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### Measurement

How far, how much, how many?

PROBLEM SOLVING

STEP 1: Understand the Problem

STEP 2: Devise a Plan

STEP 3: Carry Out the Plan

STEP 4: Look Back

A Number

A Quantity

An implied precision

15

1000000000

0.00056

A Unit

A meaning

pound

Liter

Gram

Hour

degree Celsius

A MeasurementImplied versus Exact

An 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 grams

1.0000000 kg = 1000.0000000 grams

Types of measurement

Quantitative- use numbers to describe measurement– test equipment, counts, etc.

Qualitative- use descriptions without numbers to descript measurement- use five senses to describe

4 feet

extra large

Hot

100ºF

Scientists Prefer

Quantitative- easy check

Easy to agree upon, no personal bias

The measuring instrument limits how good the measurement is

Uncertainty in Measurement

All measurements contain some uncertainty.

- We make errors
- Tools have limits

Uncertainty is measured with

Accuracy How close to the true value

Precision How close to each other

Accuracy

Measures how close the experimental measurement is to the accepted, true or book value for that measurement

Precision

Is the description of how good that measurement is, how many significant figures it has and how repeatable the measurement is.

Differences

Accuracy can be true of an individual measurement or the average of several

Precision requires several measurements before anything can be said about it

Significant figures

The number of significant digits is independent of the decimal point.

25500

2550

255

25.5

2.55

0.255

0.0255

These numbers

All have three

significant figures!

Significant Figures

Imply how the quantity is measured and to what precision.

Are always dependant upon the equipment or scale used when making the measurement

Digits

Uncertain

Digit

Significant figuresMethod 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

Significant figures: Rules for zeros

Leading zeros are notsignificant.

0.00421 - three significant figures

4.21 x 10-3

Leading zero

Notice zeros are not written in scientific notation

Captive zeros are significant.

4012 - four significant figures

4.012 x 103

Captive zero

Notice zero is written in scientific notation

Significant figures: Rules for zeros

Trailing zeros before the decimal are notsignificant.

4210000 - three significant figures

Trailing zero

Trailing zeros after the decimal are significant.

114.20 - five significant figures

Trailing zero

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

How Many Significant figures?Significant figures

Zeros are what will give you a headache!

They are used/misused all of the time.

Example

The 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

Significant figures:Rules for zeros

Scientific 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.

0.003210 = 3.210 x 10-3

Four Significant

Figures

Experimental Error

The 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?

Percent 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.

Scientific 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 10

large positive exponentials imply numbers much greater than 1

negative exponentials imply numbers smaller than 1

Scientific notation

- Method to express really big or small numbers.

Format is Mantissa x Base Power

Decimal part of

original number

Decimals

you moved

We just move the decimal point around.

Scientific notation

If 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

Scientific notation

If 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

Scientific notation

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 EE

button. This is to enter in the exponent.

1.44939 E-2

+ 234.11 g

357.56987 g

357.57 g

- 805.4 g
- 721.67912 g
- 83.72088 g
- 83.7 g

Addition and subtraction

Report your answer with the same number of digits to the right of the decimal point as the number having the fewest to start with.

Significant figures and calculations

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.

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)

Significant figuresand calculations

An answer can’t have more significant figures than the quantities used to produce it.

Example

How fast did the man run

if he went 11 km in

23.2 minutes?

0.474137931

- speed = 11 km / 23.2 min
- = 0.47 km / min

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

2.1 m

2.351 m

18.5 m

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

Scientific Notation (Division)

2.0 x 106

=

1.0 x 104

1.0 x 104

=

2.0 x 106

2.0 x 102

0.50 x 10-2

= 5.0 x 10-3

(1.400 x 105) + (3.200 x 103) =

(140.0 x 103) + (3.200 x 103) =

143.2 x 103

1.432 x 105

=

Rounding 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.

Rounding off

If a set of calculations gave you the following numbers and you knew each was supposed to have four significant figures then -

2.5795035 becomes 2.580

34.204221 becomes 34.20

1st insignificant digit

Measurements

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.

Units are important

45 has little meaning, just a number

45 g has some meaning - mass

45 g /mL more meaning - density

Units

Metric Units One base unit for each type of measurement. Use a prefix to change the size of unit.

Some common base units.

Type NameSymbol

Mass gram g

Length meter m

Volume liter L

Time second s

Temperature Kelvin K

Energy joule J

Metric prefixes

Prefix Symbol Factor

giga G 109 1 000 000 000

mega M 106 1 000 000

kilo k 103 1 000

hecto h 102100

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

Changing the prefix alters the size of a unit.

Powers of Ten http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html

Measuring mass

Mass - the quantity of matter in an object.

Weight - 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). However, in the lab, the gram (g) is more commonly used.

Derived 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

Measuring volume

Volume - the amount of space that an object occupies.

- The base metric unit is the liter (L).
- The common unit used in the lab is the milliliter (mL).
- One milliliter is exactly equal to one cm3 & cc.
- The derived SI unit for volume is the m3 which is too large for convenient use.

Example.Density calculation

- What is the density of 5.00 mL of a fluid if it
- has 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 this
- sample?

g

L

mL

Example.Density calculationWhat 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

Scientific Method

Is a way of solving problems or answering questions.

Starts with observations and recording facts.

Hypothesis- an educated guess as to the cause of the problem or poses an answer to the question.

Scientific Method

Experiment- designed to test the hypothesis

only two possible answers

hypothesis is right

hypothesis is wrong

Generates data observations from experiments.

Modify hypothesis - repeat the cycle

Cycle repeats many times.

The hypothesis gets more and more certain.

Becomes a theory

A thoroughly tested model that explains why things behave a certain way.

Observations

Hypothesis

Experiment

Theory can never be proven.

Useful because they predict behavior

Help us form mental pictures of processes (models)

Observations

Observations

Hypothesis

Hypothesis

Experiment

Experiment

Another outcome is that certain behavior is repeated many times

Scientific Law is developed

Description of how things behave

Law - how

Theory- why

Observations

Observations

Hypothesis

Hypothesis

Experiment

Experiment

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