Transcript Document

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 m

1.
2.
3.
0.0156 hm
15.6 dm = _________
3000 ms
3.0 s = _________
0.254
254 g = _________
kg
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.
• For Example:
12 inches = 1 foot
Written as an “equality” or “ratio” it looks like:
=1
or
=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 have
The number &
units you start
with
The conversion factor
(The equality that
looks like a fraction)
= What units you want
The units you
want to end
with
EXAMPLE PROBLEM #1 (CONT)
• You need a conversion factor. Something that will
change inches into feet: 12 inches = 1 foot
• 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
x
=
5 feet
Do the math:
1. Multiply all of the numerators first: 60 x 1 = 60
2. Multiply all of the denominators: 12 x 1 = 12
3. Divide the product of the numerators by the product of the
denominators: 60 ÷ 12 = 5
EXAMPLE PROBLEM #1 (CONT)
• The previous problem can also be written to
look like this:
• 60 inches
1 foot
= 5 feet
12 inches
• Using this format, the vertical lines mean
“multiply” and the horizontal bars mean
“divide.”
CONVERSION PRACTICE 1
Let’s practice setting up dimensional analysis problems
using nonsense units:
Conversion Factors:
3 bops = 5 yips
1. How many bleeps are in 12 cams?

12 cams
x 1 bleep
= 6 cams
2 cams
20 nerds = 8 cams
2 cams = 1 bleep
2 nerds = 3 tongs
1 bop = 5 cams
2. How many nerds are in 6 tongs?
6 tongs
x 2 nerds
= 4 nerds
3 tongs
3. How many yips are in 15 cams? (Hint: Use 2 conversion factors!)
15 cams x
1 bop
5 cams
x 5 yips
3 bops
= 5 yips
COMMON CONVERSION FACTORS
Units of Length
12 inches = 1 foot
3 feet = 1 yard
5280 feet = 1 mile
1 inch = 2.54 centimeters
1 foot = 0.305 meters
1 mile = 1.609 kilometers
1 mile = 1609 meters
Units of Mass
16 ounces = 1 pound
2000 pounds = 1 ton
1 ounce = 28.35 grams
1 pound = 0.454 kilograms
Units of Time
1 hour = 60 minutes
1 minute = 60 seconds
1 hour = 3600 seconds
Units of Volume
2 cups = 1 pint
2 pints = 1 quart
4 quarts = 1 gallon
16 fluid ounces = 1 pint
1 gallon = 3.79 liters
1 fluid ounce = 29.6 milliliters
CONVERSION PRACTICE 2
Now let’s practice conversions with real units:
1. How many centimeters is 8.72 in?
equality:
2.54 cm = 1 in
applicable conversion factors:
________
2.54 cm
1 in
8.72 in x
or
(
2.54 cm
______
1 in
______
1 in
2.54 cm
)
=
22.1 cm
Again, the units must cancel.
2. How many feet is 39.37 inches?
equality: 1 ft = 12 in
applicable conversion factors:
______
1 ft
12 in
39.37 in x
______
12 in
1 ft
or
( )
____
1 ft
12 in
=
3.28 ft
Again, the units must cancel.
3. Convert 65 meters/second into miles per hour.
(2 part units!)
equalities: 1 mile = 1609 meters
3600 s = 1 hour
1. Convert your distance from meters to miles:
65 meters x 1 mile
= 0.0404 miles
1609 meters
2. Convert your seconds into hours:
1 second x 1 hour
= 0.000278 hrs
3600 seconds
3. Divide your miles by hours:
0.0404 miles
0.000278 hrs.
= 145 mi/hr