9781107697690 Sang, Woodside, Jones & Chadha: Physics (AS and A Level) CVR C M Y K Cambridge International AS and A Level Sang, Jones, Chadha and Woodside Cambridge International AS and A Level Physics Coursebook Physics Coursebook Second Edition David Sang, Graham Jones, Gurinder Chadha and Richard Woodside Cambridge International AS and A Level Physics Coursebook Second edition David Sang, Graham Jones, Gurinder Chadha and Richard Woodside “The worked examples are excellent and there are lots of them which is very helpful for the students” Jackie Robinson, Physics teacher, Spain “I like the explanations and derivations. Very clear.” Head of Science, Worcester, UK This revised and updated coursebook is tailored to the new AS and A Level Physics syllabus (9702) and is endorsed by Cambridge International Examinations. Features: • Self-assessment questions to test your progress. • Exam-style questions at the end of every chapter to thoroughly prepare for examinations. • Added focus on practical procedures and greater emphasis on real world applications and skills. • Detailed Worked Examples throughout illustrate how to tackle different question types. • Easy navigation with eye-catching and engaging Introductions and straightforward Summaries in every chapter. • Accessible language and globally relevant examples to make this book ideal for international learners. Bonus accompanying CD-ROM containing: • Answers to all of the questions in the book. • Advice about how to revise and how to approach examinations. • Lists of recommended resources such as further reading and web links which are ideal for further study and special projects. Also available: Teacher’s Resource CD-ROM ISBN 978-1-107-66300-8 Completely Cambridge – Cambridge resources for Cambridge qualif cations Cambridge International Examinations is the world’s largest provider of programmes and qualif cations for 5-19 year olds. Cambridge University Press is the oldest publishing house in the world, having been operating continuously since 1584, and is one of the largest academic publishers globally. Cambridge University Press works with Cambridge International Examinations and experienced authors to produce high-quality endorsed textbooks and software that support Cambridge Teachers and encourage Cambridge Learners. Visit education.cambridge.org/cie for information on our full range of Cambridge International AS and A Level titles including e-books and supporting digital resources. David Sang, Graham Jones, Gurinder Chadha and Richard Woodside Cambridge International AS and A Level Coursebook Second Edition Physics notice to teachers in the uk It is illegal to reproduce any part of this book in material form (including photocopying and electronic storage) except under the following circumstances: (i) where you are abiding by a licence granted to your school or institution by the Copyright Licensing Agency; (ii) where no such licence exists, or where you wish to exceed the terms of a licence, and you have gained the written permission of Cambridge University Press; (iii) where you are allowed to reproduce without permission under the provisions of Chapter 3 of the Copyright, Designs and Patents Act 1988, which covers, for example, the reproduction of short passages within certain types of educational anthology and reproduction for the purposes of setting examination questions. Example answers and all other end-of-chapter questions were written by the authors. 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Paperback with M for Windows and MAC iii Introduction vii How to use this book viii Chapter 1: Kinematics – describing motion 1 Speed 2 Distance and displacement, scalar and vector 4 Speed and velocity 5 Displacement–time graphs 6 Combining displacements 8 Combining velocities 10 Chapter 2: Accelerated motion 14 The meaning of acceleration 15 Calculating acceleration 15 Units of acceleration 16 Deducing acceleration 17 Deducing displacement 17 Measuring velocity and acceleration 18 Determining velocity and acceleration in the laboratory 18 The equations of motion 20 Deriving the equations of motion 22 Uniform and non-uniform acceleration 24 Acceleration caused by gravity 25 Determining g 25 Motion in two dimensions – projectiles 28 Understanding projectiles 29 Chapter 3: Dynamics – explaining motion 37 Calculating the acceleration 38 Understanding SI units 39 The pull of gravity 41 Mass and inertia 43 Top speed 44 Moving through fluids 45 Identifying forces 47 Newton’s third law of motion 49 Chapter 4: Forces – vectors and moments 53 Combining forces 54 Components of vectors 56 Centre of gravity 59 The turning effect of a force 59 The torque of a couple 63 Chapter 5: Work, energy and power 69 Doing work, transferring energy 71 Gravitational potential energy 75 Kinetic energy 76 g.p.e.–k.e. transformations 76 Down, up, down – energy changes 77 Energy transfers 78 Power 80 Chapter 6: Momentum 85 The idea of momentum 86 Modelling collisions 86 Understanding collisions 89 Explosions and crash-landings 91 Collisions in two dimensions 93 Momentum and Newton’s laws 95 Understanding motion 96 Chapter 7: Matter and materials 101 Density 102 Pressure 102 Compressive and tensile forces 104 Stretching materials 105 Elastic potential energy 108 Chapter 8: Electric fields 116 Attraction and repulsion 117 The concept of an electric field 118 Electric field strength 119 Force on a charge 122 Chapter 9: Electric current, potential difference and resistance 127 Circuit symbols and diagrams 128 Electric current 129 An equation for current 132 The meaning of voltage 134 Electrical resistance 135 Electrical power 136 Chapter 10: Kirchhoff’s laws 143 Kirchhoff’s first law 144 Kirchhoff’s second law 145 Applying Kirchhoff’s laws 146 Resistor combinations 148 Contents iv Cambridge International AS and A Level Physics P1: Practical skills at AS level 239 Practical work in physics 240 Using apparatus and following instructions 240 Gathering evidence 241 Precision, accuracy, errors and uncertainties 241 Finding the value of an uncertainty 243 Percentage uncertainty 245 Recording results 246 Analysing results 246 Testing a relationship 248 Identifying limitations in procedures and suggesting improvements 250 Chapter 17: Circular motion 258 Describing circular motion 259 Angles in radians 260 Steady speed, changing velocity 261 Angular velocity 261 Centripetal forces 262 Calculating acceleration and force 264 The origins of centripetal forces 265 Chapter 18: Gravitational fields 271 Representing a gravitational field 272 Gravitational field strength g 274 Energy in a gravitational field 276 Gravitational potential 276 Orbiting under gravity 277 The orbital period 278 Orbiting the Earth 279 Chapter 19: Oscillations 285 Free and forced oscillations 286 Observing oscillations 287 Describing oscillations 288 Simple harmonic motion 289 Representing s.h.m. graphically 291 Frequency and angular frequency 292 Equations of s.h.m. 293 Energy changes in s.h.m. 296 Damped oscillations 297 Resonance 299 Chapter 20: Communications systems 309 Radio waves 310 Analogue and digital signals 314 Channels of communication 317 Comparison of different channels 319 Chapter 11: Resistance and resistivity 156 The I–V characteristic for a metallic conductor 157 Ohm’s law 158 Resistance and temperature 159 Resistivity 162 Chapter 12: Practical circuits 168 Internal resistance 169 Potential dividers 172 Potentiometer circuits 172 Chapter 13: Waves 178 Describing waves 179 Longitudinal and transverse waves 181 Wave energy 182 Wave speed 183 The Doppler effect 184 Electromagnetic waves 185 Electromagnetic radiation 186 Orders of magnitude 187 The nature of electromagnetic waves 188 Chapter 14: Superposition of waves 192 The principle of superposition of waves 193 Diffraction of waves 194 Interference 196 The Young double-slit experiment 200 Diffraction gratings 203 Chapter 15: Stationary waves 210 From moving to stationary 211 Nodes and antinodes 212 Formation of stationary waves 212 Determining the wavelength and speed of sound 216 Chapter 16: Radioactivity 222 Looking inside the atom 223 Alpha-particle scattering and the nucleus 223 A simple model of the atom 225 Nucleons and electrons 226 Forces in the nucleus 229 Fundamental particles? 229 Families of particles 230 Discovering radioactivity 231 Radiation from radioactive substances 231 Discovering neutrinos 232 Fundamental families 232 Fundamental forces 232 Properties of ionising radiation 233 v Contents Chapter 27: Charged particles 422 Observing the force 423 Orbiting charges 423 Electric and magnetic fields 427 The Hall effect 428 Discovering the electron 429 Chapter 28: Electromagnetic induction 435 Observing induction 436 Explaining electromagnetic induction 437 Faraday’s law of electromagnetic induction 441 Lenz’s law 443 Using induction: eddy currents, generators and transformers 445 Chapter 29: Alternating currents 451 Sinusoidal current 452 Alternating voltages 453 Power and a.c. 455 Why use a.c. for electricity supply? 457 Transformers 458 Rectification 460 Chapter 30: Quantum physics 466 Modelling with particles and waves 467 Particulate nature of light 468 The photoelectric effect 471 Line spectra 475 Explaining the origin of line spectra 476 Photon energies 477 Electron energies in solids 478 The nature of light – waves or particles? 480 Electron waves 480 Chapter 31: Nuclear physics 489 Balanced equations 490 Mass and energy 491 Energy released in radioactive decay 494 Binding energy and stability 494 Randomness and decay 496 The mathematics of radioactive decay 497 Decay graphs and equations 499 Decay constant and half-life 501 Chapter 21: Thermal physics 327 Changes of state 328 Energy changes 329 Internal energy 331 The meaning of temperature 332 Thermometers 334 Calculating energy changes 336 Chapter 22: Ideal gases 345 Particles of a gas 346 Explaining pressure 348 Measuring gases 348 Boyle’s law 349 Changing temperature 350 Ideal gas equation 351 Modelling gases – the kinetic model 352 Temperature and molecular kinetic energy 354 Chapter 23: Coulomb’s law 359 Electric fields 360 Coulomb’s law 360 Electric field strength for a radial field 362 Electric potential 363 Comparing gravitational and electric fields 366 Chapter 24: Capacitance 372 Capacitors in use 373 Energy stored in a capacitor 375 Capacitors in parallel 377 Capacitors in series 378 Comparing capacitors and resistors 379 Capacitor networks 380 Chapter 25: Electronics 386 Components of an electronic sensing system 387 The operational amplifier (op-amp) 393 The inverting amplifier 397 The non-inverting amplifier 398 Output devices 398 Chapter 26: Magnetic fields and electromagnetism 406 Producing and representing magnetic fields 407 Magnetic force 409 Magnetic flux density 411 Measuring magnetic flux density 411 Currents crossing fields 413 Forces between currents 415 Relating SI units 416 Comparing forces in magnetic, electric and gravitational fields 417 vi Cambridge International AS and A Level Physics Chapter 32: Medical imaging 506 The nature and production of X-rays 507 X-ray attenuation 509 Improving X-ray images 511 Computerised axial tomography 513 Using ultrasound in medicine 516 Echo sounding 518 Ultrasound scanning 520 Magnetic resonance imaging 522 P2: Planning, analysis and evaluation 529 Planning 530 Analysis of the data 532 Treatment of uncertainties 536 Conclusions and evaluation of results 538 Appendix 1: Physical quantities and units 542 Prefixes 542 Estimation 542 Appendix 2: Data, formulae and relationships 543 Data 543 Conversion factors 543 Mathematical equations 544 Formulae and relationships 544 Appendix 3: The Periodic Table 545 Glossary 546 Index 555 Acknowledgements 564 Advice on how to revise for and approach examinations Introduction to the examination and changes to the syllabus Answers to self-assessment questions Answers to end-of-chapter questions Recommended resources CD1 CD9 CD12 CD70 CD136 CD-ROM CD1 Introduction This book covers the entire syllabus of Cambridge International Examinations AS and A Level Physics. It is designed to work with the syllabus that will be examined from 2016. It is in three parts: ■ ■ Chapters 1–16 and P1: the AS level content, covered in the first year of the course, including a chapter (P1) dedicated to the development of your practical skills ■ ■ Chapters 17–32 and P2: the remaining A level content, including a chapter (P2) dedicated to developing your ability to plan, analyse and evaluate practical investigations ■ ■ Appendices of useful formulae, a Glossary and an Index. The main tasks of a textbook like this are to explain the various concepts of physics that you need to understand and to provide you with questions that will help you to test your understanding and prepare for your examinations. You will find a visual guide to the structure of each chapter and the features of this book on the next two pages. When tackling questions, it is a good idea to make a first attempt without referring to the explanations in this Coursebook or to your notes. This will help to reveal any gaps in your understanding. By working out which concepts you find most challenging, and by spending more time to understand these concepts at an early stage, you will progress faster as the course continues. The CD-ROM that accompanies this Coursebook includes answers with workings for all the questions in the book, as well as suggestions for revising and preparing for any examinations you take. There are also lists of recommended further reading, which in many cases will take you beyond the requirements of the syllabus, but which will help you deepen your knowledge and explain more of the background to the physics concepts covered in this Coursebook. In your studies, you will find that certain key concepts come up again and again, and that these concepts form ‘themes’ that link the different areas of physics together. It will help you to progress and gain confidence in tackling problems if you take note of these themes. For this Coursebook, these key concepts include: ■ ■ Models of physical systems ■ ■ Testing predictions against evidence ■ ■ Mathematics as a language and problem-solving tool ■ ■ Matter, energy and waves ■ ■ Forces and fields In this Coursebook, the mathematics has been kept to the minimum required by the Cambridge International Examinations AS and A Level Physics syllabus. If you are also studying mathematics, you may find that more advanced techniques such as calculus will help you with many aspects of physics. Studying physics can be a stimulating and worthwhile experience. It is an international subject; no single country has a monopoly on the development of the ideas. It can be a rewarding exercise to discover how men and women from many countries have contributed to our knowledge and well-being, through their research into and application of the concepts of physics. We hope not only that this book will help you to succeed in your future studies and career, but also that it will stimulate your curiosity and fire your imagination. Today’s students become the next generation of physicists and engineers, and we hope that you will learn from the past to take physics to ever-greater heights. vii viii Cambridge International AS and A Level Physics Each chapter begins with a short list of the facts and concepts that are explained in it. There is a short context at the beginning of each chapter, containing an example of how the material covered in the chapter relates to the ‘real world’. Questions throughout the text give you a chance to check that you have understood the topic you have just read about. You can find the answers to these questions on the CD-ROM. Important equations and other facts are shown in highlight boxes. This book does not contain detailed instructions for doing particular experiments, but you will find background information about the practical work you need to do in these Boxes. There are also two chapters, P1 and P2, which provide detailed information about the practical skills you need to develop during your course. Learning outcomes You should be able to: ■ define displacement, speed and velocity ■ draw and interpret displacement–time graphs ■ describe laboratory methods for determining speed ■ use vector addition to add two or more vectors Chapter 1: Kinematics – describing motion Describing movement Our eyes are good at detecting movement. We notice even quite small movements out of the corners of our eyes. It’s important for us to be able to judge movement – think about crossing the road, cycling or driving, or catching a ball. Figure 1.1 shows a way in which movement can be recorded on a photograph. This is a stroboscopic photograph of a boy juggling three balls. As he juggles, a bright lamp flashes several times a second so that the camera records the positions of the balls at equal intervals of time. If we knew the time between flashes, we could measure the photograph and calculate the speed of a ball as it moves through the air. Figure 1.1 This boy is juggling three balls. A stroboscopic lamp flashes at regular intervals; the camera is moved to one side at a steady rate to show separate images of the boy. Figure 13.3 or a similar graph of displacement against time illustrates the following important definitions about waves and wave motion: ■ ■ The distance of a point on the wave from its undisturbed position or equilibrium position is called the displacement x. ■ ■ The maximum displacement of any point on the wave from its undisturbed position is called the amplitude A. The amplitude of a wave on the sea is measured in units of distance, e.g. metres. The greater the amplitude of the wave, the louder the sound or the rougher the sea! ■ ■ The distance from any point on a wave to the next exactly similar point (e.g. crest to crest) is called the wavelength λ (the Greek letter lambda). The wavelength of a wave on the sea is measured in units of distance, e.g. metres. ■ ■ The time taken for one complete oscillation of a point in a wave is called the period T. It is the time taken for a point to move from one particular position and return to that same position, moving in the same direction. It is measured in units of time, e.g. seconds.. ■ ■ The number of oscillations per unit time of a point in a wave is called its frequency f. For sound waves, the higher the frequency of a musical note, the higher is its pitch. Frequency is measured in hertz (Hz), where 1 Hz = one oscillation per second (1 kHz = 103 Hz and 1 MHz = 106 Hz). The frequency f of a wave is the reciprocal of the period T: f = 1 T Waves are called mechanical waves if they need a substance (medium) through which to travel. Sound is one example of such a wave. Other cases are waves on strings, seismic waves and water waves (Figure 13.4). Some properties of typical waves are given on page 183 in Table 13.1. Figure 13.4 The impact of a droplet on the surface of a liquid creates a vibration, which in turn gives rise to waves on the surface. 1 Determine the wavelength and amplitude of each of the two waves shown in Figure 13.5. Displacement / cm 6 4 2 –2 0 –4 –6 a b 5 10 15 20 25 30 35 Distance / cm Figure 13.5 Two waves – for Question 1. BOX 13.1: Measuring frequency You can measure the frequency of sound waves using a cathode-ray oscilloscope (c.r.o.). Figure 13.6 shows how. A microphone is connected to the input of the c.r.o. Sound waves are captured by the microphone and converted into a varying voltage which has the same frequency as the sound waves. This voltage is displayed on the c.r.o. screen. It is best to think of a c.r.o. as a voltmeter which is capable of displaying a rapidly varying voltage. To do this, its spot moves across the screen at a steady speed, set by the time-base control. At the same time, the spot moves up and down according to the voltage of the input. Hence the display on the screen is a graph of the varying voltage, with time on the (horizontal) x-axis. If we know the horizontal scale, we can determine the period and hence the frequency of the sound wave. Worked example 1 shows how to do this. (In Chapter 15 we will look at one method of measuring the wavelength of sound waves.) Figure 13.6 Measuring the frequency of sound waves from a tuning fork. QUESTION 76 g.p.e.–k.e. transformations A motor drags the roller-coaster car to the top of the first hill. The car runs down the other side, picking up speed as it goes (see Figure 5.12). It is moving just fast enough to reach the top of the second hill, slightly lower than the first. It accelerates downhill again. Everybody screams! The motor provides a force to pull the roller-coaster car to the top of the hill. It transfers energy to the car. But where is this energy when the car is waiting at the top of the hill? The car now has gravitational potential energy; as soon as it is given a small push to set it moving, it accelerates. It gains kinetic energy and at the same time it loses g.p.e. Kinetic energy As well as lifting an object, a force can make it accelerate. Again, work is done by the force and energy is transferred to the object. In this case, we say that it has gained kinetic energy, Ek. The faster an object is moving, the greater its kinetic energy (k.e.). For an object of mass m travelling at a speed v, we have: kinetic energy = 1 2 × mass × speed2 Ek = 1 2 mv2 Deriving the formula for kinetic energy The equation for k.e., Ek = 1 2mv2, is related to one of the equations of motion. We imagine a car being accelerated from rest (u = 0) to velocity v. To give it acceleration a, it is pushed by a force F for a distance s. Since u = 0, we can write the equation v2 = u2 + 2as as: v2 = 2as Multiplying both sides by 1 2m gives: 1 2 mv2 = mas Now, ma is the force F accelerating the car, and mas is the force × the distance it moves, that is, the work done by the force. So we have: 1 2mv 2 = work done by force F This is the energy transferred to the car, and hence its kinetic energy. 3 Calculate the increase in kinetic energy of a car of mass 800 kg when it accelerates from 20 m s−1 to 30 m s−1. Step 1 Calculate the initial k.e. of the car: Ek = 1 2 mv2 = 1 2 × 800 × (20)2 = 160 000 J = 160 kJ Step 2 Calculate the final k.e. of the car: Ek = 1 2 mv2 = 1 2 × 800 × (30)2 = 360 000 J = 360 kJ Step 3 Calculate the change in the car’s k.e.: change in k.e. = 360 − 160 = 200 kJ Hint: Take care! You can’t calculate the change in k.e. by squaring the change in speed. In this example, the change in speed is 10 m s−1, and this would give an incorrect value for the change in k.e. 7 Calculate how much gravitational potential energy is gained if you climb a flight of stairs. Assume that you have a mass of 52 kg and that the height you lift yourself is 2.5 m. 8 A climber of mass 100 kg (including the equipment she is carrying) ascends from sea level to the top of a mountain 5500 m high. Calculate the change in her gravitational potential energy. 9 a A toy car works by means of a stretched rubber band. What form of potential energy does the car store when the band is stretched? b A bar magnet is lying with its north pole next to the south pole of another bar magnet. A student pulls them apart. Why do we say that the magnets’ potential energy has increased? Where has this energy come from? 10 Which has more k.e., a car of mass 500 kg travelling at 15 m s−1 or a motorcycle of mass 250 kg travelling at 30 m s−1? 11 Calculate the change in kinetic energy of a ball of mass 200 g when it bounces. Assume that it hits the ground with a speed of 15.8 m s−1 and leaves it at 12.2 m s−1. QUESTIONS QUESTIONS WORKED EXAMPLE How to use this book The text and illustrations describe and explain all of the facts and concepts that you need to know. The chapters, and oft en the content within them as well, are arranged in a similar sequence to your syllabus, but with AS and A Level content clearly separated into the two halves of the book. ix How to use this book Wherever you need to know how to use a formula to carry out a calculation, there are worked example boxes to show you how to do this. Key words are highlighted in the text when they are first introduced. You will also find definitions of these words in the Glossary. There is a summary of key points at the end of each chapter. You might find this helpful when you are revising. Questions at the end of each chapter begin with shorter answer questions, then move on to more demanding exam-style questions, some of which may require use of knowledge from previous chapters. Answers to these questions can be found on the CD–ROM. Summary ■ Forces are vector quantities that can be added by means of a vector triangle. Their resultant can be determined using trigonometry or by scale drawing. ■ Vectors such as forces can be resolved into components. Components at right angles to one another can be treated independently of one another. For a force F at an angle θ to the x-direction, the components are: x-direction: F cos θ y-direction: F sin θ ■ The moment of a force = force × perpendicular distance of the pivot from the line of action of the force. ■ The principle of moments states that, for any object that is in equilibrium, the sum of the clockwise moments about any point provided by the forces acting on the object equals the sum of the anticlockwise moments about that same point. ■ A couple is a pair of equal, parallel but opposite forces whose ef ect is to produce a turning ef ect on a body without giving it linear acceleration. torque of a couple = one of the forces × perpendicular distance between the forces ■ For an object to be in equilibrium, the resultant force acting on the object must be zero and the resultant moment must be zero. 40° B A 40° Cambridge International AS Level Physics AS Level Physics 1 In Figure 6.5, trolley A of mass 0.80 kg travelling at a velocity of 3.0 m s−1 collides head-on with a stationary trolley B. Trolley B has twice the mass of trolley A. The trolleys stick together and have a common velocity of 1.0 m s−1 after the collision. Show that momentum is conserved in this collision. Step 1 Make a sketch using the information given in the question. Notice that we need two diagrams to show the situations, one before and one after the collision. Similarly, we need two calculations – one for the momentum of the trolleys before the collision and one for their momentum after the collision. Step 2 Calculate the momentum before the collision: momentum of trolleys before collision = mA × uA + mB × uB = (0.80 × 3.0) + 0 = 2.4 kg m s−1 Trolley B has no momentum before the collision, because it is not moving. Step 3 Calculate the momentum after the collision: momentum of trolleys after collision = (mA + mB) × vA+B = (0.80 + 1.60) × 1.0 = 2.4 kg m s−1 So, both before and after the collision, the trolleys have a combined momentum of 2.4 kg m s−1. Momentum has been conserved. uA = 3.0 m s–1 uB = 0 vA+B = 1.0 m s–1 0.80 kg 0.80 kg 0.80kg positive direction before afer A B A B 0.80 kg 0.80 kg 0.80kg Figure 6.5 The state of trolleys A and B, before and after the collision. 2 Calculate the momentum of each of the following objects: a a 0.50 kg stone travelling at a velocity of 20 m s−1 b a 25 000 kg bus travelling at 20 m s−1 on a road c an electron travelling at 2.0 × 107 m s−1. (The mass of the electron is 9.1 × 10−31 kg.) 3 Two balls, each of mass 0.50 kg, collide as shown in Figure 6.6. Show that their total momentum before the collision is equal to their total momentum after the collision. Figure 6.6 For Question 3. QUESTIONS WORKED EXAMPLE 219 Chapter 15: Stationary waves End-of-chapter questions 1 Figure 15.19 shows a stationary wave on a string. Figure 15.19 For End-of-chapter Question 1. a On a copy of Figure 15.19, label one node (N) and one antinode (A). [1] b Mark on your diagram the wavelength of the standing wave and label it λ. [1] c The frequency of the vibrator is doubled. Describe the changes in the standing wave pattern. [1] 2 A tuning fork which produces a note of 256 Hz is placed above a tube which is nearly filled with water. The water level is lowered until resonance is first heard. a Explain what is meant by the term resonance. [1] b The length of the column of air above the water when resonance is first heard is 31.2 cm. Calculate the speed of the sound wave. [2] 3 a State two similarities and two dif erences between progressive waves and stationary waves. [4] b Figure 15.20 shows an experiment to measure the speed of a sound in a string. The frequency of the vibrator is adjusted until the standing wave shown in Figure 15.20 is formed. Figure 15.20 For End-of-chapter Question 3. i On a copy of the diagram, mark a node (label it N) and an antinode (label it A). [2] ii The frequency of the vibrator is 120 Hz. Calculate the speed at which a progressive wave would travel along the string. [3] c The experiment is now repeated with the load on the string halved. In order to get a similar standing wave the frequency has to be decreased to 30 Hz. Explain, in terms of the speed of the wave in the string, why the frequency must be adjusted. [2] vibrator AS Level Physics 40 Other SI units Using only seven base units means that only this number of quantities have to be defined with great precision. There would be confusion and possible contradiction if more units were also defined. For example, if the density of water were defined as exactly 1 g cm−3, then 1000 cm3 of a sample of water would have a mass of exactly 1 kg. However, it is unlikely that the mass of this volume of water would equal exactly the mass of the standard kilogram. The standard kilogram, which is kept in France, is the one standard from which all masses can ultimately be measured. All other units can be derived from the base units. This is done using the definition of the quantity. For example, speed is defined as distance time , and so the base units of speed in the SI system are m s−1. Since the defining equation for force is F = ma, the base units for force are kg m s−2. Equations that relate different quantities must have the same base units on each side of the equation. If this does not happen the equation must be wrong. When each term in an equation has the same base units the equation is said to be homogeneous. Base units, derived units The metre, kilogram and second are three of the seven SI base units. These are defined with great precision so that every standards laboratory can reproduce them correctly. Other units, such as units of speed (m s−1) and acceleration (m s−2) are known as derived units because they are combinations of base units. Some derived units, such as the newton and the joule, have special names which are more convenient to use than giving them in terms of base units. The definition of the newton will show you how this works. Defining the newton Isaac Newton (1642–1727) played a significant part in developing the scientific idea of force. Building on Galileo’s earlier thinking, he explained the relationship between force, mass and acceleration, which we now write as F = ma. For this reason, the SI unit of force is named after him. We can use the equation F = ma to define the newton (N). One newton is the force that will give a 1 kg mass an acceleration of 1 m s−2 in the direction of the force. 1 N = 1 kg × 1 m s−2 or 1 N = 1 kg m s−2 The seven base units In mechanics (the study of forces and motion), the units we use are based on three base units: the metre, kilogram and second. As we move into studying electricity, we will need to add another base unit, the ampere. Heat requires another base unit, the kelvin (the unit of temperature). Table 3.2 shows the seven base units of the SI system. Remember that all other units can be derived from these seven. The equations that relate them are the equations that you will learn as you go along (just as F = ma relates the newton to the kilogram, metre and second). The unit of luminous intensity is not part of the A/AS course. Base unit Symbol Base unit length x, l, s etc. m (metre) mass m kg (kilogram) time t s (second) electric current I A (ampere) thermodynamic temperature T K (kelvin) amount of substance n mol (mole) luminous intensity I cd (candela) Table 3.2 SI base quantities and units. In this course, you will learn about all of these except the candela. 4 The pull of the Earth’s gravity on an apple (its weight) is about 1 newton. We could devise a new international system of units by defining our unit of force as the weight of an apple. State as many reasons as you can why this would not be a very useful definition. 5 Determine the base units of: a pressure ( = force area ) b energy ( = force × distance ) c density ( = mass volume ) 6 Use base units to prove that the following equations are homogeneous. a pressure = density × acceleration due to gravity × depth b distance travelled = initial speed × time + 1 2 acceleration × time2 (s = ut + 1 2 at2) QUESTIONS QUESTION 546 absolute scale of temperature; see thermodynamic scale. absolute zero The temperature at which a system has minimum internal energy; equivalent to −273.15°C. absorption line spectrum A dark line of a unique wavelength seen in a continuous spectrum. acceleration The rate of change of an object’s velocity: a = Δv Δt Unit: ms−2. accuracy An accurate value of a measured quantity is one which is close to the true value of the quantity. acoustic impedance Acoustic impedance Z is the product of the density ρ of a substance and the speed c of sound in that substance (Z=ρc). Unit: kgm−2s−1. activity The rate of decay or disintegration of nuclei in a radioactive sample. ampere The SI unit of electric current (abbreviated A). amplitude The maximum displacement of a particle from its equilibrium position. amplitude modulation A form of modulation in which the signal causes variations in the amplitude of a carrier wave. analogue signal A signal that is continuously variable, having a continuum of possible values. analogue-to-digital conversion (ADC) Conversion of a continuous analogue signal to discrete digital numbers. angular displacement The angle through which an object moves in a circle. angular frequency The frequency of a sinusoidal oscillation expressed in radians per second: angular frequency ω = 2π T angular velocity The rate of change of the angular position of an object as it moves along a curved path. antinode A point on a stationary wave with maximum amplitude. atomic mass unit A unit of mass (symbol u) approximately equal to 1.661×10−27kg. The mass of an atom of 12 6 C = 12.000u exactly. attenuation The gradual loss in strength or intensity of a signal. average speed The total distance travelled by an object divided by the total time taken. Avogadro constant The number of particles in one mole of any substance approximately (6.02×1023 mol−1), denoted NA. band theory The idea that electrons in a solid or liquid can have energies within certain ranges or bands, between which are forbidden values. bandwidth (communications) A measure of the width of a range of frequencies being transmitted. base units Defined units of the SI system from which all other units are derived. best fit line A straight line drawn as closely as possible to the points of a graph so that similar numbers of points lie above and below the line. binding energy The minimum external energy required to separate all the neutrons and protons of a nucleus. bit A basic unit of information storage, the amount of information stored by a device that exists in only two distinct states, usually given as the binary digits 0 and 1. Boltzmann constant A fundamental constant given by k = R NA , where R is the ideal gas constant and NA is the Avogadro constant. Boyle’s law The pressure exerted by a fixed mass of gas is inversely proportional to its volume, provided the temperature of the gas remains constant. braking radiation X-rays produced when electrons are decelerated (also called Bremsstrahlung radiation). capacitance The ratio of charge stored by a capacitor to the potential difference across it. carrier wave A waveform (usually sinusoidal) which is modulated by an input signal to carry information. centre of gravity The point where the entire weight of an object appears to act. centripetal force The resultant force acting on an object moving in a circle; it is always directed towards the centre of the circle. characteristic radiation Very intense X-rays produced in an X-ray tube, having specific wavelengths that depend on the target metal. charge carrier Any charged particle, such as an electron, responsible for a current. Charles’s law The volume occupied by a gas at constant pressure is directly proportional to its thermodynamic (absolute) temperature. Glossary 1 Learning outcomes You should be able to: ■ ■ define displacement, speed and velocity ■ ■ draw and interpret displacement–time graphs ■ ■ describe laboratory methods for determining speed ■ ■ use vector addition to add two or more vectors Chapter 1: Kinematics – describing motion 2 Cambridge International AS Level Physics Describing movement Our eyes are good at detecting movement. We notice even quite small movements out of the corners of our eyes. It’s important for us to be able to judge movement – think about crossing the road, cycling or driving, or catching a ball. Figure 1.1 shows a way in which movement can be recorded on a photograph. This is a stroboscopic photograph of a boy juggling three balls. As he juggles, a bright lamp flashes several times a second so that the camera records the positions of the balls at equal intervals of time. If we knew the time between flashes, we could measure the photograph and calculate the speed of a ball as it moves through the air. If you look at the speedometer in a car, it doesn’t tell you the car’s average speed; rather, it tells you its speed at the instant when you look at it. This is the car’s instantaneous speed. Speed We can calculate the average speed of something moving if we know the distance it moves and the time it takes: average speed = distance time In symbols, this is written as: v = d t where v is the average speed and d is the distance travelled in time t. The photograph (Figure 1.2) shows Ethiopia’s Kenenisa Bekele posing next to the scoreboard after breaking the world record in a men’s 10000 metres race. The time on the clock in the photograph enables us to work out his average speed. If the object is moving at a constant speed, this equation will give us its speed during the time taken. If its speed is changing, then the equation gives us its average speed. Average speed is calculated over a period of time. Figure 1.1 This boy is juggling three balls. A stroboscopic lamp flashes at regular intervals; the camera is moved to one side at a steady rate to show separate images of the boy. Figure 1.2 Ethiopia’s Kenenisa Bekele set a new world record for the 10000 metres race in 2005. Units In the Système Internationale d’Unités (the SI system), distance is measured in metres (m) and time in seconds (s). Therefore, speed is in metres per second. This is written as ms−1 (or as m/s). Here, s−1 is the same as 1/s, or ‘per second’. There are many other units used for speed. The choice of unit depends on the situation. You would probably give the speed of a snail in different units from the speed of a racing car. Table 1.1 includes some alternative units of speed. Note that in many calculations it is necessary to work in SI units (ms−1). ms−1 metres per second cms−1 centimetres per second kms−1 kilometres per second kmh−1 or km/h kilometres per hour mph miles per hour Table 1.1 Units of speed. 1 Look at Figure 1.2. The runner ran 10000m, and the clock shows the total time taken. Calculate his average speed during the race. QUESTION 3 Chapter 1: Kinematics – describing motion 2 Here are some units of speed: ms−1 mms−1 kms−1 kmh−1 Which of these units would be appropriate when stating the speed of each of the following? a a tortoise b a car on a long journey c light d a sprinter. 3 A snail crawls 12cm in one minute. What is its average speed in mms−1? Determining speed You can find the speed of something moving by measuring the time it takes to travel between two fixed points. For example, some motorways have emergency telephones every 2000m. Using a stopwatch you can time a car over this distance. Note that this can only tell you the car’s average speed between the two points. You cannot tell whether it was increasing its speed, slowing down, or moving at a constant speed. BOX 1.1: Laboratory measurements of speed Here we describe four different ways to measure the speed of a trolley in the laboratory as it travels along a straight line. Each can be adapted to measure the speed of other moving objects, such as a glider on an air track, or a falling mass. Measuring speed using two light gates The leading edge of the card in Figure 1.3 breaks the light beam as it passes the first light gate. This starts the timer. The timer stops when the front of the card breaks the second beam. The trolley’s speed is calculated from the time interval and the distance between the light gates. Measuring speed using a ticker-timer The ticker-timer (Figure 1.5) marks dots on the tape at regular intervals, usually s (i.e. 0.02s). (This is because it works with alternating current, and in most countries the frequency of the alternating mains is 50Hz.) The pattern of dots acts as a record of the trolley’s movement. Measuring speed using one light gate The timer in Figure 1.4 starts when the leading edge of the card breaks the light beam. It stops when the trailing edge passes through. In this case, the time shown is the time taken for the trolley to travel a distance equal to the length of the card. The computer software can calculate the speed directly by dividing the distance by the time taken. Figure 1.3 Using two light gates to find the average speed of a trolley. stop timer light gates start Figure 1.4 Using a single light gate to find the average speed of a trolley. stop start light gate timer Figure 1.5 Using a ticker-timer to investigate the motion of a trolley. ticker-timer power supply 0 1 2 3 4 5 start trolley QUESTIONS