PHYSICS BOOK PART-1

CK PART I TEXTBOOK FOR CLASS XI 2020-21 CK

CK FFeiFrbisrrtusatErdEyid2tii0to0ino6nPhalguna 1927 ISIBSNBN818-17-475405-05-0580-83-3 February 2006 Phalguna 1927 ALL RIGHTS RESERVED JJJMDRONaaaoeRMNODJJeJcannntvcaaaporeecoaeuuuecnnntrvcpbrmohmaaaeiuuuecrebnrrrmmhaaa2bibryyyetnrrr0ee22bbreyyy222trr1e0e02de000222222rr100d1100000022260P01910101006h09172KPPP01aaaa27alKPguuuraautsssirnuaaaktKKPPPPPasiak111aaaahaaa1a9991uuuuarr19333tt9ssssl1gii93012aaaakk29u23aa811113n99999411a332399109212398433 ALL RIGHTS RESERVED ‰ No part of this publication may be reproduced, stored in a retrieval system or ‰tranNsompitatertdo, finthainsypfuobrmlicoartiboynamnyaymbeeanresp, eroledcutrcoendic, ,smtoerechdainniacarle, tprhieovtoacl soypsytinegm, or rectorardnisnmgiottreodt,hinerawniysefowrmithoorubtythaenyprmioerapnesr,meilsescitoronnoicf ,thmeepcuhbalnisichaelr,.photocopying, ‰ Thirsebcooordkinisgsoorldotshuebrjwecistetowthitheocuotntdhietiopnriothrapteitrsmhiaslslinoont,obfythweapyuobflitsrahdeer., be lent, ‰re-sTohlids,bhoiroekdiosustooldr ostuhbejerwctisteo dthisepcoosnedditoiofnwtihthaotuittsthhealpl nuobtl,isbhyewr’saycoonf streandte, ,inbaenlyent, formre-osfobldin, dhinregdooructoovreorthoethrewristheadnistphoasteindwohf wicihthiot iustpthuebplisuhbelisdh. er’s consent, in any ‰ Thefocrmorroefcbtipnrdicinegoof rthciosvpeurbolitchaetriotnhaisntthheapt irnicwehpircinhteitdisopnutbhliisshpeadg.e, Any revised ‰pricTehiendciocrarteecdt bpyricaeruobfbtheirssptaumblipcaotriobny aisstthicekperricoer bpyriannteydoothnetrhmisepaangsei,sAinncyorrerevcisted andprsicheouinldibceatuendabcycaeprutabbbler. stamp or by a sticker or by any other means is incorrect and should be unacceptable. November 2013 Kartika 1935 PDMDea3cy5e2m00Tb1eR6r N20B14 Pausa 1936 OFFICES OF THE PUBLICATION Vaishakha 1938 DIVOIFSFIOICNE, SNCOEFRTTHE PUBLICATION ©ReJDFNsaeeebancareutuimraoacrbnrhyyea2ra2l02n0C101do917u7TnraciiPPMlnahoaiuanfgslghgEa,uad1n21u9a093c014a9960t3io8nal DIVISION, NCERT Phone : 011-26562708 Phone : 011-26562708 October 2019 Ashwina 1941 NCERT Campus SriNACuEroRbTindCoamMpaurgs Phone : 080-26725740 PD 450T BS NeSwriDAeulrhoib1i1n0do01M6arg Phone : 080-26725740 © National Council of Educational New Delhi 110 016 Phone : 079-27541446 Research and Training, 2006 Phone : 079-27541446 108, 100 Feet Road Ho1s0d8a,k1e0re0HFaelelitERxoteadnsion Phone : 033-25530454 BanHaosshdaankkearrei IHIIaSlltiaEgxetension Phone : 033-25530454 BaBnganaalosrhea5n6ka0r0i I8II5Stage Phone : 0361-2674869 Bengaluru 560 085 Phone : 0361-2674869 Navjivan Trust Building P.ON.NavajvivjiavannTrust Building AhPm.Oe.dNaabvajidva3n80 014 Ahmedabad 380 014 CWC Campus OpCp.WDChaCnakaml pBuuss Stop PanOihpapt.iDhankal Bus Stop KoPlkaantiaha7t0i 0 114 Kolkata 700 114 CWC Complex MaCligWaConComplex GuMwalhigaatio7n81 021 Guwahati 781 021 ` 105.00 Publication Team : Ashok Srivastava : Anup Kumar Rajput ` 150.00 HPeuabdl,icPautbiolincaTteioanm DHiveiasdio, nPublication : Shiv Kumar Printed on 80 GSM paper with NCERT CDhiiveifsiPornoduction : Shveta Uppal wPrPainrtietnertmdedaornokn8080GGSMSMpappaeprewr withithNCNECRERT T OCfhficieefr Editor Pwuwabtaelitrsemhrmaedrakarkt the Publication Division by the SPuePbculrbieslihtsaehdreyda,tatthtehNPeauPtbiuolbincliaactaliotinonDCDioviuvsinsocinoinlbybythotehf e CChhieieffEPdriotdour ction :: NAarruenshCYhaitdkaavra ESeSdceucrcreeatttaairroyn, aNlatRNioeanstaieloaCnroacuhl ncainlCdoofuETndrcauiiclnaitnioognf,al (IOnfcfhicaerrge) ESrRdi euAscuearaotrbci ohinnadanoldMRTarerasgi,neNianergwc,hSDreialAhnuid1ro1Tb0rin0ad1i no6iMannagdr,g, pSrNineAtwuerdoDbaient ldhPoai Mn1kaar1gj0, P0Nr1eiw6ntDinaegnlhdiP1r1eps0rsi0,n1tD6e-ad2n8d,at CChhieieffBBuussinineessss :: GBaiubtaasmh GKaunmgaurlyDas pInrKidnautlyesdatnraiatEl nPateAnrrkpearajis,PersiS,nitDtine--2gA0P,,reSsMesca,ttoDhr-u2Br8-a,3, MMaannaaggeerr (IUnTtdrtauornsPtirciaaadleCshiAt)yreIan, duSsittrei-aAl , ArMea,thLuornai, (UGtthaarzPiarbaaddes-h2) 01 102 (U.P.) EAdsistoisrtant Editor :: RR.N.N. .BBhhaarrddwwaaj j APsrsoidstuacnttioPnroAdsusicsttiaont :: RParjaeknadserhCVheaeur hSainngh Officer CoCvoevrear nadndIllIullsutsrtartaiotinosns ShSwhweteataRaRoao 2020-21 CK

CK FOREWORD The National Curriculum Framework (NCF), 2005 recommends that children’s life at school must be linked to their life outside the school. This principle marks a departure from the legacy of bookish learning which continues to shape our system and causes a gap between the school, home and community. The syllabi and textbooks developed on the basis of NCF signify an attempt to implement this basic idea. They also attempt to discourage rote learning and the maintenance of sharp boundaries between different subject areas. We hope these measures will take us significantly further in the direction of a child-centred system of education outlined in the National Policy on Education (1986). The success of this effort depends on the steps that school principals and teachers will take to encourage children to reflect on their own learning and to pursue imaginative activities and questions. We must recognise that, given space, time and freedom, children generate new knowledge by engaging with the information passed on to them by adults. Treating the prescribed textbook as the sole basis of examination is one of the key reasons why other resources and sites of learning are ignored. Inculcating creativity and initiative is possible if we perceive and treat children as participants in learning, not as receivers of a fixed body of knowledge. These aims imply considerable change is school routines and mode of functioning. Flexibility in the daily time-table is as necessary as rigour in implementing the annual calendar so that the required number of teaching days are actually devoted to teaching. The methods used for teaching and evaluation will also determine how effective this textbook proves for making children’s life at school a happy experience, rather than a source of stress or boredom. Syllabus designers have tried to address the problem of curricular burden by restructuring and reorienting knowledge at different stages with greater consideration for child psychology and the time available for teaching. The textbook attempts to enhance this endeavour by giving higher priority and space to opportunities for contemplation and wondering, discussion in small groups, and activities requiring hands-on experience. The National Council of Educational Research and Training (NCERT) appreciates the hard work done by the textbook development committee responsible for this book. We wish to thank the Chairperson of the advisory group in science and mathematics, Professor J.V. Narlikar and the Chief Advisor for this book, Professor A.W. Joshi for guiding the work of this committee. Several teachers contributed to the development of this textbook; we are grateful to their principals for making this possible. We are indebted to the institutions and organisations which have generously permitted us to draw upon their resources, material and personnel. We are especially grateful to the members of the National Monitoring Committee, appointed by the Department of Secondary and Higher Education, Ministry of Human Resource Development under the Chairpersonship of Professor Mrinal Miri and Professor G.P. Deshpande, for their valuable time and contribution. As an organisation committed to systemic reform and continuous improvement in the quality of its products, NCERT welcomes comments and suggestions which will enable us to undertake further revision and refinement. New Delhi Director 20 December 2005 National Council of Educational Research and Training 2020-21 CK

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CK TEXTBOOK DEVELOPMENT COMMITTEE CHAIRPERSON, ADVISORY GROUP FOR TEXTBOOKS IN SCIENCE AND MATHEMATICS J.V. Narlikar, Emeritus Professor, Chairman, Advisory Committee, Inter University Centre for Astronomy and Astrophysics (IUCAA), Ganeshkhind, Pune University, Pune CHIEF ADVISOR A.W. Joshi, Professor, Honorary Visiting Scientist, NCRA, Pune (Formerly at Department of Physics, University of Pune) MEMBERS Anuradha Mathur, PGT , Modern School, Vasant Vihar, New Delhi Chitra Goel, PGT, Rajkiya Pratibha Vikas Vidyalaya, Tyagraj Nagar, Lodhi Road, New Delhi Gagan Gupta, Reader, DESM, NCERT, New Delhi H.C. Pradhan, Professor, Homi Bhabha Centre of Science Education, Tata Institute of Fundamental Research, V.N. Purav Marg, Mankhurd, Mumbai N. Panchapakesan, Professor (Retd.), Department of Physics and Astrophysics, University of Delhi, Delhi P.K. Srivastava, Professor (Retd.), Director, CSEC, University of Delhi, Delhi P.K. Mohanty, PGT, Sainik School, Bhubaneswar P.C. Agarwal, Reader, Regional Institute of Education, NCERT, Sachivalaya Marg, Bhubaneswar R. Joshi, Lecturer (S.G.), DESM, NCERT, New Delhi S. Rai Choudhary, Professor, Department of Physics and Astrophysics, University of Delhi, Delhi S.K. Dash, Reader, DESM, NCERT, New Delhi Sher Singh, PGT, NDMC Navyug School, Lodhi Road, New Delhi S.N. Prabhakara, PGT, DM School, Regional Institute of Education, NCERT, Mysore Thiyam Jekendra Singh, Professor, Department of Physics, University of Manipur, Imphal V.P. Srivastava, Reader, DESM, NCERT, New Delhi MEMBER-COORDINATOR B.K. Sharma, Professor, DESM, NCERT, New Delhi 2020-21 CK

CK ACKNOWLEDGEMENTS The National Council of Educational Research and Training acknowledges the valuable contribution of the individuals and organisations involved in the development of Physics textbook for Class XI. The Council also acknowledges the valuable contribution of the following academics for reviewing and refining the manuscripts of this book: Deepak Kumar, Professor, School of Physical Sciences, Jawaharlal Nehru University, New Delhi; Pankaj Sharan, Professor, Jamia Millia Islamia, New Delhi; Ajoy Ghatak, Emeritus Professor, Indian Institute of Technology, New Delhi; V. Sundara Raja, Professor, Sri Venkateswara University, Tirupati, Andhra Pradesh; C.S. Adgaonkar, Reader (Retd), Institute of Science, Nagpur, Maharashtra; D.A. Desai, Lecturer (Retd), Ruparel College, Mumbai, Maharashtra; F.I. Surve, Lecturer, Nowrosjee Wadia College, Pune, Maharashtra; Atul Mody, Lecturer (SG), VES College of Arts, Science and Commerce, Chembur, Mumbai, Maharashtra; A.K. Das, PGT, St. Xavier’s Senior Secondary School, Delhi; Suresh Kumar, PGT, Delhi Public School, Dwarka, New Delhi; Yashu Kumar, PGT, Kulachi Hansraj Model School, Ashok Vihar, Delhi; K.S. Upadhyay, PGT, Jawahar Navodaya Vidyalaya, Muzaffar Nagar (U.P.); I.K. Gogia, PGT, Kendriya Vidyalaya, Gole Market, New Delhi; Vijay Sharma, PGT, Vasant Valley School, Vasant Kunj, New Delhi; R.S. Dass, Vice Principal (Retd), Balwant Ray Mehta Vidya Bhawan, Lajpat Nagar, New Delhi and Parthasarthi Panigrahi, PGT, D.V. CLW Girls School, Chittranjan, West Bengal. The Council also gratefully acknowledges the valuable contribution of the following academics for the editing and finalisation of this book: A.S. Mahajan, Professor (Retd), Indian Institute of Technology, Mumbai, Maharashtra; D.A. Desai, Lecturer (Retd), Ruparel College, Mumbai, Maharashtra; V.H. Raybagkar, Reader, Nowrosjee Wadia College, Pune, Maharashtra and Atul Mody, Lecturer (SG), VES College of Arts, Science and Commerce, Chembur, Mumbai, Maharashtra. The council also acknowledges the valuable contributions of the following academics for reviewing and refining the text in 2017: A.K. Srivastava, DESM, NCERT, New Delhi; Arnab Sen, NERIE, New Delhi; L.S. Chauhan, RIE, Bhopal; O.N. Awasthi (Retd.), RIE., Bhopal; Rachna Garg, DESM, NCERT, New Delhi; Raman Namboodiri, RIE, Mysuru; R.R. Koireng, DCS, NCERT, New Delhi; Shashi Prabha, DESM, NCERT, New Delhi; and S.V. Sharma, RIE, Ajmer. Special thanks are due to M. Chandra, Professor and Head, DESM, NCERT for her support. The Council also acknowledges the efforts of Deepak Kapoor, Incharge, Computer Station, Inder Kumar, DTP Operator; Saswati Banerjee, Copy Editor; Abhimanu Mohanty and Anuradha, Proof Readers in shaping this book. The contributions of the Publication Department in bringing out this book are also duly acknowledged. 2020-21 CK

CK PREFACE More than a decade ago, based on National Policy of Education (NPE-1986), National Council of Educational Research and Training published physics textbooks for Classes XI and XII, prepared under the chairmanship of Professor T. V. Ramakrishnan, F.R.S., with the help of a team of learned co-authors. The books were well received by the teachers and students alike. The books, in fact, proved to be milestones and trend-setters. However, the development of textbooks, particularly science books, is a dynamic process in view of the changing perceptions, needs, feedback and the experiences of the students, educators and the society. Another version of the physics books, which was the result of the revised syllabus based on National Curriculum Framework for School Education-2000 (NCFSE-2000), was brought out under the guidance of Professor Suresh Chandra, which continued up to now. Recently the NCERT brought out the National Curriculum Framework-2005 (NCF-2005), and the syllabus was accordingly revised during a curriculum renewal process at school level. The higher secondary stage syllabus (NCERT, 2005) has been developed accordingly. The Class XI textbook contains fifteen chapters in two parts. Part I contains first eight chapters while Part II contains next seven chapters. This book is the result of the renewed efforts of the present Textbook Development Team with the hope that the students will appreciate the beauty and logic of physics. The students may or may not continue to study physics beyond the higher secondary stage, but we feel that they will find the thought process of physics useful in any other branch they may like to pursue, be it finance, administration, social sciences, environment, engineering, technology, biology or medicine. For those who pursue physics beyond this stage, the matter developed in these books will certainly provide a sound base. Physics is basic to the understanding of almost all the branches of science and technology. It is interesting to note that the ideas and concepts of physics are increasingly being used in other branches such as economics and commerce, and behavioural sciences too. We are conscious of the fact that some of the underlying simple basic physics principles are often conceptually quite intricate. In this book, we have tried to bring in a conceptual coherence. The pedagogy and the use of easily understandable language are at the core of our effort without sacrificing the rigour of the subject. The nature of the subject of physics is such that a certain minimum use of mathematics is a must. We have tried to develop the mathematical formulations in a logical fashion, as far as possible. Students and teachers of physics must realise that physics is a branch which needs to be understood, not necessarily memorised. As one goes from secondary to higher secondary stage and beyond, physics involves mainly four components, (a) large amount of mathematical base, (b) technical words and terms, whose normal English meanings could be quite different, (c) new intricate concepts, and (d) experimental foundation. Physics needs mathematics because we wish to develop objective description of the world around us and express our observations in terms of measurable quantities. Physics discovers new properties of particles and wants to create a name for each one. The words are picked up normally from common English or Latin or Greek, but gives entirely different meanings to these words. It would be illuminating to look up words like energy, force, power, charge, spin, and several others, in any standard English dictionary, and compare their 2020-21 CK

CK viii meanings with their physics meanings. Physics develops intricate and often weird- looking concepts to explain the behaviour of particles. Finally, it must be remembered that entire physics is based on observations and experiments, without which a theory does not get acceptance into the domain of physics. This book has some features which, we earnestly hope, will enhance its usefulness for the students. Each chapter is provided with a Summary at its end for a quick overview of the contents of the chapter. This is followed by Points to Ponder which points out the likely misconceptions arising in the minds of students, hidden implications of certain statements/principles given in the chapter and cautions needed in applying the knowledge gained from the chapter. They also raise some thought-provoking questions which would make a student think about life beyond physics. Students will find it interesting to think and apply their mind on these points. Further, a large number of solved examples are included in the text in order to clarify the concepts and/or to illustrate the application of these concepts in everyday real-life situations. Occasionally, historical perspective has been included to share the excitement of sequential development of the subject of physics. Some Boxed items are introduced in many chapters either for this purpose or to highlight some special features of the contents requiring additional attention of the learners. Finally, a Subject Index has been added at the end of the book for ease in locating keywords in the book. The special nature of physics demands, apart from conceptual understanding, the knowledge of certain conventions, basic mathematical tools, numerical values of important physical constants, and systems of measurement units covering a vast range from microscopic to galactic levels. In order to equip the students, we have included the necessary tools and database in the form of Appendices A-1 to A-9 at the end of the book. There are also some other appendices at the end of some chapters giving additional information or applications of matter discussed in that chapter. Special attention has been paid for providing illustrative figures. To increase the clarity, the figures are drawn in two colours. A large number of Exercises are given at the end of each chapter. Some of these are from real-life situations. Students are urged to solve these and in doing so, they may find them very educative. Moreover, some Additional Exercises are given which are more challenging. Answers and hints to solve some of these are also included. In the entire book, SI units have been used. A comprehensive account of ‘units and measurement’ is given in Chapter 2 as a part of prescribed syllabus/curriculum as well as a help in their pursuit of physics. A box-item in this chapter brings out the difficulty in measuring as simple a thing as the length of a long curved line. Tables of SI base units and other related units are given here merely to indicate the presently accepted definitions and to indicate the high degree of accuracy with which measurements are possible today. The numbers given here are not to be memorised or asked in examinations. There is a perception among students, teachers, as well as the general public that there is a steep gradient between secondary and higher secondary stages. But a little thought shows that it is bound to be there in the present scenario of education. Education up to secondary stage is general education where a student has to learn several subjects – sciences, social sciences, mathematics, languages, at an elementary level. Education at the higher secondary stage and beyond, borders on acquiring professional competence, in some chosen fields of endeavour. You may like to compare this with the following situation. Children play cricket or badminton in lanes and small spaces outside (or inside) their homes. But then 2020-21 CK

CK ix some of them want to make it to the school team, then district team, then State team and then the National team. At every stage, there is bound to be a steep gradient. Hard work would have to be put in whether students want to pursue their education in the area of sciences, humanities, languages, music, fine arts, commerce, finance, architecture, or if they want to become sportspersons or fashion designers. Completing this book has only been possible because of the spontaneous and continuous support of many people. The Textbook Development Team is thankful to Dr. V. H. Raybagkar for allowing us to use his box item in Chapter 4 and to Dr. F. I. Surve for allowing us to use two of his box items in Chapter 15. We express also our gratitude to the Director, NCERT, for entrusting us with the task of preparing this textbook as a part of national effort for improving science education. The Head, Department of Education in Science and Mathematics, NCERT, was always willing to help us in our endeavour in every possible way. The previous text got excellent academic inputs from teachers, students and experts who sincerely suggested improvement during the past few years. We are thankful to all those who conveyed these inputs to NCERT. We are also thankful to the members of the Review Workshop and Editing Workshop organised to discuss and refine the first draft. We thank the Chairmen and their teams of authors for the text written by them in 1988, which provided the base and reference for developing the 2002 version as well as the present version of the textbook. Occasionally, substantial portions from the earlier versions, particularly those appreciated by students/teachers, have been adopted/adapted and retained in the present book for the benefit of coming generation of learners. We welcome suggestions and comments from our valued users, especially students and teachers. We wish our young readers a happy journey to the exciting realm of physics. A. W. JOSHI Chief Advisor Textbook Development Committee 2020-21 CK

CK A NOTE FOR THE TEACHERS To make the curriculum learner -centred, students should be made to participate and interact in the learning process directly. Once a week or one out of every six classes would be a good periodicity for such seminars and mutual interaction. Some suggestions for making the discussion participatory are given below, with reference to some specific topics in this book. Students may be divided into groups of five to six. The membership of these groups may be rotated during the year, if felt necessary. The topic for discussion can be presented on the board or on slips of paper. Students should be asked to write their reactions or answers to questions, whichever is asked, on the given sheets. They should then discuss in their groups and add modifications or comments in those sheets. These should be discussed either in the same or in a different class. The sheets may also be evaluated. We suggest here three possible topics from the book. The first two topics suggested are, in fact, very general and refer to the development of science over the past four centuries or more. Students and teachers may think of more such topics for each seminar. 1. Ideas that changed civilisation Suppose human beings are becoming extinct. A message has to be left for future generations or alien visitors. Eminent physicist R P Feynmann wanted the following message left for future beings, if any. “Matter is made up of atoms” A lady student and teacher of literature, wanted the following message left: “Water existed, so human beings could happen”. Another person thought it should be: “Idea of wheel for motion” Write down what message each one of you would like to leave for future generations. Then discuss it in your group and add or modify, if you want to change your mind. Give it to your teacher and join in any discussion that follows. 2. Reductionism Kinetic Theory of Gases relates the Big to the Small, the Macro to the Micro. A gas as a system is related to its components, the molecules. This way of describing a system as a result of the properties of its components is usually called Reductionism. It explains the behaviour of the group by the simpler and predictable behaviour of individuals. Macroscopic observations and microscopic properties have a mutual interdependence in this approach. Is this method useful? This way of understanding has its limitations outside physics and chemistry, may be even in these subjects. A painting cannot be discussed as a collection of the properties of chemicals used in making the canvas and the painting. What emerges is more than the sum of its components. Question: Can you think of other areas where such an approach is used? Describe briefly a system which is fully describable in terms of its components. Describe one which is not. Discuss with other members of the group and write your views. Give it to your teacher and join in any discussion that may follow. 3. Molecular approach to heat Describe what you think will happen in the following case. An enclosure is separated by a porous wall into two parts. One is filled with nitrogen gas (N2) and the other with CO2. Gases will diffuse from one side to the other. Question 1: Will both gases diffuse to the same extent? If not, which will diffuse more. Give reasons. Question 2: Will the pressure and temperature be unchanged? If not, what will be the changes in both. Give reasons. Write down your answers. Discuss with the group and modify them or add comments. Give to the teacher and join in the discussion. Students and teachers will find that such seminars and discussions lead to tremendous understanding, not only of physics, but also of science and social sciences. They also bring in some maturity among students. 2020-21 CK

CK CONTENTS FOREWORD iii PREFACE v A NOTE FOR THE TEACHER x CHAPTER 1 1 2 PHYSICAL WORLD 5 6 1.1 What is physics ? 10 1.2 Scope and excitement of physics 1.3 Physics, technology and society 16 1.4 Fundamental forces in nature 16 1.5 Nature of physical laws 18 21 CHAPTER 2 22 22 UNITS AND MEASUREMENTS 27 31 2.1 Introduction 31 2.2 The international system of units 32 2.3 Measurement of length 2.4 Measurement of mass 39 2.5 Measurement of time 39 2.6 Accuracy, precision of instruments and errors in measurement 42 2.7 Significant figures 43 2.8 Dimensions of physical quantities 45 2.9 Dimensional formulae and dimensional equations 47 2.10 Dimensional analysis and its applications 51 CHAPTER 3 65 65 MOTION IN A STRAIGHT LINE 67 67 3.1 Introduction 69 3.2 Position, path length and displacement 3.3 Average velocity and average speed 3.4 Instantaneous velocity and speed 3.5 Acceleration 3.6 Kinematic equations for uniformly accelerated motion 3.7 Relative velocity CHAPTER 4 MOTION IN A PLANE 4.1 Introduction 4.2 Scalars and vectors 4.3 Multiplication of vectors by real numbers 4.4 Addition and subtraction of vectors – graphical method 4.5 Resolution of vectors 2020-21 CK

CK xii 4.6 Vector addition – analytical method 71 4.7 Motion in a plane 72 4.8 Motion in a plane with constant acceleration 75 4.9 Relative velocity in two dimensions 76 4.10 Projectile motion 77 4.11 Uniform circular motion 79 CHAPTER 5 89 90 LAWS OF MOTION 90 91 5.1 Introduction 93 5.2 Aristotle’s fallacy 96 5.3 The law of inertia 98 5.4 Newton’s first law of motion 99 5.5 Newton’s second law of motion 100 5.6 Newton’s third law of motion 104 5.7 Conservation of momentum 105 5.8 Equilibrium of a particle 5.9 Common forces in mechanics 114 5.10 Circular motion 116 5.11 Solving problems in mechanics 116 117 CHAPTER 6 118 119 WORK, ENERGY AND POWER 120 121 6.1 Introduction 123 6.2 Notions of work and kinetic energy : The work-energy theorem 126 6.3 Work 128 6.4 Kinetic energy 129 6.5 Work done by a variable force 6.6 The work-energy theorem for a variable force 141 6.7 The concept of potential energy 144 6.8 The conservation of mechanical energy 148 6.9 The potential energy of a spring 149 6.10 Various forms of energy : the law of conservation of energy 150 6.11 Power 152 6.12 Collisions 154 158 CHAPTER 7 163 164 SYSTEM OF PARTICLES AND ROTATIONAL MOTION 7.1 Introduction 7.2 Centre of mass 7.3 Motion of centre of mass 7.4 Linear momentum of a system of particles 7.5 Vector product of two vectors 7.6 Angular velocity and its relation with linear velocity 7.7 Torque and angular momentum 7.8 Equilibrium of a rigid body 7.9 Moment of inertia 7.10 Theorems of perpendicular and parallel axes 2020-21 CK

CK xiii 7.11 Kinematics of rotational motion about a fixed axis 167 7.12 Dynamics of rotational motion about a fixed axis 169 7.13 Angular momentum in case of rotations about a fixed axis 171 7.14 Rolling motion 173 CHAPTER 8 183 184 GRAVITATION 185 189 8.1 Introduction 189 8.2 Kepler’s laws 190 8.3 Universal law of gravitation 191 8.4 The gravitational constant 193 8.5 Acceleration due to gravity of the earth 194 8.6 Acceleration due to gravity below and above the surface of earth 195 8.7 Gravitational potential energy 196 8.8 Escape speed 197 8.9 Earth satellite 8.10 Energy of an orbiting satellite 207 8.11 Geostationary and polar satellites 8.12 Weightlessness 223 APPENDICES ANSWERS 2020-21 CK

CK COVER DESIGN (Adapted from the website of the Nobel Foundation http://www.nobelprize.org) The strong nuclear force binds protons and neutrons in a nucleus and is the strongest of nature’s four fundamental forces. A mystery surrounding the strong nuclear force has been solved. The three quarks within the proton can sometimes appear to be free, although no free quarks have ever been observed. The quarks have a quantum mechanical property called ‘colour’ and interact with each other through the exchange of particles called ‘gluons’ — nature glue. BACK COVER (Adapted from the website of the ISRO http://www.isro.gov.in) CARTOSAT-1 is a state-of-the-art Remote Sensing Satellite, being eleventh one in the Indian Remote Sensing (IRS) Satellite Series, built by ISRO. CARTOSAT-1, having mass of 156 kg at lift off, has been launched into a 618 km high polar Sun Synchronous Orbit (SSO) by ISRO’s Polar Satellite Launch Vehicle, PSLV-C6. It is mainly intended for cartographic applications. 2020-21 CK

CHAPTER ONE PHYSICAL WORLD 1.1 What is physics ? 1.1 WHAT IS PHYSICS ? 1.2 Scope and excitement of Humans have always been curious about the world around physics them. The night sky with its bright celestial objects has 1.3 Physics, technology and fascinated humans since time immemorial. The regular repetitions of the day and night, the annual cycle of seasons, society the eclipses, the tides, the volcanoes, the rainbow have always 1.4 Fundamental forces in been a source of wonder. The world has an astonishing variety of materials and a bewildering diversity of life and behaviour. nature The inquiring and imaginative human mind has responded 1.5 Nature of physical laws to the wonder and awe of nature in different ways. One kind of response from the earliest times has been to observe the Summary physical environment carefully, look for any meaningful Exercises patterns and relations in natural phenomena, and build and use new tools to interact with nature. This human endeavour led, in course of time, to modern science and technology. The word Science originates from the Latin verb Scientia meaning ‘to know’. The Sanskrit word Vijñãn and the Arabic word Ilm convey similar meaning, namely ‘knowledge’. Science, in a broad sense, is as old as human species. The early civilisations of Egypt, India, China, Greece, Mesopotamia and many others made vital contributions to its progress. From the sixteenth century onwards, great strides were made in science in Europe. By the middle of the twentieth century, science had become a truly international enterprise, with many cultures and countries contributing to its rapid growth. What is Science and what is the so-called Scientific Method ? Science is a systematic attempt to understand natural phenomena in as much detail and depth as possible, and use the knowledge so gained to predict, modify and control phenomena. Science is exploring, experimenting and predicting from what we see around us. The curiosity to learn about the world, unravelling the secrets of nature is the first step towards the discovery of science. The scientific method involves several interconnected steps : Systematic observations, controlled experiments, qualitative and 2020-21

2 PHYSICS quantitative reasoning, mathematical Physics is a basic discipline in the category modelling, prediction and verification or of Natural Sciences, which also includes other falsification of theories. Speculation and disciplines like Chemistry and Biology. The word conjecture also have a place in science; but Physics comes from a Greek word meaning ultimately, a scientific theory, to be acceptable, nature. Its Sanskrit equivalent is Bhautiki that must be verified by relevant observations or is used to refer to the study of the physical world. experiments. There is much philosophical A precise definition of this discipline is neither debate about the nature and method of science possible nor necessary. We can broadly describe that we need not discuss here. physics as a study of the basic laws of nature and their manifestation in different natural The interplay of theory and observation (or phenomena. The scope of physics is described experiment) is basic to the progress of science. briefly in the next section. Here we remark on Science is ever dynamic. There is no ‘final’ two principal thrusts in physics : unification theory in science and no unquestioned and reduction. authority among scientists. As observations improve in detail and precision or experiments In Physics, we attempt to explain diverse yield new results, theories must account for physical phenomena in terms of a few concepts them, if necessary, by introducing modifications. and laws. The effort is to see the physical world Sometimes the modifications may not be drastic as manifestation of some universal laws in and may lie within the framework of existing different domains and conditions. For example, theory. For example, when Johannes Kepler the same law of gravitation (given by Newton) (1571-1630) examined the extensive data on describes the fall of an apple to the ground, the planetary motion collected by Tycho Brahe motion of the moon around the earth and the (1546-1601), the planetary circular orbits in motion of planets around the sun. Similarly, the heliocentric theory (sun at the centre of the basic laws of electromagnetism (Maxwell’s solar system) imagined by Nicolas Copernicus equations) govern all electric and magnetic (1473–1543) had to be replaced by elliptical phenomena. The attempts to unify fundamental orbits to fit the data better. Occasionally, forces of nature (section 1.4) reflect this same however, the existing theory is simply unable quest for unification. to explain new observations. This causes a major upheaval in science. In the beginning of A related effort is to derive the properties of a the twentieth century, it was realised that bigger, more complex, system from the properties Newtonian mechanics, till then a very and interactions of its constituent simpler parts. successful theory, could not explain some of the This approach is called reductionism and is most basic features of atomic phenomena. at the heart of physics. For example, the subject Similarly, the then accepted wave picture of light of thermodynamics, developed in the nineteenth failed to explain the photoelectric effect properly. century, deals with bulk systems in terms of This led to the development of a radically new macroscopic quantities such as temperature, theory (Quantum Mechanics) to deal with atomic internal energy, entropy, etc. Subsequently, the and molecular phenomena. subjects of kinetic theory and statistical mechanics interpreted these quantities in terms Just as a new experiment may suggest an of the properties of the molecular constituents alternative theoretical model, a theoretical of the bulk system. In particular, the advance may suggest what to look for in some temperature was seen to be related to the average experiments. The result of experiment of kinetic energy of molecules of the system. scattering of alpha particles by gold foil, in 1911 by Ernest Rutherford (1871–1937) established 1.2 SCOPE AND EXCITEMENT OF PHYSICS the nuclear model of the atom, which then became the basis of the quantum theory of We can get some idea of the scope of physics by hydrogen atom given in 1913 by Niels Bohr looking at its various sub-disciplines. Basically, (1885–1962). On the other hand, the concept of there are two domains of interest : macroscopic antiparticle was first introduced theoretically by and microscopic. The macroscopic domain Paul Dirac (1902–1984) in 1930 and confirmed includes phenomena at the laboratory, terrestrial two years later by the experimental discovery of and astronomical scales. The microscopic domain positron (antielectron) by Carl Anderson. includes atomic, molecular and nuclear 2020-21

PHYSICAL WORLD 3 phenomena*. Classical Physics deals mainly chemical process, etc., are problems of interest in thermodynamics. with macroscopic phenomena and includes subjects like Mechanics, Electrodynamics, The microscopic domain of physics deals with Optics and Thermodynamics. Mechanics the constitution and structure of matter at the founded on Newton’s laws of motion and the law minute scales of atoms and nuclei (and even of gravitation is concerned with the motion (or lower scales of length) and their interaction with equilibrium) of particles, rigid and deformable different probes such as electrons, photons and bodies, and general systems of particles. The other elementary particles. Classical physics is propulsion of a rocket by a jet of ejecting gases, inadequate to handle this domain and Quantum propagation of water waves or sound waves in Theory is currently accepted as the proper air, the equilibrium of a bent rod under a load, framework for explaining microscopic etc., are problems of mechanics. Electrodynamics phenomena. Overall, the edifice of physics is deals with electric and magnetic phenomena beautiful and imposing and you will appreciate associated with charged and magnetic bodies. it more as you pursue the subject. Its basic laws were given by Coulomb, Oersted, Fig. 1.1 Theory and experiment go hand in hand in physics and help each other’s progress. The alpha scattering experiments of Rutherford gave the nuclear model of the atom. Ampere and Faraday, and encapsulated by You can now see that the scope of physics is Maxwell in his famous set of equations. The motion of a current-carrying conductor in a truly vast. It covers a tremendous range of magnetic field, the response of a circuit to an ac voltage (signal), the working of an antenna, the magnitude of physical quantities like length, propagation of radio waves in the ionosphere, etc., are problems of electrodynamics. Optics deals mass, time, energy, etc. At one end, it studies with the phenomena involving light. The working of telescopes and microscopes, colours exhibited phenomena at the very small scale of length by thin films, etc., are topics in optics. (10-14 m or even less) involving electrons, protons, Thermodynamics, in contrast to mechanics, does not deal with the motion of bodies as a whole. etc.; at the other end, it deals with astronomical Rather, it deals with systems in macroscopic equilibrium and is concerned with changes in phenomena at the scale of galaxies or even the internal energy, temperature, entropy, etc., of the system through external work and transfer of entire universe whose extent is of the order of heat. The efficiency of heat engines and 1026 m. The two length scales differ by a factor of refrigerators, the direction of a physical or 1040 or even more. The range of time scales can be obtained by dividing the length scales by the speed of light : 10–22 s to 1018 s. The range of masses goes from, say, 10–30 kg (mass of an electron) to 1055 kg (mass of known observable universe). Terrestrial phenomena lie somewhere in the middle of this range. * Recently, the domain intermediate between the macroscopic and the microscopic (the so-called mesoscopic physics), dealing with a few tens or hundreds of atoms, has emerged as an exciting field of research. 2020-21

4 PHYSICS Physics is exciting in many ways. To some people Hypothesis, axioms and models the excitement comes from the elegance and universality of its basic theories, from the fact that One should not think that everything can be proved a few basic concepts and laws can explain with physics and mathematics. All physics, and also phenomena covering a large range of magnitude mathematics, is based on assumptions, each of of physical quantities. To some others, the challenge which is variously called a hypothesis or axiom or in carrying out imaginative new experiments to postulate, etc. unlock the secrets of nature, to verify or refute theories, is thrilling. Applied physics is equally For example, the universal law of gravitation demanding. Application and exploitation of proposed by Newton is an assumption or hypothesis, physical laws to make useful devices is the most which he proposed out of his ingenuity. Before him, interesting and exciting part and requires great there were several observations, experiments and ingenuity and persistence of effort. data, on the motion of planets around the sun, motion of the moon around the earth, pendulums, What lies behind the phenomenal progress bodies falling towards the earth etc. Each of these of physics in the last few centuries? Great required a separate explanation, which was more progress usually accompanies changes in our or less qualitative. What the universal law of basic perceptions. First, it was realised that for gravitation says is that, if we assume that any two scientific progress, only qualitative thinking, bodies in the universe attract each other with a though no doubt important, is not enough. force proportional to the product of their masses Quantitative measurement is central to the and inversely proportional to the square of the growth of science, especially physics, because distance between them, then we can explain all the laws of nature happen to be expressible in these observations in one stroke. It not only explains precise mathematical equations. The second these phenomena, it also allows us to predict the most important insight was that the basic laws results of future experiments. of physics are universal — the same laws apply in widely different contexts. Lastly, the strategy A hypothesis is a supposition without assuming of approximation turned out to be very that it is true. It would not be fair to ask anybody successful. Most observed phenomena in daily to prove the universal law of gravitation, because life are rather complicated manifestations of the it cannot be proved. It can be verified and basic laws. Scientists recognised the importance substantiated by experiments and observations. of extracting the essential features of a phenomenon from its less significant aspects. An axiom is a self-evident truth while a model It is not practical to take into account all the is a theory proposed to explain observed complexities of a phenomenon in one go. A good phenomena. But you need not worry at this stage strategy is to focus first on the essential features, about the nuances in using these words. For discover the basic principles and then introduce example, next year you will learn about Bohr’s model corrections to build a more refined theory of the of hydrogen atom, in which Bohr assumed that an phenomenon. For example, a stone and a feather electron in the hydrogen atom follows certain rules dropped from the same height do not reach the (postutates). Why did he do that? There was a large ground at the same time. The reason is that the amount of spectroscopic data before him which no essential aspect of the phenomenon, namely free other theory could explain. So Bohr said that if we fall under gravity, is complicated by the assume that an atom behaves in such a manner, presence of air resistance. To get the law of free we can explain all these things at once. fall under gravity, it is better to create a situation wherein the air resistance is Einstein’s special theory of relativity is also negligible. We can, for example, let the stone and based on two postulates, the constancy of the speed the feather fall through a long evacuated tube. of electromagnetic radiation and the validity of In that case, the two objects will fall almost at physical laws in all inertial frame of reference. It the same rate, giving the basic law that would not be wise to ask somebody to prove that acceleration due to gravity is independent of the the speed of light in vacuum is constant, mass of the object. With the basic law thus independent of the source or observer. found, we can go back to the feather, introduce corrections due to air resistance, modify the In mathematics too, we need axioms and existing theory and try to build a more realistic hypotheses at every stage. Euclid’s statement that parallel lines never meet, is a hypothesis. This means that if we assume this statement, we can explain several properties of straight lines and two or three dimensional figures made out of them. But if you don’t assume it, you are free to use a different axiom and get a new geometry, as has indeed happened in the past few centuries and decades. 2020-21

PHYSICAL WORLD 5 theory of objects falling to the earth under A most significant area to which physics has gravity. and will contribute is the development of alternative energy resources. The fossil fuels of 1.3 PHYSICS, TECHNOLOGY AND SOCIETY the planet are dwindling fast and there is an urgent need to discover new and affordable The connection between physics, technology sources of energy. Considerable progress has and society can be seen in many examples. The already been made in this direction (for discipline of thermodynamics arose from the example, in conversion of solar energy, need to understand and improve the working of geothermal energy, etc., into electricity), but heat engines. The steam engine, as we know, much more is still to be accomplished. is inseparable from the Industrial Revolution in England in the eighteenth century, which had Table1.1 lists some of the great physicists, great impact on the course of human their major contribution and the country of civilisation. Sometimes technology gives rise to origin. You will appreciate from this table the new physics; at other times physics generates multi-cultural, international character of the new technology. An example of the latter is the scientific endeavour. Table 1.2 lists some wireless communication technology that followed important technologies and the principles of the discovery of the basic laws of electricity and physics they are based on. Obviously, these magnetism in the nineteenth century. The tables are not exhaustive. We urge you to try to applications of physics are not always easy to add many names and items to these tables with foresee. As late as 1933, the great physicist the help of your teachers, good books and Ernest Rutherford had dismissed the possibility websites on science. You will find that this of tapping energy from atoms. But only a few exercise is very educative and also great fun. years later, in 1938, Hahn and Meitner And, assuredly, it will never end. The progress discovered the phenomenon of neutron-induced of science is unstoppable! fission of uranium, which would serve as the basis of nuclear power reactors and nuclear Physics is the study of nature and natural weapons. Yet another important example of phenomena. Physicists try to discover the rules physics giving rise to technology is the silicon that are operating in nature, on the basis of ‘chip’ that triggered the computer revolution in observations, experimentation and analysis. the last three decades of the twentieth century. Physics deals with certain basic rules/laws governing the natural world. What is the nature Table 1.1 Some physicists from different countries of the world and their major contributions Name Major contribution/discovery Country of Origin Archimedes Galileo Galilei Principle of buoyancy; Principle of the lever Greece Christiaan Huygens Isaac Newton Law of inertia Italy Michael Faraday Wave theory of light Holland James Clerk Maxwell Universal law of gravitation; Laws of motion; U.K. Heinrich Rudolf Hertz Reflecting telescope J.C. Bose W.K. Roentgen Laws of electromagnetic induction U.K. J.J. Thomson Marie Sklodowska Curie Electromagnetic theory; Light-an U.K. electromagnetic wave Albert Einstein Generation of electromagnetic waves Germany Ultra short radio waves India X-rays Germany Electron U.K. Discovery of radium and polonium; Studies on Poland natural radioactivity Explanation of photoelectric effect; Germany Theory of relativity 2020-21

6 PHYSICS Name Major contribution/discovery Country of Origin Victor Francis Hess Cosmic radiation Austria R.A. Millikan Measurement of electronic charge U.S.A. Ernest Rutherford Nuclear model of atom New Zealand Niels Bohr Quantum model of hydrogen atom Denmark C.V. Raman Inelastic scattering of light by molecules India Louis Victor de Borglie Wave nature of matter France M.N. Saha Thermal ionisation India S.N. Bose Quantum statistics India Wolfgang Pauli Exclusion principle Austria Enrico Fermi Controlled nuclear fission Italy Werner Heisenberg Quantum mechanics; Uncertainty principle Germany Paul Dirac Relativistic theory of electron; U.K. Quantum statistics Edwin Hubble Expanding universe U.S.A. Ernest Orlando Lawrence Cyclotron U.S.A. James Chadwick Neutron U.K. Hideki Yukawa Theory of nuclear forces Japan Homi Jehangir Bhabha Cascade process of cosmic radiation India Lev Davidovich Landau Theory of condensed matter; Liquid helium Russia S. Chandrasekhar Chandrasekhar limit, structure and evolution India of stars John Bardeen Transistors; Theory of super conductivity U.S.A. C.H. Townes Maser; Laser U.S.A. Abdus Salam Unification of weak and electromagnetic Pakistan interactions of physical laws? We shall now discuss the ideas about it. The correct notion of force was nature of fundamental forces and the laws that arrived at by Isaac Newton in his famous laws of govern the diverse phenomena of the physical motion. He also gave an explicit form for the force world. for gravitational attraction between two bodies. We shall learn these matters in subsequent 1.4 FUNDAMENTAL FORCES IN NATURE* chapters. We all have an intuitive notion of force. In our In the macroscopic world, besides the experience, force is needed to push, carry or gravitational force, we encounter several kinds throw objects, deform or break them. We also of forces: muscular force, contact forces between experience the impact of forces on us, like when bodies, friction (which is also a contact force a moving object hits us or we are in a merry-go- parallel to the surfaces in contact), the forces round. Going from this intuitive notion to the exerted by compressed or elongated springs and proper scientific concept of force is not a trivial taut strings and ropes (tension), the force of matter. Early thinkers like Aristotle had wrong buoyancy and viscous force when solids are in * Sections 1.4 and 1.5 contain several ideas that you may not grasp fully in your first reading. However, we advise you to read them carefully to develop a feel for some basic aspects of physics. These are some of the areas which continue to occupy the physicists today. 2020-21

PHYSICAL WORLD 7 Table 1.2 Link between technology and physics Technology Scientific principle(s) Steam engine Nuclear reactor Laws of thermodynamics Radio and Television Controlled nuclear fission Generation, propagation and detection Computers of electromagnetic waves Lasers Digital logic Light amplification by stimulated emission of Production of ultra high magnetic radiation fields Superconductivity Rocket propulsion Electric generator Newton’s laws of motion Hydroelectric power Faraday’s laws of electromagnetic induction Conversion of gravitational potential energy into Aeroplane electrical energy Particle accelerators Bernoulli’s principle in fluid dynamics Motion of charged particles in electromagnetic Sonar fields Optical fibres Reflection of ultrasonic waves Non-reflecting coatings Total internal reflection of light Thin film optical interference Electron microscope Wave nature of electrons Photocell Photoelectric effect Fusion test reactor (Tokamak) Magnetic confinement of plasma Detection of cosmic radio waves Giant Metrewave Radio Telescope (GMRT) Trapping and cooling of atoms by laser beams and Bose-Einstein condensate magnetic fields. contact with fluids, the force due to pressure of to the net attraction/repulsion between the a fluid, the force due to surface tension of a liquid, neighbouring atoms of the spring when the and so on. There are also forces involving charged spring is elongated/compressed. This net and magnetic bodies. In the microscopic domain attraction/repulsion can be traced to the again, we have electric and magnetic forces, (unbalanced) sum of electric forces between the nuclear forces involving protons and neutrons, charged constituents of the atoms. interatomic and intermolecular forces, etc. We shall get familiar with some of these forces in later In principle, this means that the laws for parts of this course. ‘derived’ forces (such as spring force, friction) are not independent of the laws of fundamental A great insight of the twentieth century forces in nature. The origin of these derived physics is that these different forces occurring forces is, however, very complex. in different contexts actually arise from only a small number of fundamental forces in nature. At the present stage of our understanding, For example, the elastic spring force arises due we know of four fundamental forces in nature, which are described in brief here : 2020-21

8 PHYSICS Albert Einstein (1879-1955) Albert Einstein, born in Ulm, Germany in 1879, is universally regarded as one of the greatest physicists of all time. His astonishing scientific career began with the publication of three path-breaking papers in 1905. In the first paper, he introduced the notion of light quanta (now called photons) and used it to explain the features of photoelectric effect that the classical wave theory of radiation could not account for. In the second paper, he developed a theory of Brownian motion that was confirmed experimentally a few years later and provided a convincing evidence of the atomic picture of matter. The third paper gave birth to the special theory of relativity that made Einstein a legend in his own life time. In the next decade, he explored the consequences of his new theory which included, among other things, the mass-energy equivalence enshrined in his famous equation E = mc2. He also created the general version of relativity (The General Theory of Relativity), which is the modern theory of gravitation. Some of Einstein’s most significant later contributions are: the notion of stimulated emission introduced in an alternative derivation of Planck’s blackbody radiation law, static model of the universe which started modern cosmology, quantum statistics of a gas of massive bosons, and a critical analysis of the foundations of quantum mechanics. The year 2005 was declared as International Year of Physics, in recognition of Einstein’s monumental contribution to physics, in year 1905, describing revolutionary scientific ideas that have since influenced all of modern physics. 1.4.1 Gravitational Force electric force between two protons, for example, is 1036 times the gravitational force between The gravitational force is the force of mutual them, for any fixed distance. attraction between any two objects by virtue of their masses. It is a universal force. Every object Matter, as we know, consists of elementary experiences this force due to every other object charged constituents like electrons and in the universe. All objects on the earth, for protons. Since the electromagnetic force is so example, experience the force of gravity due to much stronger than the gravitational force, it the earth. In particular, gravity governs the dominates all phenomena at atomic and motion of the moon and artificial satellites around molecular scales. (The other two forces, as we the earth, motion of the earth and planets shall see, operate only at nuclear scales.) Thus around the sun, and, of course, the motion of it is mainly the electromagnetic force that bodies falling to the earth. It plays a key role in governs the structure of atoms and molecules, the large-scale phenomena of the universe, such the dynamics of chemical reactions and the as formation and evolution of stars, galaxies and mechanical, thermal and other properties of galactic clusters. materials. It underlies the macroscopic forces like ‘tension’, ‘friction’, ‘normal force’, ‘spring 1.4.2 Electromagnetic Force force’, etc. Electromagnetic force is the force between Gravity is always attractive, while charged particles. In the simpler case when electromagnetic force can be attractive or charges are at rest, the force is given by repulsive. Another way of putting it is that mass Coulomb’s law : attractive for unlike charges and comes only in one variety (there is no negative repulsive for like charges. Charges in motion mass), but charge comes in two varieties : produce magnetic effects and a magnetic field positive and negative charge. This is what gives rise to a force on a moving charge. Electric makes all the difference. Matter is mostly and magnetic effects are, in general, electrically neutral (net charge is zero). Thus, inseparable – hence the name electromagnetic electric force is largely zero and gravitational force. Like the gravitational force, force dominates terrestrial phenomena. Electric electromagnetic force acts over large distances force manifests itself in atmosphere where the and does not need any intervening medium. It atoms are ionised and that leads to lightning. is enormously strong compared to gravity. The 2020-21

PHYSICAL WORLD 9 Satyendranath Bose (1894-1974) Satyendranath Bose, born in Calcutta in 1894, is among the great Indian physicists who made a fundamental contribution to the advance of science in the twentieth century. An outstanding student throughout, Bose started his career in 1916 as a lecturer in physics in Calcutta University; five years later he joined Dacca University. Here in 1924, in a brilliant flash of insight, Bose gave a new derivation of Planck’s law, treating radiation as a gas of photons and employing new statistical methods of counting of photon states. He wrote a short paper on the subject and sent it to Einstein who immediately recognised its great significance, translated it in German and forwarded it for publication. Einstein then applied the same method to a gas of molecules. The key new conceptual ingredient in Bose’s work was that the particles were regarded as indistinguishable, a radical departure from the assumption that underlies the classical Maxwell- Boltzmann statistics. It was soon realised that the new Bose-Einstein statistics was applicable to particles with integers spins, and a new quantum statistics (Fermi-Dirac statistics) was needed for particles with half integers spins satisfying Pauli’s exclusion principle. Particles with integers spins are now known as bosons in honour of Bose. An important consequence of Bose-Einstein statistics is that a gas of molecules below a certain temperature will undergo a phase transition to a state where a large fraction of atoms populate the same lowest energy state. Some seventy years were to pass before the pioneering ideas of Bose, developed further by Einstein, were dramatically confirmed in the observation of a new state of matter in a dilute gas of ultra cold alkali atoms - the Bose-Eintein condensate. If we reflect a little, the enormous strength strength. It is charge-independent and acts of the electromagnetic force compared to equally between a proton and a proton, a gravity is evident in our daily life. When we neutron and a neutron, and a proton and a hold a book in our hand, we are balancing the neutron. Its range is, however, extremely small, gravitational force on the book due to the huge of about nuclear dimensions (10–15m). It is mass of the earth by the ‘normal force’ responsible for the stability of nuclei. The provided by our hand. The latter is nothing electron, it must be noted, does not experience but the net electromagnetic force between the this force. charged constituents of our hand and the book, at the surface in contact. If Recent developments have, however, electromagnetic force were not intrinsically so indicated that protons and neutrons are built much stronger than gravity, the hand of the out of still more elementary constituents called strongest man would crumble under the quarks. weight of a feather ! Indeed, to be consistent, in that circumstance, we ourselves would 1.4.4 Weak Nuclear Force crumble under our own weight ! The weak nuclear force appears only in certain 1.4.3 Strong Nuclear Force nuclear processes such as the β-decay of a nucleus. In β-decay, the nucleus emits an The strong nuclear force binds protons and electron and an uncharged particle called neutrons in a nucleus. It is evident that without neutrino. The weak nuclear force is not as weak some attractive force, a nucleus will be as the gravitational force, but much weaker unstable due to the electric repulsion between than the strong nuclear and electromagnetic its protons. This attractive force cannot be forces. The range of weak nuclear force is gravitational since force of gravity is negligible exceedingly small, of the order of 10–16 m. compared to the electric force. A new basic force must, therefore, be invoked. The strong nuclear 1.4.5 Towards Unification of Forces force is the strongest of all fundamental forces, about 100 times the electromagnetic force in We remarked in section 1.1 that unification is a basic quest in physics. Great advances in physics often amount to unification of different 2020-21

10 PHYSICS Table 1.3 Fundamental forces of nature Name Relative Range Operates among strength Gravitational force Weak nuclear force 10–39 Infinite All objects in the universe 10–13 Very short, Sub-nuclear Electromagnetic force size (∼10–16m) Some elementary particles, Strong nuclear force 10–2 particularly electron and 1 Infinite neutrino Short, nuclear size (∼10–15m) Charged particles Nucleons, heavier elementary particles theories and domains. Newton unified terrestrial 1.5 NATURE OF PHYSICAL LAWS and celestial domains under a common law of gravitation. The experimental discoveries of Physicists explore the universe. Their Oersted and Faraday showed that electric and investigations, based on scientific processes, magnetic phenomena are in general range from particles that are smaller than inseparable. Maxwell unified electromagnetism atoms in size to stars that are very far away. In and optics with the discovery that light is an addition to finding the facts by observation and electromagnetic wave. Einstein attempted to experimentation, physicists attempt to discover unify gravity and electromagnetism but could the laws that summarise (often as mathematical not succeed in this venture. But this did not equations) these facts. deter physicists from zealously pursuing the goal of unification of forces. In any physical phenomenon governed by different forces, several quantities may change Recent decades have seen much progress on with time. A remarkable fact is that some special this front. The electromagnetic and the weak physical quantities, however, remain constant nuclear force have now been unified and are in time. They are the conserved quantities of seen as aspects of a single ‘electro-weak’ force. nature. Understanding these conservation What this unification actually means cannot principles is very important to describe the be explained here. Attempts have been (and are observed phenomena quantitatively. being) made to unify the electro-weak and the strong force and even to unify the gravitational For motion under an external conservative force with the rest of the fundamental forces. force, the total mechanical energy i.e. the sum Many of these ideas are still speculative and of kinetic and potential energy of a body is a inconclusive. Table 1.4 summarises some of the constant. The familiar example is the free fall of milestones in the progress towards unification an object under gravity. Both the kinetic energy of forces in nature. of the object and its potential energy change continuously with time, but the sum remains fixed. If the object is released from rest, the initial Table 1.4 Progress in unification of different forces/domains in nature 2020-21

PHYSICAL WORLD 11 potential energy is completely converted into the the other end, all kinds of violent phenomena kinetic energy of the object just before it hits occur in the universe all the time. Yet the total the ground. This law restricted for a conservative energy of the universe (the most ideal isolated force should not be confused with the general system possible !) is believed to remain law of conservation of energy of an isolated unchanged. system (which is the basis of the First Law of Thermodynamics). Until the advent of Einstein’s theory of relativity, the law of conservation of mass was The concept of energy is central to physics regarded as another basic conservation law of and the expressions for energy can be written nature, since matter was thought to be for every physical system. When all forms of indestructible. It was (and still is) an important energy e.g., heat, mechanical energy, electrical principle used, for example, in the analysis of energy etc., are counted, it turns out that energy chemical reactions. A chemical reaction is is conserved. The general law of conservation of basically a rearrangement of atoms among energy is true for all forces and for any kind of different molecules. If the total binding energy transformation between different forms of of the reacting molecules is less than the total energy. In the falling object example, if you binding energy of the product molecules, the include the effect of air resistance during the difference appears as heat and the reaction is fall and see the situation after the object hits exothermic. The opposite is true for energy the ground and stays there, the total absorbing (endothermic) reactions. However, mechanical energy is obviously not conserved. since the atoms are merely rearranged but not The general law of energy conservation, however, destroyed, the total mass of the reactants is the is still applicable. The initial potential energy same as the total mass of the products in a of the stone gets transformed into other forms chemical reaction. The changes in the binding of energy : heat and sound. (Ultimately, sound energy are too small to be measured as changes after it is absorbed becomes heat.) The total in mass. energy of the system (stone plus the surroundings) remains unchanged. According to Einstein’s theory, mass m is equivalent to energy E given by the relation The law of conservation of energy is thought E = mc2, where c is speed of light in vacuum. to be valid across all domains of nature, from the microscopic to the macroscopic. It is In a nuclear process mass gets converted to routinely applied in the analysis of atomic, energy (or vice-versa). This is the energy which nuclear and elementary particle processes. At is released in a nuclear power generation and nuclear explosions. Sir C.V. Raman (1888-1970) Chandrashekhara Venkata Raman was born on 07 Nov 1888 in Thiruvanaikkaval. He finished his schooling by the age of eleven. He graduated from Presidency College, Madras. After finishing his education he joined financial services of the Indian Government. While in Kolkata, he started working on his area of interest at Indian Asso- ciation for Cultivation of Science founded by Dr. Mahendra Lal Sirkar, during his evening hours. His area of interest included vibrations, variety of musical instru- ments, ultrasonics, diffraction and so on. In 1917 he was offered Professorship at Calcutta University. In 1924 he was elected ‘Fellow’ of the Royal Society of London and received Nobel prize in Physics in 1930 for his discovery, now known as Raman Effect. The Raman Effect deals with scattering of light by molecules of a medium when they are excited to vibrational energy levels. This work opened totally new avenues for research for years to come. He spent his later years at Bangalore, first at Indian Institute of Science and then at Raman Re- search Institute. His work has inspired generation of young students. 2020-21

12 PHYSICS Energy is a scalar quantity. But all conserved Conservation laws in physics quantities are not necessarily scalars. The total linear momentum and the total angular Conservation of energy, momentum, angular momentum (both vectors) of an isolated system momentum, charge, etc are considered to be are also conserved quantities. These laws can fundamental laws in physics. At this moment, be derived from Newton’s laws of motion in there are many such conservation laws. Apart from mechanics. But their validity goes beyond the above four, there are others which mostly deal mechanics. They are the basic conservation with quantities which have been introduced in laws of nature in all domains, even in those nuclear and particle physics. Some of the where Newton’s laws may not be valid. conserved quantities are called spin, baryon number, strangeness, hypercharge, etc, but you Besides their great simplicity and generality, need not worry about them. the conservation laws of nature are very useful in practice too. It often happens that we cannot A conservation law is a hypothesis, based on solve the full dynamics of a complex problem observations and experiments. It is important to involving different particles and forces. The remember that a conservation law cannot be conservation laws can still provide useful proved. It can be verified, or disproved, by results. For example, we may not know the experiments. An experiment whose result is in complicated forces that act during a collision conformity with the law verifies or substantiates of two automobiles; yet momentum the law; it does not prove the law. On the other conservation law enables us to bypass the hand, a single experiment whose result goes complications and predict or rule out possible against the law is enough to disprove it. outcomes of the collision. In nuclear and elementary particle phenomena also, the It would be wrong to ask somebody to prove conservation laws are important tools of the law of conservation of energy. This law is an analysis. Indeed, using the conservation laws outcome of our experience over several centuries, of energy and momentum for β-decay, Wolfgang and it has been found to be valid in all Pauli (1900-1958) correctly predicted in 1931 experiments, in mechanics, thermodynamics, the existence of a new particle (now called electromagnetism, optics, atomic and nuclear neutrino) emitted in β-decay along with the physics, or any other area. electron. Some students feel that they can prove the Conservation laws have a deep connection conservation of mechanical energy from a body with symmetries of nature that you will explore falling under gravity, by adding the kinetic and in more advanced courses in physics. For potential energies at a point and showing that it example, an important observation is that the turns out to be constant. As pointed out above, laws of nature do not change with time! If you this is only a verification of the law, not its proof. perform an experiment in your laboratory today and repeat the same experiment (on the same because of differing conditions at different objects under identical conditions) after a year, locations. For example, the acceleration due to the results are bound to be the same. It turns gravity at the moon is one-sixth that at the earth, out that this symmetry of nature with respect to but the law of gravitation is the same both on translation (i.e. displacement) in time is the moon and the earth.) This symmetry of the equivalent to the law of conservation of energy. laws of nature with respect to translation in Likewise, space is homogeneous and there is no space gives rise to conservation of linear (intrinsically) preferred location in the universe. momentum. In the same way isotropy of space To put it more clearly, the laws of nature are the (no intrinsically preferred direction in space) same everywhere in the universe. (Caution : the underlies the law of conservation of angular phenomena may differ from place to place momentum*. The conservation laws of charge and other attributes of elementary particles can also be related to certain abstract symmetries. Symmetries of space and time and other abstract symmetries play a central role in modern theories of fundamental forces in nature. * See Chapter 7 2020-21

PHYSICAL WORLD 13 SUMMARY 1. Physics deals with the study of the basic laws of nature and their manifestation in different phenomena. The basic laws of physics are universal and apply in widely different contexts and conditions. 2. The scope of physics is wide, covering a tremendous range of magnitude of physical quantities. 3. Physics and technology are related to each other. Sometimes technology gives rise to new physics; at other times physics generates new technology. Both have direct impact on society. 4. There are four fundamental forces in nature that govern the diverse phenomena of the macroscopic and the microscopic world. These are the ‘gravitational force’, the ‘electromagnetic force’, the ‘strong nuclear force’, and the ‘weak nuclear force’. Unification of different forces/domains in nature is a basic quest in physics. 5. The physical quantities that remain unchanged in a process are called conserved quantities. Some of the general conservation laws in nature include the laws of conservation of mass, energy, linear momentum, angular momentum, charge, parity, etc. Some conservation laws are true for one fundamental force but not for the other. 6. Conservation laws have a deep connection with symmetries of nature. Symmetries of space and time, and other types of symmetries play a central role in modern theories of fundamental forces in nature. EXERCISES Note for the student The exercises given here are meant to enhance your awareness about the issues surrounding science, technology and society and to encourage you to think and formulate your views about them. The questions may not have clear-cut ‘objective’ answers. Note for the teacher The exercises given here are not for the purpose of a formal examination. 1.1 Some of the most profound statements on the nature of science have come from Albert Einstein, one of the greatest scientists of all time. What do you think did Einstein mean when he said : “The most incomprehensible thing about the world is that it is comprehensible”? 1.2 “Every great physical theory starts as a heresy and ends as a dogma”. Give some examples from the history of science of the validity of this incisive remark. 1.3 “Politics is the art of the possible”. Similarly, “Science is the art of the soluble”. Explain this beautiful aphorism on the nature and practice of science. 1.4 Though India now has a large base in science and technology, which is fast expanding, it is still a long way from realising its potential of becoming a world leader in science. Name some important factors, which in your view have hindered the advancement of science in India. 1.5 No physicist has ever “seen” an electron. Yet, all physicists believe in the existence of electrons. An intelligent but superstitious man advances this analogy to argue that ‘ghosts’ exist even though no one has ‘seen’ one. How will you refute his argument ? 1.6 The shells of crabs found around a particular coastal location in Japan seem mostly to resemble the legendary face of a Samurai. Given below are two explanations of this observed fact. Which of these strikes you as a scientific explanation ? (a) A tragic sea accident several centuries ago drowned a young Samurai. As a tribute to his bravery, nature through its inscrutable ways immortalised his face by imprinting it on the crab shells in that area. 2020-21

14 PHYSICS (b) After the sea tragedy, fishermen in that area, in a gesture of honour to their dead hero, let free any crab shell caught by them which accidentally had a shape resembling the face of a Samurai. Consequently, the particular shape of the crab shell survived longer and therefore in course of time the shape was genetically propagated. This is an example of evolution by artificial selection. [Note : This interesting illustration taken from Carl Sagan’s ‘The Cosmos’ highlights the fact that often strange and inexplicable facts which on the first sight appear ‘supernatural’ actually turn out to have simple scientific explanations. Try to think out other examples of this kind]. 1.7 The industrial revolution in England and Western Europe more than two centuries ago was triggered by some key scientific and technological advances. What were these advances ? 1.8 It is often said that the world is witnessing now a second industrial revolution, which will transform the society as radically as did the first. List some key contemporary areas of science and technology, which are responsible for this revolution. 1.9 Write in about 1000 words a fiction piece based on your speculation on the science and technology of the twenty-second century. 1.10 Attempt to formulate your ‘moral’ views on the practice of science. Imagine yourself stumbling upon a discovery, which has great academic interest but is certain to have nothing but dangerous consequences for the human society. How, if at all, will you resolve your dilemma ? 1.11 Science, like any knowledge, can be put to good or bad use, depending on the user. Given below are some of the applications of science. Formulate your views on whether the particular application is good, bad or something that cannot be so clearly categorised : (a) Mass vaccination against small pox to curb and finally eradicate this disease from the population. (This has already been successfully done in India). (b) Television for eradication of illiteracy and for mass communication of news and ideas. (c) Prenatal sex determination (d) Computers for increase in work efficiency (e) Putting artificial satellites into orbits around the Earth (f ) Development of nuclear weapons (g) Development of new and powerful techniques of chemical and biological warfare). (h) Purification of water for drinking (i) Plastic surgery (j ) Cloning 1.12 India has had a long and unbroken tradition of great scholarship — in mathematics, astronomy, linguistics, logic and ethics. Yet, in parallel with this, several superstitious and obscurantistic attitudes and practices flourished in our society and unfortunately continue even today — among many educated people too. How will you use your knowledge of science to develop strategies to counter these attitudes ? 1.13 Though the law gives women equal status in India, many people hold unscientific views on a woman’s innate nature, capacity and intelligence, and in practice give them a secondary status and role. Demolish this view using scientific arguments, and by quoting examples of great women in science and other spheres; and persuade yourself and others that, given equal opportunity, women are on par with men. 1.14 “It is more important to have beauty in the equations of physics than to have them agree with experiments”. The great British physicist P. A. M. Dirac held this view. Criticize this statement. Look out for some equations and results in this book which strike you as beautiful. 1.15 Though the statement quoted above may be disputed, most physicists do have a feeling that the great laws of physics are at once simple and beautiful. Some of the notable physicists, besides Dirac, who have articulated this feeling, are : Einstein, Bohr, Heisenberg, Chandrasekhar and Feynman. You are urged to make special efforts to get 2020-21

PHYSICAL WORLD 15 access to the general books and writings by these and other great masters of physics. (See the Bibliography at the end of this book.) Their writings are truly inspiring ! 1.16 Textbooks on science may give you a wrong impression that studying science is dry and all too serious and that scientists are absent-minded introverts who never laugh or grin. This image of science and scientists is patently false. Scientists, like any other group of humans, have their share of humorists, and many have led their lives with a great sense of fun and adventure, even as they seriously pursued their scientific work. Two great physicists of this genre are Gamow and Feynman. You will enjoy reading their books listed in the Bibliography. 2020-21

CHAPTER TWO UNITS AND MEASUREMENT 2.1 Introduction 2.1 INTRODUCTION Measurement of any physical quantity involves comparison 2.2 The international system of with a certain basic, arbitrarily chosen, internationally units accepted reference standard called unit. The result of a measurement of a physical quantity is expressed by a 2.3 Measurement of length number (or numerical measure) accompanied by a unit. 2.4 Measurement of mass Although the number of physical quantities appears to be very large, we need only a limited number of units for 2.5 Measurement of time expressing all the physical quantities, since they are inter- related with one another. The units for the fundamental or 2.6 Accuracy, precision of base quantities are called fundamental or base units. The instruments and errors in units of all other physical quantities can be expressed as measurement combinations of the base units. Such units obtained for the derived quantities are called derived units. A complete set 2.7 Significant figures of these units, both the base units and derived units, is known as the system of units. 2.8 Dimensions of physical quantities 2.2 THE INTERNATIONAL SYSTEM OF UNITS In earlier time scientists of different countries were using 2.9 Dimensional formulae and different systems of units for measurement. Three such dimensional equations systems, the CGS, the FPS (or British) system and the MKS system were in use extensively till recently. 2.10 Dimensional analysis and its applications The base units for length, mass and time in these systems were as follows : Summary • In CGS system they were centimetre, gram and second Exercises Additional exercises respectively. • In FPS system they were foot, pound and second respectively. • In MKS system they were metre, kilogram and second respectively. The system of units which is at present internationally accepted for measurement is the Système Internationale d’ Unites (French for International System of Units), abbreviated as SI. The SI, with standard scheme of symbols, units and abbreviations, developed by the Bureau International des Poids et measures (The International Bureau of Weights and Measures, BIPM) in 1971 were recently revised by the General Conference on Weights and Measures in November 2018. The scheme is now for 2020-21

UNITS AND MEASUREMENT 17 international usage in scientific, technical, industrial and commercial work. Because SI units used decimal system, conversions within the system are quite simple and convenient. We shall follow the SI units in this book. In SI, there are seven base units as given in Table 2.1. Besides the seven base units, there are two more (a) units that are defined for (a) plane angle dθ as the ratio of length of arc ds to the radius r and (b) solid angle dΩ (b) as the ratio of the intercepted area dA of the spherical Fig. 2.1 Description of (a) plane angle dθ surface, described about the apex O as the centre, to the square of its radius r, as shown in Fig. 2.1(a) and and (b) solid angle dΩ . (b) respectively. The unit for plane angle is radian with the symbol rad and the unit for the solid angle is steradian with the symbol sr. Both these are dimensionless quantities. Table 2.1 SI Base Quantities and Units* Base SI Units quantity Name Symbol Definition Length metre m The metre, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299792458 when expressed in the unit m s–1, where the second is defined in terms of the caesium frequency ∆νcs. Mass kilogram kg The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.62607015×10–34 when expressed in the unit J s, which is equal to kg m2 s–1, where the metre and the second are defined in terms of c and ∆νcs. Time second s The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency ∆νcs, the unperturbed ground- state hyperfine transition frequency of the caesium-133 atom, to be 9192631770 when expressed in the unit Hz, which is equal to s–1. Electric ampere A The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge e to be 1.602176634×10–19 when expressed in the unit C, which is equal to A s, where the second is defined in terms of ∆νcs. Thermo kelvin K The kelvin, symbol K, is the SI unit of thermodynamic temperature. dynamic It is defined by taking the fixed numerical value of the Boltzmann constant Temperature k to be 1.380649×10–23 when expressed in the unit J K–1, which is equal to kg m2 s–2 k–1, where the kilogram, metre and second are defined in terms of h, c and ∆νcs. Amount of mole mol The mole, symbol mol, is the SI unit of amount of substance. One mole substance contains exactly 6.02214076×1023 elementary entities. This number is the fixed numerical value of the Avogadro ncounmstbaenr.t,TNhAe, when expressed in the unit mol–1 and is called the Avogadro amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles. Luminous candela cd The candela, symbol cd, is the SI unit of luminous intensity in given direction. intensity It is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of firseqeuquenaclyto54c0d×s1r01W2 H–1z,,oKrcdc,dtosbrek6g8–13mw–2hse3n, wexhperreessthede in the unit lm W–1, which kilogram, metre and second are defined in terms of h, c and ∆νcs. * The values mentioned here need not be remembered or asked in a test. They are given here only to indicate the extent of accuracy to which they are measured. With progress in technology, the measuring techniques get improved leading to measurements with greater precision. The definitions of base units are revised to keep up with this progress. 2020-21

18 PHYSICS Table 2.2 Some units retained for general use (Though outside SI) Note that when mole is used, the elementary 2.3.1 Measurement of Large Distances entities must be specified. These entities may be atoms, molecules, ions, electrons, Large distances such as the distance of a planet other particles or specified groups of such or a star from the earth cannot be measured particles. directly with a metre scale. An important method in such cases is the parallax method. We employ units for some physical quantities that can be derived from the seven base units When you hold a pencil in front of you against (Appendix A 6). Some derived units in terms of some specific point on the background (a wall) the SI base units are given in (Appendix A 6.1). and look at the pencil first through your left eye Some SI derived units are given special names A (closing the right eye) and then look at the (Appendix A 6.2 ) and some derived SI units make pencil through your right eye B (closing the left use of these units with special names and the eye), you would notice that the position of the seven base units (Appendix A 6.3). These are pencil seems to change with respect to the point given in Appendix A 6.2 and A 6.3 for your ready on the wall. This is called parallax. The reference. Other units retained for general use distance between the two points of observation are given in Table 2.2. is called the basis. In this example, the basis is the distance between the eyes. Common SI prefixes and symbols for multiples and sub-multiples are given in Appendix A2. To measure the distance D of a far away General guidelines for using symbols for physical planet S by the parallax method, we observe it quantities, chemical elements and nuclides are from two different positions (observatories) A and given in Appendix A7 and those for SI units and B on the Earth, separated by distance AB = b some other units are given in Appendix A8 for at the same time as shown in Fig. 2.2. We your guidance and ready reference. measure the angle between the two directions along which the planet is viewed at these two 2.3 MEASUREMENT OF LENGTH points. The ∠ASB in Fig. 2.2 represented by symbol θ is called the parallax angle or You are already familiar with some direct methods for the measurement of length. For example, a parallactic angle. metre scale is used for lengths from 10–3 m to 102 m. A vernier callipers is used for lengths to an As the planet is very far away, b << 1, and accuracy of 10–4 m. A screw gauge and a D spherometer can be used to measure lengths as less as to 10–5m. To measure lengths beyond these therefore, θ is very small. Then we ranges, we make use of some special indirect approximately take AB as an arc of length b of a methods. circle with centre at S and the distance D as 2020-21

UNITS AND MEASUREMENT 19 the radius AS = BS so that AB = b = D θ where AO; but he finds the line of sight of C shifted θ is in radians. from the original line of sight by an angle θ = 400 (θ is known as ‘parallax’) estimate b (2.1) the distance of the tower C from his original D= position A. θ Fig. 2.2 Parallax method. Fig. 2.3 Having determined D, we can employ a similar Answer We have, parallax angle θ = 400 From Fig. 2.3, AB = AC tan θ method to determine the size or angular diameter AC = AB/tanθ = 100 m/tan 400 = 100 m/0.8391 = 119 m of the planet. If d is the diameter of the planet and α the angular size of the planet (the angle subtended by d at the earth), we have Example 2.3 The moon is observed from α = d/D (2.2) two diametrically opposite points A and B The angle α can be measured from the same on Earth. The angle θ subtended at the location on the earth. It is the angle between moon by the two directions of observation the two directions when two diametrically is 1o 54′. Given the diameter of the Earth to opposite points of the planet are viewed through be about 1.276 × 107 m, compute the distance of the moon from the Earth. the telescope. Since D is known, the diameter d Answer We have θ = 1° 54′ = 114′ of the planet can be determined using Eq. (2.2). Example 2.1 Calculate the angle of ( )= (114 × 60)′′ × 4.85 ×10-6 rad (a) 10 (degree) (b) 1′ (minute of arc or arcmin) and (c) 1″(second of arc or arc second) in = 3.32 ×10−2 rad, radians. Use 3600=2π rad, 10=60′ and 1′ = 60 ″ since 1\" = 4.85 × 10−6rad. Answer (a) We have 3600 = 2π rad Also b = AB = 1.276 ×107 m 10 = (π /180) rad = 1.745×10–2 rad Hence from Eq. (2.1), we have the earth-moon (b) 10 = 60′ = 1.745×10–2 rad distance, 1′ = 2.908×10–4 rad 2.91×10–4 rad D =b/θ (c) 1′ = 60″ = 2.908×10–4 rad 1″ = 4.847×10–4 rad 4.85×10–6 rad = 1.276 × 107 3.32 × 10-2 Example 2.2 A man wishes to estimate the distance of a nearby tower from him. = 3.84 × 108 m He stands at a point A in front of the tower C and spots a very distant object O in line Example 2.4 The Sun’s angular diameter with AC. He then walks perpendicular to is measured to be 1920′′. The distance D of AC up to B, a distance of 100 m, and looks the Sun from the Earth is 1.496 ×1011 m. at O and C again. Since O is very distant, the direction BO is practically the same as What is the diameter of the Sun ? 2020-21

20 PHYSICS Answer Sun’s angular diameter α of this solution and dilute it to 20 cm3, using alcohol. So, the concentration of the solution is = 1920\" equal to  1  cm 3 of oleic acid/cm3 of = 1920 × 4.85 × 10−6 rad × 20 20 = 9.31×10−3 rad Sun’s diameter solution. Next we lightly sprinkle some d =α D lycopodium powder on the surface of water in a =  9.31×10−3  × 1.496 ×1011  m large trough and we put one drop of this solution = 1.39 × 109 m in the water. The oleic acid drop spreads into a 2.3.2 Estimation of Very Small Distances: thin, large and roughly circular film of molecular Size of a Molecule thickness on water surface. Then, we quickly To measure a very small size, like that of a molecule (10–8 m to 10–10 m), we have to adopt measure the diameter of the thin film to get its special methods. We cannot use a screw gauge or similar instruments. Even a microscope has area A. Suppose we have dropped n drops in certain limitations. An optical microscope uses the water. Initially, we determine the visible light to ‘look’ at the system under investigation. As light has wave like features, approximate volume of each drop (V cm3). the resolution to which an optical microscope can be used is the wavelength of light (A detailed Volume of n drops of solution explanation can be found in the Class XII = nV cm3 Physics textbook). For visible light the range of wavelengths is from about 4000 Å to 7000 Å Amount of oleic acid in this solution (1 angstrom = 1 Å = 10-10 m). Hence an optical microscope cannot resolve particles with sizes = nV  1 20  cm3 smaller than this. Instead of visible light, we can  20 ×  use an electron beam. Electron beams can be   focussed by properly designed electric and magnetic fields. The resolution of such an This solution of oleic acid spreads very fast electron microscope is limited finally by the fact on the surface of water and forms a very thin that electrons can also behave as waves ! (You layer of thickness t. If this spreads to form a will learn more about this in class XII). The film of area A cm2, then the thickness of the wavelength of an electron can be as small as a film fraction of an angstrom. Such electron microscopes with a resolution of 0.6 Å have been t = Volume of the film built. They can almost resolve atoms and Area of the film molecules in a material. In recent times, tunnelling microscopy has been developed in or, t = 20 nV A cm (2.3) which again the limit of resolution is better than × 20 an angstrom. It is possible to estimate the sizes of molecules. If we assume that the film has mono-molecular thickness, then this becomes the size or diameter A simple method for estimating the molecular size of oleic acid is given below. Oleic acid is a of a molecule of oleic acid. The value of this soapy liquid with large molecular size of the thickness comes out to be of the order of 10–9 m. order of 10–9 m. Example 2.5 If the size of a nucleus (in The idea is to first form mono-molecular layer the range of 10–15 to 10–14 m) is scaled up of oleic acid on water surface. to the tip of a sharp pin, what roughly is the size of an atom ? Assume tip of the pin We dissolve 1 cm3 of oleic acid in alcohol to make a solution of 20 cm3. Then we take 1 cm3 to be in the range 10–5m to 10–4m. Answer The size of a nucleus is in the range of 10–15 m and 10–14 m. The tip of a sharp pin is taken to be in the range of 10–5 m and 10–4 m. Thus we are scaling up by a factor of 1010. An atom roughly of size 10–10 m will be scaled up to a size of 1 m. Thus a nucleus in an atom is as small in size as the tip of a sharp pin placed at the centre of a sphere of radius about a metre long. 2020-21

UNITS AND MEASUREMENT 21 2.3.3 Range of Lengths While dealing with atoms and molecules, the kilogram is an inconvenient unit. In this case, The sizes of the objects we come across in the there is an important standard unit of mass, called the unified atomic mass unit (u), which universe vary over a very wide range. These may has been established for expressing the mass of atoms as vary from the size of the order of 10–14 m of the tiny nucleus of an atom to the size of the order of 1026 m of the extent of the observable universe. Table 2.3 gives the range and order of lengths 1 unified atomic mass unit = 1u and sizes of some of these objects. ( )= (1/12) of the mass of an atom of carbon-12 We also use certain special length units for isotope 12 C including the mass of electrons 6 short and large lengths. These are = 1.66 × 10–27 kg 1 fermi = 1 f = 10–15 m 1 angstrom = 1 Å = 10–10 m Mass of commonly available objects can be determined by a common balance like the one 1 astronomical unit = 1 AU (average distance used in a grocery shop. Large masses in the universe like planets, stars, etc., based on of the Sun from the Earth) Newton’s law of gravitation can be measured by using gravitational method (See Chapter 8). For = 1.496 × 1011 m measurement of small masses of atomic/sub- atomic particles etc., we make use of mass 1 light year = 1 ly= 9.46 × 1015 m (distance spectrograph in which radius of the trajectory is proportional to the mass of a charged particle that light travels with velocity of moving in uniform electric and magnetic field. 3 × 108 m s–1 in 1 year) 1 parsec = 3.08 × 1016 m (Parsec is the distance at which average radius of earth’s orbit subtends an angle of 1 arc second) 2.4 MEASUREMENT OF MASS 2.4.1 Range of Masses Mass is a basic property of matter. It does not The masses of the objects, we come across in depend on the temperature, pressure or location the universe, vary over a very wide range. These of the object in space. The SI unit of mass is may vary from tiny mass of the order of 10-30 kg kilogram (kg). It is defined by taking the fixed of an electron to the huge mass of about 1055 kg numerical value of the Plank Constant h to be of the known universe. Table 2.4 gives the range 6.62607015×10–34 when expressed in the unit of and order of the typical masses of various Js which is equal to kg m2s–1, where the objects. metre and the second are defined is terms of C and ∆νcs. Table 2.3 Range and order of lengths 2020-21

22 PHYSICS Table 2.4 Range and order of masses ± 1 × 10–15, i.e. 1 part in 1015. This implies that the uncertainty gained over time by such a 2.5 MEASUREMENT OF TIME device is less than 1 part in 1015; they lose or To measure any time interval we need a clock. gain no more than 32 µs in one year. In view of We now use an atomic standard of time, which the tremendous accuracy in time measurement, is based on the periodic vibrations produced in the SI unit of length has been expressed in terms a cesium atom. This is the basis of the caesium the path length light travels in certain interval clock, sometimes called atomic clock, used in of time (1/299, 792, 458 of a second) (Table 2.1). the national standards. Such standards are available in many laboratories. In the caesium The time interval of events that we come atomic clock, the second is taken as the time across in the universe vary over a very wide needed for 9,192,631,770 vibrations of the range. Table 2.5 gives the range and order of radiation corresponding to the transition some typical time intervals. between the two hyperfine levels of the ground state of caesium-133 atom. The vibrations of the You may notice that there is an interesting caesium atom regulate the rate of this caesium coincidence between the numbers appearing atomic clock just as the vibrations of a balance in Tables 2.3 and 2.5. Note that the ratio of the wheel regulate an ordinary wristwatch or the longest and shortest lengths of objects in our vibrations of a small quartz crystal regulate a universe is about 1041. Interestingly enough, quartz wristwatch. the ratio of the longest and shortest time intervals associated with the events and objects The caesium atomic clocks are very accurate. in our universe is also about 1041. This number, In principle they provide portable standard. The 1041 comes up again in Table 2.4, which lists national standard of time interval ‘second’ as typical masses of objects. The ratio of the well as the frequency is maintained through four largest and smallest masses of the objects in cesium atomic clocks. A caesium atomic clock our universe is about (1041)2. Is this a curious is used at the National Physical Laboratory coincidence between these large numbers (NPL), New Delhi to maintain the Indian purely accidental ? standard of time. 2.6 ACCURACY, PRECISION OF INSTRUMENTS In our country, the NPL has the responsibility of maintenance and improvement of physical AND ERRORS IN MEASUREMENT standards, including that of time, frequency, etc. Note that the Indian Standard Time (IST) is Measurement is the foundation of all linked to this set of atomic clocks. The efficient experimental science and technology. The result caesium atomic clocks are so accurate that they of every measurement by any measuring impart the uncertainty in time realisation as instrument contains some uncertainty. This uncertainty is called error. Every calculated quantity which is based on measured values, also has an error. We shall distinguish between two terms: accuracy and precision. The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. Precision tells us to what resolution or limit the quantity is measured. The accuracy in measurement may depend on several factors, including the limit or the resolution of the measuring instrument. For example, suppose the true value of a certain length is near 3.678 cm. In one experiment, using a measuring instrument of resolution 0.1 cm, the measured value is found to be 3.5 cm, while in another experiment using a measuring device of greater resolution, say 0.01 cm, the length is determined to be 3.38 cm. The first measurement has more accuracy (because it is 2020-21

UNITS AND MEASUREMENT 23 Table 2.5 Range and order of time intervals closer to the true value) but less precision (its of a human body, a thermometer placed resolution is only 0.1 cm), while the under the armpit will always give a second measurement is less accurate but temperature lower than the actual value of more precise. Thus every measurement is the body temperature. Other external approximate due to errors in measurement. In conditions (such as changes in temperature, general, the errors in measurement can be humidity, wind velocity, etc.) during the broadly classified as (a) systematic errors and experiment may systematically affect the (b) random errors. measurement. (c) Personal errors that arise due to an Systematic errors individual’s bias, lack of proper setting of the apparatus or individual’s carelessness The systematic errors are those errors that in taking observations without observing tend to be in one direction, either positive or proper precautions, etc. For example, if you, negative. Some of the sources of systematic by habit, always hold your head a bit too far errors are : to the right while reading the position of a needle on the scale, you will introduce an (a) Instrumental errors that arise from the error due to parallax. errors due to imperfect design or calibration of the measuring instrument, zero error in Systematic errors can be minimised by the instrument, etc. For example, the improving experimental techniques, selecting temperature graduations of a thermometer better instruments and removing personal bias may be inadequately calibrated (it may read as far as possible. For a given set-up, these 104 °C at the boiling point of water at STP errors may be estimated to a certain extent and whereas it should read 100 °C); in a vernier the necessary corrections may be applied to the callipers the zero mark of vernier scale may readings. not coincide with the zero mark of the main scale, or simply an ordinary metre scale may Random errors be worn off at one end. The random errors are those errors, which occur (b) Imperfection in experimental technique irregularly and hence are random with respect or procedure To determine the temperature 2020-21

24 PHYSICS to sign and size. These can arise due to random as to underestimate the true value of the and unpredictable fluctuations in experimental quantity. conditions (e.g. unpredictable fluctuations in temperature, voltage supply, mechanical The magnitude of the difference vibrations of experimental set-ups, etc), personal (unbiased) errors by the observer taking between the individual measurement and readings, etc. For example, when the same person repeats the same observation, it is very the true value of the quantity is called the likely that he may get different readings absolute error of the measurement. This everytime. is denoted by |∆a |. In absence of any other method of knowing true value, we considered Least count error arithmatic mean as the true value. Then the errors in the individual measurement values The smallest value that can be measured by the from the true value, are measuring instrument is called its least count. All the readings or measured values are good only ∆a1 = a1 – amean, up to this value. ∆a2 = .a..2. amean, – The least count error is the error .... .... associated with the resolution of the instrument. For example, a vernier callipers has the least .... .... .... count as 0.01 cm; a spherometer may have a least count of 0.001 cm. Least count error ∆a n = an – amean belongs to the category of random errors but within a limited size; it occurs with both The ∆a calculated above may be positive in systematic and random errors. If we use a metre scale for measurement of length, it may have certain cases and negative in some other graduations at 1 mm division scale spacing or cases. But absolute error |∆a| will always interval. be positive. Using instruments of higher precision, (b) The arithmetic mean of all the absolute errors improving experimental techniques, etc., we can is taken as the final or mean absolute error reduce the least count error. Repeating the of the value of the physical quantity a. It is observations several times and taking the arithmetic mean of all the observations, the represented by ∆amean. mean value would be very close to the true value of the measured quantity. Thus, 2.6.1 Absolute Error, Relative Error and ∆amean = (|∆a1|+|∆a2 |+|∆a3|+...+ |∆an|)/n Percentage Error (2.6) (a) Suppose the values obtained in several ∑n (2.7) amreitahsmueretimc emnetasnaorfethae1s, eav2a, lau3e.s...i,s atank. enThase the best possible value of the quantity under = |∆ai|/n the given conditions of measurement as : i =1 If we do a single measurement, the value we get may be in the range amean ± ∆amean i.e. =a amean ± ∆amean or, amean – ∆amean ≤ a ≤ amean + ∆amean amean = (a1+a2+a3+...+an ) / n (2.4) (2.8) or, This implies that any measurement of the ∑n (2.5) physical quantity a is likely to lie between amean = a i / n (amean+ ∆amean) and (amean− ∆amean). i =1 (c) Instead of the absolute error, we often use This is because, as explained earlier, it is the relative error or the percentage error reasonable to suppose that individual (δa). The relative error is the ratio of the mean absolute error ∆amean to the mean measurements are as likely to overestimate value amean of the quantity measured. 2020-21

UNITS AND MEASUREMENT 25 Relative error = ∆amean/amean (2.9) = 2.624 s = 2.62 s When the relative error is expressed in per As the periods are measured to a resolution cent, it is called the percentage error (δa). of 0.01 s, all times are to the second decimal; it Thus, Percentage error is proper to put this mean period also to the δa = (∆amean/amean) × 100% Let us now consider an example. second decimal. (2.10) The errors in the measurements are Example 2.6 Two clocks are being tested 2.63 s – 2.62 s = 0.01 s against a standard clock located in a 2.56 s – 2.62 s = – 0.06 s national laboratory. At 12:00:00 noon by 2.42 s – 2.62 s = – 0.20 s the standard clock, the readings of the two 2.71 s – 2.62 s = 0.09 s clocks are : 2.80 s – 2.62 s = 0.18 s Clock 1 Clock 2 Note that the errors have the same units as the quantity to be measured. Monday 12:00:05 10:15:06 Tuesday 12:01:15 10:14:59 The arithmetic mean of all the absolute errors Wednesday 11:59:08 10:15:18 (for arithmetic mean, we take only the Thursday 12:01:50 10:15:07 magnitudes) is Friday 11:59:15 10:14:53 Saturday 12:01:30 10:15:24 ∆Τmean = [(0.01+ 0.06+0.20+0.09+0.18)s]/5 Sunday 12:01:19 10:15:11 = 0.54 s/5 = 0.11 s If you are doing an experiment that requires That means, the period of oscillation of the precision time interval measurements, which simple pendulum is (2.62 ± 0.11) s i.e. it lies of the two clocks will you prefer ? between (2.62 + 0.11) s and (2.62 – 0.11) s or between 2.73 s and 2.51 s. As the arithmetic Answer The range of variation over the seven mean of all the absolute errors is 0.11 s, there days of observations is 162 s for clock 1, and is already an error in the tenth of a second. 31 s for clock 2. The average reading of clock 1 Hence there is no point in giving the period to a is much closer to the standard time than the hundredth. A more correct way will be to write average reading of clock 2. The important point is that a clock’s zero error is not as significant T = 2.6 ± 0.1 s for precision work as its variation, because a ‘zero-error’ can always be easily corrected. Note that the last numeral 6 is unreliable, since Hence clock 2 is to be preferred to clock 1. it may be anything between 5 and 7. We indicate this by saying that the measurement has two Example 2.7 We measure the period of significant figures. In this case, the two oscillation of a simple pendulum. In significant figures are 2, which is reliable and successive measurements, the readings 6, which has an error associated with it. You turn out to be 2.63 s, 2.56 s, 2.42 s, 2.71s will learn more about the significant figures in and 2.80 s. Calculate the absolute errors, section 2.7. relative error or percentage error. For this example, the relative error or the percentage error is Answer The mean period of oscillation of the δa = 0.1 ×100 = 4 % pendulum 2.6 T = ( 2.63 + 2.56 + 2.42 + 2.71 + 2.80)s 2.6.2 Combination of Errors 5 If we do an experiment involving several = 13.12 s measurements, we must know how the errors 5 in all the measurements combine. For example, 2020-21

26 PHYSICS How will you measure the length of a line? (a) Error of a sum or a difference What a naïve question, at this stage, you might Suppose two physical quantities A and B have say! But what if it is not a straight line? Draw measured values A ± ∆A, B ± ∆B respectively a zigzag line in your copy, or on the blackboard. where ∆A and ∆B are their absolute errors. We Well, not too difficult again. You might take a wish to find the error ∆Z in the sum thread, place it along the line, open up the thread, and measure its length. Z = A + B. We have by addition, Z ± ∆Z Now imagine that you want to measure the length of a national highway, a river, the railway = (A ± ∆A) + (B ± ∆B). track between two stations, or the boundary The maximum possible error in Z between two states or two nations. If you take a string of length 1 metre or 100 metre, keep it ∆Z = ∆A + ∆B along the line, shift its position every time, the For the difference Z = A – B, we have arithmetic of man-hours of labour and expenses on the project is not commensurate with the Z ± ∆ Z = (A ± ∆A) – (B ± ∆B) outcome. Moreover, errors are bound to occur = (A – B) ± ∆A ± ∆B in this enormous task. There is an interesting fact about this. France and Belgium share a or, ± ∆Z = ± ∆A ± ∆B common international boundary, whose length mentioned in the official documents of the two The maximum value of the error ∆Z is again countries differs substantially! ∆A + ∆B. Go one step beyond and imagine the Hence the rule : When two quantities are coastline where land meets sea. Roads and rivers added or subtracted, the absolute error in the have fairly mild bends as compared to a final result is the sum of the absolute errors coastline. Even so, all documents, including our in the individual quantities. school books, contain information on the length of the coastline of Gujarat or Andhra Pradesh, Example 2.8 The temperatures of two or the common boundary between two states, bodies measured by a thermometer are etc. Railway tickets come with the distance t1 = 20 0C ± 0.5 0C and t2 = 50 0C ± 0.5 0C. between stations printed on them. We have Calculate the temperature difference and ‘milestones’ all along the roads indicating the the error theirin. distances to various towns. So, how is it done? Answer t′ = t2–t1 = (50 0C±0.5 0C)– (200C±0.5 0C) One has to decide how much error one can t′ = 30 0C ± 1 0C tolerate and optimise cost-effectiveness. If you want smaller errors, it will involve high (b) Error of a product or a quotient technology and high costs. Suffice it to say that Suppose Z = AB and the measured values of A it requires fairly advanced level of physics, and B are A ± ∆A and B ± ∆B. Then mathematics, engineering and technology. It belongs to the areas of fractals, which has lately Z ± ∆Z = (A ± ∆A) (B ± ∆B) become popular in theoretical physics. Even = AB ± B ∆A ± A ∆B ± ∆A ∆B. then one doesn’t know how much to rely on the figure that props up, as is clear from the Dividing LHS by Z and RHS by AB we have, story of France and Belgium. Incidentally, this 1±(∆Z/Z) = 1 ± (∆A/A) ± (∆B/B) ± (∆A/A)(∆B/B). story of the France-Belgium discrepancy Since ∆A and ∆B are small, we shall ignore their appears on the first page of an advanced Physics product. book on the subject of fractals and chaos! Hence the maximum relative error ∆Z/ Z = (∆A/A) + (∆B/B). mass density is obtained by deviding mass by You can easily verify that this is true for division the volume of the substance. If we have errors also. in the measurement of mass and of the sizes or dimensions, we must know what the error will Hence the rule : When two quantities are be in the density of the substance. To make such multiplied or divided, the relative error in the estimates, we should learn how errors combine result is the sum of the relative errors in the in various mathematical operations. For this, multipliers. we use the following procedure. 2020-21

UNITS AND MEASUREMENT 27 Example 2.9 The resistance R = V/I where Then, V = (100 ± 5)V and I = (10 ± 0.2)A. Find the ∆Z/Z = (∆A/A) + (∆A/A) = 2 (∆A/A). percentage error in R. Hence, the relative error in A2 is two times the Answer The percentage error in V is 5% and in error in A. I it is 2%. The total error in R would therefore In general, if Z = Ap Bq/Cr be 5% + 2% = 7%. Then, Example 2.10 Two resistors of resistances ∆Z/Z = p (∆A/A) + q (∆B/B) + r (∆C/C). cRo1n=n1e0c0te±d3(ao)hinmsaenridesR, 2(b=) 200 ± 4 ohm are Hence the rule : The relative error in a in parallel. Find physical quantity raised to the power k is the k times the relative error in the individual the equivalent resistance of the (a) series quantity. combination, (b) parallel combination. Use Example 2.11 Find the relative error in Z, if Z = A4B1/3/CD3/2. for (a) the relation R = R1 + R2, and for (b) Answer The relative error in Z is ∆Z/Z = and 4(∆A/A) +(1/3) (∆B/B) + (∆C/C) + (3/2) (∆D/D). Answer (a) The equivalent resistance of series Example 2.12 The period of oscillation of combination a simple pendulum is T = 2π L/g. R = R1 + R2 = (100 ± 3) ohm + (200 ± 4) ohm Measured value of L is 20.0 cm known to 1 = 300 ± 7 ohm. mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using (b) The equivalent resistance of parallel a wrist watch of 1 s resolution. What is the combination accuracy in the determination of g ? R′ = R1R2 = 200 = 66.7 ohm R1 + R2 3 Then, from 1 = 1 +1 Answer g = 4π2L/T2 R′ R1 R2 t ∆t ∆T ∆t we get, Here, T = n and ∆T = n . Therefore, T = t . ∆R ′ = ∆R1 + ∆R2 The errors in both L and t are the least count R ′2 R12 R 2 errors. Therefore, 2 (∆g/g) = (∆L/L) + 2(∆T/T ) ∆R1 ∆R2 ( ) ( )∆R′ =R ′2R12 + R ′2 R22 = 0.1 + 2  910 = 0.027 20.0 =  66.7 2 3 +  66.7 2 4 Thus, the percentage error in g is  100   200  100 (∆g/g) = 100(∆L/L) + 2 × 100 (∆T/T ) = 3% = 1.8 2.7 SIGNIFICANT FIGURES Then, R′ = 66.7 ±1.8 ohm As discussed above, every measurement (Here, ∆R is expresed as 1.8 instead of 2 to involves errors. Thus, the result of keep in confirmity with the rules of significant measurement should be reported in a way that figures.) indicates the precision of measurement. Normally, the reported result of measurement (c) Error in case of a measured quantity is a number that includes all digits in the raised to a power number that are known reliably plus the first digit that is uncertain. The reliable digits plus Suppose Z = A2, 2020-21

28 PHYSICS the first uncertain digit are known as • The trailing zero(s) in a number with a significant digits or significant figures. If we say the period of oscillation of a simple decimal point are significant. pendulum is 1.62 s, the digits 1 and 6 are [The numbers 3.500 or 0.06900 have four reliable and certain, while the digit 2 is significant figures each.] uncertain. Thus, the measured value has three significant figures. The length of an object (2) There can be some confusion regarding the reported after measurement to be 287.5 cm has trailing zero(s). Suppose a length is reported to four significant figures, the digits 2, 8, 7 are be 4.700 m. It is evident that the zeroes here certain while the digit 5 is uncertain. Clearly, are meant to convey the precision of reporting the result of measurement that measurement and are, therefore, significant. [If includes more digits than the significant digits these were not, it would be superfluous to write is superfluous and also misleading since it would them explicitly, the reported measurement give a wrong idea about the precision of would have been simply 4.7 m]. Now suppose measurement. we change units, then The rules for determining the number of 4.700 m = 470.0 cm = 4700 mm = 0.004700 km significant figures can be understood from the following examples. Significant figures indicate, Since the last number has trailing zero(s) in a as already mentioned, the precision of number with no decimal, we would conclude measurement which depends on the least count erroneously from observation (1) above that the of the measuring instrument. A choice of number has two significant figures, while in fact, it has four significant figures and a mere change of different units does not change the change of units cannot change the number of significant figures. number of significant digits or figures in a measurement. This important remark makes (3) To remove such ambiguities in most of the following observations clear: (1) For example, the length 2.308 cm has four determining the number of significant significant figures. But in different units, the same value can be written as 0.02308 m or 23.08 figures, the best way is to report every mm or 23080 µm. measurement in scientific notation (in the power of 10). In this notation, every number is All these numbers have the same number of expressed as a × 10b, where a is a number significant figures (digits 2, 3, 0, 8), namely four. between 1 and 10, and b is any positive or This shows that the location of decimal point is negative exponent (or power) of 10. In order to of no consequence in determining the number get an approximate idea of the number, we may of significant figures. round off the number a to 1 (for a ≤ 5) and to 10 The example gives the following rules : (for 5<a ≤ 10). Then the number can be expressed approximately as 10b in which the • All the non-zero digits are significant. exponent (or power) b of 10 is called order of • All the zeros between two non-zero digits magnitude of the physical quantity. When only an estimate is required, the quantity is of the are significant, no matter where the order of 10b. For example, the diameter of the earth (1.28×107m) is of the order of 107m with decimal point is, if at all. the order of magnitude 7. The diameter of hydrogen atom (1.06 ×10–10m) is of the order of • If the number is less than 1, the zero(s) 10–10m, with the order of magnitude –10. Thus, the diameter of the earth is 17 orders on the right of decimal point but to the of magnitude larger than the hydrogen atom. left of the first non-zero digit are not It is often customary to write the decimal after significant. [In 0.00 2308, the underlined the first digit. Now the confusion mentioned in zeroes are not significant]. (a) above disappears : • The terminal or trailing zero(s) in a 4.700 m = 4.700 × 102 cm = 4.700 × 103 mm = 4.700 × 10–3 km number without a decimal point are not The power of 10 is irrelevant to the significant. determination of significant figures. However, all [Thus 123 m = 12300 cm = 123000 mm has three significant figures, the trailing zero(s) being not significant.] However, you can also see the next observation. 2020-21

UNITS AND MEASUREMENT 29 zeroes appearing in the base number in the following rules for arithmetic operations with scientific notation are significant. Each number significant figures ensure that the final result of in this case has four significant figures. a calculation is shown with the precision that is consistent with the precision of the input Thus, in the scientific notation, no confusion measured values : arises about the trailing zero(s) in the base (1) In multiplication or division, the final number a. They are always significant. result should retain as many significant (4) The scientific notation is ideal for reporting measurement. But if this is not adopted, we use figures as are there in the original number the rules adopted in the preceding example : with the least significant figures. • For a number greater than 1, without any Thus, in the example above, density should decimal, the trailing zero(s) are not be reported to three significant figures. significant. D ens ity = 4.237g = 1.69 g cm -3 • For a number with a decimal, the trailing 2.51 cm 3 zero(s) are significant. Similarly, if the speed of light is given as 3 × 108 m s-1 (one significant figure) and one (5) The digit 0 conventionally put on the left of a year (1y = 365.25 d) has 3.1557 × 107 s (five decimal for a number less than 1 (like 0.1250) significant figures), the light year is 9.47 × 1015 m is never significant. However, the zeroes at the (three significant figures). end of such number are significant in a measurement. (2) In addition or subtraction, the final result (6) The multiplying or dividing factors which are should retain as many decimal places as are neither rounded numbers nor numbers representing measured values are exact and there in the number with the least decimal have infinite number of significant digits. For places. example in r = d or s = 2πr, the factor 2 is an For example, the sum of the numbers 2 436.32 g, 227.2 g and 0.301 g by mere arithmetic exact number and it can be written as 2.0, 2.00 addition, is 663.821 g. But the least precise measurement (227.2 g) is correct to only one or 2.0000 as required. Similarly, in T =t , n is decimal place. The final result should, therefore, n be rounded off to 663.8 g. an exact number. Similarly, the difference in length can be expressed as : 2.7.1 Rules for Arithmetic Operations with 0.307 m – 0.304 m = 0.003 m = 3 × 10–3 m. Significant Figures Note that we should not use the rule (1) The result of a calculation involving approximate applicable for multiplication and division and measured values of quantities (i.e. values with write 664 g as the result in the example of limited number of significant figures) must reflect addition and 3.00 × 10–3 m in the example of the uncertainties in the original measured values. subtraction. They do not convey the precision It cannot be more accurate than the original of measurement properly. For addition and measured values themselves on which the result subtraction, the rule is in terms of decimal is based. In general, the final result should not places. have more significant figures than the original data from which it was obtained. Thus, if mass of 2.7.2 Rounding off the Uncertain Digits an object is measured to be, say, 4.237 g (four significant figures) and its volume is measured to The result of computation with approximate be 2.51 cm3, then its density, by mere arithmetic numbers, which contain more than one division, is 1.68804780876 g/cm3 upto 11 decimal uncertain digit, should be rounded off. The rules places. It would be clearly absurd and irrelevant for rounding off numbers to the appropriate to record the calculated value of density to such a significant figures are obvious in most cases. A precision when the measurements on which the number 2.746 rounded off to three significant value is based, have much less precision. The figures is 2.75, while the number 2.743 would be 2.74. The rule by convention is that the preceding digit is raised by 1 if the 2020-21

30 PHYSICS insignificant digit to be dropped (the Example 2.14 5.74 g of a substance occupies 1.2 cm3. Express its density by underlined digit in this case) is more than keeping the significant figures in view. 5, and is left unchanged if the latter is less Answer There are 3 significant figures in the than 5. But what if the number is 2.745 in measured mass whereas there are only 2 which the insignificant digit is 5. Here, the significant figures in the measured volume. convention is that if the preceding digit is Hence the density should be expressed to only 2 significant figures. even, the insignificant digit is simply Density = 5.74 g cm−3 dropped and, if it is odd, the preceding digit 1.2 is raised by 1. Then, the number 2.745 rounded off to three significant figures becomes 2.74. On = 4.8 g cm--3 . the other hand, the number 2.735 rounded off to three significant figures becomes 2.74 since 2.7.3 Rules for Determining the Uncertainty the preceding digit is odd. in the Results of Arithmatic Calculations In any involved or complex multi-step calculation, you should retain, in intermediate The rules for determining the uncertainty or steps, one digit more than the significant digits error in the number/measured quantity in and round off to proper significant figures at the arithmetic operations can be understood from end of the calculation. Similarly, a number the following examples. known to be within many significant figures, (1) If the length and breadth of a thin such as in 2.99792458 × 108 m/s for the speed rectangular sheet are measured, using a metre of light in vacuum, is rounded off to an scale as 16.2 cm and, 10.1 cm respectively, there approximate value 3 × 108 m/s , which is often are three significant figures in each employed in computations. Finally, remember measurement. It means that the length l may that exact numbers that appear in formulae like be written as 2 π in T = 2π L , have a large (infinite) number l = 16.2 ± 0.1 cm g = 16.2 cm ± 0.6 %. of significant figures. The value of π = Similarly, the breadth b may be written as 3.1415926.... is known to a large number of b = 10.1 ± 0.1 cm significant figures. You may take the value as = 10.1 cm ± 1 % 3.142 or 3.14 for π, with limited number of significant figures as required in specific Then, the error of the product of two (or more) cases. experimental values, using the combination of errors rule, will be Example 2.13 Each side of a cube is measured to be 7.203 m. What are the l b = 163.62 cm2 + 1.6% total surface area and the volume of the cube to appropriate significant figures? = 163.62 + 2.6 cm2 Answer The number of significant figures in This leads us to quote the final result as the measured length is 4. The calculated area and the volume should therefore be rounded off l b = 164 + 3 cm2 to 4 significant figures. Here 3 cm2 is the uncertainty or error in the Surface area of the cube = 6(7.203)2 m2 estimation of area of rectangular sheet. (2) If a set of experimental data is specified = 311.299254 m2 to n significant figures, a result obtained by combining the data will also be valid to n = 311.3 m2 significant figures. However, if data are subtracted, the number of Volume of the cube = (7.203)3 m3 significant figures can be reduced. = 373.714754 m3 = 373.7 m3 2020-21

UNITS AND MEASUREMENT 31 For example, 12.9 g – 7.06 g, both specified to three square brackets [ ]. Thus, length has the significant figures, cannot properly be evaluated dimension [L], mass [M], time [T], electric current as 5.84 g but only as 5.8 g, as uncertainties in [A], thermodynamic temperature [K], luminous subtraction or addition combine in a different intensity [cd], and amount of substance [mol]. fashion (smallest number of decimal places rather than the number of significant figures in any of The dimensions of a physical quantity are the the number added or subtracted). powers (or exponents) to which the base (3) The relative error of a value of number specified to significant figures depends not quantities are raised to represent that only on n but also on the number itself. quantity. Note that using the square brackets [ ] round a quantity means that we are dealing For example, the accuracy in measurement of with ‘the dimensions of’ the quantity. mass 1.02 g is ± 0.01 g whereas another measurement 9.89 g is also accurate to ± 0.01 g. In mechanics, all the physical quantities can The relative error in 1.02 g is be written in terms of the dimensions [L], [M] and [T]. For example, the volume occupied by = (± 0.01/1.02) × 100 % an object is expressed as the product of length, = ± 1% breadth and height, or three lengths. Hence the Similarly, the relative error in 9.89 g is dimensions of volume are [L] × [L] × [L] = [L]3 = [L3]. = (± 0.01/9.89) × 100 % As the volume is independent of mass and time, = ± 0.1 % it is said to possess zero dimension in mass [M°], Finally, remember that intermediate results in zero dimension in time [T°] and three dimensions in length. a multi-step computation should be Similarly, force, as the product of mass and calculated to one more significant figure in acceleration, can be expressed as Force = mass × acceleration every measurement than the number of = mass × (length)/(time)2 digits in the least precise measurement. These should be justified by the data and then The dimensions of force are [M] [L]/[T]2 = the arithmetic operations may be carried out; [M L T–2]. Thus, the force has one dimension in otherwise rounding errors can build up. For mass, one dimension in length, and –2 example, the reciprocal of 9.58, calculated (after dimensions in time. The dimensions in all other rounding off) to the same number of significant base quantities are zero. figures (three) is 0.104, but the reciprocal of 0.104 calculated to three significant figures is Note that in this type of representation, the 9.62. However, if we had written 1/9.58 = 0.1044 magnitudes are not considered. It is the quality and then taken the reciprocal to three significant of the type of the physical quantity that enters. figures, we would have retrieved the original Thus, a change in velocity, initial velocity, value of 9.58. average velocity, final velocity, and speed are all equivalent in this context. Since all these This example justifies the idea to retain one quantities can be expressed as length/time, more extra digit (than the number of digits in their dimensions are [L]/[T] or [L T–1]. the least precise measurement) in intermediate steps of the complex multi-step calculations in 2.9 DIMENSIONAL FORMULAE AND order to avoid additional errors in the process DIMENSIONAL EQUATIONS of rounding off the numbers. The expression which shows how and which of 2.8 DIMENSIONS OF PHYSICAL QUANTITIES the base quantities represent the dimensions of a physical quantity is called the dimensional The nature of a physical quantity is described formula of the given physical quantity. For by its dimensions. All the physical quantities example, the dimensional formula of the volume represented by derived units can be expressed is [M° L3 T°], and that of speed or velocity is in terms of some combination of seven [M° L T-1]. Similarly, [M° L T–2] is the dimensional fundamental or base quantities. We shall call formula of acceleration and [M L–3 T°] that of these base quantities as the seven dimensions mass density. of the physical world, which are denoted with An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical 2020-21

32 PHYSICS quantity. Thus, the dimensional equations are from the thermodynamic temperature. This the equations, which represent the dimensions simple principle called the principle of of a physical quantity in terms of the base homogeneity of dimensions in an equation is quantities. For example, the dimensional extremely useful in checking the correctness of equations of volume [V ], speed [v], force [F ] and an equation. If the dimensions of all the terms are not same, the equation is wrong. Hence, if mass density [ρ] may be expressed as we derive an expression for the length (or distance) of an object, regardless of the symbols [V] = [M0 L3 T0] appearing in the original mathematical relation, [v] = [M0 L T–1] when all the individual dimensions are [F] = [M L T–2] simplified, the remaining dimension must be that of length. Similarly, if we derive an equation [ρ] = [M L–3 T0] of speed, the dimensions on both the sides of equation, when simplified, must be of length/ The dimensional equation can be obtained time, or [L T–1]. from the equation representing the relations between the physical quantities. The Dimensions are customarily used as a dimensional formulae of a large number and preliminary test of the consistency of an wide variety of physical quantities, derived from equation, when there is some doubt about the the equations representing the relationships correctness of the equation. However, the among other physical quantities and expressed dimensional consistency does not guarantee in terms of base quantities are given in correct equations. It is uncertain to the extent Appendix 9 for your guidance and ready of dimensionless quantities or functions. The reference. arguments of special functions, such as the trigonometric, logarithmic and exponential 2.10 DIMENSIONAL ANALYSIS AND ITS functions must be dimensionless. A pure APPLICATIONS number, ratio of similar physical quantities, such as angle as the ratio (length/length), The recognition of concepts of dimensions, which refractive index as the ratio (speed of light in guide the description of physical behaviour is vacuum/speed of light in medium) etc., has no of basic importance as only those physical dimensions. quantities can be added or subtracted which have the same dimensions. A thorough Now we can test the dimensional consistency understanding of dimensional analysis helps us or homogeneity of the equation in deducing certain relations among different physical quantities and checking the derivation, x = x0 + v0 t + (1/2) a t 2 accuracy and dimensional consistency or homogeneity of various mathematical for the distance x travelled by a particle or body expressions. When magnitudes of two or more aainncctieinmlieteriaatltwvioehnloicachitayslotvan0rgattstthfirmeodemitrt=ehc0etiaopnnodsoihftiamosnoutxnio0infwo.ritmh physical quantities are multiplied, their units should be treated in the same manner as The dimensions of each term may be written as ordinary algebraic symbols. We can cancel [x] = [L] identical units in the numerator and denominator. The same is true for dimensions [v[0x0t]]===[[[LLL]]T–1] [T] of a physical quantity. Similarly, physical [(1/2) a t2] = [L T–2] [T2] quantities represented by symbols on both sides of a mathematical equation must have the same = [L] dimensions. As each term on the right hand side of this equation has the same dimension, namely that 2.10.1 Checking the Dimensional of length, which is same as the dimension of Consistency of Equations left hand side of the equation, hence this equation is a dimensionally correct equation. The magnitudes of physical quantities may be added together or subtracted from one another It may be noted that a test of consistency of only if they have the same dimensions. In other dimensions tells us no more and no less than a words, we can add or subtract similar physical quantities. Thus, velocity cannot be added to force, or an electric current cannot be subtracted 2020-21

UNITS AND MEASUREMENT 33 test of consistency of units, but has the (b) and (d); [M L T–2] for (c). The quantity on the advantage that we need not commit ourselves right side of (e) has no proper dimensions since to a particular choice of units, and we need not two quantities of different dimensions have been worry about conversions among multiples and added. Since the kinetic energy K has the sub-multiples of the units. It may be borne in dimensions of [M L2 T–2], formulas (a), (c) and (e) mind that if an equation fails this consistency are ruled out. Note that dimensional arguments test, it is proved wrong, but if it passes, it is cannot tell which of the two, (b) or (d), is the not proved right. Thus, a dimensionally correct correct formula. For this, one must turn to the equation need not be actually an exact actual definition of kinetic energy (see Chapter (correct) equation, but a dimensionally wrong 6). The correct formula for kinetic energy is given (incorrect) or inconsistent equation must be by (b). wrong. 2.10.2 Deducing Relation among the Example 2.15 Let us consider an equation Physical Quantities 1 m v2 = m g h The method of dimensions can sometimes be 2 used to deduce relation among the physical where m is the mass of the body, v its quantities. For this we should know the velocity, g is the acceleration due to dependence of the physical quantity on other gravity and h is the height. Check quantities (upto three physical quantities or whether this equation is dimensionally linearly independent variables) and consider it correct. as a product type of the dependence. Let us take an example. Answer The dimensions of LHS are [M] [L T–1 ]2 = [M] [ L2 T–2] Example 2.17 Consider a simple = [M L2 T–2] pendulum, having a bob attached to a string, that oscillates under the action of The dimensions of RHS are the force of gravity. Suppose that the period [M][L T–2] [L] = [M][L2 T–2] of oscillation of the simple pendulum = [M L2 T–2] depends on its length (l), mass of the bob (m) and acceleration due to gravity (g). The dimensions of LHS and RHS are the same and Derive the expression for its time period hence the equation is dimensionally correct. using method of dimensions. Example 2.16 The SI unit of energy is Answer The dependence of time period T on J = kg m2 s–2; that of speed v is m s–1 and the quantities l, g and m as a product may be of acceleration a is m s–2. Which of the written as : formulae for kinetic energy (K) given below can you rule out on the basis of T = k lx gy mz dimensional arguments (m stands for the mass of the body) : where k is dimensionless constant and x, y (a) K = m2 v3 and z are the exponents. (b) K = (1/2)mv2 (c) K = ma By considering dimensions on both sides, we (d) K = (3/16)mv2 have (e) K = (1/2)mv2 + ma [Lo Mo T1 ] =[L1 ]x [L1 T –2 ]y [M1 ]z Answer Every correct formula or equation must = Lx+y T–2y Mz have the same dimensions on both sides of the equation. Also, only quantities with the same On equating the dimensions on both sides, physical dimensions can be added or we have subtracted. The dimensions of the quantity on the right side are [M2 L3 T–3] for (a); [M L2 T–2] for x + y = 0; –2y = 1; and z = 0 So that x = 1 ,y = – 1 , z =0 2 2 Then, T = k l½ g–½ 2020-21

34 PHYSICS or, T = k l Dimensional analysis is very useful in deducing g relations among the interdependent physical quantities. However, dimensionless constants Note that value of constant k can not be obtained cannot be obtained by this method. The method by the method of dimensions. Here it does not of dimensions can only test the dimensional matter if some number multiplies the right side validity, but not the exact relationship between of this formula, because that does not affect its physical quantities in any equation. It does not dimensions. distinguish between the physical quantities having same dimensions. Actually, k = 2π so that T = 2π l g A number of exercises at the end of this chapter will help you develop skill in dimensional analysis. SUMMARY 1. Physics is a quantitative science, based on measurement of physical quantities. Certain physical quantities have been chosen as fundamental or base quantities (such as length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity). 2. Each base quantity is defined in terms of a certain basic, arbitrarily chosen but properly standardised reference standard called unit (such as metre, kilogram, second, ampere, kelvin, mole and candela). The units for the fundamental or base quantities are called fundamental or base units. 3. Other physical quantities, derived from the base quantities, can be expressed as a combination of the base units and are called derived units. A complete set of units, both fundamental and derived, is called a system of units. 4. The International System of Units (SI) based on seven base units is at present internationally accepted unit system and is widely used throughout the world. 5. The SI units are used in all physical measurements, for both the base quantities and the derived quantities obtained from them. Certain derived units are expressed by means of SI units with special names (such as joule, newton, watt, etc). 6. The SI units have well defined and internationally accepted unit symbols (such as m for metre, kg for kilogram, s for second, A for ampere, N for newton etc.). 7. Physical measurements are usually expressed for small and large quantities in scientific notation, with powers of 10. Scientific notation and the prefixes are used to simplify measurement notation and numerical computation, giving indication to the precision of the numbers. 8. Certain general rules and guidelines must be followed for using notations for physical quantities and standard symbols for SI units, some other units and SI prefixes for expressing properly the physical quantities and measurements. 9. In computing any physical quantity, the units for derived quantities involved in the relationship(s) are treated as though they were algebraic quantities till the desired units are obtained. 10. Direct and indirect methods can be used for the measurement of physical quantities. In measured quantities, while expressing the result, the accuracy and precision of measuring instruments along with errors in measurements should be taken into account. 11. In measured and computed quantities proper significant figures only should be retained. Rules for determining the number of significant figures, carrying out arithmetic operations with them, and ‘rounding off ‘ the uncertain digits must be followed. 12. The dimensions of base quantities and combination of these dimensions describe the nature of physical quantities. Dimensional analysis can be used to check the dimensional consistency of equations, deducing relations among the physical quantities, etc. A dimensionally consistent equation need not be actually an exact (correct) equation, but a dimensionally wrong or inconsistent equation must be wrong. 2020-21

UNITS AND MEASUREMENT 35 EXERCISES Note : In stating numerical answers, take care of significant figures. 2.1 Fill in the blanks (a) The volume of a cube of side 1 cm is equal to .....m3 (b) The surface area of a solid cylinder of radius 2.0 cm and height 10.0 cm is equal to ...(mm)2 (c) A vehicle moving with a speed of 18 km h–1 covers....m in 1 s (d) The relative density of lead is 11.3. Its density is ....g cm–3 or ....kg m–3. 2.2 Fill in the blanks by suitable conversion of units (a) 1 kg m2 s–2 = ....g cm2 s–2 (b) 1 m = ..... ly (c) 3.0 m s–2 = .... km h–2 (d) G = 6.67 × 10–11 N m2 (kg)–2 = .... (cm)3 s–2 g–1. 2.3 A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where 1J = 1 kg m2 s–2. Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, the unit of time is γ s. Show that a calorie has a magnitude 4.2 α –1 β –2 γ 2 in terms of the new units. 2.4 Explain this statement clearly : “To call a dimensional quantity ‘large’ or ‘small’ is meaningless without specifying a standard for comparison”. In view of this, reframe the following statements wherever necessary : (a) atoms are very small objects (b) a jet plane moves with great speed (c) the mass of Jupiter is very large (d) the air inside this room contains a large number of molecules (e) a proton is much more massive than an electron (f) the speed of sound is much smaller than the speed of light. 2.5 A new unit of length is chosen such that the speed of light in vacuum is unity. What is the distance between the Sun and the Earth in terms of the new unit if light takes 8 min and 20 s to cover this distance ? 2.6 Which of the following is the most precise device for measuring length : (a) a vernier callipers with 20 divisions on the sliding scale (b) a screw gauge of pitch 1 mm and 100 divisions on the circular scale (c) an optical instrument that can measure length to within a wavelength of light ? 2.7 A student measures the thickness of a human hair by looking at it through a microscope of magnification 100. He makes 20 observations and finds that the average width of the hair in the field of view of the microscope is 3.5 mm. What is the estimate on the thickness of hair ? 2.8 Answer the following : (a)You are given a thread and a metre scale. How will you estimate the diameter of the thread ? (b)A screw gauge has a pitch of 1.0 mm and 200 divisions on the circular scale. Do you think it is possible to increase the accuracy of the screw gauge arbitrarily by increasing the number of divisions on the circular scale ? (c) The mean diameter of a thin brass rod is to be measured by vernier callipers. Why is a set of 100 measurements of the diameter expected to yield a more reliable estimate than a set of 5 measurements only ? 2.9 The photograph of a house occupies an area of 1.75 cm2 on a 35 mm slide. The slide is projected on to a screen, and the area of the house on the screen is 1.55 m2. What is the linear magnification of the projector-screen arrangement. 2.10 State the number of significant figures in the following : (a) 0.007 m2 (b) 2.64 × 1024 kg (c) 0.2370 g cm–3 2020-21

36 PHYSICS (d) 6.320 J (e) 6.032 N m–2 (f) 0.0006032 m2 2.11 The length, breadth and thickness of a rectangular sheet of metal are 4.234 m, 1.005 m, and 2.01 cm respectively. Give the area and volume of the sheet to correct significant figures. 2.12 The mass of a box measured by a grocer’s balance is 2.30 kg. Two gold pieces of masses 20.15 g and 20.17 g are added to the box. What is (a) the total mass of the box, (b) the difference in the masses of the pieces to correct significant figures ? 2.13 A physical quantity P is related to four observables a, b, c and d as follows : ( )P = a 3b2/ c d The percentage errors of measurement in a, b, c and d are 1%, 3%, 4% and 2%, respectively. What is the percentage error in the quantity P ? If the value of P calculated using the above relation turns out to be 3.763, to what value should you round off the result ? 2.14 A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion : (a) y = a sin 2π t/T (b) y = a sin vt (c) y = (a/T) sin t/a (d) y = (a 2 ) (sin 2πt / T + cos 2πt / T ) (a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds. 2.15 A famous relation in physics relates ‘moving mass’ m to the ‘rest mass’ mo of a particle in terms of its speed v and the speed of light, c. (This relation first arose as a consequence of special relativity due to Albert Einstein). A boy recalls the relation almost correctly but forgets where to put the constant c. He writes : m0 1 − v2 1/2 . ( )m = Guess where to put the missing c. 2.16 The unit of length convenient on the atomic scale is known as an angstrom and is denoted by Å: 1 Å = 10–10 m. The size of a hydrogen atom is about 0.5 Å. What is the total atomic volume in m3 of a mole of hydrogen atoms ? 2.17 One mole of an ideal gas at standard temperature and pressure occupies 22.4 L (molar volume). What is the ratio of molar volume to the atomic volume of a mole of hydrogen ? (Take the size of hydrogen molecule to be about 1 Å). Why is this ratio so large ? 2.18 Explain this common observation clearly : If you look out of the window of a fast moving train, the nearby trees, houses etc. seem to move rapidly in a direction opposite to the train’s motion, but the distant objects (hill tops, the Moon, the stars etc.) seem to be stationary. (In fact, since you are aware that you are moving, these distant objects seem to move with you). 2.19 The principle of ‘parallax’ in section 2.3.1 is used in the determination of distances of very distant stars. The baseline AB is the line joining the Earth’s two locations six months apart in its orbit around the Sun. That is, the baseline is about the diameter of the Earth’s orbit ≈ 3 × 1011m. However, even the nearest stars are so distant that with such a long baseline, they show parallax only of the order of 1” (second) of arc or so. A parsec is a convenient unit of length on the astronomical scale. It is the distance of an object that will show a parallax of 1” (second of arc) from opposite ends of a baseline equal to the distance from the Earth to the Sun. How much is a parsec in terms of metres ? 2020-21