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CHAPTER 2
The Metric System
Conversions
Measurement
Significant Digits
Graphing
Measure The Room Lab
Measure the length and width of the room
in “shoe” units.
Discuss differences.
Why doe we use standards?
Name US units volume, distance, …etc
Name Metric “Base” units.
Name Metric prefixes
Place prefixes in order smallest to largest.
2.1 The Metric System and SI
Why use the Metric System
Based on powers of 10, convenient to use
The Syste’me Interationale d’Unite’s =
SI = The Metric System
This is the standard system used
throughout the world by scientists,
engineers, and everyone else everywhere
except US.
Why don’t we use the Metric
System in the United States?
Good question.
Base Units
Length = meter (m)
Mass = gram (g) → “standard” unit = kg
Volume = liter (l)
Time = second (s)
Temperature = Kelvin (K)
Amount of a substance = mole (mol)
…etc
Definition of a kilogram
The mass of a small
platinum-iridium metal
cylinder kept at a very
controlled temperature
and humidity.
Definition of a meter
The distance traveled by light in a vacuum
during a time interval of 1/299 792 458
seconds.
Definition of a second
The fraquency of one type of radiation
emitted by a cesium-133 atom.
SI Prefixes
Giga – G
Mega –M
kilo – k
hecto – h
deka – da
BASE UNIT
deci – d
centi – c
milli – m
micro – μ
nano – n
pico – p
109
1 000 000 000
106
1 000 000
103
1000
102
100
10
10
meters/liters/grams/…etc
10-1
0.1
10-2
0.01
10-3
0.001
10-6
0.000 001
10-9
0.000 000 001
10-12
0.000 000 000 001
Introduction to meter stick
Meter (m) – dm – cm - mm
METRIC “STEP” SYSTEM &
CONVERSIONS
Convert 102m → mm
102m = 102 000mm
Convert 427 693μm → m
427 693μm = 0.427 693m
Metric Conversion Hand Out #1
METRIC “Step
System”
G
M
k
BASE
h
da
d
c
m
For each step you go up,
move the decimal point one
place to the left.
For each step you go down,
move the decimal point one
place to the right.
μ
n
p
Conversions Cont.
Length
1 inch = 2.54 cm (exactly)
Volume
1 liter = 1.0576 qt
Mass
1 kg = 2.21 lbs
Weight
1 lb = 4.45N (Newtons)
More useful conversions on back cover of
text book
Dimensional Analysis Method
= Factor Label Method
Example #1
4km → in
4km x 1000m x 100cm x 1in
=
1
1km
1m
2.54cm
157 480in
Example #2
26dam → yds
26dam x 1000cm x 1in x 1ft x 1yd =
1
1dam
2.54cm 12in 3ft
284.3yds
More Example Problems
3)
37hl → gal
37hl = 978.28gal
4)
439 672 107mg → tons
439 672 107mg = 0.4858tons
5)
467 223 921 732 oz → Gg
467 223 921 732 oz = 13,269.16Gg
6)
937 456 737mg → tons
937 456 737mg = 1.03tons
Hint:
Insert the units first to ensure that the units
will cancel out leaving only the unit that
you want to end up with.
Handouts #3 & #4
When using the dimensional analysis
method it is very helpful to insert units first,
then the proper numbers.
SCIENTIFIC NOTATION
(and Calculators)
Convert 276Gl → pl
276Gl = 276000000000000000000000pl
Convert 146ng → Mg
146ng = 0.000000000000146Mg
Q: Is it convenient to use these
types of numbers?
A: NO!!!!!!
Scientific Notation is used to
represent these very large/small
numbers.
Rules for Scientific Notation
The numerical part of the quantity is
written as a number between 1 and 10
multiplied by a whole-number power of 10.
M = 10n where: 1 ≤ M < 10
n is an integer
If the decimal point must be moved to the
right to achieve 1 ≤ M < 10, then n is
negative (-).
If the decimal point must be moved to the
left to achieve 1 ≤ M < 10, then n is
positive (+).
100 = 1
Therefore written in proper
scientific notation:
276000000000000000000000 pl
= 2.76 x 1023pl
0.000000000000146ng = 1.46 x 10-13Mg
Calculator Buttons
In class examples of E, EE, and
positive/negative exponents.
Addition & Subtraction
If the numbers have the same exponent,
n, add or subtract the values of M and
keep the same n.
3.7 x 104 + 6.2 x 104
= (3.7 + 6.2) x 104
= 9.9 x 104
Example-2
9.3 x 107 - 4.1 x 107
= (9.3 – 4.1) x 107
= 5.2 x 107
If the exponents are not the same, move
the decimal point to the left or right until
the exponents are the same. Then add or
subtract M.
Example-1
2.1 x 108 + 7.9 x 105
= 2.1 x 108 + 0.0079 x 108
= (2.1 + 0.0079) x 108
= 2.1079 x 108
or
Example – 2
2.1 x 108 + 7.9 x 105
= 2100 x 105 + 7.9 x 105
= (2100 + 7.9) x 105
= 2107.9 x 105
= 2.1079 x 108
Exactly the same as previous example
If the magnitude of one number is very
small compared to the other number, its
effect on the larger number is insignificant.
The smaller number can be treated as
zero. (9.99 x 103 = 9999)
7.98 x 1012 - 9.99 x 103
= 7980000000 x 103 - 9.99 x 103
= (7980000000 - 9.99) x 103
= 7979999990.01 x 103
= 7.98 x 1012
Multiplication
Multiply the values of M and add the
exponents, n. Multiply the units.
4.37 x 107m x 6.17 x 1013s
= (4.37 x 6.17) x 10 (7 + 13) (m x s)
= 26.9629 x 1020ms
= 2.69629 x 1021ms
Division
Divide the values of M and subtract the
exponents of the divisor from the exponent of
the dividend. Divide the units.
7.9 x 109 m4
3.1 x 106 m3
7.9
= 3.1 x 10 (9 -6) m (4-3)
= 2.548 x 103m
Challenging Addition
8.9 x 105m + 7.6 103km
= 8.9 x 105m + 7600 x 103m
= 8.9 x 105m + 76 x 105m
= (8.9 + 76) x 105m
= 84.9 x 105m
= 8.49 x 106m
or
Challenging Addition Cont.
8.9 x 105m + 7.6 x 103km
= 0.0089 x 105km + 7.6 x 103km
= 0.89 x 103km + 7.6 x 103km
= 8.49 x 103km
8.49 x 103km = 8.49 x 106m
Challenging Multiplication
2.7 x 1010μl X 4.3 x 10-4cl
= 0.00027 x 1010cl X 4.3 x 10-4cl
= (0.00027 x 4.3) x 10 (10-4) (cl x cl)
= 0.001161 x 106cl2
= 1.161 x 103cl2
Challenging Division
6.2 x 108kg
4.2 x 10-5Mg
6.2 x 108kg
= 4200 x 10-5kg
6.2
= 4200 x 10 (8- -5)
= 0.00147 x 1013
= 1.47 x 1010
SECTION 2.2
Measurement Uncertainties
Comparing Results
Three students measure the width of a sheet of paper
multiple times.
#1 18.5cm→19.1cm, avg=18.8cm ∴(18.8 ± 0.3)cm
#2 18.8cm→19.2cm, avg=19.0cm ∴(19.0 ± 0.2)cm
#3 18.2cm→18.4cm, avg=18.3cm ∴(18.3 ± 0.1)cm
Q: Are the three measurements in agreement?
A: Students #1 & #2 have measurements that overlap, both
have measurements between 18.8cm→19.1cm
∴ #1 and #2 are in agreement.
However, student #3 does not have any overlap with #1 or #2, ∴
there is no agreement between student #3 and/or #1 & #2.
Accuracy and Precission
Precision =
The degree of exactness with which a
quantity is measured using a given
instrument.
Q: Which student had the most precise
measurement?
A: #3 18.2cm–18.4cm, all measurements
are within ± 0.1cm.
Generally when measuring quantities, the
device that has the finest divisions on its
scale yields the most precise
measurement.
The precision of a measurement is ½ the
smallest division of the instrument.
Q: How precise is a meter stick?
A: The smallest division on a meter stick is
a millimeter(mm)  you can measure an
object to within 0.5 mm.
Accuracy =
How well the results of an experiment or
measurement agree with an accepted
standard value.
If the accepted/standard value of the sheet
of paper was 19.0cm wide, which student
was the most accurate, least accurate?
Most accurate = #2.
Least accurate = #3.
When checking the accuracy of a
measuring device use the Two-Point
calibration method.
#1 Make sure the instrument reads 0 when
it should.
#2 Make sure the instrument yields the
correct measurement on some accepted
standard.
Techniques of Good
Measurements
Measurements must be made carefully.
Common source of error = reading an
instrument when looking at it from an
angle  read the instrument from directly
above.
Parallax = the apparent shift in position of
an object when viewed from different
angles.
Significant digits
Significant Digits = the valid digits in a
measurement.
The last (estimated) digit is called the
uncertain digit.
All non zero digits in a measurement are
significant.
A = 1.24m
B = 0.23cm
How many significant digits for A & B?
A=3
B=2
Which is a more precise measurement?
A is to the nearest cm
B is to the nearest 1/100cm
 B is the more precise measurement
ZEROS
Q: Are all zeros significant?
A: No
Q: Which zeros are significant?
0.0860m
# of significant digits =?
A: 3 significant digits, first 2 zeros only
show the decimal place, the last one is
significant, it indicates the degree of
precision of the measuring device.
186 000 m
Q: How many significant digits?
A: ???????????? Cannot tell, it is
ambiguous, you do not know what
instrument was used to achieve this
measurement, possibly 3, 4, 5 or 6
significant digits.
To avoid confusion rewrite #
186 km
186.000 km
186.0 km
1.86 x 105 m
1.86000 x 105 m
0.186 Mm
0.000186 Gm
= 3 sig dig
= 6 sig dig
= 4 sig dig
= 3 sig dig
= 6 sig dig
= 3 sig dig
= 3 sig dig
Rules to Determine # of Sig Dig
1. Nonzeros are always significant.
2. All final zeros after the decimal point are
significant.
3. Zeros between two other sig dig are
always significant.
4. Zeros used solely as placeholders are
not significant.
EXAMPLES
450 026 =
6
0.123 =
3
100 258 =
6
0.000 009 =
1
0.000 090 =
2
# Sig Dig
Addition & Subtraction
Perform the operation, then round off the least
precise value involved.
64.0324
9.641
+ 129 458.1
= 129 531.7734
129 458.1 is the least precise value
 round off to 129 531.8 ,one digit past the
decimal point.
Multiplication & Division
Perform the calculation, round the product
or quotient to the factor with the least
significant digit.
4.631cm x 7.2cm = 33.3432cm2
 33cm2
3.67 x 1.9 = 6.973
 7.0
More Examples
29.4m ÷ 2.431s = 12. 09378856m/s
 12.1m/s
143 004 + 16.235 + 7.04 + 98.0357 + 0.1
= 143 135. 4107
 143 135
Examples Cont.
142.65 - 73.976 = 68.674
 68.67
15.003 x 29.745 x 0.62 x 145
= 40 119.15473
 40 000 (ambiguous)
More correct
4.0 x 104
Examples Cont.
62 579 ÷ 0.37 = 169 132.4324
 170 000
More correct 1.7 x 105
Rounding with “5”
Rounding off when “5” is the last digit
142.55
&
142.65
142.55 = 142.6
142.65 = 142.6
When an even number precedes the 5,
round “down”, when odd number precedes
the 5 round “up”
Section 2.3 Visualizing Data
Graphing Data
Independent Variable =
The variable that is changed or manipulated
directly by the experimenter.
Dependent Variable =
A result of a Δ the independent variable,
AKA, the responding variable. The value
of the dependent variable “DEPENDS” on
the Δ the independent variable.
Rules for Plotting Line Graphs
1. Identify the IV and DV. The IV is plotted
on the horizontal, x-axis, and the DV is
plotted on the vertical, y-axis.
2. Determine the range of the IV to be
plotted.
3. Decide where the graph begins, the
origin (0,0) is NOT ALWAYS the starting
point.
4. Spread the data out as much as
possible. Let each division on the graph
paper stand for a convenient unit.
5. Number and LABEL the horizontal axis.
6. Repeat steps 2-5 for the DV.
7. Plot the data points on the graph.
8. Draw the “BEST FIT
LINE”, straight or a smooth
curve, that passes through
as many data points as
possible. Keep in mind the
best fit line may not pass
through any points. Do not
draw a series of straight
lines that simply “connect
the dots”.
9. Give the graph a title that
CLEARLY tells what the
graph represents.
Linear Relationships
A graph where a straight line can be
drawn through ALL the points.
The two variables are directly proportional;
as x increases, y also increases by the
same % and as x decreases, y decreases
by the same %.
Example of Linear Graph
Slope = Linear Graph
y = mx + b
m = slope, the ratio of vertical Δ to
horizontal Δ
b = y-intercept, the point at which the line
crosses the y-axis and it is the y value
when x = 0.
rise Δ y
m = run = Δx
=
yf - yi
xf – xi
Slope can be negative, y gets smaller as x
gets bigger.
Example
TRANSPARENCY EXAMPLES
Calculate slope
Which slope is greater
Nonlinear Relationships
A graph that produces a smooth curved
line.
Sometimes a parabola where the two
variables are related by a quadratic
relationship:
y = ax2 + bx + c
Also expressed as ax2 + bx + c = 0
One variable depends
on the square of the
other.
Sometimes produces
a graph that is
inversely proportional,
the graph is a
hyperbola.
Inverse relationship
y = a/x or xy = a
As one variable
increases the other
variable decreases.