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Chapter 2

Scientific measurement

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Types of measurement Quantitative- uses or refers to a

standard (numerical measurements) Qualitative- use description without

reference to a standard 40 cm large Hot 100ºC

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Scientists prefer Quantitative- easy to check Easy to agree upon, no personal bias The measuring instrument limits how

good the measurement is

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How good are the measurements?

Scientists use two word to describe how good measurements are

Accuracy- how close the measurement is to an accepted value

Precision- how well can the measurement be repeated (are the readings closely grouped)

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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

examples

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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? No

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Accurate? Yes

Precise? We cant say!

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In terms of measurement Three students measure

the room to be 10.2 m, 10.3 m and 10.4 m across.

Were they precise? Were they accurate?

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Significant figures (sig figs) Are the digits in a numerical measurement

that have meaning (were measured) When we measure, we always estimate

between the smallest divisions.

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The length may be recorded as: 4.5, 4.6 or 4.7 cm (estimated digit)

What is the length?

(Variation is .1 cm) The value of the estimate is not known with certainty

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Significant figures (sig figs) The smaller the divisions the better we can

estimate (readings are more closely grouped). Scientist understand that the last digit in a

measurement is an estimate

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What is the length?

Length may be recorded as: 4.53, 4.54 or 4.55 cm (estimate)

(Variation is .01 cm) Compare this grouping to the previous slide.

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Sig Figs What is the smallest mark on the ruler that measures

142.15 cm? 142 cm? 140 cm? All nonzero digits in a measurement are significant

(have meaning – were measured or estimated) There is a problem, does a zero count or not? Was

the zero measured or is it a place holder? You need a set of rules to decide which zeroes count

as measured (are significant digits) and which are place holders (do not count as significant digits).

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Which zeros count? If the measurement is a number with a decimal

point count from left to right starting at the first nonzero. (DR)

0.045

Three sig figs (two zeroes are place holders)

If a measurement is expressed as a number with no decimal point shown, you start to count the number of significant digits from right to left. The count starts at the first nonzero digit. (NDL)12400

Two sig figs (two zeroes are place holders)

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Which zeros count?

1002 m 45.8300 cm 0.0000001500 m 15020100 km

State the number of sigfigs in each of the following:

4 sig fig

6 sig fig

4 sig fig

6 sig fig

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Sig FigsAll measurements have two components: numerical (sig figs.) and the dimension (unit).

Sig fig rules do not apply to:counting numbers or defined numbers.

Counted numbers are exact. A dozen is exactly 12.

Defined numbers are exact. 1 m is 100 cm.

Being able to locate, and count significant figures is an important skill.

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Sig figs.How many sig figs in the following measurements?

458 g

4085 g

4850 g

0.048 g

4.0485 g

40.40 g

3 sig fig

4 sig fig

3 sig fig

2 sig fig

5 sig fig

4 sig fig

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Sig Figs. 405.0 g 4050 g 0.450 g 4050.05 g 0.5060 g

Next we learn the rules for calculations

4 sig fig

3 sig fig

3 sig fig

6 sig fig

4 sig fig

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Problems 50 is only 1 significant figure But if it really has two, how can it be written to

show that both digits are significant? A zero at the end only counts after the decimal

place. If we use Scientific notation

5.0 x 101 the zero counts. (2 sig figs)

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Adding and subtracting with sig figs The last sig fig in a measurement is an

estimate (not known with certainty). Measurements can only have one estimated digit.

The answer, when you add or subtract, can not be better than your worst estimate.

have to round the answer to the place value of the measurement (in the problem) with the greatest uncertainty.

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For example

27.93 6.4+ First line up the decimal places

27.936.4+

Then do the adding

34.33Find the estimated numbers in the problem

27.936.4

This answer must be rounded to the tenths place

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Rounding rules Look at the digit in the place value following the one

you’re rounding. If the first digit to be cut is 0 to 4 don’t change it (round

down) If the first digit to be cut is 6 to 9 make it one bigger

(round up) If the first digit to be cut is exactly 5 (followed by

nothing or zeros), round the number so that the preceding digit will be even.

Round 45.462 cm to:

four sig figs

to three sig figs

to two sig figs

to one sig fig

45.46 cm

45.5 cm45 cm

50 cm

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Practice4.8 + 6.8765

520 + 94.98

0.0045 + 2.113

6.0 x 103 - 3.8 x 102

5.4 - 3.28

6.7 - .542

500 -126

6.0 x 10-2 - 3.8 x 10-3

11.6765 = 11.7

614.98 = 610

2.1175 = 2.118

6.0x103-.38x103=5.62x103=5.6x103

2.12 = 2.1

6.158 = 6.2

374 = 400

56.2x10-3 = 5.6x10-2

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Multiplication and Division Rule is simpler Same number of sig figs in the answer as

the least number of s.f. in the question 3.6 cm x 653 cm = 2350.8 cm2

3.6 cm has 2 s.f. 653 cm has 3 s.f. answer can only have 2 s.f. The rounded answer is 2400 cm2

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The LEAST number of significant figures in any number of the problem determines the number of significant figures in the answer.

425 3 sf x 11 2 sf 425 425 4675 = 4700 2 sf Round answer to have 1 uncertain digit =4700 The product has the same number of significant

digit as the least number in the multiplication

Multiplication and Division

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Multiplication and Division The rule is the same rules for division. Practice

4.5 / 6.245

4.5 x 6.245

9.8764 x 0.43

3.876 / 1983

16547 / 714

0.72056 = 0.72

28.1025 = 28

4.246852 = 4.2

0.001955

23.17507 = 23.2

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The Metric System

An easy way to measure

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Measuring

Measurements involve two components: a number and a unit.

The number is only part of a measurement

It is 10 long

10 what.

Numbers without units are meaningless.

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The Metric Systemis used because it is a decimal system

Every unit conversion is some power of 10.

A metric unit has two parts A prefix and a base unit.

The prefix tells you how many times to divide or multiply by 10.

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Base Units Length - meter - m Mass - grams – (about a raisin) – g Kg Time - second - s Temperature - Kelvin orºCelsius K orºC Energy - Joules- J Volume - Litre - L Force Newton (N)

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Prefixes kilo k 1000 times 103

deci d 1/10 10-1

centi c 1/100 10-2

milli m 1/1000 10-3

kilometer – 1000 m centimeter - 1/100 m (100 cm = 1 m) millimeter - 1/1000 m (1000 mm = 1 m)

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Volume 1 L = 1000 cm3 = 1000mL 1/1000 L = 1 cm3 1 mL = 1 cm3

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Mass is the amount of matter. 1gram is defined as the mass of 1 cm3

of water at 4 ºC. 1000 g = 1000 cm3 of water 1 kg = 1 L of water

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Converting

k h D d c m how far you have to move on this chart,

tells you the power of ten, and which sign to use with the power of ten.

The box is the base unit, meters, Liters, grams, etc.

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Conversion Factors

Change 5.6 m to millimeters

k h D d c m

start at the base unit and move three to the right. The power of ten is +3 = 1000. We want to change the units not the value, so we must multiply by 1.

5.6m x 1000 mm = 5600mm (the ratio=1) 1m

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Conversion FactorsThe units of measurement are not always

convenient dimensions and it may become necessary to change units. In a lab the distance could only be measured in cm. To calculate the speed the cm must be converted to m without changing the value of the measurement.

Distance in cm x [conversion factor] = distance in m

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Conversion FactorsThe only number that can multiply any other number

without changing the number’s value is 1. The conversion factor is a ratio. The value of the ratio is 1. For the ratio to have a value of one the top term has to equal the bottom term.

Start with 1255cm, want to find the number of m, then:By definition 1m = 100 cm

1 m = 1100cm

 1255 cm x 1 m = 12.55m 100cm

The conversion factor must cancel the present unit and introduce the desired unit

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Conversion Factors

convert 25 mg to grams

convert 0.45 km to mm

convert 35 mL to liters

k h D d c m

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The solutions for some problems contain multi-steps (require more than one calculation to solve).

Using Dimensional Analysis can solve this type of problem.Dimensional Analysis1.   Identify the given or known data (information).2.   Identify the unknown. 3. Plan the solution or calculations by either:

i. setting up a series of conversion factors ORii. using a formula.

4. Check your work by canceling out units.   1. Calculate the number of seconds of Physics class

there is in a week.2. The density of gold is 19.3 g.

cm3

What is the density of gold expressed in kg? m3