DIMENSIONAL ANALYSIS(FACTOR-LABEL METHOD)

How can we convert units?

MEASUREMENTS

Every measurement needs to have a value (number) and a unit (label). Without units, we have no way of knowing what the

actual measurement is Sometimes the units that something is measured

in, need to be converted into a comparable unit for a calculation

So how do we convert our units into new units?

METRIC CONVERSIONS REVIEW

When we are converting from one metric unit to another, all we need to do it move the decimal point

Convert the following: k h da _ d c m1. 15.6 dm = _________ hm2. 3.0 s = _________ ms3. 254 g = _________ kg

0.0156

3000

0.254

OTHER CONVERSIONS

Not every type of conversion that you will encounter will be a metric conversion where you can just move the decimal

Dimensional Analysis (Factor-Label Method) is the process that we can use to mathematically convert units from one unit system to another

GETTING STARTED

Before we can look at examples of dimensional analysis, let’s review some basic math principles: What happens when you divide a number by itself? What happens when you divide a unit by itself?

In both cases, you get the number 1.

Dimensional analysis involves multiplication and division using conversion factors. Conversion factors : two numbers with their units that

are equivalent to each other i.e. 1 foot = 12 inches, 12 eggs = 1 dozen

CONVERSION FACTORS• Conversion factors can be written as ratios because both values

equal each other• Because they equal each other, if we divide the quantities they

would be equal to one.

or

• For Example: 12 inches = 1 foot

Written as an “equality” or “ratio” it looks like:

= 1 = 1

•When a value is multiplied by a conversion factor the units behave like numbers do when you multiply fractions: If you have the same units in both the numerator and the denominator, they cancel!

EXAMPLE PROBLEM #1• How many feet are in 60 inches?

Solve using dimensional analysis.

• All dimensional analysis problems are set up the same way. They follow this same pattern:

What units you have x What units you want = What units you want What units you have

The number & units you start with

The conversion factor(The equality that

looks like a fraction)

The units you want to end with

• Write this conversion factor as a ratio, making sure that the number on the bottom of the ratio has units that match the units of your starting units so that they will cancel

60 inches

EXAMPLE PROBLEM #1 (CONT)• You need a conversion factor. Something that will

change inches into feet: 12 inches = 1 foot

x = 5 feetDo the math: 1. Multiply all of the numerators first: 60 x 1 = 602. Multiply all of the denominators: 12 x 1 = 123. Divide the product of the numerators by the product of the denominators: 60 ÷ 12 = 5

• Using this format, the vertical lines mean “multiply” and the horizontal bars mean “divide.”

EXAMPLE PROBLEM #1 (CONT)

• The previous problem can also be written to look like this:

• 60 inches 1 foot = 5 feet

12 inches

CONVERSION PRACTICE 1 Let’s practice setting up dimensional analysis problems

using nonsense units:1. How many bleeps are in 12 cams?

2. How many nerds are in 6 tongs?

3. How many yips are in 15 cams? (Hint: Use 2 conversion factors!)

Conversion Factors:3 bops = 5 yips

20 nerds = 8 cams2 cams = 1 bleep2 nerds = 3 tongs1 bop = 5 cams

12 cams x 1 bleep

2 cams

6 tongs x 2 nerds

3 tongs

15 cams x 1 bop

5 cams

= 6 cams

= 4 nerds

= 5 yipsx 5 yips

3 bops

COMMON CONVERSION FACTORSUnits of Length12 inches = 1 foot3 feet = 1 yard5280 feet = 1 mile1 inch = 2.54 centimeters1 foot = 0.305 meters1 mile = 1.609 kilometers1 mile = 1609 meters

Units of Mass16 ounces = 1 pound2000 pounds = 1 ton1 ounce = 28.35 grams1 pound = 0.454 kilograms

Units of Volume2 cups = 1 pint2 pints = 1 quart4 quarts = 1 gallon16 fluid ounces = 1 pint1 gallon = 3.79 liters1 fluid ounce = 29.6 milliliters

Units of Time1 hour = 60 minutes1 minute = 60 seconds1 hour = 3600 seconds

______2.54 cm1 in

CONVERSION PRACTICE 2Now let’s practice conversions with real units:

1. How many centimeters is 8.72 in?

applicable conversion factors:

equality:

or

8.72 in x =

2.54 cm = 1 in

________2.54 cm 1 in

Again, the units must cancel.

( )______ 22.1 cm2.54 cm1 in

2. How many feet is 39.37 inches?

applicable conversion factors:

equality:

or

39.37 in x =

1 ft = 12 in

______1 ft 12 in

Again, the units must cancel.

( )____ 3.28 ft1 ft12 in

______1 ft

12 in

3. Convert 65 meters/second into miles per hour.(2 part units!)

1. Convert your distance from meters to miles:

2. Convert your seconds into hours:

3. Divide your miles by hours:

equalities: 1 mile = 1609 meters3600 s = 1 hour

65 meters x 1 mile

1609 meters

= 0.0404 miles

1 second x 1 hour

3600 seconds

= 0.000278 hrs

0.0404 miles

0.000278 hrs.= 145 mi/hr