1 chapter 2 - measurements section 2.1 units of measurement
TRANSCRIPT
1
Chapter 2 - Measurements
Section 2.1
Units of Measurement
2
Stating a Measurement
In a measurement, a number is followed by a unit.
Observe the following examples of measurements:
Number Unit
35 m
0.25 L
225 lb
3.4 hr
3
The Metric System (SI)
The metric system or SI (international system) is
• A decimal system based on 10.
• Used in most of the world.
• Used everywhere by scientists.
4
Units in the Metric System
In the metric (SI) system, one unit is used for each
type of measurement:
Measurement Metric SI
length meter (m) meter (m)
volume liter (L) cubic meter (m3)
mass gram (g) kilogram (kg)
time second (s) second (s)
temperature Celsius (C) Kelvin (K)
5
Length Measurement
Length
• Is measured using a meter stick.
• Has the unit of meter (m) in the metric (SI) system.
6
Volume Measurement
Volume
• Is the space occupied by a substance.
• Has the unit liter (L) in metric system.
1 L = 1.057 qt
• Uses the unit m3(cubic meter) in the SI system.
• Is measured using a graduated cylinder.
7
Mass Measurement
The mass of an object
• Is the quantity of material it contains.
• Is measured on a balance.
• Has the unit gram(g) in the metric system.
• Has the unit kilogram(kg) in the SI system.
8
Temperature Measurement
The temperature of a substance
• Indicates how hot or cold it is.
• Is measured on the Celsius (C) scale in the metric system.
• In the SI system uses the Kelvin(K) scale.
9
Time Measurement
Time measurement
• Has the unit second(s) in both the metric and SI systems.
• Is based on an atomic clock that uses a frequency emitted by cesium atoms.
10
Chapter 2 - Measurements
Section 2.2
Scientific Notation
11
Scientific NotationScientific notation • Is used to write very large or very small
numbers.• For the width of a human hair of 0.000 008 m
is written as
8 x 10-6 m• For a large number such as 2 500 000 s is
written as
2.5 x 106 s
12
Scientific Notation
• A number written in scientific notation contains a coefficient and a power of 10.
coefficient power of ten coefficient power of ten
1.5 x 102 7.35 x 10-4
• To write a number in scientific notation, the decimal point is moved after the first digit.
• The spaces moved are shown as a power of ten.
52 000. = 5.2 x 104 0.00378 = 3.78 x 10-
3
4 spaces left 3 spaces right
13
Comparing Numbers in Standard and Scientific Notation
Number in Standard Format Scientific NotationDiameter of the Earth 12 800 000 m 1.28 x 107 mMass of a human 68 kg 6.8 x 101 kgMass of a hummingbird 0.002 kg 2 x 10-3 kg Length of a pox virus 0.000 000 3 cm 3 x 10-7 cm
14
Learning Check
Write each number using the correct scientific notation for each.
A. 0.000 008
B. 72 000
15
Learning Check
Write each as a standard number.
A. 2.0 x 10-2
B. 1.8 x 105
16
Section 2.3
Measured Numbers
and Significant Figures
Chapter 2 - Measurements
17
Measured Numbers
A measuring tool • Is used to determine
a quantity such as height or the mass of an object.
• Provides numbers for a measurement called measured numbers.
18
. l2. . . . l . . . . l3 . . . . l . . . . l4. . cm
• The markings on the meter stick at the end of the orange line are read as
The first digit 2 plus the second digit 2.7
• The last digit is obtained by estimating. • The end of the line might be estimated between
2.7–2.8 as half-way (0.5) or a little more (0.6), which gives a reported length of 2.75 cm or 2.76 cm.
Reading a Meter Stick
19
Known + Estimated Digits
In the length reported as 2.76 cm,
• The digits 2 and 7 are certain (known)
• The final digit 6 was estimated (uncertain)
• All three digits (2.76) are significant including the estimated digit
20
Significant Figuresin Measured Numbers
Significant figures
• Obtained from a measurement include all of the known digits plus the estimated digit.
• Reported in a measurement depend on the measuring tool.
21
Significant Figures
22
All non-zero numbers in a measured number are significant.
Measurement Number of Significant Figures
38.15 cm 4
5.6 ft 2
65.6 lb 3
122.55 m 5
Counting Significant Figures
23
Sandwiched zeros• Occur between nonzero numbers.• Are significant.
Measurement Number of Significant Figures
50.8 mm 32001 min 40.0702 lb 30.40505 m 5
Sandwiched Zeros
24
Trailing zeros
• Follow non-zero numbers in numbers without decimal points.
• Are usually place holders.
• Are not significant.
Measurement Number of Significant Figures
25 000 cm 2 200 kg 1 48 600 mL 3
25 005 000 g 5
Trailing Zeros
25
Leading zeros • Precede non-zero digits in a decimal number.• Are not significant.
Measurement Number of Significant
Figures0.008 mm 10.0156 oz 30.0042 lb 20.000262 mL 3
Leading Zeros
26
Significant Figures in Scientific Notation
In scientific notation• All digits including zeros in the coefficient are
significant.
Scientific Notation Number of
Significant Figures
8 x 104 m 1
8.0 x 104 m 2
8.00 x 104 m 3
27
State the number of significant figures in each
of the following measurements:
A. 0.030 m
B. 4.050 L
C. 0.0008 g
D. 2.80 m
Learning Check
28
A. Which answer(s) contain 3 significant figures? 1) 0.4760 2) 0.00476 3) 4.76 x 103
B. All the zeros are significant in
1) 0.00307 2) 25.300 3) 2.050 x 103
C. The number of significant figures in 5.80 x 102 is
1) one 3) two 3) three
Learning Check
29
In which set(s) do both numbers contain the
same number of significant figures?
1) 22.0 and 22.00
2) 400.0 and 40
3) 0.000015 and 150 000
Learning Check
30
Examples of Exact Numbers
An exact number is obtained • When objects are counted.
Counting objects2 soccer balls
4 pizzas• From numbers in a defined relationship.
Defined relationships1 foot = 12 inches1 meter = 100 cm
31
Exact Numbers
32
Learning Check
Classify each of the following as (1) exact or (2) measured numbers.
A.__Gold melts at 1064°C.
B.__1 yard = 3 feet
C.__The diameter of a red blood cell is 6 x 10-4 cm.
D.__There are 6 hats on the shelf.
E.__A can of soda contains 355 mL of soda.
33
Chapter 2 - Measurements
Section 2.4Significant Figures in
Calculations
34
Rounding Off Answers
Often, the correct answer
• Requires that a calculated answer be rounded off.
• Uses rounding rules to obtain the correct answer with the proper number of significant figures.
35
Rounding Off Calculated Answers
When the first digit dropped is 4 or less,
• The retained numbers remain the same.
45.832 rounded to 3 significant figures
drops the digits 32 = 45.8
When the first digit dropped is 5 or greater,
• The last retained digit is increased by 1.
2.4884 rounded to 2 significant figures
drops the digits 884 = 2.5 (increase by 0.1)
36
Adding Significant Zeros
• Occasionally, a calculated answer requires more significant digits. Then one or more zeros are added.
Calculated Zeros added to answer give 3 significant figures
4 4.001.5 1.500.2 0.200
12 12.0
37
Learning Check
Adjust the following calculated answers to give
answers with three significant figures:
A. 824.75 cm
B. 0.112486 g
C. 8.2 L
38
Calculations with Measured Numbers
In calculations with measured numbers: • Significant figures are counted or place
values are determined to give the correct number of figures in the final answer.
39
When multiplying or dividing
• The answer has the same number of significant figures as in the measurement with the fewest significant figures.
• Rounding rules are used to obtain the correct number of significant figures.
Example:
110.5 x 0.048 = 5.304 = 5.3 (rounded)
4 SF 2 SF calculator 2 SF
Multiplication and Division
40
Give an answer for each with the correct
number of significant figures:
A. 2.19 x 4.2 =
B. 4.311 ÷ 0.07 =
C. 2.54 x 0.0028 =
0.0105 x 0.060
Learning Check
41
When adding or subtracting use
• The same number of digits as the measurement whose last digit has the highest place value.
• Use rounding rules to adjust the number of digits in the answer.
25.2 one decimal place
+ 1.34 two decimal places
26.54 calculated answer
26.5 answer with one decimal place
Addition and Subtraction
42
For each calculation, round the answer to givethe correct number of digits.
A. 235.05 + 19.6 + 2 =
B. 58.925 - 18.2 =
Learning Check
43
Chapter 2 - Measurements
Section 2.5
Prefixes and Equalities
44
PrefixesA prefix
• In front of a unit increases or decreases the size of that unit.
• Make units larger or smaller than the initial unit by one or more factors of 10.
• Indicates a numerical value.
prefix = value
1 kilometer = 1000 meters
1 kilogram = 1000 grams
45
Metric and SI Prefixes
46
Indicate the unit that matches the description:
1. A mass that is 1000 times greater than 1 gram.
2. A length that is 1/100 of 1 meter.
3. A unit of time that is 1/1000 of a second.
Learning Check
47
An equality• States the same measurement in two different units.• Can be written using the relationships between two
metric units.
Example:
1 meter is the same as 100 cm and 1000 mm.
1 m = 100 cm
1 m = 1000 mm
100 cm = 1000 mm
Metric Equalities
48
Metric Equalities for Volume
49
Metric Equalities for Mass
• Several equalities can be written for mass in the metric (SI) system
1 kg = 1000 g
1 g = 1000 mg
1 mg = 0.001 g
1 mg = 1000 µg
50
Indicate the unit that completes each of the followingequalities:
A. 1000 m =
B. 0.001 g =
C. 0.1 s =
D. 0.01 m =
Learning Check
51
Chapter 2 - Measurements
Section 2.6Writing Conversion Factors
52
Equalities
• Use two different units to describe the same measured amount.
• Are written for relationships between units of the metric system, U.S. units or between metric and U.S. units.
• For example,
1 m = 1000 mm
1 lb = 16 oz
2.205 lb = 1 kg
Equalities
53
Exact and Measured Numbers in Equalities
Equalities between units of
• The same system are definitions and use exact numbers.
• Different systems (metric and U.S.) use measured numbers and count as significant figures.
54
Some Common Equalities
55
A conversion factor • Is a fraction obtained from an equality.
Equality: 1 in. = 2.54 cm• Is written as a ratio with a numerator and
denominator.
• Can be inverted to give two conversion factors for every equality.
1 in. and 2.54 cm 2.54 cm 1 in.
Conversion Factors
56
Write conversion factors for each pair of units:
A. liters and mL
B. hours and minutes
C. meters and kilometers
Learning Check
57
Factors with Powers
A conversion factor • Can be squared or cubed on both sides of the
equality.Equality 1 in. = 2.54 cm
1 in. and 2.54 cm 2.54 cm 1 in.
Squared (1 in.)2 = (2.54 cm)2
(1 in.)2 and (2.54 cm)2
(2.54 cm)2 (1 in.)2
Cubed (1 in.)3 = (2.54 cm)3
(1 in.)3 and (2.54 cm)3
(2.54 cm)3 (1 in.)3
58
A conversion factor • May be obtained from information in a word
problem.• Is written for that problem only.
Example:
The price of one pound (1 lb) of red peppers
is $2.39.1 lb red peppers and $2.39$2.39 1 lb red peppers
Conversion Factors in a Problem
59
Chapter 2 - Measurements
Section 2.7Problem Solving
60
To solve a problem• Identify the given unit.• Identify the needed unit.
Problem: A person has a height of 2.0 meters. What is thatheight in inches? The given unit is the initial unit of height. given unit = meters (m)
The needed unit is the unit for the answer. needed unit = inches (in.)
Given and Needed Units
61
Setting up a Problem
How many minutes are 2.5 hours?
given unit =
needed unit =
plan =
62
A rattlesnake is 2.44 m long. How long is the snake
in centimeters?
Learning Check
63
• Often, two or more conversion factors are required to obtain the unit needed for the answer.
Unit 1 Unit 2 Unit 3
• Additional conversion factors can be placed in the setup to cancel each preceding unitGiven unit x factor 1 x factor 2 = needed unitUnit 1 x Unit 2 x Unit 3 = Unit 3
Unit 1 Unit 2
Using Two or More Factors
64
How many minutes are in 1.4 days?
Example: Problem Solving
65
• Be sure to check your unit cancellation in the setup.
• The units in the conversion factors must cancel to give the correct unit for the answer.
What is wrong with the following setup?1.4 day x 1 day x 1 hr
24 hr 60 min
Units = day2/min is Not the needed unit
Units don’t cancel properly.
Check the Unit Cancellation
66
Learning Check
What is 165 lb in kg?
67
A bucket contains 4.65 L water. How many gallons of water is that?
Learning Check
68
If a ski pole is 3.0 feet in length, how long is the ski pole in mm?
Learning Check
69
If your pace on a treadmill is 65 meters per
minute, how many minutes will it take for you to
walk a distance of 7500 feet?
Learning Check
70
Section 2.8
Density
Chapter 2 - Measurements
71
Density• Compares the mass of an object to its
volume.• Is the mass of a substance divided by its
volume.• Note: 1 mL = 1 cm3
• Is expressed as
D = mass = g or g = g/cm3 volume mL cm3
Density
72
Densities of Common Substances
73
Osmium is a very dense metal. What is its
density In g/cm3 if 50.0 g of osmium has a
volume of 2.22 cm3?
Learning Check
74
Volume by Displacement
• A solid completely submerged in water displaces its own volume of water.
• The volume of the solid is calculated from the volume difference.
45.0 mL - 35.5 mL
= 9.5 mL = 9.5 cm3
75
What is the density (g/cm3) of 48.0 g of a metal if the level of water in a graduated cylinder rises from 25.0 mL to 33.0 mL after the metal is added?
25.0 mL 33.0 mL
object
Learning Check
76
Sink or Float
• Ice floats in water because the density of ice is less than the density of water.
• Aluminum sinks because its density is greater than the density of water.
77
Which diagram correctly represents the liquid layers in the cylinder? Karo (K) syrup (1.4 g/mL), vegetable (V) oil (0.91 g/mL,) water (W) (1.0 g/mL)
1 2 3
K
K
W
W
W
V
V
V
K
Learning Check
78
Density can be written as an equality. • For a density of 3.8 g/mL, the equality is:
3.8 g = 1 mL
• From this equality, two conversion factors can be written:
Conversion 3.8 g and 1 mL factors 1 mL 3.8 g
Density as a Conversion Factor
79
The density of octane, a component of gasoline, is 0.702 g/mL. What is the mass, in kg, of 875 mL of octane?
Learning Check
80
If olive oil has a density of 0.92 g/mL, how many liters of olive oil are in 285 g of olive oil?
Learning Check
81
A group of students collected 125 empty aluminum cans to take to the recycling center. If 21 cans make 1.0 lb aluminum, how many liters of aluminum (D=2.70 g/cm3) are obtained from the cans?
Learning Check
82
Which of the following samples of metals will displace the greatest volume of water?
1 2 3
25 g of aluminum2.70 g/mL
45 g of gold19.3 g/mL
75 g of lead11.3 g/mL
Learning Check