Dimensional Analysis

1 foot = 12 inches 1 mile = 5280 ft

1000 mL = 1 L 4 quarts = 1 gal

Dimension Analysis makes use of equivalent statements. What are some examples of equivalent statements?

Why are these equivalent statements?

Both values represent the same quantity of material.

Equivalent statements can be written as a fraction for use in calculations:

1ft 5280

mile 1mile 1

ft 52801

in 12foot 1

foot 1

in 12

Written as fractions these will be called conversion factors.

They will allow us to “convert” a number with one unit into another number with another unit WITHOUT changing the VALUE of the original number.

We can dimensional analysis to determine the equivalent number of feet in 100 inches

orthe equivalent number of liters in 50 gallons

orthe mass of one material based on an equivalent amount of another material.

A typical example: How many feet are in 136 inches?

There are many approaches to doing this. The one that I will employ is to start with what I know and work toward what I want using dimensional analysis…

We start with 136 inches and we need a conversion factor…

We know:

in 12foot 1

or foot 1

in 12

Which will we choose to use?

In order to get the inches to cancel mathematically we must have inches on the bottom, so we will choose the second conversion factor.

in 136

in 12ft 1

in 12ft 1 in 136

ft 11.3

Now suppose that we want to determine how many seconds there are in one day…

We will need several equivalent statements to work this problem…

1 min = 60 sec 1 hour = 60 min 1 day = 24 hr

Keep in mind that we can write these as fractional conversion factors.

hr 24day 1

or day 1hr 24

hr 1min 60

or min 60hr 1

min 1sec 60

or sec 60

min 1

Lets begin with what we know. We know that we are dealing with one day…

day 1

day 1hrs 24

day 1hrs 24 day 1

hrs 24We put hrs on top and day on bottom to cancel the units.

hrs 24

Convert 1 day to hours:

Convert hours to minutes:

hr 1min 60

hr 1min 60hr 24 min 1440

min 1440

Convert minutes to seconds:

min 1sec 60

min 1sec 60 min 1440 sec 86,400

Therefore, 1 day is equivalent to 86,400 seconds.

Instead of doing individual steps, we can do the same calculation using one conversion factor right after the other forming one continuous math expression:

day 1

day 1hrs 24

hr 1min 60

min 1sec 60

1 day 24 hr 60 min 60 sec

1 day 1 hr 1 min

1 24 60 60 sec 86,400 sec86,400 sec

1 1 1 1

Notice that the only unit that we are left with is seconds and that is the unit that we wanted. If you end up with the correct units, most of the time you get the correct numerical answer.

You try a couple on your own:

1. How many inches are in 5.2 miles?

2. How many minutes does it take for light to travel from the Sun to the Earth? (The speed of light is 186,000 miles per sec and the Earth is 93,000,000 miles from the Sun.)

1. miles 5.2

mile 1ft 5280

ft 1in 12 330,000 in = 3.3 E 5 in

2. miles 93,000,000

miles 186,000sec 1

sec 60min 1 min 8.3

Questions????