MPC 095 1 Dimensional Analysis
Dimensional Analysis
Dimensional analysis converts one set of units to another just by multiplying by 1. In
mathematics, you can always multiply by 1 and not change the value of a number. Further,
if two numbers are equal to each other, then either number divided by the other is also
equal to 1. So,
1 = 4
4 =
1
2
2
4 = 100 centimeters
1 meter = 1 pound
16 ounces = 1 hour
60 minutes = 60 minutes
1hour .
If 1 mile = 5280 feet, then
1 mile
5280 feet = 1 and
5280 feet
1 mile = 1.
These “funny looking ones” are called conversion factors. We can use them to convert
from feet to miles or vice versa. For instance, how many feet are there in 17.37 miles?
17.37 miles ×
5280 feet
1 mile = 91713.6 feet.
Notice that I multiplied by a conversion factor that made the “miles” in the numerator divide
out with the “miles” in the denominator. If I had multiplied by the other conversion factor, the
units wouldn’t have divided out, and I would have ended up with
17.37 miles ×
1 mile
5280 feet = 0.003290 (miles)2
feet .
The result is equivalent to 17.37 miles, but I wouldn’t have answered the original question.
We treat the units of a measurement just the same as we treat the number part of a
measurement. If the same unit appears in the numerator and the denominator, the
common unit can be divided out; the unit of miles reduced out in the first example.
This method of converting units can be used in a string to convert more than one unit. For
instance, the speed of a good fastball in professional baseball is about 90 miles per hour.
To express this number in feet per second, do the following:
90 miles
hour ×
1 hour
3600 seconds ×
5280 feet
1 mile = 132 feet
second
Notice how the conversion factors were chosen so that the unwanted units divided out, and
didn’t multiply each other.
We can treat the units just as we would variables in an equation. If we were asked to
convert 16.2 ft2 (square feet) to in2 (square inches), we could just square the appropriate
conversion factor.
12 in.
1 ft
2 =
12 in.
1 ft ×
12 in.
1 ft =
144 in2
1 ft2
Now we have an appropriate conversion factor to convert square feet to square inches.
16.2 ft2 ×
144 in2
1 ft2 = 2332.8 in2
This same method can be used to convert units of volume such as cubic yards (yd3) and
cubic centimeters (cm3). Dimensional analysis can be used to convert any unit to any other
appropriate unit.
