Dr. Chin Chu
Measurements
 Temperature demonstration
 What have been learned here?
 Human senses are not reliable indicator of physical
properties.
 We need instruments to give us unbiased determination
of physical properties.
 A system must be established to properly quantify the
measurements. Scales and units.
Measurements
 Definition – comparison between measured quantity
and accepted, defined standards (SI).
 Quantity:
 Property that can be measured and described by a pure
number and a unit that refers to the standard.
Measurement Requirements
 Know what to measure.
 Have a definite agreed upon standard.
 Know how to compare the standard to the measured
quantity. Tools such as ruler, graduated cylinder,
thermometer, balances and etc…
Measurements – Units
 SI Units – (the metric system)
 Universally accepted
 Scaling with 10
 Base Units:
 Time (second, s)
 Length (meter, m)
 Mass (kilogram, kg)
 Temperature (Kelvin, K)
 Amount of a substance (mole, mol)
 Electric current (ampere, A)
 Luminous intensity (candela, cd)
Measurements - Temperature
 Temperature Scales:
 Celsius (°C, centigrade)


Water freezing: 0 °C
Water boiling: 100 °C
 Kelvin (K, SI base unit of temperature)


Same spacing as in Celsius scale.
Conversion: Celsius + 273 = Kelvin
 Fahrenheit (°F)


Not the same spacing as the other two.
Conversion: Fahrenheit = (5/9)(Celsius -32)
Measurements – Units
 Derived Units:
 Volume
volume  length  height  depth
Units: (length)3, such as cm3,m3, dm3 (liter)
 Density

Defined as mass per unit volume the substance occupies.
mass
density 
volume
Problem Solving Process
THE PROBLEM
1. Read the problem carefully.
2. Be sure to understand what
it is asking you.
SOLVE THE UNKNOWN
1. Determine whether you need a
sketch to solve the problem.
2. If the solution is mathematical,
write the equation and isolate the
unknowns.
3. Substitute the known quantities
into the equation.
4. Solve the equation.
5. Continue the solution process
until you solve the problem..
ANALYZE THE PROBLEM
1. Read the problem again.
2. Identify what you are given and list
the known data.
3. Identify and list the unknowns.
4. Gather information you need from
graphs, tables or figures.
5. Plan the steps you will follow to
find the answer.
EVALUATE THE ANSWER
1. Re-read the problem. Is
the answer reasonable.
2. Check your math. Are the
units and the significant
figures correct? (Sect. 2.2
and 2.3)
Problem Solving Process – Example
THE PROBLEM:
A metal cube is 2 cm on each edge and has a mass of 20 g. Is the cube
made of pure aluminum? Density of pure aluminum is 2.7 g/cm3.
THE APPROACH:
Determining the nature of the metal, density is the parameter to compare.
WHAT’S KNOWN:
Density of pure Al
WHAT’S UNKNOWN:
THE MATH:
Density = Mass/Volume
WHAT ARE
NEEDED:
Mass and Volume
Density of the metal cube
THE MATH:
Volume=(Edge)3
The edge of the cube
is known.
HOW TO
CALCULATE THE
VOLUME OF A
CUBE?
WHAT’S KNOWN:
Mass
WHAT’S UNKNOWN:
Volume
Problem Solving Process – Example
THE SOLUTION:
1. Construct the logic flow chart.
2. Write backwards from the flow chart, the last step first then the 2nd
last and so on.
Actual solution of the problem:
1) Volume of the cube = (edge of the cube)3 = (2 cm) 3 = 8 cm3
2) Density of the cube = (mass of the cube)/(volume of the cube) = 20 g/8
cm3 = 2.5 g/cm3
3) Comparison between the densities: density of the metal cube (2.5
g/cm3) is less than the density of the pure Al (2.7 g/cm3)
4) Conclusion: the metal cube is not made of pure Al.
Problem Solving Process – Challenge
 What determines whether the object will float or sink in
water?
 Density of the object relative to water (1 g/cm3).
 Sink if density of the object is higher than water.
 Float if density of the object is smaller than water.
For a solid piece of aluminum (Al),the density is 2.7 g/cm3. Given a
piece of Al that weights 27.0 grams.
•Will it float or sink in water? Why?
•If your answer is sink, what would you do to make it float?
Dr. Chin Chu
Significant Figures
 Measurements are always done against a standard.
 When we measure something, we can (and do) always
estimate between the smallest marks.
4.5 mm
mm
1
2
3
4
5
Significant Figures
 The better marks the better we can estimate.
 Scientist always understand that the last number
measured is actually an estimate, where the level of
uncertainty is defined.
4.55 mm
object
mm
1
2
3
4
5
Significant Digits and Measurement
 Measurement
 Done with tools.
 The value depends on the smallest subdivision on
the measuring tool.
 Significant Digits (Figures):
 consist of all the definitely known digits plus one
final digit that is estimated in between the divisions.
Significant Figures
 Only measurements have significant figures.
 Counted numbers and defined constants are exact and
have infinite number of significant figures.
 A dozen is exactly 12
 1000 mL = 1 L
 Being able to locate, and count significant figures is an
important skill.
Significant Figures - Examples
Measured
Value
Uncertainty
Ruler
Division
Known
digits
Estimated
digit
1.07 cm
+/-0.01
cm
0.1 cm
1, 0
7
3.576 cm
+/-0.001
cm
0.01 cm
3,5,7
6
22.7 cm
+/- 0.1 cm
1 cm
2, 2
7
Significant Figures: Examples
 What is the smallest mark on the ruler that
measures 142.15 cm?
 ____________________
 142 cm?
 ____________________
 140 cm?
 ____________________
 Does the zero count?
 We need rules!!!
Rules of Significant Figures
 If there is a decimal point present
start counting from the left to right until
encountering the first nonzero digit.
All digits thereafter are significant.
 If the decimal point is absent
start counting from the right to left until
encountering the first nonzero digit.
All digits are significant.
Rules of Significant Figures - Examples
Pacific Ocean
Example 1
Atlantic Ocean
78638
0.00078638
decimal
point
Example 2
78638
78638000
No decimal
point
Significant Figures - Exercise
In the following measurements , identify the number of
significant figures, uncertainty level and estimated digit:
 120 cm
 0.00347 kg
 0.23400 L
 11.24 s
 1100. km
 4.560 x 10-3 m
 0.09720 g/mL
Significant Figures - Answers
In the following measurements , identify the number of
significant figures, uncertainty level and estimated digit:
 120 cm [2 sig. fig.; ±10cm; 2]
 0.00347 kg [3 sig. fig.; ±0.00001kg; 7]
 0.23400 L [5 sig. fig.; ±0.00001L; the last “0”]
 11.24 s [4 sig. fig.; ±0.01s; 4]
 1100. km [4 sig. fig.; ±1km; the last “0”]
 4.560 x 10-3 m [4 sig. fig.; ±0.001x10-3m; 0]
 0.09720 g/mL [4 sig. fig.; ±0.00001g/mL; 0]
Rounding Rules
 Rounding is always from right to left.
 Look at the number next to the one you’re rounding.
0 - 4 : leave it
5 - 9 : round up
With one exception: when the number next to
the one you’re rounding is 5 and not followed
by nonzero digits (a.k.a. followed by all zeros) –
round up if the number (rounding to) is odd;
don’t do anything if it is even.
Rounding Rules
 Further explanation for the special rule regarding the
last digit to be exactly 5:
Rounding Rules - Examples
Example 1
2.532
<5
leave it
2.53
Last significant digit
Example 2
2.536
>5
2.54
round up
Last significant digit
Example 3
2.5351
>5
2.54
round up
Last significant digit
Example 4
2.5350
the
2.54
exception odd round up
Last significant digit
Rounding – Exercise
Round the following measurements to the specified
number of significant figures:
 4.256 cm to 2 sig. fig.
 123500 g to 3 sig. fig.
 0.00374 L to 2 sig. fig.
 2.3451 s to 3 sig. fig.
 5.675 miles to 3 sig. fig.
 0.34625 mm to 4 sig. fig.
Rounding – Answers
Round the following measurements to the specified
number of significant figures:
 4.256 cm to 2 sig. fig. [4.3 cm]
 123500 g to 3 sig. fig. [124000 g]
 0.00374 L to 2 sig. fig. [0.0037 L]
 2.3451 s to 3 sig. fig. [2.35 s]
 5.675 miles to 3 sig. fig. [5.68 miles]
 0.34625 mm to 4 sig. fig. [0.3462 mm]
Mathematical Operations
Involving Significant Figures
Addition and Subtraction
The answer must have the same number of digits to the
right of the decimal point as the value with the fewest
digits to the right of the decimal point.
Why?
The result from the addition or subtraction would have
the same precision as the least precise measurement.
Mathematical Operations
Involving Significant Figures
Addition and Subtraction
Example: 28.0 cm 23.538 cm 25.68 cm
28.0 cm
23.538 cm
25.68 cm
77.218 cm
77.2 cm
1. Arrange the values so that
decimal points line up.
2. Do the sum or subtraction.
3. Identify the value with
fewest places after decimal
point.
4. Round the answer to the
same number of places.
Mathematical Operations
Involving Significant Figures
Multiplication and Division
The answer must have the same number of significant
figures as the measurement with the fewest significant
figures.
Mathematical Operations
Involving Significant Figures
Multiplication and Division
Example: 28.0 cm 23.538 cm 25.68 cm
28.0 cm
23.538 cm
25.68 cm
16924.76352 cm3
16900 cm3
3
1. Carry out the operation.
2. Identify the value with
fewest significant figures.
3. Round the answer to the
same significant figures.
Math Operation – Exercise
Complete the following math calculations with proper
significant figures:
 12.45 m + 34 m = _____ m
 1100 g + 123 g + 823.6 g = _______ g
 23.45 L - 5.572 L = ______ L
 24.1 mm x 2.7 mm = _______ mm2
 0.965 m x 2.63 m x 0.5472 m = _______ m3
 45.76 kg ÷ 25.67L = _________ kg/L
Math Operation – Answers
Complete the following math calculations with proper
significant figures:
 12.45 m + 34 m = [46] m
 1200 g + 123 g + 823.6 g = [2100] g
 23.45 L - 5.572 L = [17.88] L
 24.1 mm x 2.7 mm = [65] mm2
 0.965 m x 2.63 m x 0.5472 m = [1.39] m3
 45.76 kg ÷ 25.67L = [1.783] kg/L
Dr. Chin Chu
How Reliable are Measurements?
 Multiple measurements are taken to ensure data
integrity.
 Assessments have to be made regarding how close
the data are to the actual value (accuracy) and how
close those multiple measurements are relative to
each other (precision).
Let’s use a golf analogy
Accurate? No
Precise?
Yes
Accurate? Yes
Precise?
Yes
Precise?
No
Accurate? Maybe?
Accurate? Yes
Precise?
We cannot say!
Accuracy vs. Precision - Exercise
 Three students measure the room to
be 10.2 m, 10.3 m and 10.4 m across.
 Were they precise?
 Were they accurate?
Accuracy vs. Precision - Answers
 Three students measure the room to
be 10.2 m, 10.3 m and 10.4 m across.
 Were they precise? [Yes]
 Were they accurate? [Not sure since
the actual width of the room was
not provided.]
Dr. Chin Chu
Scientific Notations
 Mass of a proton is




0.00000000000000000000000000167262 kg
Mass of an electron is
0.000000000000000000000000000000910939 kg
Which one has more mass?
Hard to handle those numbers, right?
A better way has to be somewhere!
Scientific Notations
 THE MATH OF 10’s
1 = 1 = 100
10 = 10 = 101
100 = 10x10 = 102
1,000 = 10x10x10 = 103
10,000 = 10x10x10x10x10 = 104
0.1 = 1/10 = 1/101 = 10-1
0.01 = 1/100 = 102 = 10-2
0.001 = 1/1000 = 1/103 = 10-3
0.0001 = 1/10000 = 1/104 = 10-4
Did you see the
pattern?
Scientific Notation
 The exponent of 10 indicate the number of digits
away from the decimal point:
 Positive value: to the left of the decimal point.
 Negative value : to the right of the decimal point.
 It provides us a way to shorten very large (or small)
numbers into manageable parts.
 Scientific Notation: multiple of two factors
 Factor 1: a number between 1 and 10.
 Factor 2: 10 raised to a power, or exponent
Scientific Notation
Example 1: 12670000
12670000 = 1267000 x 10 = 1267 00 x 10 x 10 = 12670 x 10 x 10 x 10 =
1267 x 10 x 10 x 10 x 10 = 126.7 x 10 x 10 x 10 x 10 x 10 = 12.67 x 10 x 10
x 10 x 10 x 10 x 10 = 1.267 x 10 x 10 x 10 x 10 x 10 x 10 x 10
= 1.267 x 107
Alternatively,
12670000
12670000.
7 6 5 4
3
2
1.267 x 107
1
Scientific Notation
Example 2: 0.0000001267
0.0000001267 = 0.000001267 x (1/10) = 0.00001267 x (1/10) x (1/10)
= 0.0001267 x (1/10) x (1/10) x (1/10) = 0.001267 x (1/10) x (1/10) x
(1/10) x (1/10) = 0.01267 x (1/10) x (1/10) x (1/10) x (1/10) x (1/10) =
12.67 x (1/10) x (1/10) x (1/10) x (1/10) x (1/10) x (1/10) = 1.267 x
(1/10) x (1/10) x (1/10) x (1/10) x (1/10) x (1/10) x (1/10)
= 1.267 x 10-7
Alternatively,
0.0000001267
1
2 3 4 5 6 7
1.267 x 10-7
Scientific Notation – Prefixes Used
with SI Units
Prefix
Symbol
Factor
Scientifi
c
Notation
Example
giga
G
1,000,000,000
109
gigameter (Gm)
mega
M
1,000,000
106
megagram (Mg)
kilo
k
1,000
103
kilometer (km)
deci
d
1/10
10-1
deciliter (dL)
centi
c
1/100
10-2
centimeter (cm)
milli
m
1/1,000
10-3
milligram (mg)
micro
m
1/1,000,000
10-6
microgram (mg)
nano
n
1/1,000,000,000
10-9
nanometer (nm)
pico
p
1/1,000,000,000,000
10-12
picometer (pm)
Scientific Notation – Operations
 Addition and Subtraction
 How does one add 34562 and 76541290?
 Add them in columns!
34562
+) 76541290
76575852
3.4562
x104
7.6541290x107
Obviously wrong answer!
11.1103290x10?
Notice that the first digit 3 of the
1st number is right on top of the
forth digit 4 of the 2nd number.
How about move digits to line up?
3.4562 x 104
+) 7654.1290 x 104
7657.5852 x 104
76575852
The right
answer!
When adding or subtracting numbers written in scientific notation,
one must be sure that the exponents are the same before doing the
arithmetic!
Scientific Notation – Operations
 Multiplication and Division
Example 1
10 x 100 = 1000
101
102
1+2=3
103
Example 2
100
102
1 = 0.01
100
10-2
0 - 2 = -2
Add exponents for multiplication and subtract for
division.
Scientific Notation – Operations
 Multiplication and Division
a x b x c x d= (a x c) x (b x d)
axb = a x a
cxd
c c
• Two-steps:
• Multiply (or divide) the first factors.
• Multiply (or divide) the 2nd exponent
factors.
Scientific Notation – Operations
 Examples – see the in-class worksheets.
Dr. Chin Chu
Percent Error
 Definitions:
 Accepted Value: a value that is considered true. Sometimes
also referred to as “expected” value.
 Experimental Value: the value measured in the experiment
 Error: difference between an experimental value and the
accepted value. Could be either positive or negative.
– Percent Error: ratio of an error to an accepted
value. Only positive values.
error
Percent Error = accepted value
100% =
exp. - accepted
accepted value
100%
Representing Data – Graphing
 What is a graph?
 A visual display of data. “A picture is worth a thousand
words.”
 Often used to make it easier to understand large
quantity of data and relationships between parts of the
data.
 Types of graphs/charts:
 Circle (a.k.a. Pie Charts)
 Bar
 Line
Representing Data – Graphing
 Common Features of a Graph/Chart:
 Typical graph/chart is graphical and contains very little
text.
 A title: most important, and generally appears above the
main graph to provide a succinct description of what the
data in the graph refers to.
 Axes: display of dimensions in the data.


Scales
Axis label: briefly describe the dimension represented. If
numerical scales, units are usually included in the axis labels.
 Grid lines: major and minor.
 Legend: for displaying data with multiple variables.
Representing Data – Graphing
Histogram of Class Scores on the 2nd Quiz
Graphing – Pie Chart
 What is this chart about?
 What is the percentage of
seats occupied by the
Conservatives?
 What is the percentage of
seats occupied by the
Liberals?
Graphing – Line Chart
Speed vs. Time
 What is this chart about?
 What happens with
respect to speed as time
elapses?
Graphing – Line Chart
Growth of a Tree
 Independent variable:
12
Height (m)
10
8
A (x1, y1)
6
4
B (x2, y2)
2
0
0
1
2
3
4
5
6
7
time (year)
 Dependent variable:
height of the tree.
 Best fit line is straight.
 Linear relationship
between the two. Direct
related.
Year
y2  y1 y
slope 

x2  x1 x
• Positive: dependent variable
increases with the
independent variable
• Negative: decreases .
Graphing – Line Graph
Steps to Use to Make a Line Graph:
1.
2.
3.
4.
5.
6.
Identify the independent and dependent variables.
Determine the range of data that needs to be plotted for
each axis. Choose intervals for the axis to spread out the
data.
Number and label each axis.
Plot the data points.
Draw a best fit line for the data. The line may be
straight or curved, and not all points may fall on the
line.
Give the graph a title.
Representing Data – Graphing
 What do you see?