Undergraduate Lecture Notes in Physics Series Editors Neil Ashby William Brantley Michael Fowler Elena Sassi Helmy S Sherif For further volumes: http://www.springer.com/series/8917 Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topics throughout pure and applied physics Each title in the series is suitable as a basis for undergraduate instruction, typically containing practice problems, worked examples, chapter summaries, and suggestions for further reading ULNP titles must provide at least one of the following: • An exceptionally clear and concise treatment of a standard undergraduate subject • A solid undergraduate-level introduction to a graduate, advanced, or nonstandard subject • A novel perspective or an unusual approach to teaching a subject ULNP especially encourages new, original, and idiosyncratic approaches to physics teaching at the undergraduate level The purpose of ULNP is to provide intriguing, absorbing books that will continue to be the reader’s preferred reference throughout their academic career Hafez A Radi John O Rasmussen • Principles of Physics For Scientists and Engineers 123 Hafez A Radi October University for Modern Sciences and Arts (MSA) 6th of October City Egypt John O Rasmussen University of California at Berkeley and Lawrence Berkeley Lab Berkeley, CA USA Solutions to the exercises are accessible to qualified instructors at springer.com on this book’s product page Instructors may click on the link additional information and register to obtain their restricted access ISSN 2192-4791 ISBN 978-3-642-23025-7 DOI 10.1007/978-3-642-23026-4 ISSN 2192-4805 (electronic) ISBN 978-3-642-23026-4 (eBook) Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012947066 Ó Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface This book on Principles of Physics is intended to serve fundamental college courses in scientific curricula Physics is one of the most important tools to aid undergraduates, graduates, and researchers in their technical fields of study Without it many phenomena cannot be described, studied, or understood The topics covered here will help students interpret such phenomena, ultimately allowing them to advance in the applied aspects of their fields The goal of this text is to present many key concepts in a clear and concise, yet interesting way, making use of practical examples and attractively colored illustrations whenever appropriate to satisfy the needs of today’s science and engineering students Some of the examples, proofs, and subsections in this textbook have been identified as optional and are preceded with an asterisk * For less intensive courses these optional portions may be omitted without significantly impacting the objectives of the chapter Additional material may also be omitted depending on the course’s requirements The first author taught the material of this book in many universities in the Middle East for almost four decades Depending on the university, he leveraged different international textbooks, resources, and references These used different approaches, but were mainly written in an expansive manner delivering a plethora of topics while targeting students who wanted to dive deeply into the subject matter In this textbook, however, the authors introduce a large subset of these topics but in a more simplified manner, with the intent of delivering these topics and their key facts to students all over the world and in particular to students in the Middle East and neighboring regions where English may not be the native language The second author went over the entire text with the background of study and/or teaching at Caltech, UC Berkeley, and Yale Instructors teaching from this textbook will be able to gain online access from the publisher to the solutions manual, which provides step-by-step solutions to all exercises contained in the book The solutions manual also contains many tips, colored illustrations, and explanations on how the solutions were derived v vi Preface Acknowledgments from Prof Hafez A Radi I owe special thanks to my wife and two sons Tarek and Rami for their ongoing support and encouragement I also owe special thanks to my colleague and friend Prof Rasmussen for his invaluable contributions to this book, and for everything that I learned from him over the years while carrying out scientific research at Lawrence Berkeley Lab Additionally, I would like to express my gratitude to Prof Ali Helmy Moussa, Prof of Physics at Ain Shams University in Egypt, for his assistance, support, and guidance over the years I also thank all my fellow professors and colleagues who provided me with valuable feedback pertaining to many aspects of this book, especially Dr Sana’a Ismail, from Dar El Tarbiah School, IGCSE section and Dr Hesham Othman from the Faculty of Engineering at Cairo University I would also like to thank Professor Mike Guidry, Professor of Physics and Astronomy at the University of Tennessee Knoxville, for his valuable recommendations I am also grateful to the CD Odessa LLC for their ConceptDraw software suite which was used to create almost all the figures in this book I finally extend my thanks and appreciation to Professor Nawal El-Degwi, Professor Khayri Abdel-Hamid, Professor Said Ashour, and the staff members and teaching assistants at the faculty of Engineering at MSA University, Egypt, for all their support and input Hafez A Radi hafez.radi@gmail.com Acknowledgments from Prof John O Rasmussen I would like to thank Prof Radi for the opportunity to join him as coauthor I am grateful to the many teachers, students, and colleagues from whom I learned various aspects of the fascinating world of the physical sciences, notably the late Drs Linus Pauling, Isadore Perlman, Stanley Thompson, Glenn Seaborg, Earl Hyde, Hilding Slätis, Aage Bohr, Gaja Alaga, and Hans-Järg Mang There are many others, still living, too numerous to list here I would also like to extend my special thanks to my wife for her support and encouragement John O Rasmussen oxras@berkeley.edu Contents Part I Fundamental Basics Dimensions and Units 1.1 The International System of Units 1.2 Standards of Length, Time, and Mass 1.3 Dimensional Analysis 1.4 Exercises 3 12 Vectors 2.1 Vectors and Scalars 2.2 Properties of Vectors 2.3 Vector Components and Unit Vectors 2.4 Multiplying Vectors 2.5 Exercises 17 17 19 22 27 33 Part II Mechanics Motion in One Dimension 3.1 Position and Displacement 3.2 Average Velocity and Average Speed 3.3 Instantaneous Velocity and Speed 3.4 Acceleration 3.5 Constant Acceleration 3.6 Free Fall 3.7 Exercises 41 41 42 44 48 52 57 62 Motion in Two Dimensions 4.1 Position, Displacement, Velocity, and Acceleration Vectors 4.2 Projectile Motion 71 71 79 vii viii Contents 4.3 4.4 4.5 4.6 Uniform Circular Motion Tangential and Radial Acceleration Non-uniform Circular Motion Exercises 87 90 91 93 Force 5.1 5.2 5.3 5.4 and Motion The Cause of Acceleration and Newton’s Laws Some Particular Forces Applications to Newton’s Laws Exercises 103 103 106 113 124 Work, 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Energy, and Power Work Done by a Constant Force Work Done by a Variable Force Work-Energy Theorem Conservative Forces and Potential Energy Conservation of Mechanical Energy Work Done by Non-conservative Forces Conservation of Energy Power Exercises 137 137 142 148 151 157 159 162 166 170 Linear 7.1 7.2 7.3 Momentum, Collisions, and Center of Mass Linear Momentum and Impulse Conservation of Linear Momentum Conservation of Momentum and Energy in Collisions 7.3.1 Elastic Collisions in One and Two Dimensions 7.3.2 Inelastic Collisions Center of Mass (CM) Dynamics of the Center of Mass Systems of Variable Mass 7.6.1 Systems of Increasing Mass 7.6.2 Systems of Decreasing Mass; Rocket Propulsion Exercises 181 181 184 187 187 194 195 199 203 204 205 209 227 227 228 232 233 233 238 240 248 7.4 7.5 7.6 7.7 Rotational Motion 8.1 Radian Measures 8.2 Rotational Kinematics; Angular Quantities 8.3 Constant Angular Acceleration 8.4 Angular Vectors 8.5 Relating Angular and Linear Quantities 8.6 Rotational Dynamics; Torque 8.7 Newton’s Second Law for Rotation 8.8 Kinetic Energy, Work, and Power in Rotation Contents 8.9 8.10 ix Rolling Motion Exercises Angular Momentum 9.1 Angular Momentum of Rotating Systems 9.1.1 Angular Momentum of a Particle 9.1.2 Angular Momentum of a System of Particles 9.1.3 Angular Momentum of a Rotating Rigid Body 9.2 Conservation of Angular Momentum 9.3 The Spinning Top and Gyroscope 9.4 Exercises 10 Mechanical Properties of Matter 10.1 Density and Relative Density 10.2 Elastic Properties of Solids 10.2.1 Young’s Modulus: Elasticity in Length 10.2.2 Shear Modulus: Elasticity of Shape 10.2.3 Bulk Modulus: Volume Elasticity 10.3 Fluids 10.4 Fluid Statics 10.5 Fluid Dynamics 10.6 Exercises Part III 252 259 269 269 269 271 271 277 285 289 303 304 306 307 310 312 314 316 328 345 357 357 360 361 362 365 371 379 379 379 380 384 390 395 396 406 416 Introductory Thermodynamics 11 Thermal Properties of Matter 11.1 Temperature 11.2 Thermal Expansion of Solids and Liquids 11.2.1 Linear Expansion 11.2.2 Volume Expansion 11.3 The Ideal Gas 11.4 Exercises 12 Heat and the First Law of Thermodynamics 12.1 Heat and Thermal Energy 12.1.1 Units of Heat, The Mechanical Equivalent of Heat 12.1.2 Heat Capacity and Specific Heat 12.1.3 Latent Heat 12.2 Heat and Work 12.3 The First Law of Thermodynamics 12.4 Applications of the First Law of Thermodynamics 12.5 Heat Transfer 12.6 Exercises Fundamental Physical Constants Quantity Symbol Approximate value Speed of light in vacuum c 3:00  108 m/s Avogadro’s number NA 6:02  1023 molÀ1 ¼ 6:02  1026 kmolÀ1 Gas constant R 8:314 J/mol ÁK ¼ 314J/kmol Á K Boltzmann’s constant k 1:38  10À23 J/K Gravitational constant G 6:67  10À11 N Á m2 =kg2 Planck’s constant h 6:63  10À34 J Á s Permittivity of free space 0 8:85  10À12 C2 =N Á m2 Permeability of free space l0 ¼ 1=ðc2 0 Þ 4p  10À7 T Á m/A Atomic mass unit 1u 1:6605  10À27 kg ¼ 931:49 MeV/c2 Electron charge -e À1:60  10À19 C Electron rest mass me 9:11  10À31 kg ¼ 0:000549 u ¼ 0:511 MeV/c2 Proton rest mass mp 1:6726  10À27 kg ¼ 1:00728 u ¼ 938:27 MeV/c2 Neutron rest mass mn 1:6749  10À27 kg ¼ 1:008665 u ¼ 939:57 MeV/c2 xv Other useful constants Acceleration due to gravity at the Earth’s surface (av.) g ¼ 9:8 m/s2 Absolute zero (0 K) À273:15 C Joule equivalent (1 kcal) 4; 186 J Speed of sound in air (20 C) 343 m/s Density of air (dry) 1:29 kg/m3 Standard atmosphere 1:01  105 Pa Electric breakdown strength  106 V/m Earth: Mass Radius (av.) 5:98  1024 kg 6:38  103 km Moon: Mass Radius (av.) 7:35  1022 kg 1:74  103 km Sun: Mass Radius (av.) 1:99  1030 kg 6:96  105 km Earth–Moon distance (av.) 3:84  105 km Earth–Sun distance (av.) 1:5  108 km The greek alphabet Alpha A a Nu M m Beta B b Xi N n Gamma C c Omicron O o Delta D d Pi P p Epsilon E e Rho Q q Zeta F f Sigma R r Eta H g Tau S s Theta H h Upsilon T t Iota I i Phi U / Kappa J j Chi V v Lambda K k Psi W w Mu L l Omega X x xvi Some SI base units and derived units Quantity Unit name Unit symbol In terms of base units Mass kilogram kg Length meter m Time second s Electric current ampere A { Force newton N kgÁm=s2 Energy and work joule J kgÁm2 =s2 Power watt W kgÁm2 =s3 Pressure pascal Pa kg=ðmÁs2 Þ Frequency hertz Hz s-1 Electric charge coulomb C AÁs Electric potential volt V kgÁm2 =ðAÁs3 Þ Electric resistance ohm X kgÁm2 =ðA2 Ás3 Þ Capacitance farad F A2 Ás4 =ðkgÁm2 Þ Magnetic field tesla T kg=ðAÁs2 Þ Magnetic flux weber Wb kgÁm2 =ðAÁs2 Þ Inductance henry H kgÁm2 =ðs2 ÁA2 Þ Base SI units xvii SI multipliers yotta Y 1024 zeta Z 1021 exa E 1018 peta P 1015 tera T 1012 giga G 109 mega M 106 kilo k 103 hecto h 102 deka da 101 deci d 10-1 centi c 10-2 milli m 10-3 micro l 10-6 nano n 10-9 pico p 10-12 femto f 10-15 atto a 10-18 zepto z 10-21 yocto y 10-24 xviii Part I Fundamental Basics Dimensions and Units The laws of physics are expressed in terms of basic quantities that require a clear definition for the purpose of measurements Among these measured quantities are length, time, mass, temperature, etc In order to describe any physical quantity, we first have to define a unit of measurement (which was among the earliest tools invented by humans), i.e a measure that is defined to be exactly 1.0 After that, we define a standard for this quantity, i.e a reference to compare all other examples of the same physical quantity 1.1 The International System of Units Seven physical quantities have been selected as base quantities in the 14th General Conference on Weights and Measurements, held in France in 1971 These quantities form the basis of the International System of Units, abbreviated SI (from its French name Système International) and popularly known as the metric system Table 1.1 depicts these quantities, their unit names, and their unit symbols Many SI derived units are defined in terms of the first three quantities of Table 1.1 For example, the SI unit for force, called the newton (abbreviated N), is defined in terms of the base units of mass, length, and time Thus, as we will see from the study of Newton’s second law, the unit of force is given by: N = kg.m/s2 (1.1) When dealing with very large or very small numbers in physics, we use the so-called scientific notation which employs powers of 10, such as: H A Radi and J O Rasmussen, Principles of Physics, Undergraduate Lecture Notes in Physics, DOI: 10.1007/978-3-642-23026-4_1, © Springer-Verlag Berlin Heidelberg 2013 Dimensions and Units 210 000 000 m = 3.21 × 109 m (1.2) 0.000 000 789 s = 7.89 × 10−7 s (1.3) Table 1.1 The seven independent SI base units Quantity Unit name Unit symbol Length Meter m Time Second s Mass Kilogram kg Temperature Kelvin K Electric current Ampere A Amount of substance Mole mol Luminous intensity Candela cd An additional convenient way to deal with very large or very small numbers in physics is to use the prefixes listed in Table 1.2 Each one of these prefixes represents a certain power of 10 Table 1.2 Prefixes for SI unitsa Factor Prefix Symbol Factor Prefix Symbol 1024 yotta- Y 10−24 yocto- y 1021 zeta- Z 10−21 zepto- z 1018 exa- E 10−18 atto- a 1015 peta- P 10−15 femto- f 1012 tera- T 10−12 pico- p 109 giga- G 10−9 nano- n 106 mega- M 10−6 micro- µ 103 kilo- k 10−3 milli- m 102 hecta- h 10−2 centi- c da 10−1 deci- d 101 a deca- The most commonly used prefixes are shown in bold face type Accordingly, we can express a particular magnitude of force as: 1.23 × 106 N = 1.23 mega newtons = 1.23 MN (1.4) 1.1 The International System of Units or a particular time interval as: 1.23 × 10−9 s = 1.23 nano seconds (1.5) = 1.23 ns We often need to change units in which a physical quantity is expressed We that by using a method called chain-link conversion, in which we multiply by a conversion factor that equals unity For example, because minute and 60 seconds are identical time intervals, then we can write: 60 s = and =1 60 s This does not mean that treated together 60 (1.6) = or 60 = 1, because the number and its unit must be Example 1.1 Convert the following: (a) kilometer per hour to meter per second, (b) mile per hour to meter per second, and (c) mile per hour to kilometer per hour [to a good approximation 1mi = 1.609 km] Solution: (a) To convert the speed from the kilometers per hour unit to meters per second unit, we write: km/h = km h × 103 m km × 1h 60 × 60 s = 0.2777 m = 0.278 m/s s (b) To convert from miles per hour to meters per second, we write: mi/h = mi 1609 m 1h × × h mi 60 × 60 s = 0.447 m = 0.447 m/s s (c) To convert from miles per hour to kilometers per hour, we write: 1mi/h = 1.2 mi h × 1.609 km mi = 1.609 km = 1.609 km/h h Standards of Length, Time, and Mass Definitions of the units of length, time, and mass are under constant review and are changed from time to time We only present in this section the latest definitions of those quantities Dimensions and Units Length (L) In 1983, the precision of the meter was redefined as the distance traveled by a light wave in vacuum in a specified time interval The reason is that the measurement of the speed of light has become extremely precise, so it made sense to adopt the speed of light as a defined quantity and to use it to redefine the meter In the words of the 17th General Conference on Weights and Measurements: One Meter One meter is the distance traveled by light in vacuum during the time interval of 1/299 792 458 of a second This time interval number was chosen so that the speed of light in vacuum c will be exactly given by: c = 299 792 458 m/s (1.7) For educational purposes we usually consider the value c = × 108 m/s Table 1.3 lists some approximate interesting lengths Table 1.3 Some approximate lengths Length Meters Distance to farthest known galaxy × 1025 Distance to nearest star × 1016 Distance from Earth to Sun 1.5 × 1011 Distance from Earth to Moon × 108 Mean radius of Earth × 106 Wave length of light × 10−7 Radius of hydrogen atom × 10−11 Radius of proton × 10−15 Time (T) Recently, the standard of time was redefined to take advantage of the high-precision measurements that could be obtained by using a device known as an atomic clock Cesium is most common element that is typically used in the construction of atomic clocks because it allows us to attain high accuracy 1.2 Standards of Length, Time, and Mass Since 1967, the International System of Measurements has been basing its unit of time, the second, on the properties of the isotope cesium-133 (133 55 Cs) One of the 133 transitions between two energy levels of the ground state of 55 Cs has an oscillation frequency of 192 631 770 Hz, which is used to define the second in SI units Using this characteristic frequency, Fig 1.1 shows the cesium clock at the National Institute of Standards and Technology The uncertainty is about × 10−16 (as of 2005) Or about part in × 1015 This means that it would neither gain nor lose a second in 64 million years One Second One second is the time taken for the cesium atom 133 Cs 55 to perform 192 631 770 oscillations to emit radiation of a specific wavelength Fig 1.1 The cesium atomic clock at the National Institute of Standards and Technology (NIST) in Boulder, Colorado (photo with permission) Table 1.4 lists some approximate interesting time intervals Table 1.4 Some approximate time intervals Time intervals Seconds Lifetime of proton (predicted) × 1039 Age of the universe × 1017 Age of the Earth 1.3 × 1017 Period of one year 3.2 × 107 Time between human heartbeats × 10−1 Period of audible sound waves × 10−3 Period of visible light waves × 10−15 Time for light to cross a proton 3.3 × 10−24 Dimensions and Units Mass (M) The Standard Kilogram A cylindrical mass of 3.9 cm in diameter and of 3.9 cm in height and made of an unusually stable platinum-iridium alloy is kept at the International Bureau of Weights and Measures near Paris and assigned in the SI units a mass of kilogram by international agreement, see Fig 1.2 One Kilogram The SI unit of mass, kilogram, is defined as the mass of a platinum-iridium alloy cylinder kept at the International Bureau of Weights and Measures in France Fig 1.2 The standard kilogram of mass is a platinum-iridium cylinder 3.9 cm in height and diameter and kept under a double bell jar at the International Bureau of Weights and Measures in France Accurate copies of this standard kilogram have been sent to standardizing laboratories in other countries Table 1.5 lists some approximate mass values of various interesting objects A Second Standard Mass Atomic masses can be compared with each other more precisely than the kilogram By international agreement, the carbon-12 atom, 126 C, has been assigned a mass of 12 atomic mass units (u), where: u = (1.660 540 ± 0.000 001 0) × 10−27 kg (1.8) Experimentally, with reasonable precision, all masses of other atoms can be measured relative to the mass of carbon-12 1.3 Dimensional Analysis Table 1.5 Mass of various objects (approximate values) 1.3 Object Kilogram Known universe (predicted) × 1053 Our galaxy the milky way (predicted) × 1041 Sun × 1030 Earth × 1024 Moon × 1022 Small mountain × 1012 Elephant × 103 Human × 101 Mosquito × 10−5 Bacterium × 10−15 Uranium atom × 10−25 Proton × 10−27 Electron × 10−31 Dimensional Analysis Throughout your experience, you have been exposed to a variety of units of length; the SI meter, kilometer, and millimeter; the English units of inches, feet, yards, and miles, etc All of these derived units are said to have dimensions of length, symbolized by L Likewise, all time units, such as seconds, minutes, hours, days, years, and centuries are said to have dimensions of time, symbolized by T The kilogram and all other mass units have dimensions of mass, symbolized by M In general, we may take the dimension (length, time, and mass) as the concept of the physical quantity From the three fundamental physical quantities of length L, time T, and mass M, we can derive a variety of useful quantities Derived quantities have different dimensions from the fundamental quantities For example, the area obtained by multiplying one length by another has the dimension L2 Volume has the dimension L3 Mass density is defined as mass per unit volume and has the dimension M/L3 The SI unit of speed is meters per second (m/s) with the dimension L/T The concept of dimensionality is important in understanding physics and in solving physics problems For example, the addition or subtraction of quantities with different dimensions makes no sense, i.e kg plus s is meaningless Actually, physical equations must be dimensionally consistent For example, the equation giving the position of a freely falling body (see Chap 3) is giving by: 10 Dimensions and Units x = v◦ t + g t (1.9) where x is the position (length), v◦ the initial speed (length/time), g is the acceleration due to gravity (length/time2 ), and t is time If we analyze the equation dimensionally, we have: L L × T + × T2 = L + L (Dimensional analysis) L= T T Note that every term of this equation has the dimension of length L Also note that numerical factors, such as 21 in Eq 1.9, are ignored in dimensional analysis because they have no dimension Dimensional analysis is useful since it can be used to catch careless errors in any physical equation On the other hand, Eq 1.9 may be correct with respect to dimensional analysis, but could still be wrong with respect to dimensionless numerical factors If we had incorrectly written Eq 1.9 as follows: x = v◦ t + g t Then, by analyzing this equation dimensionally, we have: (1.10) L L × T2 + × T T T (Dimensional analysis) L L ×T× T+ ×T = T T× T L= and finally we get: L = LT+ L T (Dimensional analysis) Dimensionally, Eq 1.10 is meaningless, and thus cannot be correct Example 1.2 Use dimensional analysis to show that the expression v = v◦ + at is dimensionally correct, where v and v◦ represent velocities, a is acceleration, and t is a time interval Solution: Since L/T is the dimension of v and v◦ , and the dimension of a is L/T2, then when we analyze the equation v = v◦ + at dimensionally, we have: L L L = + ×T T T T L L = + ×T T T× T (Dimensional analysis) 1.3 Dimensional Analysis 11 L L L = + T T T and finally we get: (Dimensional analysis) Thus, the expression v = v◦ + at is dimensionally correct Example 1.3 A particle moves with a constant speed v in a circular orbit of radius r, see the figure below Given that the magnitude of the acceleration a is proportional to some power of r, say r m , and some power of v,say v n , then determine the powers of r and v v a r Solution: Assume that the variables of the problem can be expressed mathematically by the following relation: a = k r m , where k is a dimensionless proportionality constant With the known dimensions of r, v, and a we analyze the dimensions of the above relation as follows: L = Lm × T2 L T n = Lm+n (Dimensional analysis) Tn This dimensional equation would be balanced, i.e the dimensions of the right hand side equal the dimensions of the left hand side only when the following two conditions are satisfied: m + n = 1, and Thus: n = m = −1 12 Dimensions and Units Therefore, we can rewrite the acceleration as follows: a = k r −1 v = k v2 r When we later introduce uniform circular motion in Chap 4, we shall see that k = if SI units are used However, if for example we choose a to be in m/s2 and v to be in km/h, then k would not be equal to one 1.4 Exercises Section 1.1 The International System of Units (1) Use the prefixes introduced in Table 1.2 to express the following: (a) 103 lambs, (b) 106 bytes, (c) 109 cars, (d) 1012 stars, (e) 10−1 Kelvin, (f) 10−2 meter, (g) 10−3 ampere, (h) 10−6 newton, (i) 10−9 kilogram, (j) 10−15 second Section 1.2 Standards of Length, Time, and Mass Length (2) The original definition of the meter was based on distance from the North pole to the Earth’s equator (measures along the surface) and was taken to be 107 m (a) What is the circumference of the Earth in meters? (b) What is the radius of the Earth in meters, (c) Give your answer to part (a) and part (b) in miles (d) What is the circumference of the Earth in meters assuming it to be a sphere of radius 6.4 × 106 m? Compare your answer to part (a) (3) The time of flight of a laser pulse sent from the Earth to the Moon was measured in order to calculate the Earth-Moon distance, and it was found to be 3.8 × 105 km (a) Express this distance in miles, meters, centimeters, and millimeters (4) A unit of area, often used in measuring land areas, is the hectare, defined as 104 m2 An open-pit coal mine excavates 75 hectares of land, down to a depth of 26 m, each year What volume of Earth, in cubic kilometers, is removed during this time? (5) The units used by astronomers are appropriate for the quantities they usually measure As an example, for planetary distances they use the astronomical