Tutorial for Dimensional Analysis

							                 Tutorial for Dimensional Analysis
Dimensional analysis is simply a fancy name for converting units.
Whenever we measure something, we measure it with units. For example,

       15 in (length)        50.0 mL (volume)               1105 mm Hg (pressure)

Often, more than one unit system will be available. Length, for example, can be
measured in inches or in centimeters.

First, remember that 1 inch = 2.54 cm. Now, we can make a conversion equality.

Divide both sides of that equality by 2.54 cm or by 1 inch, and you find that:

                                 1 inch   2.54 cm
                                        =         = 1
                                2.54 cm    1 inch

Notice how all of those equal 1. Thus, multiplying anything by either of those fractions
would just be multiplying by one.

If you measure this line:


You would find that it is 6.0 inches long. However, what if you wanted to know how
long it was in centimeters? That’s when unit conversions help.

In order to convert 6.0 inches into centimeters, we would multiply 6.0 inches by 1.
Not just any 1, though. We would multiply it by something equal to one:

                                          ⎛ 2.54 cm ⎞          ⎛ 2.54 cm ⎞
           6.0 in = 6.0 in x (1) = 6.0 in ⎜         ⎟ = 6.0 in ⎜
                                                             /           ⎟ = 15.24 cm
                                          ⎝ 1 in ⎠                   /
                                                               ⎝ 1 in ⎠

Notice how we just multiplied “6.0 in” with a different form of “1”. Then we canceled
the units.

We can use this same process in any equality. For example, let’s say that for every
mechanical pencil, there are 3 pieces of lead. Therefore, 1 pencil = 3 lead. Again
dividing both sides by either 1 pencil or 3 lead, we get

                              1 pencil    3 lead
                                       =          = 1
                               3 lead    1 pencil
With that equality, if we have 22 pencils, we can find out how many pieces of lead we
have. We just start with 22 pencils:

                                              ⎛ 3 lead ⎞                ⎛ 3 lead ⎞
                                              ⎜ 1 pencil ⎟ = 22 pencils ⎜ 1 pencil ⎟ = 66 lead
   22 pencils = 22 pencils x (1) = 22 pencils ⎜          ⎟              ⎜          ⎟
                                              ⎝          ⎠              ⎝          ⎠
    Unit conversions get easier with practice. Try these problems:

            Convert:                                               Helpful Equalities

1           5 dogs into legs                                       1 dog = 4 legs



2           15.2 days into hours                                   24.0 hours = 1 day



3           30.0 centimeters into inches                           1 inch = 2.54 centimeters



4           3.5 pounds into newtons                                1 newton = 0.225 pounds



5           1.23 x 10-3 kilometers into miles                      1 km = 0.621 miles



6           1.204 x 1024 molecules into moles                      1 mole = 6.022 x 1023 molecules



For problems 7-9, multiple steps are required.
7           16 meters/second into miles/hour                       1 meter/second = 3.60 km/h
                                                                   1 km/h = 0.621 mi/h



8           2.1 light years into feet                              1 light-year = 9.46 x 1015 meters
                                                                   1 foot = 0.31 meters



9           14.6 kilometers into inches                            1 km = 0.621 miles
                                                                   1 mile = 5280 feet
                                                                   1 foot = 12 inches




    Answer Key:
                                                        5.   0.00076383 miles or 7.64 x 10–4 miles (w/sig. figs.)
       1.   20 legs                                     6.   2.000 moles (w/sig. figs.)
       2.   364.8 or 365 hours (w/sig. figs.)           7.   35.8 mi/h or 36 mi/h (w/sig. figs.)
       3.   11.8 or 12 inches (w/sig. figs.)            8.   6.5 x 1016 feet (w/sig. figs.)
       4.   15.6 newtons or 16 newtons (w/sig. figs.)   9.   5.74 x 105 inches (w/sig. figs.)