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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

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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