### Dimensional Analysis

```CHAPTER 2
ANALYZING DATA
SI MEASUREMENT

SI (def): Le Systeme International d’
Unites (International System of Units)

SI has 7 base units and almost all other
units are derived from these.
SI MEASUREMENT
QUANTITY QUANTITY
SYMBOL
UNIT
NAME
UNIT
ABBREVIATION
Length
l
meter
m (not italicized)
Mass
m
(italicized)
kilogram
kg
Time
t
second
s
Temperature
T
Kelvin
K
SI MEASUREMENT
QUANTITY QUANTITY
SYMBOL
UNIT
NAME
UNIT
ABBREVIATION
Amount of
substance
n
mole
mol
Electric
current
I
ampere
A
Luminous
intensity
IV
candela
cd
SI MEASUREMENT

Prefixes are added to the base units to
represent larger or smaller quantities.

Table 2.2: SI Prefixes, pg. 33
 MUST
MEMORIZE
SI MEASUREMENT
SI MEASUREMENT

SI units are defined in terms of
standards of measurement. They are
either objects or consistent natural
phenomena.

International organizations monitor the
defining process. In the US, the
National Institute of Standards and
Technology plays a major role in setting
standards
DERIVED UNITS

1) Derived SI units: combinations of SI
base units
Examples:
density = mass
volume
DERIVED UNITS

2) volume: amount of space occupied
by an object

non-SI volume unit:
liter, L 1 L = 1000 cm3
SI volume unit: m3
DERIVED UNITS

3) density: mass per unit volume
d = m/V

Mass and volume change proportionately,
meaning that the ratio of m to V is
constant. Density is an intensive property.

Density and temperature: at high T, most
objects expand.
SCIENTIFIC NOTATION
Scientific Notation: numbers written in the
form M x 10n, where the factor M is a
number greater than or equal to 1 but less
than 10, and n is a whole number.
65 000 km  6.5 x 104 km
0.0012 mm  1.2 x 10-3 mm
Scientific Notation Rules
To find M: Move the decimal point in the
original # to the left or right so that only
one nonzero digit remains to the left of the
decimal point
To find n: Count the # of places that you
moved the decimal point
(Moved left, n = + Moved right, n = - )
SCIENTIFIC NOTATION
Addition and Subtraction: Values must have
same value exponent before you can do these
operations
Multiplication: M factors are multiplied and
Division: M factors divided and exponent of
denominator subtracted from exponent of
numerator
Dimensional Analysis
 The “Factor-Label” Method
 Units, or “labels” are canceled, or
“factored” out
cm 
3
g
cm
3
 g
Dimensional Analysis
 Steps:
1. Identify starting & ending units.
2. Line up conversion factors so units
cancel.
3. Multiply all top numbers & divide by
each bottom number.
Dimensional Analysis
 Lining up conversion factors:
1 in = 2.54 cm
=1
2.54 cm 2.54 cm
1 in = 2.54 cm
1=
1 in
1 in
Dimensional Analysis
 How many milliliters are in 1.00 quart of
milk?
qt
mL
1.00 qt

1L
1000 mL
1.057 qt
1L
= 946 mL
Dimensional Analysis
 You have 1.5 pounds of gold. Find its
volume in cm3 if the density of gold is
19.3 g/cm3.
cm3
lb
1.5 lb 1 kg 1000 g 1 cm3
2.2 lb
1 kg
19.3 g
= 35 cm3
Dimensional Analysis
 How many liters of water would fill a
container that measures 75.0 in3?
in3
L
75.0 in3 (2.54 cm)3
(1 in)3
1L
1000 cm3
= 1.23 L
Dimensional Analysis
5) Your European hairdresser wants to cut
your hair 8.0 cm shorter. How many
inches will he be cutting off?
cm
in
8.0 cm 1 in
2.54 cm
= 3.2 in
Dimensional Analysis
6) Taft football needs 550 cm for a 1st
down. How many yards is this?
cm
550 cm
yd
1 in
1 ft 1 yd
2.54 cm 12 in 3 ft
= 6.0 yd
Dimensional Analysis
7) A piece of wire is 1.3 m long. How many
1.5-cm pieces can be cut from this wire?
cm
pieces
1.3 m 100 cm
1m
1 piece
1.5 cm
= 86 pieces
A. Accuracy vs. Precision
Accuracy - how close a measurement is to
the accepted value
Precision - how close a series of
measurements are to each other
ACCURATE = CORRECT
PRECISE = CONSISTENT
ERROR
B. Percent Error
Indicates accuracy of a measurement
% error 
experiment
al  literature
literature
accepted value
 100
% Error Problems
Try the two practice problems on the
outline.
Percent Error Examples
a.
What is the % error for a mass measurement
of 17.7 g if the correct value is 21.2 g?
17.7 g – 21.2 g x 100 =
21.2 g
b. A volume is measured experimentally to be
4.26 mL. What is the % error if the correct
value is 4.15 mL?
4.26 mL – 4.15 mL x 100 =
4.15 mL
ERROR IN MEASUREMENT
In any measurement, some error or
uncertainty exists
Measuring instruments themselves place
limitations in precision
Estimate the final questionable digit.
D. Significant Figures
Indicate precision of a measurement.
Recording Sig Figs
– Sig figs in a measurement include the known
digits plus a final estimated digit
2.35 cm
SIGNIFICANT FIGURES
Significant ≠ Certain
Must memorize the rules for recognizing
significant figures!
SIG. FIGS. RULES
RULE
EXAMPLE
1. No zeros, All sig.
852 m
97.25 mL
2. Zeros between
nonzero digits = sig.
3. Zeros at front of
nonzero digits ≠ sig.
40.7 L
87009 km
0.095897 m
0.0009 kg
4. Zeros at end of # and 85.00 g
to right of decimal = sig. 9.000 000 000 mm
5. Decimal after zeros,
2000 m
sig. Zeros with no
2000. m
Atlantic-Pacific Check
Pacific,
Decimal is
Present
Atlantic,
Decimal is
Absent
Significant figures practice
Try the practice problems on the outline
Sig. Figs. Practice
a) 804.05 g
b) 0.0144030 km
c) 1002 m
d) 400 mL
e) 30000. cm
f)
0.000625000 kg
ROUNDING RULES
Digit after last digit Last digit should:
to be kept:
Examples (3 sig.
Figs)
>5
Increase by 1
42.68 g  42.7 g
<5
Stay the same
17.32 m  17.3 m
5, followed by
nonzero
5, preceded by
odd
5, preceded by
even
Increase by 1
2.7851 cm 
2.79 cm
4.635 kg  4.64
kg
78.65 mL  78.6
mL
Increase by 1
Stay the same
C. Significant Figures
Calculating with Sig Figs (con’t)
– Add/Subtract - The # with the lowest decimal
value determines the place of the last sig fig in the
3.75 mL
+ 4.1 mL
7.85 mL  7.8 mL
C. Significant Figures
Calculating with Sig Figs
– Multiply/Divide - The # with the fewest sig figs
determines the # of sig figs in the answer.
(13.91g/cm3)(23.3cm3) = 324.103g
4 SF
3 SF
3 SF
324 g
Sig. Figs./Rounding Practice
Try the practice problems on the outline.
Practice Problems
1. What is the sum of 2.099 and 0.05681?
2. Calculate the quantity 87.3 cm – 1.655 cm
3. Polycarbonate has a density of 1.2 g/cm3.
A photo frame is constructed from two 3.0
mm sheets. Each side measures 28 cm
by 22 cm. What is the mass of the frame?
Conversion Factors
Conversion factors are typically exact.
Do not count when determining # of
E. Proportions
Direct Proportion
y x
y
x
 Inverse Proportion
y
1
y
x
x
Direct and Indirect Proportions
Direct: 2 quantities are directly proportional if
dividing one by the other gives a constant value;
graph is a straight line, y/x = k
Indirect: 2 quantities are indirectly proportional if
their product is constant, graph curved, xy = k or
y α 1/x
GRAPHS
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