Group Activity: Dimensional Analysis

Introduction

Dimensional Analysis is a technique for solving problems. Particularly, this technique is suited to converting one kind of unit into another.

Converting one unit to another changes nothing: it just expresses a distance, a volume, a mass (etc) in different units. In effect, you are multiplying by one. Using it is the easiest way to find out how many miles someone from Canada means when they say that the hockey rink is about 42 km away. Here’s what you do:

          1 mi
42 km × ———————— = 26 mi  This works because 1 mi = 1.61 km
         1.61 km

Multiplying by any conversion factor is just like multiplying by one. You change the number but not the quantity.

The number of miles is directly proportional to the number of kilometers. The algebraic equation that leads to the math above is:

 x mi        1 mi
———————  =  ——————— which is:
 42 km       1.61 km

           1 mi
 x mi  =  ——————— × 42 km    so x mi = 26 mi
           1.61 km
Notice especially the way the units cancel each other out so 
that you are left with the units you want and you cancel the units you have.

These proportions are called unit factors because they are equal to the number one. Remember, multiplying by the number one does not change the numerical value! The unit factor is used to convert one unit to another unit as in the examples on this page. Pick the unit factor you use carefully as the units must cancel out. If the unit to be cancelled is not part of a fraction or is in the numerator then the unit factor must be written with that unit in the denominator. Here are more examples:

How many seconds are in one year?
        365 day    24 hr    60 min      60 s
1 yr ×  ——————— × ——————— × ——————— × ——————— = 31,536,000 s
         1 yr      1 day     1 hr      1 min
         
Notice that the unit you want to cancel goes on the opposite side of the
conversion factor from the unit you want to replace it: we want the unit
‘yr’ to cancel to be replaced by ‘day’.

Express 25 miles per hour as meters per second
   mi     1 km     1000 m     1 hr     1 min
25 ——— × ——————— × ——————— × ——————— × ——————— = 11 m/s
   hr   0.621 mi    1 km     60 min     60 s

Key Point

The math for all of this is really quite simple: multiply by the top, divide by the bottom. Do this for each conversion factor, cancelling units as you go. Do not record intermediate values from your calculator: do all calculations as one long series.

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Unit Factor Basic Skills

When it comes to knowing how to use the Unit Factor Process (also known as dimensional analisys) there are two basic skills. First, building a unit factor. Second, setting up and carrying out the calculation.

Building a Unit Factor

One quantity (like how much milk is left in the jug) can be expressed using different units: saying 1 quart is left is the same as saying 2 pints are left or saying 4 cups are left or even 950 mL are left. A unit equality gives the relationship between two units. It can be used to make two unit factors.

For example:

1 lb = 0.454 kg
can be written as either of the following:

      1 lb                  0.454 kg
     ———————                ———————
    0.454 kg                1 lb

This works because of some simple algebra:

                   1 lb          0.454 kg            1 lb
1 lb = 0.454 kg  ----------- = -----------       ----------- = 1
                   0.454 kg      0.454 kg           0.454 kg 
                divide both sides by 0.454 kg    0.454 kg/0.454 kg = 1!

The math above gives the unit factor on the left, but what about the one on the right? Can you figure out the necessary math to get that one?

Setting up Calculations

Which of the unit factors do you choose to solve a problem? It depends.
If you have a quantity expressed in kg then choose the unit factor on the left.

          1 lb
5.0 kg × ——————— = 11 lb
         0.454 kg

         0.454 kg 
11 lb × ——————— = 5.0 kg
          1 lb

Round to a convenient number of digits in the answer: only use as many digits as in the number you started with.

If you have a quantity expressed in lb then choose the unit factor on the right.

Why do you choose the unit factors as described in the examples above? You choose them that way so that you cancel out the units you are changing from and introduce the units you are changing to.

Here is a bigger example.

Conversion Steps
Start
yr
↓
day
↓
hr
↓
min
↓
s
↓
ms
Finish

Convert 2 years to milliseconds (ms).
2 yr = 2 yr/1

 2 yr     365 day    24 hr      60 min      60 s      1,000 ms    
------ x -------- x -------- x -------- x -------- x ---------- = 6.3072 × 10¹⁰  ms
  1        1 yr      1 day       1 hr      1 min         1s     
Enter this into your calculator as 2 × 365 × 24 × 60 × 60 × 1,000 = 6.3072 × 10¹⁰

If it helps you to keep track of what you are doing then make a small chart like the one at left. The important thing to note in this example is that the unit to be cancelled out is always on the opposite side of the fraction bar.

Here is an example in which you must cancel units both in the numerator and the denominator:
Convert 2.88 × 10⁴ km/hr to mi/s

 28,800 km     1 hr       1 min     0.621 mi
------------ x -------- x -------- x -------- = 4.97 mi/s
  1 hr          60 min      60 s       1 km

Type into your calculator: 2.88e4 ÷ 60 ÷ 60 × 0.621 =

Do the following conversions using the unit factor process. Show your work for each step and cancel out units as you work. See the examples on the first page for how you should show your work! Express your answers in scientific notation when appropriate. Do all work on a separate piece of paper and number each problem clearly.

Basic Conversions

Thinking about Conversions

The following problems ask you to convert from one metric unit from another. Even if you know the proper metric conversion, do the calculation using the units given in the tables above. After you have finished all of the math, answer the questions below.

Units with Remainders

Some traditional units are not given in decimal form when there are whole units and parts of units. For example, weights for babies and produce are often given in this form—6 lbs 12 oz—rather than this one—6.75 lbs. Write the answers to the following conversions using this convention, which is also used for time units. For example, 2.5 hours is 2 hours 30 minutes.

One reason this is important is that it enables much quicker arithmetic. Try the following problems to see why. Show your work for each step of each calculation. Do not try to do it in your head!

Trivia

Combining Units

So far we have concentrated on cancelling units when using dimesional analysis. Units can also be combined according to the rules of algebra. For example, 12x multiplied by 9/4x equals 27 because the x cancels out and 12/4 × 9 is 27. If the problem had been 2x multiplied by 5x then the answer is 10x². The x variables combine to be written together as x² instead of x · x. Here are some examples with actual units:

Areas and Volumes

Areas and volumes are examples of combined units, for the most part. An acre is an example of a unit of area which is not a combined unit. A gallon is similarly a unit of volume which is not a combined unit. Square feet (ft²) are really (feet) times (feet) because that’s how area is calculated. Similarly, a cubic meter (m³) can be imagined as a cubic space one meter on a side. For area units you must convert with two conversion factors. For volume, use three conversion factors.

Speeds

Common units of speed are miles per hour (mi/hr), kilometers per hour (km/hr), meters per second (m/s), centimeters per second (cm/s), kilometers per second (km/s), and miles per second (mi/s). Show work as always. There is an example of this type of calculation in the introduction.

Density

Density is often taught using the formula D = m/V. Since students frequently make mistakes using this formula a better way to proceed is to use dimensional analysis, as in the example shown above. Some useful data: the density of aluminum is 2.70 g/mL; the density of lead is 11.3 g/cm³; the density of water is 1 g/mL.

Linear Measure
2.54 cm	=	1 in
12 in	=	1 ft
5280 ft	=	1 mi
0.3048 m	=	1 ft
1 km	=	0.621 mi
1 furlong	=	660 ft
3 ft	=	1 yd
0.0394 in	=	1 mm

Weight/Mass
1 stone	=	14 lbs
1 lb	=	0.4536 kg
1 oz	=	16 drams
1 oz	=	28.3 g
16 oz	=	1 lb
2,000 lbs	=	1 ton
1 oz	=	28,350 mg
1 carat	=	0.2 g

Volumetric Measure
1 cup	=	16 tablespoons
2 cups	=	1 pint
2 pints	=	1 quart
1 gal	=	4 quarts
1 gal	=	3.78 L
1 mL	=	1 cm³
1 m³	=	1000 L
1 cup	=	236.6 mL

Time
1,000 ms	=	1 s
60 s	=	1 min
60 min	=	1 hr
24 hr	=	1 day
365 day	=	1 yr
14 days	=	1 fortnight
10 yr	=	1 decade
100 yr	=	1 century