Dimensional Analysis
Dimensional Analysis, also known as the factor-label method, involves comparing the dimensions of
physical quantities (such as length, mass, time, etc.) occurring in a problem to find relationships between
those quantities and perform calculations. Fields such as physics and chemistry use this method to easily
convert between units of measurement.
When using dimensional analysis, it is important to be familiar with common conversions between units of
measurements, such as:
1 mile = 5280 feet = 1609 meters
1 lb. (pound) = 16 oz. (ounces)
1 inch = 2.54 centimeters
1 ton = 2000 lbs.
1 meter = 3.28 feet
1 oz. = 28.35 g (grams)
1 kilo[gram, meter, liter] = 1000 [grams, meters, liters]
1 [gram, meter, liter] = 1000 milli[grams, meters, liters]
Simplest Case: Single Units
When solving dimensional analysis problems, conversion factors are expressed as fractions and arranged so
any dimensional unit that appears in both the numerator and denominator can be cancelled out. After all
cancellations, the ending calculation should be in the desired dimensional units.
Example: How many millimeters are in 2 kilometers?
2 kilometers
1
X
1000 meters
1 kilometer
X
1000 millimeters
1 meter
= ?
To solve the problem, cross out the cancelled units. Then multiply all of the numerators and divide that
number by the product of the denominator.
2 kilometers
1
X
1000 meters
1 kilometer
X
1000 millimeters (2x1000x1000)
1 meter
= (1x1x1)
= 2,000,000 millimeters
Difficult Case: Multiple Units
Example: Joe travels 10 miles per hour by bicycle. How many meters per second does he travel?
10 miles
1 hour
10 miles
1 hour
X
X
1609 meters
1 mile
X
1609 meters
1 mile
X
1 hour
1 minute
60 minutes X 60 seconds
1 hour
1 minute
60 minutes X 60 seconds
= ?
(10x1609x1x1)
16090
= (1x1x60x60) = 3600 = 4.47 m/s