mathrm{in}}\right)=(2.5 \times 36 \times 2.54) \mathrm{cm}=228.6 \mathrm{~cm}\)
Notice that yards (yd) appear in the numerator and the denominator, and so get canceled out, as does inches. We’re left with just cm. All we did was multiply the value 2.5 yd by 1, so we didn’t change it at all; 228.6 cm is another way of saying 2.500 yd.
Another example: suppose I tell you that the surface area of the Sun is 2.4×1019 square meters. How many square miles is that?
\(\ \left(2.4 \times 10^{19} \mathrm{~m}^{2}\right)\left(\frac{100 \mathrm{~cm}}{1 \mathrm{~m}}\right)^{2}\left(\frac{1 \mathrm{in}}{2.54 \mathrm{~cm}}\right)^{2}\left(\frac{1 \mathrm{ft}}{12 \mathrm{in}}\right)^{2}\left(\frac{1 \mathrm{mi}}{5280 \mathrm{ft}}\right)^{2}\)
Two things to notice about this. First, notice how all the unit factors are squared. That’s because we started with meters squared at the beginning, which is meters times meters. If we’re going to get rid of both of them, we have to divide by meters twice. The same then goes for all of the other units. Next, notice that everything except for the left-over miles squared cancel out. We’re left with a bunch of numbers we can punch into our calculator (remembering to square things) to get
\(\ \frac{\left(2.4 \times 10^{19}\right)\left(100^{2}\right)}{\left(2.54^{2}\right)\left(12^{2}\right)\left(5280^{2}\right)} \mathrm{mi}^{2}=9.3 \times 10^{12} \mathrm{mi}^{2}\)
One more example. Sometimes you have more than one unit to convert. If I tell you that a car moves 60 miles per hour, how many meters per second is it going? (Notice here that instead of arduously multiplying out the conversion between meters and miles as I did in the previous example, I’ve looked up that there are about 1609 meters in one mile.
\(\ \left(60 \frac{\mathrm{mi}}{\mathrm{h}}\right)\left(\frac{1609 \mathrm{~m}}{1 \mathrm{mi}}\right)\left(\frac{1 \mathrm{~h}}{60 \mathrm{~min}}\right)\left(\frac{1 \mathrm{~min}}{60 \mathrm{~s}}\right)=27 \frac{\mathrm{m}}{\mathrm{s}}\)
Note that since hours was originally in the denominator, we had to make sure to put it in the numerator in a later unit factor to make it go away (since we didn’t want any hours in our final answer).
With this simple method, you can convert any quantity from one set of units to another set of units, keeping track of all the conversions as you do so