physics_for_the_ib_diploma__fifth_edition____cambridge_education___cambridge_university_press_samples.pdf...

Physicsfor the IB Diploma

K. A. Tsokos

This fifth edition of Physics for the IB Diploma covers, in onevolume, the entire new International Baccalaureate (IB) syllabus(for first examinations in May 2009), including all the options atboth Standard Level and Higher Level. It features a chapter onthe role of physics in the context of the Theory of Knowledge(TOK) course, along with many discussion questions for TOK.The student-friendly design makes this comprehensive book easyto use and the accessible language ensures that the material isalso suitable for students whose first language is not English.

The division of this book into chapters and sections usually followsthe organisation of the syllabus. However, since the sections arereasonably independent of each other, alternative teachingsequences may be used. Each chapter opens with a list of learningobjectives that are directly related to the syllabus. The questionsat the end of each chapter have been expanded and answers tothose involving calculation, plus some others, have been suppliedat the back of the book. Worked examples throughout the texthighlight important results, laws, definitions and formulae, andprovide students with the necessary skills to solve problems.

Features include:

x answers to the end-of-chapter questions

x supplementary material for specific topics

x glossary of key terms.

The author has been teaching physics for over twenty years.

Physicsfor the

IB Diploma

Fifth editionK. A. Tsokos

9780

5211

3821

5 T

soko

s:Ph

ysic

s for

IB D

iplo

ma.

Cov

erC

MY

K

Learning

Standard andHigher Level

Physics

for the IB Diplom

aK

. A. Tsokos

Fifth edition

9780521138215cvr.qxd 8/12/09 10:28 Page 1

Contents

Preface xiA note to the reader xii

Part I Core and AHL 1

Topic 1: Physics and physical measurement 2

1.1 The realm of physics Core 2Orders of magnitude and units 2Fundamental interactions 5

1.2 Uncertainties and errors Core 8Errors of measurement 8Significant digits 11Line of best fit 12

1.3 Mathematical and graphical techniques Core 14

Multiplicative changes 14Straight-line graphs 15Getting a linear graph 16Interpreting graphs 17Sine curves 18Making assumptions 18

1.4 Vectors and scalars Core 21Vectors 21Multiplication of a vector by a scalar 22Addition of vectors 22Subtraction of vectors 24Components of a vector 25

1.5 Graphical analysis and uncertainties Core 31

Logarithmic functions 31Propagation of errors 33

Topic 2: Mechanics 38

2.1 Kinematic concepts Core 38Displacement and velocity 38Frames of reference 42Graphs for uniform motion 44

2.2 Motion with constant acceleration Core 48

Acceleration 48Measuring speed and acceleration 54More on graphs 56

2.3 The concept of force Core 63Forces and their direction 64Hookes law 67

2.4 Newtons first law Core 69Newtons first law 69Equilibrium 70

2.5 Newtons second and third laws Core 76

Newtons second law 77The inclined plane 82Newtons third law 83

2.6 Linear momentum Core 87The concept of momentum 87Impulse 89The law of conservation of

momentum 91Proof of momentum conservation 93Two-dimensional collisions 95

2.7 Work, energy and power Core 99Work done by a force 99Gravitational potential energy 102The workkinetic energy relation 103Conservation of energy 104Frictional forces 107Power 108Kinetic energy and momentum 110The problem of least time 112

2.8 Circular motion Core 119Circular motion and centripetal

acceleration 119Centripetal forces 122Angular momentum 123

www.cambridge.org in this web service Cambridge University Press

Cambridge University Press978-0-521-13821-5 - Physics for the IB Diploma, Fifth EditionK. A. TsokosTable of ContentsMore information

2.9 The law of gravitation Core 127Newtons law of gravitation 127Gravitational field strength 129

2.10 Projectile motion AHL 132Parabolic motion 132Launch at an arbitrary angle 134Effect of air resistance forces 138

2.11 Motion in a gravitational field AHL 142Gravitational potential energy 143Escape velocity 145Orbital motion 146Equipotential surfaces 149The binary star system 151

Topic 3: Thermal properties of matter 158

3.1 Thermal concepts Core 158Temperature 158Heat as energy 159The atomic model of matter 160

3.2 Thermal properties Core 163Specific heat capacity 163Change of state 165Measuring specific heats 167Evaporation 168The kinetic theory of gases 169

3.3 Ideal gases AHL 174Pressure 174The BoyleMariotte law 175The volumetemperature law 176The pressuretemperature law 177The equation of state 178

3.4 Thermodynamics AHL 183Internal energy 183Work done on or by a gas 184The first law of thermodynamics 187The second law of thermodynamics 189Degradation of energy 192

Topic 4: Oscillations and waves 195

4.1 Simple harmonic motion Core 195Oscillations 195Kinematics of simple harmonic motion 196Energy in simple harmonic motion 204Damping 207Forced oscillations and resonance 208

iv Contents

4.2 Travelling-wave characteristics Core 216What is a wave? 216Transverse and longitudinal waves 217Wave pulses 218Travelling waves 218Wavefronts 224

4.3 Wave phenomena I: reflection andrefraction Core 228The principle of superposition 228Reflection and refraction of waves 231Huygens principle 234

4.4 Wave phenomena II: diffraction andinterference Core 238Diffraction 238Interference 240

4.5 The Doppler effect AHL / SL Option A 244

The Doppler effect 244

4.6 Standing waves AHL / SL Option A 251Standing waves on strings and tubes 251Resonance and the speed of sound 255

4.7 Diffraction AHL / SL Option A 259Diffraction 259Single-slit diffraction 261

4.8 Resolution AHL / SL Option A 267The Rayleigh criterion 267

4.9 Polarization AHL / SL Option A 271What is polarization? 271Maluss law 272Polarizers and analysers 273Polarization by reflection 274Optical activity 275Practical applications of

polarization 276

Topic 5: Electricity and magnetism 280

5.1 Electric charge Core 280Properties of electric charge 280Coulombs law for the electric force 285

5.2 Electric field and electric potential Core 289

Electric field 289Electric potential 292The electronvolt 295



5.3 Electric field and electric potential AHL 299

Electric fields 299Electric potential and energy 300Equipotential surfaces 304The connection between electric

potential and electric field 305Similarities between electricity

and gravitation 306

5.4 Electric current and electric resistance Core 310

Electric current 310Electric resistance 312Electric power 314

5.5 Electric circuits Core 318Emf 318Simple electric circuits 320Ammeters and voltmeters 325Sensors based on the potential

divider 327

5.6 Magnetic fields Core 336Magnetic field 336The magnetic force on a current 338The magnetic force on a

moving charge 339rsteds discovery 341The force between two

current-carrying wires 343

5.7 Electromagnetic induction AHL 350A wire moving in a magnetic field 350Faradays law 351Lenzs law 354Faradays disc 356

5.8 Alternating current AHL 360The AC generator 360Power in AC circuits 361The transformer 363Transformers and power

transmission 365

Topic 6: Atomic and nuclear physics 367

6.1 The atom and its nucleus Core 367The discovery of the nuclear atom 367Consequences of the Rutherford

(GeigerMarsden) experiment 368

Contents v

The Rutherford model of the atom 369The Bohr model 369Nuclear structure 370The forces within the nucleus 371

6.2 Radioactivity Core 373The nature of alpha, beta and

gamma radiations 373Radioactive decay equations 376The law of radioactive decay 377

6.3 Nuclear reactions Core 380The unified mass unit 380The mass defect and binding energy 381Nuclear reactions 384Nuclear fission 385Nuclear fusion 385

6.4 Interactions of matter with energy AHL / SL Option B 389

The photoelectric effect 389De Broglies wavelength 394

6.5 Quantum theory and the uncertainty principle AHL / SL Option B 398

Atomic spectra 398The electron in a box model 399The Schrdinger theory 400The Heisenberg uncertainty principle 402

6.6 Nuclear physics AHL / SL Option B 407Scattering experiments and distance

of closest approach 407The mass spectrometer 408Beta decay and the neutrino 409Nuclear energy levels 410The radioactive decay law 411

Topic 7: Energy, power and climate change 415

7.1 Energy degradation and powergeneration Core 415Degradation of energy 415Electricity production 417Energy sources 417Fossil fuels 418Nuclear power 420Solar power 423Hydroelectric power 425Wind power 427Wave power 428



7.2 The greenhouse effect and global warming Core 434

The black-body law 434Solar radiation 436The greenhouse effect 438Global warming 444Sea level 447Effects of global warming on climate 448

Topic 8: Digital technology

8.1 Analogue and digital signals AHL / SL Option C / SL and HL Option F 454

Binary numbers 454Analogue and digital signals 455Compact disks 458Other storage devices 459Advantages of digital storage 461

8.2 Digital imaging with charge-coupled devices AHL / SL Option C 463

Capacitance 463The charge-coupled device 464CCD imaging characteristics 466Medical uses of CCDs 467

Part II Options 471The content of Options AD (except Option A1) is identical to that in theappropriate chapter in the Core and AHL material, to which the reader isreferred (details are given in the text).

SL Option A Sight and wave phenomena

A1 The eye and sight SL 472The structure of the human eye 472Depth of vision 473Accommodation 474Scotopic and photopic vision 474Colour 475Colour blindness 476Colour addition 476Colour subtraction 477Perception of colour and light 478

A2 The Doppler effect SL 480

A3 Standing waves SL 480

A4 Diffraction SL 480

vi Contents

A5 Resolution SL 481

A6 Polarization SL 481

SL Option B Quantum physics

B1 Quantum physics SL 482

B2 Nuclear physics SL 482

SL Option C Digital technology

C1 Analogue and digital signals SL 483

C2 Data capture and imaging using CCDs SL 483

C3 Electronics SL 484

C4 The mobile phone system SL 484

SL Option D Relativity and particle physics

D1 Introduction to relativity SL 485

D2 Concepts and postulates of special relativity SL 485

D3 Relativistic kinematics SL 485

D4 Particles and interactions SL 486

D5 Quarks SL 486

SL and HL Option E Astrophysics

E1 Introduction to the universe SLand HL 487

The solar system 487Beyond the solar system 489The motion of the stars 491

E2 Stellar radiation SL and HL 494The energy source of stars 494Luminosity 495Black-body radiation 496Stellar spectra 498The HertzsprungRussell diagram 499Types of stars 500

E3 Stellar objects SL and HL 506The parallax method 506Absolute and apparent magnitudes 507Spectroscopic parallax 510The Cepheids 511



E4 Cosmology SL and HL 514Olbers paradox 514The expanding universe 516The Big Bang: the creation of space

and time 517The development of the universe 517

E5 Stellar evolution Extension HL 521Nucleosynthesis 521Evolutionary paths and stellar

processes 525Pulsars and quasars 529

E6 Galaxies Extension HL 533Types of galaxy 533Galactic motion 536Hubbles law 537The evolution of the

universe 539

SL and HL Option F Communications

F1 Radio communication SL and HL 544

Modulation 544Amplitude modulation (AM) 546Frequency modulation (FM) 548Comparing AM and FM 550The AM radio receiver 550

F2 Analogue and digital signals SL and HL 554

Binary numbers 554Analogue and digital signals 554Bit rate of a digital signal 554The transmission and reception of

digital signals 557Time division multiplexing 558The advantages of digital

communication 558

F3 Optic fibre transmission SL and HL 562

Total internal reflection 562Optical fibres 564Dispersion 565Attenuation 567Detection 569Noise 570Regeneration 571

Contents vii

F4 Channels of communication SL and HL 574

Copper wires 574Wire pairs 575Coaxial cables 575Optic fibres 575Radio waves 576Microwave transmission through

free space 577Satellite communications 578

F5 Electronics Extension HL 584The operational amplifier (op-amp) 584The non-inverting amplifier 588Reshaping digital pulses the

Schmitt trigger 590

F6 The mobile phone system Extension HL 596

SL and HL Option G Electromagnetic waves

G1 Light SL and HL 598The speed of light 598Electromagnetic waves 599Properties of EM waves 601The laser 603

G2 Optical instruments SL and HL 607Lenses 607Optical instruments 617Lens aberrations 620

G3 Interference and diffraction SL and HL 624

Two-source interference 624Youngs two-slit experiment 626Intensity in two-slit interference 628Multiple-slit diffraction 629The diffraction grating 631

G4 X-rays Extension HL 634The production of X-rays 634X-ray diffraction 635

G5 Thin-film interference Extension HL 640

Parallel films 640Thin air wedges 641Measuring small distances 642



HL Option H Special and general relativity

H1 The principle of special relativity HL 644

Frames of reference 645The speed of light 647The principle of special relativity 648

H2 The effects of special relativity HL 652Time dilation 652Length contraction 657Addition of velocities 659

H3 Consequences of and evidence for special relativity HL 663

Relativistic energy 663Evidence for special relativity 666The MichelsonMorley experiment 667The constancy of the speed of

light 669

H4 Relativistic mechanics HL 671Momentum and energy

(momenergy) 671A point about units 673A free electron cannot absorb

(or emit) a photon 674

H5 General relativity HL 677The principle of equivalence 677The tests of general relativity 680The structure of the theory 682Black holes 683Time dilation in general

relativity 685

HL Option I Biomedical physics

I1 The functioning of the ear HL 690The ear 690Intensity of sound 693Hearing defects 697

I2 Medical imaging HL 700Properties of radiation 700X-ray imaging 702Other imaging techniques 705Ultrasound 707Diagnostic uses of radioactive

sources 709

I3 Radiation in medicine HL 712Biological effects of radiation

and dosimetry 712Radiation therapy 716

HL Option J Particle physics

J1 Particles and interactions HL 718Particles and antiparticles 718The elementary particles 719Quantum numbers 720Antiparticles 720Spin 721The Heisenberg uncertainty

principle for time and energy 722Virtual particles 723Interactions and exchange particles 723Feynman diagrams 725The range of an interaction 728

J2 Detectors and accelerators HL 731The need for high energies 731Resolution 732Accelerators 733Synchrotron radiation 738Available energy 738Detectors 740

J3 Quarks and leptons HL 746Hadrons baryons and mesons 746Baryon number 748Strangeness 748The spin of hadrons 749Colour 749Gluons 750Confinement 751The interaction between nucleons 752Quarks, leptons and the standard

model 752Lepton number 753The Higgs particle 753

J4 Experimental evidence for the standard model HL 758

Gell-Manns prediction of the omega-minus 758

The direct evidence for quarks 758Asymptotic freedom 761The discovery of the Z0 and

neutral currents 761

viii Contents



J5 Cosmology and strings HL 764The Boltzmann equation 764Matter and antimatter 765Strings 766

Physics and the theory of knowledge (TOK) SL and HL 770

Appendices

1 Physical constants 777

2 Masses of elements and selected isotopes 778

3 Astronomical data 780

4 Some important mathematical results 781

5 Nobel prize winners in physics 782

Answers to questions 788

Glossary of selected terms 819

Index 832

Contents ix



Core and AHLPart ICore and AHL


Cambridge University Press978-0-521-13821-5 - Physics for the IB Diploma, Fifth EditionK. A. TsokosExcerptMore information

CHAPTE R 1.1 Core Physics and physical measurement

The realm of physics

The point of this exercise is that, first, we needunits to express the magnitude of physicalquantities. We must have a consistent set ofunits we all agree upon. One such set is theInternational System (SI system), which hasseven basic or fundamental units. The unitsof all other physical quantities are combinationsof these seven. These units are presented laterin this section. The second point is that wehave been able to answer a fairly complicatedsounding question without too much detailedknowledge a few simplifying assumptionsand general knowledge have been enough. Thethird point you may already have experienced.How close was your guess for the number ofmolecules in the sun? By how much did yourexponent differ from 56? Many of you will haveguessed a number around 101000 and that is wayoff. The number 101000 is a huge number youcannot find anything real to associate withsuch a number. The mass of the universe is about1053 kg and so repeating the calculation above wefind that the number of hydrogen molecules inthe entire universe (assuming it is all hydrogen)is about 1079 a big number to be sure butnowhere near 101000. Part of learning physics is

Physics is an experimental science in which measurements made must be expressed in

units. In the International System of units used throughout this book, the SI system, there

are seven fundamental units, which are dened in this chapter. All quantities are expressed

in terms of these units directly or as a combination of them.

Objec t i vesBy the end of this chapter you should be able to: appreciate the order of magnitude of various quantities; perform simple order-of-magnitude calculations mentally; state the fundamental units of the SI system.

Orders of magnitude and unitsHow many molecules are there in the sun? Thismay sound like a very difficult question withwhich to start a physics textbook, but very basicphysics can give us the answer. Before we try towork out the answer, guess what you think theanswer is by giving a power of 10. The numberof molecules in the sun is 10 to the power . . . ?

To answer the question we must first have anidea of the mass of the sun. You may know this,or you can easily look it up (to save you doingthis for this example, we can tell you that it isabout 1030 kg). Next, you will need to knowwhat the chemical composition of the sun is. Itis made up of 75% hydrogen and 25% helium,but as we are only making a rough estimate, wemay assume that it is made out of hydrogenentirely. The molar mass of hydrogen is 2 gmol1 and so the sun contains 1033/2 mol 5 1032 mol. The number of molecules in one moleof any substance is given by the Avogadroconstant, which is about 6 1023, so the sunhas around 5 1032 6 1023 3 1056

molecules. How close was your guess?



to appreciate the magnitude of things whetherthey are masses, times, distances, forces or justpure numbers such as the number of hydrogenmolecules in the universe. Hopefully, you will beable to do that after finishing this course.

The SI systemThe seven basic SI units are:

1 The metre (m). This is the unit of distance. It isthe distance travelled by light in a vacuum in atime of 1/299 792 458 seconds.

2 The kilogram (kg). This is the unit of mass. It isthe mass of a certain quantity of a platinumiridium alloy kept at the Bureau Internationaldes Poids et Mesures in France.

3 The second (s). This is the unit of time. A second isthe duration of 9 192 631 770 full oscillations ofthe electromagnetic radiation emitted in atransition between the two hyperfine energylevels in the ground state of a caesium-133 atom.

4 The ampere (A). This is the unit of electriccurrent. It is defined as that current which,when flowing in two parallel conductors 1 mapart, produces a force of 2 107 N on alength of 1 m of the conductors.

5 The kelvin (K). This is the unit of temperature. Itis 1273 .16 of the thermodynamic temperature ofthe triple point of water.

6 The mole (mol). One mole of a substancecontains as many molecules as there are atomsin 12 g of carbon-12. This special number ofmolecules is called Avogadros number and isapproximately 6.02 1023 .

7 The candela (cd). This is a unit of luminous inten-sity. It is the intensity of a source of frequency5.40 1014 Hz emitting 1683 W per steradian.

The details of these definitions should not bememorized.

In this book we will use all of the basic unitsexcept the last one. Some of these definitionsprobably do not make sense right now buteventually they will.

Physical quantities other than those above haveunits that are combinations of the seven

fundamental units. They have derived units. Forexample, speed has units of distance over time,metres per second (i.e. m/s or, preferably, m s1).Acceleration has units of metres per secondsquared (i.e. m/s2, which we write as m s2). Inother words, we treat the symbols for units asalgebraic quantities. Similarly, the unit of forceis the newton (N). It equals the combinationkg m s2. Energy, a very important quantity inphysics, has the joule (J) as its unit. The joule isthe combination N m and so equals (kg m s2 m),or kg m2 s2. The quantity power has units ofenergy per unit of time and so is measured inJ s1. This combination is called a watt. Thus,1 W (1 N m s1) (1 kg m s2 m s1)

1 kg m2 s3.

Occasionally, small or large quantities can beexpressed in terms of units that are related tothe basic ones by powers of 10. Thus, ananometre (symbol nm) is 109 m, a microgram(g) is 106 g = 109 kg, a gigaelectron volt(GeV) equals 109 eV, etc. The most commonprefixes are given in Table 1.1.

1.1 The realm of physics 3

Power Prefix Symbol Power Prefix Symbol

1018 atto- a 101 deka- da*

1015 femto- f 102 hecto- h*

1012 pico- p 103 kilo- k

109 nano- n 106 mega- M

106 micro- 109 giga- G

103 milli- m 1012 tera- T

102 centi- c 1015 peta- P*

101 deci- d 1018 exa- E*

*Rarely used.

Table 1.1 Common prefixes.

When we write an equation in physics, we haveto make sure that the units of the quantity onthe left-hand side of the equation are the sameas the units on the right-hand side. If the unitsdo not match, the equation cannot be right. Forexample, the period T (a quantity with units oftime) of a pendulum is related to the lengthof the pendulum l (a quantity with units of



length) and the acceleration due to gravity g(units of acceleration) through

T = 2

lg

The units on the right-hand side must reduce tounits of time. Indeed, the right-hand side unitsare

m

ms2=

s2 = s

as required (note that 2 is a dimensionlessconstant). The fact that the units on both sidesof an equation must match actually offers apowerful method for guessing equations.

For example, the velocity of a wave on a string isrelated to the length l and mass m of the string,and the tension force F the string is subjected to.How exactly does the velocity depend on thesethree variables? One guess is to write

v = c F x l ymz

where c is a numerical constant (a pure numberwithout units) and x, y and z are numbers to bedetermined. There could be some confusionhere because m stands for mass but we also usethe symbol m for the metre. To avoid this wewill use the notation [M] to stand for the unit ofmass, [L] for the unit of length, [T] for the unitof time, etc. Then, looking at the units of thelast equation we have that

[L ][T ]

= ([M][L ][T ]2)x [L ] y [M]z[L ][T ]1 = [M]x+z [L ]x+y [T ]2x

The two equations match if the exponents of [L],[M] and [T] match that is, if

x + z = 0x + y = 12x = 1

These equations imply that

x = 12, y = 1

2and z = 1

2

In other words, the original formula becomes

v = c F 1/2 l1/2 m1/2 = c

F lm

Obviously this method cannot give the value ofthe dimensionless constant c. To do that wehave to learn some physics!

Tables 1.21.4 give approximate values for someinteresting sizes, masses and time intervals.

Expressing a quantity as a plain power of 10gives what is called the order of magnitude ofthat quantity. Thus, the mass of the universe

4 Core Physics and physical measurement

Length/m

Distance to edge of observable universe 1026

Distance to the Andromeda galaxy 1022

Diameter of the Milky Way galaxy 1021

Distance to nearest star 1016

Diameter of solar system 1013

Distance to sun 1011

Radius of the earth 107

Size of a cell 105

Size of a hydrogen atom 1010

Size of a nucleus 1015

Size of a proton 1017

Planck length 1035

Table 1.2 Some interesting sizes.

Mass/kg

The universe 1053

The Milky Way galaxy 1041

The sun 1030

The earth 1024

Boeing 747 (empty) 105

An apple 0.25

A raindrop 106

A bacterium 1015

Smallest virus 1021

A hydrogen atom 1027

An electron 1030

Table 1.3 Some interesting masses.



has an order of magnitude of 1053 kg and themass of the Milky Way galaxy has an order ofmagnitude of 1041 kg. The ratio of the twomasses is then simply 1012.

Fundamental interactionsThere are four basic or fundamentalinteractions in physics. However, in 1972, theelectromagnetic and weak interactions wereunified into one the electroweak interaction.In this sense, then, we may speak of just threefundamental interactions (see Figure 1.1).

Q1How many grains of sand are required to ll theearth? (This is a classic problem that goes back toAristotle.)

Answer

The radius of the earth is about 6400 km, which wemay approximate to 10 000 km. The volume of theearth is thus approximately 8 (10 106)3 m3 8 1021 m3. We are assuming a cubical earth ofside equal to twice the radius. This is a simplifyingassumption. The true volume is 43R

3 = 1.1 1021 m3, which agrees with our estimate (we areonly interested in the power of 10 not the numberin front). The diameter of a grain of sand varies ofcourse but we will take 1 mm as a fair estimate.Then the number of grains of sand required to llthe earth is

8 1021 m3(1 103)3 m3 = 8 10

30 1031

Q2Estimate the speed with which human hair grows.

Answer

I cut my hair every 2 months and the barber cutsa length of about 2 cm. The speed is thus

2 1022 30 24 60 60 m s

1 102

3 2 36 104

106

6 40 =106

240 4 109 m s1

Q3If all the people on earth were to hold hands in astraight line, how long would the line be? Howmany times would it wrap around the earth?

Answer

Assume that each person has his or her handsstretched out to a distance of 1.5 m and that thepopulation of earth is 6 109 people. Then thelength would be 6 109 1.5 m = 9 109 m.The circumference of the earth is 2R 66 106 m 4 107 m and so the line wouldwrap 910

9

4107 200 times around the equator.


Time/s

Age of the universe 1017

Age of the earth 1017

Time of travel by light to nearby star 108

One year 107

One day 105

Period of a heartbeat 1

Period of red light 1015

Time of passage of light across a nucleus 1024

Planck time 1043

Table 1.4 Some interesting time intervals.

Example questions

Let us close this chapter with a few problemssimilar to the one we started with. Theseproblems are sometimes known as Fermiproblems, after the great physicist Enrico Fermi,who was a master in this kind of estimation.

Figure 1.1 The fundamental interactions ofphysics. Since 1972, the electromagnetic andweak interactions have been shown to be part ofa generalized interaction called the electroweakinteraction.

Gravitational Gravitational

ElectromagneticElectroweak

Weak (nuclear force)

Colour (or strong nuclear force) Colour



Q4How many revolutions do the wheels of a carmake before it is junked?

Answer

We assume that the car runs 250 000 km beforeit is junked and that the wheels have a radius of30 cm. Then the number of revolutions is

2.5 1082 0.3

2.52 1 10

8 108

Q5What depth of car tyre wears off with each turn?(This is another classic problem.)

Answer

We assume that a depth of 5 mm wears off every60 000 km. (These numbers are standard forpeople who own cars.) Then, for a wheel ofradius 30 cm the number of revolutions is (seeprevious problem) 6 10

7

2 0.3 62 1 107 3 107and so the wear per revolution is 5

3 107 mm/rev 107 mm/rev.

8 The molar mass of water is 18 g mol1. Howmany molecules of water are there in a glassof water (of volume 0.3 L)?

9 Assuming that the mass of a person is madeup entirely of water, how many molecules arethere in a human body (of mass 60 kg)?

10 Assuming the entire universe to be made up ofhydrogen gas, how many molecules ofhydrogen are there?

11 Give an order-of-magnitude estimate of thedensity of a proton.

12 How long does light from the sun take toarrive on earth?

13 How many apples do you need to make upthe mass of an average elephant?

14 How many bricks are used to build anaverage two-storey family house?

15 (a) How many metres are there in 5.356 nm?(b) How many in 1.2 fm?(c) How many in 3.4 mm?

16 (a) How many joules of energy are there in4.834 MJ?

(b) How many in 2.23 pJ?(c) How many in 364 GJ?

17 (a) How many seconds are there in 4.76 ns? (b) How many in 24.0 ms? (c) How many in 8.5 as?

18 What is the velocity of an electron that coversa distance of 15.68 mm in 87.50 ns?

19 An electron volt (eV) is a unit of energy equalto 1.6 1019 J. An electron has a kineticenergy of 2.5 eV. (a) How many joules is that?(b) What is the energy in eV of an electron

that has an energy of 8.6 1018 J?20 What is the volume in cubic metres of a cube

of side 2.8 cm?

21 What is the side in metres of a cube that has avolume of 588 cubic millimetres?

22 One inch is 2.54 cm and one foot has12 inches. The acceleration due to gravity isabout 9.8 m s2. What is it in feet per squaresecond?

23 One uid ounce is a volume of about2.96 105 m3. What is the side, in inches, of


Have a look through these questions and answerany that you can. However, dont worry aboutany you cant answer; leave them for now andcome back to them when you reach the end ofthe course.1 How long does light take to travel across a

proton?

2 How many hydrogen atoms does it take tomake up the mass of the earth?

3 What is the age of the universe expressed inunits of the Planck time?

4 What is the radius of the earth (6380 km)expressed in units of the Planck length?

5 How many heartbeats are there in the lifetimeof a person (75 years)?

6 What is the mass of our galaxy in terms of asolar mass?

7 What is the diameter of our galaxy in terms ofthe astronomical unit, i.e. the distancebetween the earth and the sun?

Q

QUESTIONS?



a cube whose volume is 125 uid ounces?(One inch is 2.54 cm.)

24 A horsepower (hp) is a unit of power equal toabout 746 W. What is the power in hp of a224 kW car engine?

25 Give an order-of-magnitude estimate for themass of: (a) an apple;(b) this physics book;(c) a soccer ball.

26 Give an order-of-magnitude estimate for thetime taken by light to travel across thediameter of the Milky Way galaxy.

27 A white dwarf star has a mass about that ofthe sun and a radius about that of the earth.Give an order-of-magnitude estimate of thedensity of a white dwarf.

28 A sports car accelerates from rest to 100 km perhour in 4.0 s. What fraction of the accelerationdue to gravity is the cars acceleration?

29 Give an order-of-magnitude estimate for thenumber of electrons in your body.

30 Give an order-of-magnitude estimate for thegravitational force of attraction between twopeople 1 m apart.

31 Give an order-of-magnitude estimate for theratio of the electric force between twoelectrons 1 m apart to the gravitational forcebetween the electrons.

32 The frequency f of oscillation (a quantity withunits of inverse seconds) of a mass m attachedto a spring of spring constant k (a quantitywith units of force per length) is related to mand k. By writing f = cmxky and matchingunits on both sides show that

f = c

km

, where c is a dimensionless

constant.

33 Without using a calculator estimate the valueof the following expressions and thencompare with the exact value using acalculator:

(a)24343

;

(b) 2.80 1.90;

(c)312 480

160;

(d)8.99 109 7 106 7 106

(8 102)2 ;

(e)6.6 1011 6 1024

(6.4 106)2 .




CHAPTE R 1.2 Core Physics and physical measurement

Uncertainties and errorsThis chapter introduces the basic methods of dealing with experimental error and

uncertainty in measured physical quantities. Physics is an experimental science and

often the experimenter will perform an experiment to test the prediction of a given

theory. No measurement will ever be completely accurate, however, and so the result of

the experiment will be presented with an experimental error. Thus, in comparing the

results of an experiment with the prediction of the theory being tested, the

experimenter will have to decide if the disagreement between theory and experiment

is due to failure of the theory or whether the disagreement falls within the bounds of

experimental error and so can be tolerated.

Objec t i vesBy the end of this chapter you should be able to: state the various types of errors that may arise in the measurement of a

physical quantity; state the difference between accuracy and precision; draw a line of best fit; appreciate the importance of significant digits.

repeated measurements, whereas errors thatare systematic cannot.

We may also consider a third class of errorscalled reading errors. This is a familiar type oferror that has to do with the fact that it is oftendifficult to read the instrument being usedwith absolute precision. This type of readingerror is inherent in the instrument being usedand cannot be improved upon by repeatedmeasurements. If a length is measured using aruler whose smallest division is a millimetreand the end of the object to be measured fallsin between two divisions on the ruler, it is easyto determine that the length is, say, between14.5 cm and 14.6 cm, but there is someguesswork involved in stating that the length is14.54 cm. It is standard practice to assume thatthe reading error is half the smallest division

Errors of measurementThere are two main types of error ofmeasurement or observation. They can begrouped into random and systematic even thoughin many cases it is not possible to sharplydistinguish between the two. We may say thatrandom errors are almost always the fault ofthe observer whereas systematic errors are dueto both the observer and the instrument beingused. In practice, all errors are a combination ofthe two.

A random error is characterized by the fact thatit is revealed by repeated measurements (themeasurements f luctuate about some value theyare sometimes larger and sometimes smaller)whereas a systematic error is not. Randomerrors can be reduced by averaging over



1.2 Uncertainties and errors 9

interval on the instrument. For the ruler, thisinterval is 1 mm, and half of this is 0.5 mm or0.05 cm. We may state the position of the rightend of the object we are measuring as(14.54 0.05) cm. In practice, though, tomeasure the length means also finding theposition of the left end of the object, and thereis a similar uncertainty in that measurement.Suppose that the left end is recorded at(1.00 0.05) cm. The length is then thedifference of the measurements for thepositions of the right and left ends of theobject, and this is 13.54 cm. As we will see later,the subtraction of the two measurementsimplies that the uncertainties will add, so thatwe end up with a length measurement that isuncertain by 0.1 cm. In that case it does notmake sense to quote the answer for the lengthto more than one decimal place, and we mayquote the length as (13.5 0.1) cm.For digital instruments we may take thereading error to be the smallest division thatthe instrument can read. So a stopwatch thatreads time to two decimal places, e.g. 25.38 s,will have a reading error of 0.01s, and aweighing scale that records a mass as 184.5 gwill have a reading error of 0.1g. The typicalreading errors for some common instrumentsare listed in Table 2.1.

error will tend to cancel. The experimentalresult of the measurement of a given quantity xwill thus be the average of the individualmeasurements, i.e.

x = x1 + x2 + + xNN

where N is the total number of measurements.We then define the deviation of each individualmeasurement from the average by xi = xi x .If the absolute magnitudes of all thesedeviations are smaller than the reading error,then we can quote the experimental result as

x reading errorHowever, if the deviations from the mean arelarger in magnitude than the reading error, theexperimental error in the quantity x will haveto include random errors as well. To estimatethe random error we calculate the quantity

e =

(x1)2 + (x2)2 + + (xN )2N 1

which is called the unbiased estimate of thestandard deviation of the N measurements xi .(With graphic calculators this can be done quiteeasily and quickly.) The result of the experimentis then expressed as

x eTo illustrate these points consider themeasurement of a length using a ruler. Thereading error according to one observer is 0.1 cm. The experimenter produces a table ofresults and, after computing the average of themeasurements, the deviation and its square arealso inserted in the table see Table 2.2.

Length/cm Deviation(0.1 cm) x/cm (x)2/cm2

14.88 0.09 0.0081

14.84 0.05 0.0025

15.02 0.23 0.0529

14.57 0.22 0.0484

14.76 0.03 0.0009

14.66 0.13 0.0169

Table 2.2 A table of results.

Instrument Reading error

Ruler 0.5 mm

Vernier calipers 0.05 mm

Micrometer 0.005 mm

Volumetric (measuring) cylinder 0.5 mL

Electronic weighing scale 0.1 g

Stopwatch 0.01 s

Table 2.1 Reading errors for some commoninstruments.

Random errorsIf a measurement is repeated many times, it canbe expected that the measurement will be toolarge as often as it will be too small. So, if anaverage of these measurements is taken, the




The average is 14.79 cm and the standarddeviation of these measurements is 0.1611 cm.The random error is larger than the reading errorand so it must be included in the result. Theresult of the measurement is thus expressed as14.8 0.2 cm. This is much more realistic thansimply quoting the average and the reading error,14.79 0.1 cm. Note that it does not make senseto quote the average to more than one decimalpoint as the error makes even the first decimalpoint uncertain. Note also that once a largenumber of measurements are accumulated,further measurements do not appreciably changethe estimate of the error. Thus, it is of little use totake, say, 50 measurements of the length of theobject in the example above.

Note also, finally, that even the calculation of astandard deviation is not all that necessary. Thelargest deviation from the mean in Table 2.2 is0.23 cm, which we may round to 0.2 cm andaccept that as a rough estimate of the error.

Systematic errorsThe most common source of a systematic erroris an incorrectly calibrated instrument. Forexample, consider a digital force sensor. Whenthe sensor is to be used for the first time, itmust be calibrated. This means that we mustapply a force whose value we are confident weknow, say 5.0 N, and then adjust the sensor sothat it too reads 5.0 N. If we apply the 5.0 N forceand then adjust the instrument to read 4.9 N,the instrument will be incorrectly calibrated. Itwill also be incorrectly calibrated if the sensor isadjusted to read 5.0 N when the known forcethat we apply is not really 5.0 N. If we use thissensor to verify Newtons second law, we would

expect to get a straight-line graph through theorigin if we plot the net force on the body versusits acceleration. Since all measurements of theforce will be off by the same amount, thestraight line will not pass through the origin.The systematic error in the force would then bethe vertical intercept (see Figure 2.1a).

A systematic error will also arise if we use aninstrument that has a zero error. For example, if anammeter shows a current of 0.1 A even before it isconnected to a circuit, it has a zero error. It mustbe adjusted to read zero. If the adjustment is notdone, every measurement of current made withthis ammeter will be larger than the true value ofthe current by 0.1 A. Thus, if this ammeter is usedto investigate the voltagecurrent characteristic ofan ohmic resistor, we will not get the expectedstraight line through the origin but a straight linethat misses the origin. The systematic error in thecurrent would then be the horizontal intercept(see Figure 2.1b).

Systematic errors are not always easy toestimate but sometimes the direction of theerror is. Thus, suppose that an experimenterassumes that no friction is present in anexperiment on an air track, where the velocityof an object sliding on it is measured afterhaving travelled a certain distance. A smallamount of friction will slow down the objectand so the velocity measurements will beconsistently lower than their true values. It isdifficult though to estimate by how much.

A systematic error will also arise if theexperimenter makes the same error for all themeasurements she takes. For example, considermeasuring a length with a ruler. The ruler is

aligned with the object to bemeasured and the experimentermust then position her eyedirectly above the ruler. If,however, the experimenterconsistently stands to the side,as shown in Figure 2.2, themeasured value will always belarger than the true length. Ifshe stands on the other side, the

Figure 2.1 The types of systematic error that arise from incorrectlycalibrated instruments and instruments that have a zero error.

F/N

a/m s2

observedexpected

I/A

V/V

expectedobserved

(a) (b)



physics_for_the_ib_diploma__fifth_edition____cambridge_education___cambridge_university_press_samples.pdf...

Documents

law core

errors core

uncertainties core

laws core

power core

scalars core

realm of physics core

linear momentum core