• Slightly rough surface

• Round

• Grayish brown

• Mass = 23.6 grams

• Volume = 38.4 ml

• Density = 0.615 g/ml

TWO METHODS OF OBSERVATION

Qualitative observations describe Quantitative observations measure

Length Mass TimeSI Unit(System International)

Meters (m) Kilograms (kg) Seconds (s)

Measuring devicesRuler

Measuring stickElectric balance, spring

scale, triple beam balance

stopwatch

Original Standard 1/10,000,000 of distance from equator to North Pole

Mass of 0.001 cubic meters of water 0.001574 average solar days

Current Standard Distance traveled by light in a vacuum in 3.34 x 10-9 s

Mass of a specific platinum-iridium alloy cylinder

9,192,631,770 times the period of a radio wave emitted from a cesium-

133 atom

QUANTITATIVE MEASUREMENT

Quantitative data requires a standard unit of measure

• SI unit: uses meters, kilograms, seconds • Principal system used in scientific work; What we will be using in class

• English units: uses feet, pounds, seconds

prefix mega kilo hecto Deka Deci Centi Milli micro

Value 106 103 102 101

MeterGram

SecondLiter

10-1 10-2 10-3 10-6

Abbrev-iation M k h da

mgsl

d c m μ (“mew”)

• In the SI units, the larger and smaller units are defined in multiples of 10 from the standard unit.

• See table 1-4 (pg. 11) for complete table

SI UNIT PREFIXES

• Based on the definitions of precision and accuracy, describe the following pictures as being: 1) precise or imprecise, 2) accurate or inaccurate

PRECISION & ACCURACY

UNDERSTANDING SCIENCE• In physics, we typically understand our world through measurement and developing

quantitative relationships (equations) between physical quantities

• There is uncertainty in all measurements:

• Blunders, human error

• Limitation in accuracy of measuring device

• Inability to read device beyond smallest division

• Manufacturing of device

• Two factors to consider in quantitative measure: the accuracy & precision

• Precision: the degree of exactness with which a measurement is made.

• Two ways to describe precision:

1. How close measured values are to one another

• Ex: Given 2 sets of data:

• Standard deviation (a measure of how spread out numbers are)

2. The number of decimal places used to express a value

• Ex: a measurement of 23.5 cm is less precise than 23.453 cm

MEASUREMENTS IN SCIENCE

Precise7.35ml 7.33ml 7.34ml 7.33ml

Imprecise7.35ml6.94ml 8.23ml 7.37ml

Precise7.35ml 7.33ml 7.34ml 7.33ml

• When measuring, the precision of a measure depends on the measuring device used.

• Precision of a measuring device = smallest division on the device

• However, we can estimate to an extra decimal past this!

• Looking at the ruler below…what is the precision of the ruler in cm? What is the measurement at the arrow?

PRECISION

Measurement is 4.34 cm

Precision of ruler is 0.1 cm

MEASURING IN SCIENCE

8.24 cmPrecision: 0.1 cm 52.8 ml

Precision: 1 ml(measure from bottom of meniscus)

MEASURING IN SCIENCE• Accuracy: How close a measured value is to the true value of the quantity

measured.

• Accuracy is affected by:

• Lack of calibration or error in measuring instrument

• Human error in measurement

• Ex: reaction time in measuring time, etc.

• Minimized by conducting multiple trials

• Relative error tells us the accuracy of a measure.

• Defined as the number of reliably known digits in a number.

• Rules for determining significant figures:

SIGNIFICANT FIGURES

Rules for determining significant figures: Examples

1. Zeros between other nonzero digits are significant 50.3 m 3.0025 s

2. Zeros in front of nonzero digits are not significant 0.893 kg0.00008 ms

3. Zeros that are at the end of a number and to the right of the decimal are significant

57.00 g2.000000 kg

4. Zeros at the end of a number to the left of a decimal are significant if they have been measured, are the first estimated digit, or are hatted; otherwise, they are not significant.

1000 m (could have 1 or 4 significant figures. We will say it has 1 sig figs)100Ō m

SIGNIFICANT FIGURESAn alternative way of determining significant figures: Atlantic / Pacific rule

• Is the decimal point present or absent?

Atlantic / Pacific Rule Pacific: Decimal is present• Start counting sig figs

from left starting with first nonzero digit. Keep counting until you run out of 1-9 digits

0.0006 => 1 sig fig0.00935 => 3 sig figs1.020 => 4 sig figs

Atlantic: Decimal is absent• Start counting from

right with first nonzero digit. Keep counting until you run out of 1-9 digits

40,000 => 1 sig fig1,040 => 3 sig figs1,200,100 => 5 sig figs

SIGNIFICANT FIGURES• Hatted zeros indicate significant zero digits

Digit Explanation

2000 - Start from first nonzero, nonhatted digit from right => 1 sig fig

2Ō00 - Start from first nonzero nonhatted digit from right => 2 sig figs

2000. - Start from first nonzero digit from left- 4 sig figs

• Try these out:

• 1020

• 3300

• .0012

• 3000

• 5.0020

• 1.00

• 80,000

• 0.100

• Answers

• 3

• 2

• 2

• 4

• 5

• 3

• 1

• 3

SIGNIFICANT FIGURES

• Throughout time, are system of measuring has evolved a number of times.

• The inch evolved from the barleycorn

• The mile evolved from the furlong (the distance a plough team could be driven without rest; 8 furlongs = 1 mile)

• Today, you will use 1 or 2 objects of your choosing to measuring an unknown distance

• You will then convert this to meters as a way to study precision & accuracy.

LAB ACTIVITY: HOW FAR DID YOU GO?

Farm-derived units of measurement:The rod is a historical unit of length equal to 5½ yards. It may have originated from the typical length of a mediaeval ox-goad.The furlong (meaning furrow length) was the distance a team of oxen could plough without resting. This was standardised to be exactly 40 rods.An acre was the amount of land tillable by one man behind one ox in one day. Traditional acres were long and narrow due to the difficulty in turning the plough.An oxgang was the amount of land tillable by one ox in a ploughing season. This could vary from village to village, but was typically around 15 acres.A virgate was the amount of land tillable by two oxen in a ploughing season.A carucate was the amount of land tillable by a team of eight oxen in a ploughing season. This was equal to 8 oxgangs or 4 virgates.Source: "Furlong." Wikipedia. Wikimedia Foundation, 09 June 2012. Web. 06 Sept. 2012. <http://en.wikipedia.org/wiki/Furlong>.

SIGNIFICANT FIGURES WITH MATH OPERATIONS

• When adding or subtracting, the precision matters!

• your answer can only have as many decimal positions as the value with the least number of decimal places

1.2003 ml + 23.25 ml = 24.45 ml

NOTE: calculators DO NOT give values in the correct number of significant digits!

SIGNIFICANT FIGURES WITH MATH OPERATIONS

• When multiplying or dividing, significant figures matter!

• your answer can only have as many significant figures as the value with the least number of significant figures.

3.6 cm x 0.01345 cm = 0.048 cm2

• Must determine the number of significant figures in the numbers being multiplied/divided

• NOTE: calculators DO NOT give values in the correct number of significant digits!

• Try these:

16 x 2

1.35 x 400.

10,002 x 0.034

300÷5.0

350.÷17.7

• Try these:

4001 + 3.8 =

24.38+0.0078 =

15.3– 4.38=

100.– 3.2=

11.40– 3.8=

MATH OPERATIONS WITH SIGNIFICANT FIGURES

20

540.

340

60

19.8

4005

24.39

10.9

97

7.6

SCIENTIFIC NOTATION • When expressing an extremely large number such as the mass of Earth, or a very small

number such as the mass of an electron, scientists use the scientific notation.

• For example, the mass of Earth is about 6,000,000,000,000,000,000,000,000 kg

and can be written as 6 X 1024 kg.

• Makes it clear which figures in a number are significant.

• Scientific notation converts a number from standard form to one digit, a decimal point, and a power of 10

• To convert from standard form to scientific notation:

• Move decimal point so that number is one decimal form

• Power of 10 = number of spaces moved

• If moved to left , exponent is +

• If moved to right , exponent is -

• Examples:

10 = 1x101

100 = 1x102

1000 = 1x103

MEASURING: SCIENTIFIC NOTATION

1 = 1x100

1/10 = 0.1 = 1x10-1

1/100 = 0.01 = 1x10-2

• Example 1: Convert 3,020 to scientific notation.

• Solution:

• Significant figures must be conserved. How many significant figures are there?

• Move the decimal so that there is one digit before the decimal.

• The decimal must be moved by 3 positions to the right

• Will the exponent be positive or negative?

• The exponent will be positive since 3,020 is greater than one.

SCIENTIFIC NOTATION

3.02 x 103

• Example 2: Convert .0003070 to scientific notation.

Solution:

• Significant figures must be conserved. How many significant figures are there?

• Move the decimal so that there is one digit before the decimal.

• Will the exponent be positive or negative?

• The exponent will be negative since the number is less than one.

SCIENTIFIC NOTATION

3.070x10-4 Notice how the zero remains at the end to show that it is significant!

• Convert the following to scientific notation with correct sig figs.

1. 346,000

2. 0.0210

3. 0.00000900

4. 500,Ō00

3.46 x 105

2.10 x 10-2

9.00 x 10-6

5.000 x 105

SCIENTIFIC NOTATION

Converting from scientific notation to standard form:

1. Identify & preserve all sig figs from scientific notation

2. Move decimal the number of spaces of the exponent

• If exponent is POSITIVE +, move decimal to RIGHT

• If exponent is NEGATIVE -, move decimal to LEFT • Examples:

SCIENTIFIC NOTATION

Scientific Notation

Operations Standard notation

8.75 x 10-2 - 3 sig figs- Negative exponent - move decimal to left

0.0875

3.635 x 105 - 4 sig figs- Positive exponent – move decimal to right

363,500

2.50 x 102 - 3 sig figs- Positive exponent – move decimal to right

250. OR 25Ō

Scientific Notation on a calculator

• Use the “EE” key to do scientific notation on a calculator

• Identify the number of significant figures and convert the number to standard notation for the following:

1.52 x 103

7.30 x 10-3

6.75 x 105

5.3030 x 10-2

3.670 x 105

Sig Figs Standard Form

3 1520

3 0.00730

3 675000

5 0.053030

4 367Ō00

MEASURING: SCIENTIFIC NOTATION

Mathematical operations with significant figures for scientific notation

Same rules applyOperation (rule) Operation Examples

Multiplication / Division(Input with lowest # of sig figs limits output to that number of sig figs )

Multiplication: 10a(10b) = 10a + b

2 sf 2sf 2 sf

Division:

2 sf / 3 sf 2 sf Note: exponent in denominator becomes negative

Addition / Subtraction(Output is as precise as least precise input)

Must change exponents to be the same before performing operation

7.4 x 10-3 7.4 x 10-3 (2 sig figs)-3.5 x 10-4 -0.35 x 10-3 (2 sig figs) 7.05 x 10-3 7.1x10-3 (2 sf)

2.35 x 105 2.35 x 105 (3 sig figs)+ 3.70 x 103 + 0.0370x105 (3 sig figs) 2.3870x105 2.39x105 (3 sf)

SCIENTIFIC NOTATION

• Try these:

16 x 2

1.35 x 400.

10,002 x 0.034

• Try these:

4001

+ 3.8

24.38

+0.0078

15.3

– 4.38

100.

– 3.2

11.40

– 3.8

MEASURING: SIGNIFICANT FIGURES

20

540.

340

60

19.8

4005

24.39

10.9

97

7.6

Mathematical operations with significant figures for scientific notation

Same rules applyOperation (rule) Operation Examples

Multiplication / Division(Input with lowest # of sig figs limits output to that number of sig figs )

Multiplication: 10a(10b) = 10a + b

2 sf 2sf 2 sf

Division:

2 sf / 3 sf 2 sf Note: exponent in denominator becomes negative

Addition / Subtraction(Output is as precise as least precise input)

Must change exponents to be the same before performing operation

7.4 x 10-3 7.4 x 10-3 (2 sig figs)-3.5 x 10-4 -0.35 x 10-3 (2 sig figs) 7.05 x 10-3 7.1x10-3 (2 sf)

2.35 x 105 2.35 x 105 (3 sig figs)+ 3.70 x 103 + 0.0370x105 (3 sig figs) 2.3870x105 2.39x105 (3 sf)

MEASURING: SIGNIFICANT FIGURES