SCIENCE AND THE SCIENTIFIC METHOD E A G › 2012 › 09 › unit-1-4th-eso.pdf · Unit 1: Science E. Aller 2 Physics is an experimental and natural science that studies the composition

Physics and Chemistry 4th ESO

Unit 1: Science E. Aller

UNIT 1: SCIENCE. THE SCIENTIFIC METHOD. QUANTITIES AND UNITS 1. SCIENCE AND THE SCIENTIFIC METHOD

In this unit we will learn how scientists work, make new discoveries and explain unknown

phenomena of nature. All this knowledge is what we call Science. Science is the systematic

knowledge of the world gained through observation, experimentation and reasoning and the

formulation of laws and theories. But Science has different branches depending on what is

being studied. We can divide Science into two major groups:

a) Natural Sciencies which study natural phenomena including biological life.

b) Social Sciences, which study human behaviour and societies.

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Natural Sciencies are empirical sciences, which means knowledge must be based on

observable phenomena and capable of being tested for its validity with expermients made by

other researchers working under the same conditions. Natural Sciencies include Physics,

Chemistry, Biology and Geology. In this subject we will learn what Physics and Chemistry

study.

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Unit 1: Science E. Aller 2

Physics is an experimental and natural science that studies the composition of matter, its general properties and its physical changes. A physical change of matter is a change

that undergoes a substance, a body or an object without transformation into other substance/s,

its chemical composition doesn’t change. Examples of physical changes are: changes in

position, in volume, in form, in temperature or state; mixtures, electrical phenomena…

Chemistry is an experimental and natural science that studies the composition, the

structure and the properties of matter and its chemical changes. A chemical change: of

matter is a change that undergoes a substance, a body or an object with transformation into

other substance/s, its chemical composition changes (iron oxidation, fruit ripening, cooking,

digestion, respiration…).

So Physics and Chemistry are two natural sciences which study matter and its

properties. The method that these two sciences use to understand matter and its changes is the

scientific method. So scientists use the scientific method to know everything about Nature. But

the scientific method is not only used by scientists. Any worker, a doctor, a policeman or a

technician, use this method to solve their unknown problems.

The scientific method is a sequence of steps that the scientist follows to solve a problem.

a) The first step is the observation. Scientists observe nature through the senses. Scientific

observation consists of receiving knowledge of the outside world through our senses. But

our senses have limits. We cannot see a distant object, or a very small object. We need to

use technological iinstruments to obtain information from nature. We use telescopes to see

distant objects. We can see small objects with the microscope. While human beings can't

detect radiations using their natural senses, many technologies exist that help detect and

identify radiations, wherever it may be, such as the use of spectrrophotometers.

b) The second step is to identify the problem and to make the right question. When a

scientist watches nature with his instruments he sometimes finds unknown or unexplained

phenomena. Scientists must make the right question to identify clearly the unsolved

problem. If I want to study the growth of a plant, it is a complex problem because there are

multiple factors that can affect it, the amount of water, oxygen carbon dioxide, light… A

scientist can’t study all the variables that affect plant growth at the same time. Even plant

growth is another variable. We must reduce the number of studied variables to two

variables. So the right question is not, what affects plant growth? The right question would

be: Does the amount of water affect the growth of a plant? We only choose one factor that

can affect plant growth. In the question there can only be two variables, the amount of

water and the growth of a plant.

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c) The third step is to obtain the background information. To answer the question a scientist

must obtain information about his problem, which can be found in scientific journals,

specialized books, in scientific congresses or by an internet research.

d) The fourth step is to define the variables of the problem. With this background

information we have to identy the variables of the problem.For example, if we are studying

the growth of a plant a problem that we can study is the following one: Does the amount of

water affect the growth of a plant? There are two variables written in the question: the

amount of water (the volume of water) and the plant growth, which can be measured with

the length of the trunk or its diameter, or the number of new leaves or new branches. Of

course, there are other variables that affect plant growth: the amount of sunlight, oxygen,

carbon dioxide... But we want to find the relationship between the amount of water and plant

growth.

A variable is a quantitative property of an object that can take on different values (length,

speed, density, volume…). Now, we must define the variables of our problem:

The independent variable is the variable that a scientist deliberately changes during the

experiment. In our example is the amount of water.

The dependent variable is the variable that responds to the changes in the independent

variable. In our case it can be plant growth.

The rest of the variables must be controlled, although these variables are changing in

nature. But we must controll them to concentrate our study on the amount of water and the

growth of the plant.

e) The next step is to formulate a hypothesis. With the background information we should

answer the question. For example, the amount of water affects plant growth, without water

plants die. But this is a possible answer and it must be tested. We call hypothesis an

educated guess, a prediction that must be tested with an experiment. It is a tentative

statement that proposes a possible explanation to some phenomenon or event.

f) Experimentation. Our hypothesis must be tested with an experiment. The experiment must

be designed, indicating the procedure and a list of materials. Then, the experiment or

experiments are made and numerical data are recorded. These numerical data should be

organized into tables and represented in graphs. Graphs represent data visually and it is

much easier their analysis to test our hypothesis.

g) In the following step of data analysis we will test if our hypothesis is correct or it is not valid.

If the hypothesis is not valid, our work hasn’t finished. A new hypothesis should be proposed

or formulated according to the new knowledge. If the hypothesis is valid, then we should

write a report with our conclusions.

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h) Laws and theories. In science, once a hypothesis has been widely accepted, it is called a

law, which is a concise verbal or mathematical statement which describes a relationship

between the variables of a determined nature phenomenon. (e.g. Newton’s second law, F =

m ⋅ a). Theories are general explanations based on a large amount of data. For example,

the theory of evolution applies to all living things and is based on wide range of

observations. A theory is an explanation of a set of observations and scientific laws of an

entire group of related phenomena of nature. It explains how it works, what causes it, and

how it behaves. For example, the kinetic theory of matter or Einstein’s General Theory of

Relativity.

Concepts SCIENCE is the systematic knowledge of the world gained through OBSERVATION,

EXPERIMENTATION and REASONING and the formulation of LAWS and THEORIES.

PHYSICAL CHANGE: Change that undergoes a substance without transformation into other

substance/s, its chemical composition doesn’t change. Examples of physical changes are:

changes in position, in volume, in form, in temperature or state; mixtures, electrical phenomena…

PHYSICS is an EXPERIMENTAL AND NATURAL SCIENCE that studies the composition of

matter, its general properties and its PHYSICAL CHANGES.

CHEMICAL CHANGE: Change that undergoes a substance with transformation into other

substance/s, its chemical composition changes (iron oxidation, fruit ripening, cooking, digestion,

respiration…).

CHEMISTRY: EXPERIMENTAL AND NATURAL SCIENCE that studies the composition, the

structure and the properties of matter and its chemical changes.

SCIENTIFIC METHOD: procedure used in scientific research to solve problems of nature.

OBSERVATION: an activity consisting of receiving information or knowledge of natural

phenomena of the outside world through the senses, recording of data using scientific instruments.

BACKGROUND INFORMATION: knowledge that you have acquired through the research in

scientific journals, Internet or books on the topic of the problem. Scientific inventions must be

published to be known by the rest of the scientific community.

VARIABLE: a quantitative property of an object that can take on different values (length,

speed, density, volume…).

INDEPENDENT VARIABLE: variable that a scientist deliberately changes during the experiment.

DEPENDENT VARIABLE: variable that responds to the changes in the independent variable.

CONTROLLED VARIABLES: variables that are constant (don’t change) during the experiment.

HYPOTHESIS: an educated guess, a prediction that must be tested with an experiment.

EXPERIMENTATION: involves carrying out experiments under controlled conditions to make

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observations and collect data to test the hypothesis. You must design the experiment before,

indicating the materials and the procedure in steps and include as much detail as possible

about measurements and techniques in each step.

EXPERIMENTAL DATA ORGANIZATION: The experimental data must be organized the

experimental data in tables and graphs for analysis and interpretation.

DATA ANALYSIS: analysis of the data to look for patterns to know if there is a relationship

between the independent variable and the dependent variable of the experiment.

SCIENTIFIC LAW: a concise verbal or mathematical statement which describes a relationship

between the variables of a determined nature phenomenon. (Newton’s second law, F = m ⋅ a).

THEORY: explanation of a set of observations and scientific laws of an entire group of

related phenomena of nature (it explains how it works, what causes it, and how it behaves)

(The kinetic theory of matter, Einstein’s General Theory of Relativity).

Activities 1. Write the steps of the scientific method.

2. Write the names of all the variables that affect plant growth.

3. Does the amount of sunlight affect plant growth? Formulate a hypothesis and design an

experiment to test it. Classify the different variables of your experiment.

4. Imagine that you have a lantern and it doesn’t work. Formulate a hypothesis and design an

experiment to test it.

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2. QUANTITIES AND UNITS The goal of science is to provide an understanding of the physical world by developing

theories based on experiments. The basic laws of physics and chemistry involve the

measurement of properties of matter and of the all observed phenomena.

A quantity is a property of a phenomenon, body or substance that can be quantified or

measured. Examples of quantities are mass, volume, temperature, speed).

A quality is property of a phenomenon, body or substance that cannot be quantified. Examples

of qualities are colour, shape., roughness, toughness…

To measure a quantity we compare it with a chosen value of that quantity to express its

value with a number followed by the chosen unit.

A unit is a definite magnitude of a particular quantity, defined and adopted by

convention, with which other particular quantities of the same kind are compared to

express their value. For example, when we measure a quantity, we always compare it with

some reference standard. When we say that a rope is 30 metres long, we mean that it is 30

times as long as a metre stick, which we define to be 1 metre long.

The result of a measurement includes a number and the chosen unit for the measurement.

The value of a quantity is generally expressed as the product of a number and a unit.

All physical quantities can be classified into two types: base quantities and derived

quantities.

Base quantities are quantities that are common to every object or phenomenon and are

assumed to be mutually independent. There are 7 base quantities used in The International System of Units (SI). Each base quantity has its own SI base units. The symbols used for the

base quantities and their base units are given as follows:

BASE QUANTITIES AND THEIR SI BASE UNITS Quantity Symbol Unit Symbol

Length l, x, r, … metre m Mass m kilogram kg Time t second s Intensity of electric I, i ampere A Temperature T kelvin K Amount of substance n mole mol Luminous intensity Iv candela cd

Base units can be too large or too small for some measurements, so the base units may

be modified by attaching prefixes, which are decimal multiples and submultiples and we obtain

secondary units. These prefixes are given the name of SI prefixes. The prefix names and

symbols are listed below:

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DECIMAL MULTIPLES AND SUBMULTIPLES OF SI UNITS. SI PREFIXES Factor Prefix Name Symbol

106 (1.000.000) mega M

103 (1.000) kilo k

102 (100) hecto h

101 (10) deca da

10-1 (0,1) deci d

10-2 (0,01) centi c

10-3 (0,001) mili m

10-6 (0,000001) micro μ

Examples: Decimetre: 1 dm = 0,1 m. Centimetre: 1 cm = 0,01 m. Hectometre: 1 hm = 100 m.

The most common units for the base quantities length, mass and time are listed below:

LENGTH UNITS

MASS UNITS

:10 :10 :10 :10 :10 :10 hg g dg cg mgkg dag

×10 ×10 ×10×10 ×10 ×10

TIME UNITS

Name Symbol Value (SI Units) Name Symbol Value (SI Units)second s hour h 1 h = 60 min = 3.600 sminute min 1 min = 60 s day d 1 d = 24 h = 86.400 s

All other quantities different than base quantities are known as derived quantities, quantities that derive from the base quantities by equations. Examples of derived quantities are

area, volume, capacity, density, speed... The most common units for the derived quantities

area, volume and capacity are listed below:

AREA UNITS

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:100 :100 :100 :100 :100:100 hm2 km2

×100 ×100 ×100 ×100 ×100 ×100m2 dm2 cm2 mm2dam2

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VOLUME UNITS. The amount of 3-dimensional space an object occupies.

:1000 :1000 :1000 :1000 :1000:1000

×1000 ×1000 ×1000 ×1000 ×1000 ×1000hm3 m3 dm3 cm3 mm3km3 dam3

CAPACITY UNITS. The amount of substance, liquid or gas, which a container can hold.

:10 :10 :10 :10 :10 :10 hL L dL cL mLkL daL

×10 ×10 ×10×10 ×10 ×10

CAPACITY AND VOLUME CONVERSION

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3. UNIT CONVERSIONS Units are multiplied and divided just like ordinary algebraic symbols. We can express the

same physical quantity in two different units and form an equality. For example, when we say

that 1 m = 100 cm we don’t mean that the number 1 is equal to the number 60, but rather that 1

m represents the same physical length interval as 100 cm. To find the number on centimetres of

25 m, we write:

( ) cm2500m1cm100m25m25 =⎟⎟

⎠

⎞⎜⎜⎝

⎛/

⋅/=

A Conversion factor is a ratio of units, which expresses a quantity expressed in some

unit or units divided by its equal expressed in some different unit or units, such as 1 m/100 cm

(or 1 m = 100 cm). The units of the quantity and the conversion factor must be combined

properly to give the desired final units.

The unit conversion method:

1. Write the quantity you want to change its unit. 2000 mL

2. Write the equality between the old unit and the new

unit. Remember the table of prefixes. 1L = 1000 mL

3. Write the conversion factor with the two units

properly combined to give the desired final unit. ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

mL1000L1mL2000

4. Multiply the quantity by the conversion factor,

cancel the old units and express the final result.

( ) L2

mL1000L1mL2000 =⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

Example. Convert 0,5 hL to mm3.

The unit conversion method can also be used to convert units of derived quantities which

are composed of two different units. In this case, you must use a conversion factor for each unit.

Example: The maximum speed limit on motorways in Spain is 120 km/h. Express this

speed in m/s.

sm33,33

s3600h1

km1m1000

hkm120 =⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

10

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4. SCIENTIFIC NOTATION It is a way of writing very large and very small numbers in standard decimal notation,

based on powers of the base number 10. The number is written as a product of a number

between 1 and 10 and a power of 10.

4.1. LARGE NUMBERS

For example, the following number is written:

87 600 000 = 8,76 ⋅ 107

The first number, 8,76, is called the coefficient. It must be greater than or equal to 1 and

less than 10.

The second number, 107, is called the base. It must always be 10 written in exponent

form.

To write a number in scientific notation:

a) Find the coefficient: Put the decimal after the first digit and drop the zeroes.

b) Find the exponent: Count the number of the places from the decimal to the end of the

number. Example:

4.2. SMALL NUMBERS

To write a number in scientific notation that is less than 1:

a) Find the coefficient: Move the decimal point right until you reach a coefficient greater than 1

but less than 10 and place the decimal point there.

b) Find the exponent: count the number of places the decimal point was moved and the

exponent is 10 raised to the negative of that number.

Examples:

0,00000087 = 8,7 ⋅ 10−7

Quantity Scientific notation Earth-Sun distance: 150 000 000 000 m 1,5 ⋅ 1011 m

Electric charge of the proton: 0,0000000000000000001602 C 1,602 ⋅ 10−19 C

Mass of the electron: 0,00000000000000000000000000000091 kg 9,1 ⋅ 10−31 kg

Activities 5. Define quantity. Write four examples of quantities and four examples of properties which are

not quantities.

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6. Define unit.

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7. Write the units of mass in a decreasing sequence.

8. Write the units of length in a decreasing sequence.

9. Write the units of area in a decreasing sequence.

10. Write the units of volume in a decreasing sequence.

11. Write the units of capacity in a decreasing sequence.

12. Complete:

a) 6 760 000 = 6,76 × 10 b) 6760 = 6,76 × 10 c) 5 000 000 = 5 × 10

d) 430 000 000 = 4,3 × 10 e) 25 700 000 = 2,57 × 10 f) 45 000 = 4,5 × 10

13. Complete:

a) 0,0000676 = 6,76 × 10 b) 0,00000089 = 8,9 × 10 c) 0,00005 = 5,0 × 10

d) 0,0000043 = e) 0,0000000257 = f) 0,00000000045 =

14. Convert the following units and indicate their corresponding quantity. Write the resulting

numbers in decimal and in scientific notations.

a. 72 hm to m b. 4 m to mm c. 3,5 m to cm d. 2,3 mL to m3 e. 5,2 m2 to mm2 f. 3 m3 to dm3 g. 0,2 m3 to cm3 h. 3 dm3 to dL i. 4300 cm3 to L j. 2340 cL to dm3 k. 220 dL to dm3 l. 0,0003 cm3 to daL m. 0,0045 mm2 to hm2 n. 0,0002 kg to mg o. 1 day to s p. 20,456 hL to dL q. 0,0012 hL to dm3 r. 14,56 mm3 to dL s. 40,56 g to hg t. 3.820 s to h u. 1,0003 mm3 to L v. 2,2 cm2 to m2 w. 7,2 mm to dam x. 0,2 cm3 to mL y. 2,34 L to dm3 z. 220 kL to m3

15. Express in metres the result of the following operation: 0,30 km + 3,6 hm + 23.45 m + 284 cm.

16. How many mL are there in 1/3 L?

17. How many g are there in 1/4 kg?

18. Convert the following units and indicate their corresponding quantity. Write the resulting

numbers in decimal and in scientific notations:

a. 55 m to cm b. 3 m2 to cm2 c. 2 mL to cm3 d. 1 Tm to kg e. 40 cm to dm f. 2 mm2 to dm2 g. 200 mL to m3 h. 20 m3 to dm3 i. 2 mm to m j. 4 cm2 to m2 k. 200 kg to g l. 200 L to mm3 m. 7 m to mm n. 20 m3 to dm3 o. 2000 g to kg p. 4,67 m2 to mm2 q. 5 km to cm r. 20 cm3 to dm3 s. 200 g to mg t. 20 dam3 to mm3 u. 3.000 mg to kg v. 3 hours to s w. 1 h to s x. 200 hL to km3 y. 180 s to min. z. 2 days to s

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5. MEASUREMENT AND ERROR All measurements are approximate values not true values. An error is a comparison

between the measured value and the true value of a measurement. Errors can be classified into

two types:

a) Systematic or determined errors, errors that produce a result that differs from the true

value by a fixed amount. They can be identified and corrected. These errors result from

biases introduced by:

1. Instrumental bias. For example, errors in the calibration of the measuring instruments.

2. Human bias. For example, the person who is measuring might read an instrument

incorrectly or might let knowledge of the expected value of a result influence the

measurement.

3. Method bias. For example, the person might make an incorrect scale reading because

of parallax error.

4. Operative bias. For example, the person knows the measurement method but might

make it wrong.

b) Random errors are caused by unknown and unpredictable changes in the measurement.

These changes may occur in the measuring instruments or in the environmental conditions.

They cannot be corrected but the can be minimized by:

1. Making several measurements.

2. Calculating the arithmetic mean.

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• Example. Several measurements of the mass of an object: 2,350 g, 2,352 g, 2,348 g,

2,350 g.

Arithmetic mean = g350,24

350,2348,2352,2350,2=

+++

An error in a measurement of a physical quantity may be represented by the actual

amount of error, or by a ratio comparing the error to the size of the measurement. We use the

absolute error and the relative error.

The absolute error is the difference between the measured value and the true or exact

value.

Ea = Vm − Vr

Vm is the measured value.

Vr is the true value.

If Ea > 0 the absolute error is positive, and the error in excess.

If Ea < 0 ⇒ absolute error is negative, it is a scant measurement.

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The relative error is computed by dividing the absolute error by the true value. It is the error per unit of measured quantity.

r

ar V

EE =

To convert to percentage, multiply by 100: 100VE

(%)Er

ar ⋅=

The relative error gives an indication of how good a measurement is relative to the size

of the thing being measured. It indicates the accuracy of a measurement. Example: Two students measure two different lengths. Student A measures a real length

of 5 m. The result of his measurement is 6 m. Student B measures a real length of 500 m. The

result of his measurement is 501 m. Although both students have the same absolute error, +1

m, student B is more precise than student A because he has measured a greater length. His

error per measured metre is lower, so is his relative error.

Student True Value Measured Value Ea Er (%) A 5 m 6 m 1 m (excess) (1/5) 100 = 20 % B 500 m 501 m 1 m (excess) (1/500) 100 = 0,2 %

The sensitivity of a measurement instrument is the

smallest amount of a quantity that the instrument can detect. For

example, ruler 1 has a sensitivity of 1 cm, and ruler 2 has a

sensitivity of 1 mm (0,1 cm).

The precision or reproducibility of a measurement instrument is the

degree to which repeated measurements of the same quantity under

unchanged conditions show the same results. This instrument is precise

because all measurements are grouped tightly together.

0,2 0,3

The accuracy of a measurement instrument is the degree of closeness of the

measurements of a determined quantity to that quantity’s true value.

A direct measurement is the determination of the value of a quantity with a measurement

instrument. For example, the measurement of the mass of an object with a balance.

Uncertainty or margin or error of a measurement is the range of values likely to

enclose the true value. It depends on the sensitivity of the measurement instrument. The

uncertainty of ruler 1 is 1 cm and the uncertainty of ruler 2 is 0,1 cm.

The result of any physical measurement has two essential components:

1. a numerical value giving the best estimate possible of the quantity measured, and

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2. the degree of uncertainty associated with this estimated value.

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measurement = best estimate ± uncertainty

Ruler 1

L = 17 ± 1 cm (ruler in cm)

Ruler 2

L = 17,6 ± 0,1 cm (ruler in mm)

Significant figures are the digits that carry meaning contributing to its precision. For

example the number 5,423 cm has four significant figures. There are some rules for counting

the significant figures in a number:

1. Nonzero integers always count as significant figures (452 ⇒ 3; 2,227 ⇒ 4).

2. Zeros appearing between two significant digits are significant (101,25 ⇒ 5).

3. Leading zeros (zeros that precede all the nonzero digits) are not significant (0,706 ⇒ 3;

0,0007080 ⇒ 4).

4. Trailing zeros (zeroes at the right end of the number) in a number containing a decimal

point are significant (13,4300 ⇒ 6; 5,030 ⇒ 4; 0,00134300 ⇒ 6; 35,0 ⇒ 3; 35,00 ⇒ 4).

5. In scientific notation, all the figures that appear before 10 are significant (1,430 ⋅ 105 ⇒ 4).

6. If you perform an arithmetic operation (addition, subtraction, multiplication or division), the

number of significant figures of the result mustn’t exceed the lowest number of significant

figures of the numbers (3,5432 + 2,531 = 6,0742 ≅ 6,074; 2,33 ⋅ 2,4 = 5,592 ≅ 5,6).

Examples:

Measurement No. of significant figures Measurement No. of significant figures 0,0005090 dg 4 6 · 108 kg 1 0,0036 g 2 4,07 · 1016 m 3 0,007050 m 4 3,05 hm 3 7,64 cm 3 18,5 s 3 248 m 3 7,35 s 3 64,01 kg 4 0,220 kg 3 0,00003 m 1 42,05 km 4 2,40 · 106 kg 3 0,075 m 2 2.500 mm 4 80,0 s 3

An indirect measurement is the determination of the value of a quantity by calculation.

For example, the determination of the density of a solid substance requires the measurement of

its mass and its volume followed by a calculation.

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Rounding is the process of replacing a number by another number of approximately the

same value but having fewer digits. If the digit following the last digit to be retained is:

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1. 5 or greater than 5 the last digit should be increased by 1.

2. less than 5 the last digit should be stay the same.

3. It is advisable to round a number only until two decimal places (Er < 1 %).

Example: π = 3,1415926...

3,141593

3,14159

3,1416

3,142

3,14

Activities 19. We measured with a chronometer the time of the free fall of an object from the roof of a

building. We obtained the following measurements: 1,43 s; 1,45 s; 1, 47 s; 1,42 s; 1,46 s;

1,43 s; 1,47 s.

a) How long does it take for the object to reach the ground?

b) What is the sensitivity of the chronometer?

c) What is the absolute error of the last measurement?

d) What is the relative error of the last measurement?

e) Indicate de number of significant figures in the second measurement.

20. We measured with a balance the mass of an object. We obtained the following

measurements: 2,430 g; 2,427 g; 2, 433 g; 2,428 g; 2,432 g.

a) What is the mass of the object?

b) What is the sensitivity of the balance?

c) What is the absolute error of the second measurement?

d) What is the relative error of the second measurement?

e) Indicate de number of significant figures in the second measurement.

21. The exact mass of an object is 2,350 g. A student measured that mass on a balance which was

2,348 g.

a) What is the absolute error of the measurement? Explain the meaning of this result.

b) What is the relative error of the second measurement? Explain the meaning of this result.

c) Indicate de number of significant figures in the exact mass.

d) What is the sensitivity of the balance?

22. The exact mass of an object is 0,70 g. A student measured that mass on a balance and

obtained the following results: 0,67 g, 0,72 g and 0,71 g.

a) What is the absolute error of the first measurement? Explain the meaning of this result.

b) What is the relative error of the first measurement? Explain the meaning of this result.

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23. The exact mass of an object is 2,050 g. A student measured that mass on a balance and

obtained the following results: 2,048 g, 2,049 g and 2,051 g.

a) What is the absolute error of the third measurement? Explain the meaning of this result.

b) What is the relative error of the third measurement? Explain the meaning of this result.



6. TABLES AND GRAPHS The experimental data obtained during a experiment must be ordered in a table in order

to find a relationship between two quantities, the independent variable and the dependent variable, indicating at the top of the table each quantity and its unit.

The next step consists in creating a graph following the next rules:

1. Graphs are done on graph paper. The graph is two-dimensional because it represents the

possible relationship between two variables.

2. The independent variable is represented with a horizontal straight line, from a central point in

the graph, called the origin. This horizontal line is known as the X-axis and should be labelled

with the corresponding quantity what it measures and its units at the end of the line.

3. The dependent variable is represented with a vertical straight line from the origin. This

vertical line is known as the Y-axis and should be labelled with the corresponding quantity

what it measures and its units at the end of the line.

4. The scale of the axes should be chosen to include all data points and to allow as much

room as possible on both axes.

5. Each axis should be evenly divided with plenty of space between divisions. Divisions should

be labelled in multiple units of 1, 2, 5, or 10.

6. The experimental data must be plotted in the graph with points corresponding to both

coordinates. You mustn’t draw lines parallel to the axes to determine the plotted points.

7. It is important to add a title to the top of the graph.

The form of the line that connects the plotted data points shows the mathematical relationship between the two variables. There are different relationships between two

variables:

1. If the plotted data points roughly form a straight line, use a ruler to draw a line that best

represents the data points, which comes closest to all the data. This straight line is represented

as a mathematical equation, the straight line equation, which quantify the relationship

between the two variables of the experiment.

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The straight line means that the two

variables are directly proportional. The equation of a straight line is

usually written this way:

y = y0 + m x

x and y are the corresponding

variables of the experiment.

m is the slope or the gradient of the

straight line. Its value is constant, and it can

be determined choosing two points of the

straight line: P1 (x1, y1) y P2 (x2, y2):

12

12

xxyym

−−

=

y0: It is simply the value of y where the line crosses the Y axis, when x = 0.

If the line passes through the origin one of the points, i.e. the point P1 (x1, y1), can be the point

(0, 0).

Example: Study of the relationship between the extension of a spring, Δx, and the weight

of an object suspended from it, F.

F and Δx: are directly proportional.

Equation of the straight line: Δx = 2 ⋅ F

The slope m is equal to 2.

Hooke’s Law: F = k ⋅ Δx

18

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Example: Study of the relationship between the potential difference, V, applied to an

electric circuit and its current intensity, I.

V (V) I (A) 0 0

2,2 0,1 5,3 0,2 7,2 0,3

10,0 0,4 13,0 0,5 14,4 0,6 18,0 0,7 19,6 0,8

Relationship between the voltage, V, and the current intensity, I

I (A)0,0 0,2 0,4 0,6 0,8

V (V)

0

5

10

15

20

V and I are directly proportional.

The equation of the straight line is: V = 0+ R ⋅ I ⇒ V = R ⋅ I

R is the slope of the straight line, a constant value, which corresponds to the electrical resistance of the circuit.

If the plotted data points do not form a straight line but appear to form a curve, we must

find the mathematical equation which gives the relationship between the two quantities of the

experiment.

19

2. If the plotted data points form a curve of an equilateral hyperbola the two quantities of the

experiment are inversely proportional. The

mathematical equation which gives the

relationship between the quantities is:

kxyxky =⋅⇒=

The product of the two variables is always

constant. k is a constant value. X

Y

0 5 10 150

2

4

6

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Example: Study of the relationship

between the pressure of a gas, P, and its

volume, V.

There is a relationship between P and V: P

and V are inversely proportional.

The equation of an equilateral hyperbola

is:

kPVPkV =⋅⇒=

The product of the two variables is always

constant. k is a constant value.

The relationship between P and V can be

found much easier if we plot V against 1/P. We

will get a straight line passing through the origin

and k (the constant) is the slope or the gradient

of the graph.

V and 1/P are directly proportional.

The equation of the straight line is:

PkV =

P(atm)

V (L)

0 5 10 150

2

4

6

3. If the plotted data points form a curve of a parabola the mathematical relationship between

two quantities of the experiment is a second

grade equation:

2xky ⋅=

k is a constant value.

X0 1 2 3 4

Y

0

20

40

60

80

20

Example: Study of the free fall of an object.

Relationship between height, s, and time, t.

The mass, m, of an object doesn’t affect its

motion during free fall; if two objects of different

masses are dropped from the same height at the

s (m) t (s) 5 1

20 2 45 3

Object Am = 10 g

80 4 5 1 Object B

m = 20 g 20 2

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same time, they hit the ground at the same time. 45 3 80 4

However, height affects the time of free fall.

The graph s-t for the object A is a

parabola which means that the mathematical

relationship between height and time is a

second grade equation.

s = k ⋅ t2

k is a constant value.

The relationship between s and t can be

found much easier if we plot s against t2. We

will get a straight line passing through the origin

and k (the constant) is the slope or the gradient

of the graph (k = 1/2 g).

s (m) t (s) t2 (s2) 5 1 1

20 2 4 45 3 9 80 4 16

s and t2 are directly proportional.

The equation of the straight line is:

s = 1/2 ⋅ g ⋅ t2

t (s)0 1 2 3 4

s (m)

0

20

40

60

80

Sustituir

Activities 24. During an investigation to find the relationship between the diameter and the circumference

of a circle, a group of students chose seven circle-shaped objects of different sizes and

measured their diameters (D) and their circumferences (C):

D (cm) 2,0 3,0 4,0 5,0 6,0 7,0

C (cm) 6,3 9,4 12,6 15,7 18,8 22,00

a) Identify the independent variable, the dependent variable and the possible controlled

variables of this investigation.

b) Verify if there is some type of mathematical relationship between variables, the

independent variable and the dependent variable (for example, calculating the ratio

between each pair of values and observing the obtained results).

21

c) Formulate a hypothesis.

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d) Do the graph on graph paper of the variable diameter and the variable circumference

and determine the mathematical equation of the graph that relates both variables.

e) Use the obtained mathematical relationship and calculate the circumference of a circle-

shaped object of 2,0 m of diameter.

25. An investigation has been made on the relationship of the masses of the substances and their

volumes. The exposition of the problem is the following one: Does the volume of a substance

affect its mass? The volumes and the masses of different marble pieces have been measured

to verify if there is dependence between the volume and the mass, and the following values

have been obtained:

V (cm3) 3,1 4,8 7,6 10,0 14,1

m (g) 9 14 22 29 41







d) Do the graph on graph paper of the variable mass and the variable volume. Determine

the mathematical equation of the graph that relates both variables.

e) A sculptor needs a cubical marble block of 2 m of edge to make a sculpture. The people

in charge of the quarry want to know the mass of the block for their transport. Use the

obtained mathematical relationship and calculate the mass of the cubical marble block.

26. During an investigation to find the relationship between the pressure and volume of an

enclosed gas at constant temperature, a glass syringe was filled of air, its opening was

sealed so that no air can escape the syringe, and the plunger was slowly pushed

decreasing the volume and at the same time measuring the air pressure inside the syringe.

The following results were obtained.

P (atm) 2,5 3,6 4,2 6,3 8,3 12,5 25

V (cm3) 7 6 5 4 3 2 1





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d) Do the graph on graph paper of the variable pressure and the variable volume. Do the

graph on graph paper of the variable pressure and the variable 1/V. Compare both

graphs and try to determine the mathematical equation of the graph that relates both

variables.

e) Use the obtained mathematical relationship and calculate the pressure of the air inside

the syringe in a volume of 5 ⋅ 10−5 m3.

27. Tests were carried out to determine the magnitude of the force of air resistance, F, based

on the speed, v, of a body which moves immersed in the air: the body takes different

speeds and for each speed the force of air resistance is measured. The following results

have been obtained:

v (m/s) 2,0 3,5 5,0 6,5 8,0

F (N) 8,4 25,9 52,8 87,4 135,1







d) Do the graph on graph paper of the variable speed and the variable force and explain if

there is some type of mathematical relationship between both variables.

28. Science project. Free fall E A G



7. DICTIONARY

science __________________ base quantity __________________

knowledge __________________ base unit __________________

reasoning __________________ multiple __________________

law __________________ submultiple __________________

theory __________________ prefix __________________

physical change __________________ derived quantity __________________

undergo __________________ length __________________

phenomenon __________________ mass __________________

phenomena __________________ volume __________________

physics __________________ time __________________

matter __________________ area __________________

chemical change __________________ capacity __________________

fruit ripening __________________ container __________________

iron __________________ conversion factor __________________

scientific method __________________ express __________________

background information __________________ scientific notation __________________

independent variable __________________ decimal __________________

dependent variable __________________ power __________________

controlled variable __________________ base __________________

hypothesis __________________ coefficient __________________

experimentation __________________ zero (zeroes) __________________

procedure __________________ decrease __________________

table __________________ systematic error __________________

graph __________________ differ __________________

scientific law __________________ amount __________________

receive __________________ bias __________________

acquire __________________ random error __________________

deliberately __________________ minimize __________________

step __________________ arithmetic mean __________________

statement __________________ absolute error __________________

quantity __________________ relative error __________________

quality __________________ accuracy __________________

measure __________________ sensitivity __________________

measurement __________________ precision __________________

unit __________________ reproducibility __________________

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magnitude __________________ uncertainty __________________

enclose __________________ scale __________________

leading zeroes __________________ plot __________________

trailing zeroes __________________ roughly __________________

significant figures __________________ slope __________________

rounding __________________ weight __________________

graph paper __________________ directly proportional __________________

axis(axes) __________________ equilateral hyperbola __________________

label __________________ inversely proportional __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

__________________ __________________ __________________ __________________

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SCIENCE AND THE SCIENTIFIC METHOD E A G › 2012 › 09 › unit-1-4th-eso.pdf · Unit 1: Science E. Aller 2 Physics is an experimental and natural science that studies the composition

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