# Dimensional Analysis,

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"Dimensional Analysis,"

```					             Dimensional Analysis,
Equivalence Statements, &Conversion Factors
I) Equivalence Statements
An equivalence statement shows two quantities with different units that are equal to
each other.

A) equivalence statements can be counted numbers:
Examples:
1 dozen eggs = 12 eggs              1 pair of shoes = 2 shoes

B) equivalence statements can be numbers with in the same measurement
system:
Examples:
1 km = 1000m                          5280 feet = 1 mile

C) equivalence statements can be numbers in different measurement systems:
Examples:
1 lb = 2.21 kg                        1 in = 2.540 cm

D) equivalence statements can be numbers that are specific to a situation:
Examples:
1 in = 50 km such as on a map   1m = 3.45cm such as in a photograph

3 students = 1 lab group such as during a particular lab

Practice:
Design four equivalence statements and write them below:

1: _______________________________        2: ______________________________

3: _______________________________        4: ______________________________
II) Conversion Factors
A) Every equivalence statement can be used to construct 2 conversion factors.

B) Conversion factors are fractions that ALWAYS equal one. The numerator
and the denominator have equal value.

Examples:
equivalence statements                           conversion factors
1 dozen eggs             12 eggs
1 dozen eggs = 12 eggs                               OR      1 dozen eggs
12 eggs

5280 feet                1 mile
5280 feet = 1 mile                              OR
1 mile                 5280 feet

Practice:
Write the two conversion factors for the equivalent statement below:
equivalence statements                         conversion factors
5)    13 steps = 1 flight of stairs                          OR

III) Using Conversion Factors for Dimensional
Analysis
A) What happens to the value of a number when you multiply it by one?

B) Because a conversion factor is a fraction that equals one, when it is used
in a calculation it does not effect the value of the number, however, it does
change the units.

C) Dimensional analysis is the process by which a conversion factor is used
to convert a value from one unit to another.

D) To decide which of the two conversion factors to use make sure the units
from the known value will cancel leaving units for the unknown value.
Examples:

convert 3.5 hours to minutes:
equivalence statement              conversion factors     dimensional analysis

1 hour = 60 minutes
1 hr
60 min
60 min
OR 1 hr                ( 601 min ( = 210 min
3.5 hr     hr

convert 375 inches to meters
1in        2.540cm
1 in = 2.540 cm               or
2.540cm          1in

100cm€          1m
100 cm = 1 meter
€                    or
1m          100cm

€        €
 2.540cm  1m 
375in              = 9.525 = 9.53m
 1in 100cm 
Practice:

6) convert 27.5 L to mL
€
7) convert 5.0 km to feet

8) convert 15 years to minutes

9) convert 65 mph to m/s
IV) Dimensional Analysis and Sig Figs.
A) Pure Numbers
a) Equivalence statements that compare numbers within the same system,
either Metric or English, consist of pure numbers.

12 inches = 1 foot     100 cm = 1 meter         10) _____ m = _____km

b) Pure numbers are exact numbers that exist by definition. There is no
uncertainty and therefore no error in the comparison of these values.

c) The conversion factors constructed from these equivalence statements
are not considered when determining the correct number of sig figs.

B) Often times a power of 10 (1,10,100,etc) will appear as part of a conversion
factor. Generally speaking this number is not considered when determining sig
figs.

C) Otherwise, the standard sig fig rules apply to dimensional analysis.

Examples:

1m = 100cm             Both are pure numbers because they exist by definition in the
Metric System, therefore they are not sig figs.

1.609 km = 1 mi        4 sig figs in 1.609 km, 1 mi is not considered because it is a
power of 10.

5280 ft = 1 mi         Both are pure numbers because they exist by definition in the
English System, therefore they are not sig figs.

Practice:

11) 3 ft = 36 inches

12) 1 gal = 3.785L

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