# Dimensional Analysis ```CHAPTER 2
UNITS OF MEASUREMENT
ACCURACY AND PRECISION
(DART BOARD EXAMPLE)

Accuracy How
close you are to the mark you are trying to hit
(bulls-eye)

Precision How
close the measurements are to each other
Three students were each asked to prepare 5 samples of
sodium chloride crystals that weigh ~10.0 g each. After
preparing the samples, each sample was weighed and the
following measurements were recorded.
Student 1
Student 2
Student 3
10.0
10.4 g
8.1 g
10.1 g
12.3 g
8.0 g
9.9 g
8.4 g
7.9 g
9.8 g
6.2 g
7.8 g
9.7 g
14.1 g
7.7 g
Which set of measurements are the most accurate? The least accurate?
Which set of measurements are the most precise? The least precise?
PERCENTAGE ERROR
Can be calculated for one experimental/measured
data point or a set of data
 Compares the experimental value/set of values to
the correct or accepted value (given or look up)
 % Error = Value experimental – Value accepted x 100%
Value accepted
 When would percentage error be negative?
Positive?
 Sample problem on p.43

HOMEWORK

Complete Measurements and Calculations:
Sample Problem C: Percentage Error worksheet
MEASUREMENT ACTIVITY
Using a ruler, measure a small object in
centimeters (pen, pencil, book thickness, etc.)
 How certain are you that _____________is
______________ long?
 Is it exactly _____________ long or is it a little
more or a little less?

RULE FOR MEASUREMENT

Report measurements to one decimal place
past the smallest graduation on the
instrument.

If you say someone is 6.0000 feet tall, you are
saying that the instrument you are using to
measure height can accurately measure to the
thousandths place!
UNITS OF MEASURE

SI (Le Syst&egrave;me International d’Unit&eacute;s)/ Metric
System
World’s most widely used system of measurement
 7 base units (length, mass, time, temperature, amount
of substance, electric current, luminous intensity- see
p.34)

U.S. Customary System (developed from English
system)
 We need to be able to convert units within each
system and between the two systems.

DIMENSIONAL ANALYSIS
(UNIT CONVERSIONS)

Conversion factors



Ratios derived from the equality between two different units
that can be used to convert from one unit to the other
See Metric Prefixes and Conversion Factors handout
1 ft = 12 in.
Write this as 1 ft or 12 in
12 in 1 ft
Convert 78.0 inches to feet.
Convert 78.0 inches to centimeters.
6.5 feet or ~198 cm
HOW DO I START?
One given: (examples to follow)

Multiple givens:
 Try to find a given with no denominator if

HOW TO GO TO 2ND, 3RD, ETC. STEPS

Each step should cancel out a previous
unwanted unit and bring in other units that are
progressing toward the desired units of the

5 gallons of water weighs _________ kg?



5 gal 4qt 1L
1000ml 1g
1 kg
------ x ------ x ---------- x ---------- x ------ x ------- =
1 1 gal 1.06qt 1L
1ml 1000g
CALCULATOR USE

The math of the previous problem is most
easily handled as a “chain operation”.
All numbers in the numerator are multipliers
and all numbers in the denominators are
divisors.
 Example from previous slide:
 5 x 4 / 1.06 x 1000 / 1000 = 18.9

CLASS WORK

Complete Measurements and Calculations:
Sample Problem B-Conversion Factors
worksheet
MASS AND WEIGHT- WHAT IS THE DIFFERENCE?

Mass
Measure of the quantity/amount of matter
 Does not depend on gravity


Weight

Measure of the gravitational pull on matter
In space there is no gravity so you are weightless, but you
still have the same mass as you do on earth.
On other planets with different gravitational pull, you
would weight more or less.
DENSITY
Density = Mass
Volume
 D = m/ V


Densities of common materials given on p.36
in Figure 2.8.
DENSITY PROBLEM
A sample of aluminum metal has a mass of
8.4g. The volume of the sample is 3.1 cm3.
Calculate the density of aluminum.
 What are we given?
 What are we trying to find?
 D = m/ V
 8.4g
= 2.7 g/cm3
3.1 cm3

VARIATIONS ON DENSITY PROBLEMS
Given the density and mass, find the volume.
 V = m/ D
 Given the density and volume, find the mass.
m=D*V

PROBLEM
The volume of a copper wire is 1000 cm3. The
density of copper is 8.92g/cm3. What is the
mass of the copper wire?
m=D*V
 1000 cm3 * 8.92g/cm3 = 8920 cm3

CLASS WORK

Complete Measurements and Calculations: Problem A- Density
worksheet
 Now we need to know how to write our answers using scientific
notation and the correct number of significant figures.
SIGNIFICANT FIGURES

Rules (from p.45 in the textbook)
Zeros appearing between nonzero digits are significant.
 Zeros appearing in front of all nonzero digits are not
significant.
 Zeros at the end of a number and to the right of a
decimal point are significant.
 Zeros at the end of a number but to the left of a
decimal point may or may not be significant. Zeros as
placeholders are not significant. Use a decimal after
the zeros if the zeros are considered significant.

HOW MANY SIGNIFICANT FIGURES ARE IN EACH
OF THESE MEASUREMENTS?
35.2 m
 10.04 g
 0.00020 kg
 2000 lb
 2000. lb
 1.25000 m3

SIGNIFICANT FIGURES (P.44-49)
Rules for Rounding- do not follow rules given in
Table 6 (too complicated!)
 Rules for Addition/ Subtraction with Decimals



Answer must have same number of digits to the right of
the decimal place as there are in the measurement
having the fewest digits to the right of the decimal point
Rules for Multiplication/ Division

Answer can have no more significant figures than are in
the measurement with the fewest number of significant
figures
EXAMPLES

5.44m – 2.6103m =
 2.83m

(2 decimal places)
2.4 g/ml x 15.82 ml =
 38g
(2 sig figs)

Complete Measurements and Calculations:
Significant Figures Worksheets D and E

Four Step Process to Solving Problems



Polycarbonate plastic has a density of 1.2
g/cm3. A photo frame is constructed from two
3.0 mm sheets of polycarbonate. Each sheet
measures 28 cm by 22 cm. What is the mass
of the photo frame?
1.2 g/cm3 x 3.0 m x 1 cm x 28 cm x 22 cm x 2
10 mm
= 440 g
SCIENTIFIC NOTATION &amp; SIGNIFICANT FIGURES
1000 = 1.00x103 This form is taught in math.
 This form is not recognized by computers, however.
Therefore, we will use the following:
 Always put the decimal after the 1st number and
using the 4th number, round back to the 3rd
number for 3 significant figures.


1400 = 1.40e3

.00001256 = 1.26e-5
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