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MeasurementMeasurement
How far, how much, how many?
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PROBLEM SOLVINGPROBLEM SOLVING
STEP 1: Understand the Problem
STEP 2: Devise a Plan
STEP 3: Carry Out the Plan
STEP 4: Look Back
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Step 1. Understand the ProblemStep 1. Understand the Problem
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Step 2. Devise a PlanStep 2. Devise a Plan
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Step 3. Carry Out the PlanStep 3. Carry Out the Plan
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Step 4. Look BackStep 4. Look Back
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A MeasurementA Measurement
A NumberA Quantity
An implied precision
15
1000000000
0.00056
A UnitA meaning
pound
Liter
Gram
Hour
degree Celsius
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Implied versus ExactImplied versus ExactAn implied or measured quantity has significant
figures associated with the measurement 1 mile = 1603 meters
Exact - defined measured - 4 sig figs
An exact number is not measured, it is defined or counted; therefore, it does not have significant figures or it has an unlimited number of significant figures.
1 kg = 1000 grams1.0000000 kg = 1000.0000000 grams
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Types of measurementTypes of measurement
Quantitative- use numbers to describe measurement– test equipment, counts, etc.
Qualitative- use descriptions without numbers to descript measurement- use five senses to describe4 feetextra largeHot100ºF
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Scientists PreferScientists Prefer
Quantitative- easy checkEasy to agree upon, no personal biasThe measuring instrument limits how good
the measurement is
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Uncertainty in MeasurementUncertainty in Measurement
All measurements contain some uncertainty.
We make errors
Tools have limits
Uncertainty is measured with
AccuracyAccuracy How close to the true value
PrecisionPrecision How close to each other
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AccuracyAccuracy
Measures how close the experimental measurement is to the accepted, true or book value for that measurement
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PrecisionPrecision
Is the description of how good that measurement is, how many significant figures it has and how repeatable the measurement is.
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DifferencesDifferences
Accuracy can be true of an individual measurement or the average of several
Precision requires several measurements before anything can be said about it
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Let’s use a golf analogy
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Accurate? No
Precise? Yes
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Accurate? Yes
Precise? Yes
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Precise? No
Accurate? Maybe?
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Accurate? Yes
Precise? We can’t say!
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Accuracy vs. Precision
Correct
True
val
ue
Single Measurement
Bul
ls e
ye!
Synonyms for Accuracy…
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Accuracy vs. Precision
Synonyms for precision…
Closely Grouped
Repeatable
MultipleMeasurements
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Significant figuresSignificant figures
The number of significant digits is independent of the decimal point.
25500 2550
255 25.5 2.55 0.255 0.0255
These numbersAll have three
significant figures!
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Significant FiguresSignificant Figures
Imply how the quantity is measured and to what precision.
Are always dependant upon the equipment or scale used when making the measurement
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SCALESSCALES
0 1
0.2, 0.3, 0.4?
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SCALESSCALES
0 1
0 1
0.26, 0.27, or 0.28?
0.2, 0.3, 0.4?
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SCALESSCALES
0 1
0.26, 0.27, or 0.28?
0 1
0.262, 0.263, 0.264?
0.2, 0.3, 0.4?
0 1
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Significant figuresSignificant figures
Method used to express accuracy and precision.
You can’t report numbers better than the method used to measure them.
67.2 units = three significant figures
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Significant figures: Significant figures: Rules for zerosRules for zeros
Leading zeros are notare not significant.
Notice zeros are not written in scientific notation
Notice zero is written in scientific notation
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Significant figures: Significant figures: Rules for zerosRules for zeros
Trailing zeros before the decimal are notare not significant.
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How Many Significant figures?How Many Significant figures?
123 grams
1005 mg
250 kg
250.0 kg
2.50 x 102 kg
0.0005 L
0.00050 L
5.00 x 10-4 L
3 significant figures
4 significant figures
2 significant figures
4 significant figures
3 significant figures
1 significant figures
2 significant figures
3 significant figures
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Significant figuresSignificant figures
Zeros are what will give you a headache!Zeros are what will give you a headache!
They are used/misused all of the time.
ExampleExampleThe press might report that the federal deficit is three trillion dollars. What did they mean?
$3 x 1012
or$3,000,000,000,000.00
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Significant figures:Significant figures:Rules for zerosRules for zeros
Scientific notationScientific notation - can be used to clearly express significant figures.
A properly written number in scientific notation always has the the proper number of significant figures.
0.0032103210 = 3.2103.210 x 10-3
Four SignificantFigures
Four SignificantFigures
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A comparison of massesA comparison of masses
Mass of a block of wood1 1.35 grams2 1.653 grams3 1.40 grams4 1.5115 grams
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Average mass calculationAverage mass calculation
Mass of a block of wood1 1.35 grams2 1.653 grams3 1.40 grams4 1.5115 grams
Average 1.48 grams
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Experimental ErrorExperimental ErrorThe accuracy is measured by comparing
the result of your experiment with a true or book value.
The block of wood is known to weigh exactly 1.5982 grams.
The average value you calculated is 1.48 g.
Is this an accurate measurement?
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Percent ErrorPercent Error
Indicates accuracy of a measurement
100literature
literaturealexperimenterror %
your value
accepted value
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Percent ErrorPercent Error
A student determines the density of a substance to be 1.40 g/mL. Find the % error if the accepted value of the density is 1.36 g/mL.
100g/mL 1.36
g/mL 1.36g/mL 1.40error %
% error = 2.94 %
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Scientific NotationScientific Notation
Is used to write very, very small numbers or very large numbers
Is used to imply a specific number of significant figures
Uses exponentials or powers of 10large positive exponentials imply
numbers much greater than 1negative exponentials imply numbers
smaller than 1
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Scientific notationScientific notation
Method to express really big or small numbers.
Format is Mantissa x Base Power
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Scientific notationScientific notationIf a number is larger than 1If a number is larger than 1
•The original decimal point is moved X places to the left.
•The resulting number is multiplied by 10X.
•The exponent is the number of places you moved the decimal point.
•The exponent is a positive value.1 2 3 0 0 0 0 0 0 = 1.23 x 108
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Scientific notationScientific notationIf a number is smaller than 1If a number is smaller than 1
•The original decimal point is moved X places to the right.
•The resulting number is multiplied by 10-X.
•The exponent is the number of places you moved the decimal point.
•The exponent is a negative value. 0. 0 0 0 0 0 0 1 2 3 = 1.23 x 10-7
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Most scientific calculators use scientific notation when the numbers get very large or small.
How scientific notation is
displayed can vary.
It may use x10n
or may be displayed
using an E or e.
They usually have an Exp or EEbutton. This is to enter in the exponent.
Scientific notationScientific notation
+
-1
/
x
0
2 3
4 5 6
7 8 9
.
CE
EE
log
ln
1/x
x2
cos tan
1.44939 E-2
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ExamplesExamples
378 000
3.78 x 10 5
8931.5
8.9315 x 10 3
0.000 593
5.93 x 10 - 4
0.000 000 40
4.0 x 10 - 7
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ExpandExpand
1 x 104
10,0005.60 x 1011
560,000,000,0001 x 10-5
0.000 015.02 x 10-8
0.000 000 0502
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Significant figures and calculationsSignificant figures and calculations
Addition and subtractionAddition and subtractionReport your answer with the same number of digits to the right of the decimal point as the number having the fewest to start with.
123.45987 g+ 234.11 g 357.56987 g 357.57 g
805.4 g- 721.67912 g 83.72088 g
83.7 g
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Significant figures and calculationsSignificant figures and calculations
Multiplication and division.Multiplication and division.Report your answer with the same number of digits as the quantity have the smallest number of significant figures.
Example. Density of a rectangular solid.Example. Density of a rectangular solid.251.2 kg / [ (18.5 m) (2.351 m) (2.1m) ]= 2.750274 kg/m3
= 2.8 kg / m3
(2.1 m - only has two significant figures)
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Significant figuresSignificant figuresand calculationsand calculations
An answer can’t have more significant figures than the quantities used to produce it.
ExampleExample How fast did the man runif he went 11 km in 23.2 minutes?
speed = 11 km / 23.2 min = 0.47 km / min +
-1
/
x
0
2 3
4 5 6
7 8 9
.
CE
EE
log
ln
1/x
x2
cos tan
0.474137931
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How many significant figures?How many significant figures?
What is the Volume of this box?
Volume = length x width x height = (18.5 m x 2.351 m x 2.1 m) = 91.33635 m3
= 91 m3
18.5 m2.351 m
2.1 m
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Scientific Notation (Multiplication)
(3.0 x 104) x (3.0 x 105) =9.0 x 109
(6.0 x 105) x (2.0 x 104) =12 x 109
But 12 x 109 =
1.2 x 1010
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Scientific Notation (Division)
2.0 x 106
1.0 x 104= 2.0 x 102
1.0 x 104
2.0 x 106= 0.50 x 10-2
= 5.0 x 10-3
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Scientific Notation Add & Subtract
6.4 x 104
(2.3 x 104) + (4.1 x 104) =
*Exponent must be the same!*
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(1.400 x 105) + (3.200 x 103) =
(140.0 x 103) + (3.200 x 103) =
143.2 x 103 = 1.432 x 105
Scientific Notation (+ and -)
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Rounding off numbersRounding off numbers
After calculations, the last thing you do is round the number to correct number of significant figures.
If the first insignificant digit is 5 or more,
- you round up
If the first insignificant digit is 4 or less,
- you round down.
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If a set of calculations gave you the following numbers and you knew each was supposed to have four significant figures then -
2.57995035 becomes 2.580
34.2004221 becomes 34.20
Rounding offRounding off
1st insignificant digit1st insignificant digit
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MeasurementsMeasurements
Many different systems for measuring the world around us have developed over the years.
People in the U.S. rely on the English System.
Scientists make use of SI units so that we all are speaking the same measurement language.
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Units are importantUnits are important
45 has little meaning, just a number
45 g has some meaning - mass
45 g /mL more meaning - density
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Metric UnitsMetric Units One base unit for each type of measurement. Use a prefix to change the size of unit.
Some common base units.TypeType NameName SymbolSymbol
Mass gram g
Length meter mVolume liter L Time second
sTemperature Kelvin KEnergy joule J
UnitsUnits
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Metric prefixesMetric prefixesChanging the prefix alters the size of a unit.Powers of Ten http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html
giga G 109 1 000 000 000
mega M 106 1 000 000
kilo k 103 1 000
hecto h 102 100
deca da 101 10
base - 100 1
deci d 10-1 0.1
centi c 10-2 0.01
milli m 10-3 0.001
micro or mc 10-6 0.000 001
nano n 10-9 0.000 000 001
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Measuring massMeasuring massMassMass - the quantity of
matter in an object.WeightWeight - the effect of
gravity on an object.
Since the Earth’s gravity is relatively constant, we can interconvert between weight and mass.
The SI unit of mass is the kilogram (kg)kilogram (kg). However, in the lab, the gram (g)gram (g) is more commonly used.
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TemperatureTemperature
Units of measurement
Fahrenheit, Celsius,
Kelvin
Method of measurement
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Derived UnitsDerived Units
Quantity Definition Derived Unit
Area length x length m2
Volume length x length x length m3
density mass per unit volume kg/m3
speed distance per unit time m/s
acceleration speed per unit time m/s2
Force mass x acceleration kg m/s2 N
Pressure force per unit area kg/m s2 Pa
Energy force x distance kg m2 / s2 J
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Measuring volumeMeasuring volume
VolumeVolume - the amount of space that an object occupies.
• The base metric unit is the liter (L)liter (L).
• The common unit used in the lab is the milliliter (mL)milliliter (mL).
• One milliliter is exactly equal to one cmcm3 3 & & cccc.
• The derived SISI unit for volume is the mm33 which is too large for convenient use.
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DensityDensity
Density is an intensive property of a substance based on two extensive properties.
Common units are g / cm3 or g / mL.
g / cm3
g / cm3
Air 0.0013 Bone 1.7 - 2.0Water 1.0 Urine 1.01 - 1.03Gold 19.3 Gasoline 0.66 - 0.69
Density = Mass
Volume
cm3 = mL cm3 = mL
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Example.Example.Density calculationDensity calculation
What is the density of 5.00 mL of a fluid if ithas a mass of 5.23 grams?
d = mass / volume
d = 5.23 g / 5.00 mL
d = 1.05 g / mL
What would be the mass of 1.00 liters of thissample?
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Example.Example.Density calculationDensity calculation
What would be the mass of 1.00 liters of the fluid sample?
The density was 1.05 g/mL.
density = mass / volume
so mass = volume x density
mass = 1.00 L x 1000 x 1.05
= 1.05 x 103 g
mlL
gmL