Download Dimensional Analysis Review Notes

11/16/2013
Chapter 2 Vocabulary
Base Unit
Second (s)
Meter (m)
Kilogram (kg)
Kelvin (K)
Derived unit
Liter
Density
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Co‐Curricular Data Analysis Review
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Scientific notation
Dimensional analysis
(Equality) – not in
book
Conversion factor
Accuracy
Precision
Significant Figures
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Chapter 2 – Analyzing Data
Scientific Notation
Units of measure
 Base units: A defined unit in a system of
measurement that is based on an object or
event in the physical world and is
independent of other units.
Examples: time (sec), mass (gram),
length (meter)
 Derived units: combinations of base units
such as volume (length3), velocity
(length/time), and density (mass/volume).
 Temperature: Either Celsius (⁰C) or Kelvin (K).
Scientific Notation – It is notation
used to express very large or very small
numbers using powers of 10.
 Proper format: (1≤Number<10) x 10n
Examples
 175 L = 1.75 x 102 L
 15,000,000 g = 1.5 x 107 g
 0.00000000530 m = 5.30 x 10-9 m
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Scientific Notation
Metric Units of Length Comparison
Scientific Notation and
Commonly Used Metric Prefixes
Prefix
Symbol
Meaning
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Factor
Unit
Makes Unit BIGGER!
kilometer
mega
M
1 million times larger than the unit it precedes
106
kilo
k
1000 times larger than the unit it precedes
103
Makes Unit smaller!
deci
d
10 times smaller than the unit it precedes
10-1
centi
c
100 times smaller than the unit it precedes
10-2
milli
m
1000 times smaller than the unit it precedes
10-3
micro
μ
1 million times smaller than the unit it
precedes
10-6
nano
n
1 billion times smaller than the unit it
precedes
10-9
Examples: 1 kilometer = 1000 meters
1 millimeter = 0.001 meters (10-3 m)
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Symbol Relationship
km
1 km = 103 m
10-3 km = 1 m
Example
Length of about 5 city blocks ≈ 1 km
meter
m
Base unit
Height of doorknob from the floor ≈ 1 m
decimeter
dm
101 dm = 1 m
1 dm = 10-1 m
Diameter of a large orange ≈ 1 dm
centimeter
cm
102 cm = 1 m
1 cm = 10-2 m
Diameter of a shirt button ≈ 1 cm
millimeter
m
103 mm = 1 m Thickness of a dime ≈ 1 mm
1 mm = 10-3 m
Micrometer
μ
106 μm = 1 m Diameter of a bacteria cell ≈ 1 μm
1 μm = 10-6 m Human hair ≈ 25 μm
nanometer
n
109 nm = 1 m
1 nm = 10-9 m
Thickness of RNA molecule ≈ 1 nm
Example: 123,000 m = 123 x 103 m = 123 km
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Scientific Notation
Metric Units of Mass Comparison
Scientific Notation
Metric Units of Volume Comparison
Unit
Symbol Relationship
Liter
L
milliliter
mL
Cubic
centimeter
microliter
cm3
or
cc
μL
Example
Base unit
Quart of milk ≈ 1 L
103 mL = 1 L
1 mL = 10-3 L
20 drops of water ≈ 1 mL
1 cm3 = 1 mL
103 cm3 = 1 L
Cube of sugar ≈ 1 cm3
106 μm = 1 L
1 μm = 10-6 L
Crystal of table salt ≈ 1 μL
Drop of water from a needle ≈ 1 μL
Unit
Symbol Relationship
Example
kilogram
kg
1 kg = 103 g
gram
g
Base unit
Dollar bill ≈ 1 g
milligram
mg
103 mg = 1 g
Ten grains of salt ≈ 1 mg
microgram
μg
106 μg = 1 g
1 particle of baking powder ≈ 1 μg
A small textbook ≈ 1 kg
Example 1: 43,000 g = 43 x 103 g = 43 kg
Notice we are not putting in proper scientific notation,
but in an order of magnitude that corresponds to a
prefix.
Example 2: 0.0000033 g = 3.3 x 10-6 g = 3.3 μg
Example: Break up number into groups of 3.
0.000005 L = 0.000 005 L = 5 x 10-6 L = 5 μL
(10-6)
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Scientific Notation
Addition and Subtraction
Scientific Notation Practice
Express in Scientific notation:
 3.8 x 104 m
 38000 m
 5.06 x 103 s
 5060 s
 5.4 x 10-3 g
 0.0054 g
Convert:
 4.5 km
 4500 m to km
 5.33 μg
 0.00000533 g to μg
 777 x 10-6 L or 7.77x10-4 L
 777 μL to L
Unit to measure:
 Thickness of a quarter?  mm (millimeter)
 Mass of a car?
 kg (kilogram)
 Volume of soda can?
 mL (milliliter)
Make sure numbers are in the same
order of magnitude (You don’t have
to have them in proper format to
add and subtract.)
2. Then add or subtract as usual, carry
down the exponent and the unit.
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Scientific Notation
Addition and Subtraction Example
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Scientific Notation
Rules of Exponents
Add 3.553 x 104 ft+ 2.22 x 103 ft
3.553 x 104 ft
+ 0.222 x 104 ft
3.775 x 104 ft
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You can use the Rules of Exponents to
multiply or divide numbers in scientific
notation.
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The Rules of Exponents:
◦ (10m)(10n) = 10m+n
◦ (10m)n = 10m*n
◦ 10m/10n = 10m-n
◦ 10-m = 1/10m
◦ 100 = 1
Subtract 4.753 x 106 m – 8.52 x 105 m
47.53 x 105 m
- 8.52 x 105 m
39.01 x 105 m
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Scientific Notation
Multiplication and Division
More Scientific Notation Practice
For multiplication, multiply the two first factors,
then add the exponents. Multiply the units too.
Ex: (6.5 x 102 ft)(2.0 x 102 ft) = 13 x 104 ft2
or 1.3 x 105 ft2
 For division, divide the two first factors, than
subtract the denominator from the numerator.
Keep units in same position.
 Ex: 1.5 x 104 g
3.0 x 103 cm3
= 1.5/3.0 x 101 g/cm3
= 0.50 x 101 or 5.0 g/cm3
Solve:
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5 x 10-5 m + 2 x 10-5 m
 1.26 x 104 kg + 2.5 x 103 kg
 5.36 x 10-1 kg – 7.40 x 10-2 kg
 (4 x 102 cm) x (1 x 108 cm)
 (6 x 102 g) / (2 x 101 cm3)
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7 x 10-5 m
1.51 x 104 kg
 4.62 x 10-1 kg
 4 x 1010 cm2
 3 x 101 g/cm3
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Dimensional Analysis
Dimensional Analysis
Dimensional
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analysis is a
method of problem solving
that focuses on the units to
describe what you are
looking at.
It uses a conversion factor, which is the
ratio of equivalent values (equalities) used
to express the same quantity with different
units.
Example: 1 mile = 5280 ft…so
1 mile = 1 and 5280 ft = 1
5280 feet
1 mile
 When
using them, put the unit given in the
denominator and the unit you want in the
numerator. New Unit ‘
Old (Given) Unit
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Dimensional Analysis
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Dimensional Analysis
If you are using a derived unit, such as
density, the equality will show the relationship
between units that measure different
properties.
 Example: If the density of iron is 7.87 g/cm3,
that really means:
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Remember that units work like variables.
◦ Example:
8 .00 a *
Every 7.87 g of iron will take up 1 cm3
Or
7.87 g = 1 cm3 (for iron)
2 .54 b
 20 .3b
a
◦ Same as:
8.00in *
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2.54cm
 20.3cm
1in
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Dimensional Analysis
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Dimensional Analysis Practice
Convert:
 12.0 feet to inches
 12.5 meters to
centimeters
 2500 cm2 to m2
 65 miles/hr to ft/s
Remember that units work like variables.
◦ Example:
1b
1b
*
32 .50 a 2 *
 0 .003250 b 2
100 a 100 a
◦ Same as:
32 .50 cm 2 *
1m
1m
 0.003250 m 2
*
100 cm 100 cm
 144
inches
 1250 cm
 0.25
m2
 95 ft/s
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Reliability of Measurements
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Reliability of Measurements
Percent Error
 How far you are off from an accepted
measurement.
Error
Percent Error 
x100
Accepted Value
 Error = |obtained value – accepted value|
 The error is the absolute difference. It
doesn’t matter if the error is above or
below the accepted value.
With scientific measurements, you want
to know accuracy, precision and certainty.
◦ Accuracy – How close a measurement is to
an accepted value
◦ Precision – How close a measurement is to
other measurements of the same thing.
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Accuracy Example
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Precision Example
A student runs an experiment three times and
obtains values of 6.54 g, 6.60 g, and 6.65 g.
Ideally, they should have gotten 6.55 g as a
result. Determine the overall accuracy of the
experiment.
The average of the three results is:
6.54 + 6.60 + 6.65 = 6.60 g
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| 6.60 – 6.55| x 100 = 0.7% error
6.55
A. One set of balances give the following
readings: 10.05 g, 9.92 g, 10.77 g for a 10.00 g
mass.
B. Another set of balances give these readings:
9.95g, 10.02 g, and 10.00 g.
Which set, A or B, is more precise?
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Significant Figures (Sig Figs)
Significant Figures!
 For
any measured value, sig figs are
all of the certain digits plus an
uncertain digit.
 The last digit is an estimate and is off
by at least +1.
Example: 5.2 really means between 5.1
and 5.3.
10.53 really means between 10.52 and
10.54
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Significant Figure RULES
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Significant Figure RULES
Example
3. Zeros between two
significant digits are  38.002 kg has 5
SF’s
significant.
Example
1. All non zero
 34.554 m has 5
digits
SF’s
(1,2,3,4,5,6,7,8,
and 9) are
significant.
2. Trailing zeros to  $2.00 has 3 SF’s,
the right of the
both trailing zero’s
decimal point are are significant.
significant.
 0.05410 has 4 SF’s
4. Zeros used as
placeholders are
not significant.
 200
m has 1 SF
 0.002 g has only
1 SF
 But, 500. s has
3 SF’s
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Significant Figure RULES
Example
5. For numbers in
scientific notation, all
of the digits before
the “x 10n” are
significant.
6. Counting numbers
and defined constants
are considered to
have an infinite
number of significant
figures.
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Significant Figure RULES
7. When you add or subtract values,
your final answer must have the
same number of digits to the right of
the decimal place as the value with
the fewest number of decimal places.
Example:
4.662 km
+10.5 km
15.162 ≈ 15.2 km
 5.23
x 105 km
has 3 SF’s
6 coins
 2.54 cm/1 in
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Significant Figure RULES
Significant Figure RULES (Cont’d)
8. When you multiply or divide,
your final answer can only have the
same number of sig figs as the
measurement with the fewest sig
figs.
Example: 3.444 m* 2.11 m= 7.2584 m2
which becomes 7.26 m2
Note: For simplicity, report 3 sig figs for
all x and / calculations.
9. Figuring out the number of
significant figures is the LAST thing
you do. If you have more than one
step, carry more digits than you
think you need while you’re doing
the calculation. This will minimize
errors due to rounding.
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Significant Figure Rules of Thumb
Rounding Rules
1.
2.
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When multiplying or dividing, report
answer to 3 sig figs.
If the digit to the right of the last
significant figure is less than five, do not
change the last significant figure.
Ex. 4.433  4.43
If the digit to the immediate right of the
last significant figure is equal to or
greater than five, round up the last
significant figure.
Ex. 5.446  5.45
Ex. 3.335  3.34
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This represents 3 orders of magnitude and your measuring
device (meter stick, balance, graduated cylinder, etc.) will give
you 3 or 4 sig figs.
Start with first nonzero digit and report 3 digits, regardless of
where the decimal point is.
Example 1: 0.212 g/53.3 mL = 0.00398 g/mL
Example 2: 45.8 mm x 53.2 mm = 2436.56 mm2 ≈ 2440 mm2
Note that 45.7 mm x 53.1 mm = 2426.67 mm2 ≈ 2430 mm2
Remember that the last digit represents an estimated value.
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Significant Figures – why they are
important!!
More Dimensional Analysis Practice
Convert and give
answer w/correct
# of sig figs
 14.2 hours to
seconds
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Densities of Metals
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51120 s
5.12 x 104 s
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1.50 ft2 to in2
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216 in2
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17.8 ft3 to gallons
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133 gallons
Zinc: 7.14 g/cm3
Iron: 7.87 g/cm3
Chromium: 7.20 g/cm3
Zirconium: 6.51 g/cm3
Tin: 7.31 g/cm3
Manganese: 7.47 g/cm3
Three
students
measure
mass and volume
of an g/cm
unknown
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Student
Mass,
g theVolume,
Density,
metal. They will try and determine
cm3 which metal it is by its
density.
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2
7
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2
14.1
2.0
7.1
3
14.14
1.96
7.21
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