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PART I
TEXTBOOK FOR CLASS XI
2018-19
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ISBN 81-7450-508-3
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2018-19
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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.
Director
New Delhi National Council of Educational
20 December 2005 Research and Training
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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
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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, Shillong; 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.
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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
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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
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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
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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 (N ) and the other with CO . Gases
2 2
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.
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CONTENTS
FOREWORD iii
PREFACE vii
A NOTE FOR THE TEACHER x
C H A P T E R 1
PHYSICAL WORLD
1.1 What is physics ? 1
1.2 Scope and excitement of physics 2
1.3 Physics, technology and society 5
1.4 Fundamental forces in nature 6
1.5 Nature of physical laws 10
C H A P T E R 2
UNITS AND MEASUREMENTS
2.1 Introduction 16
2.2 The international system of units 16
2.3 Measurement of length 18
2.4 Measurement of mass 21
2.5 Measurement of time 22
2.6 Accuracy, precision of instruments and errors in measurement 22
2.7 Significant figures 27
2.8 Dimensions of physical quantities 31
2.9 Dimensional formulae and dimensional equations 31
2.10 Dimensional analysis and its applications 32
C H A P T E R 3
MOTION IN A STRAIGHT LINE
3.1 Introduction 39
3.2 Position, path length and displacement 39
3.3 Average velocity and average speed 42
3.4 Instantaneous velocity and speed 43
3.5 Acceleration 45
3.6 Kinematic equations for uniformly accelerated motion 47
3.7 Relative velocity 51
C H A P T E R 4
MOTION IN A PLANE
4.1 Introduction 65
4.2 Scalars and vectors 65
4.3 Multiplication of vectors by real numbers 67
4.4 Addition and subtraction of vectors – graphical method 67
4.5 Resolution of vectors 69
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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
C H A P T E R 5
LAWS OF MOTION
5.1 Introduction 89
5.2 Aristotle’s fallacy 90
5.3 The law of inertia 90
5.4 Newton’s first law of motion 91
5.5 Newton’s second law of motion 93
5.6 Newton’s third law of motion 96
5.7 Conservation of momentum 98
5.8 Equilibrium of a particle 99
5.9 Common forces in mechanics 100
5.10 Circular motion 104
5.11 Solving problems in mechanics 105
C H A P T E R 6
WORK, ENERGY AND POWER
6.1 Introduction 114
6.2 Notions of work and kinetic energy : The work-energy theorem 116
6.3 Work 116
6.4 Kinetic energy 117
6.5 Work done by a variable force 118
6.6 The work-energy theorem for a variable force 119
6.7 The concept of potential energy 120
6.8 The conservation of mechanical energy 121
6.9 The potential energy of a spring 123
6.10 Various forms of energy : the law of conservation of energy 126
6.11 Power 128
6.12 Collisions 129
C H A P T E R 7
SYSTEM OF PARTICLES AND ROTATIONAL MOTION
7.1 Introduction 141
7.2 Centre of mass 144
7.3 Motion of centre of mass 148
7.4 Linear momentum of a system of particles 149
7.5 Vector product of two vectors 150
7.6 Angular velocity and its relation with linear velocity 152
7.7 Torque and angular momentum 154
7.8 Equilibrium of a rigid body 158
7.9 Moment of inertia 163
7.10 Theorems of perpendicular and parallel axes 164
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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
C H A P T E R 8
GRAVITATION
8.1 Introduction 183
8.2 Kepler’s laws 184
8.3 Universal law of gravitation 185
8.4 The gravitational constant 189
8.5 Acceleration due to gravity of the earth 189
8.6 Acceleration due to gravity below and above the surface of earth 190
8.7 Gravitational potential energy 191
8.8 Escape speed 193
8.9 Earth satellite 194
8.10 Energy of an orbiting satellite 195
8.11 Geostationary and polar satellites 196
8.12 Weightlessness 197
APPENDICES 207
ANSWERS 223
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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.org)
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.
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CHAPTER ONE
PHYSICAL WORLD
1.1 WHAT IS PHYSICS ?
Humans have always been curious about the world around
them. The night sky with its bright celestial objects has
fascinated humans since time immemorial. The regular
1.1 What is physics ?
repetitions of the day and night, the annual cycle of seasons,
1.2 Scope and excitement of
the eclipses, the tides, the volcanoes, the rainbow have always
physics
been a source of wonder. The world has an astonishing variety
1.3 Physics, technology and of materials and a bewildering diversity of life and behaviour.
society The inquiring and imaginative human mind has responded
1.4 Fundamental forces in to the wonder and awe of nature in different ways. One kind
nature of response from the earliest times has been to observe the
1.5 Nature of physical laws physical environment carefully, look for any meaningful
Summary patterns and relations in natural phenomena, and build and
use new tools to interact with nature. This human endeavour
Exercises
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 Vijnan 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
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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
The interplay of theory and observation (or and their manifestation in different natural
experiment) is basic to the progress of science. phenomena. The scope of physics is described
Science is ever dynamic. There is no ‘final’ briefly in the next section. Here we remark on
theory in science and no unquestioned two principal thrusts in physics : unification
authority among scientists. As observations and reduction.
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
heliocentric theory (sun at the centre of the motion of planets around the sun. Similarly, the
solar system) imagined by Nicolas Copernicus basic laws of electromagnetism (Maxwell’s
(1473–1543) had to be replaced by elliptical equations) govern all electric and magnetic
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
to explain new observations. This causes a quest for unification.
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
Just as a new experiment may suggest an mechanics interpreted these quantities in terms
alternative theoretical model, a theoretical of the properties of the molecular constituents
advance may suggest what to look for in some of the bulk system. In particular, the
experiments. The result of experiment of temperature was seen to be related to the average
scattering of alpha particles by gold foil, in 1911 kinetic energy of molecules of the system.
by Ernest Rutherford (1871–1937) established
the nuclear model of the atom, which then 1.2 SCOPE AND EXCITEMENT OF PHYSICS
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
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PHYSICAL WORLD 3
phenomena*. Classical Physics deals mainly chemical process, etc., are problems of interest
with macroscopic phenomena and includes in thermodynamics.
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
bodies, and general systems of particles. The different probes such as electrons, photons and
propulsion of a rocket by a jet of ejecting gases, other elementary particles. Classical physics is
propagation of water waves or sound waves in inadequate to handle this domain and Quantum
air, the equilibrium of a bent rod under a load, Theory is currently accepted as the proper
etc., are problems of mechanics. Electrodynamics framework for explaining microscopic
deals with electric and magnetic phenomena phenomena. Overall, the edifice of physics is
associated with charged and magnetic bodies. beautiful and imposing and you will appreciate
Its basic laws were given by Coulomb, Oersted, it more as you pursue the subject.
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 truly vast. It covers a tremendous range of
motion of a current-carrying conductor in a magnitude of physical quantities like length,
magnetic field, the response of a circuit to an ac mass, time, energy, etc. At one end, it studies
voltage (signal), the working of an antenna, the phenomena at the very small scale of length
propagation of radio waves in the ionosphere, etc., -14
(10 m or even less) involving electrons, protons,
are problems of electrodynamics. Optics deals
etc.; at the other end, it deals with astronomical
with the phenomena involving light. The working
phenomena at the scale of galaxies or even the
of telescopes and microscopes, colours exhibited entire universe whose extent is of the order of
by thin films, etc., are topics in optics. 26
Thermodynamics, in contrast to mechanics, does 10 m. The two length scales differ by a factor of
40
not deal with the motion of bodies as a whole. 10 or even more. The range of time scales can
Rather, it deals with systems in macroscopic be obtained by dividing the length scales by the
–22
18
equilibrium and is concerned with changes in speed of light : 10 s to 10 s. The range of
–30
internal energy, temperature, entropy, etc., of the masses goes from, say, 10 kg (mass of an
55
system through external work and transfer of electron) to 10 kg (mass of known observable
heat. The efficiency of heat engines and universe). Terrestrial phenomena lie somewhere
refrigerators, the direction of a physical or 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.
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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 For example, the universal law of gravitation
theories, is thrilling. Applied physics is equally proposed by Newton is an assumption or hypothesis,
demanding. Application and exploitation of which he proposed out of his ingenuity. Before him,
physical laws to make useful devices is the most there were several observations, experiments and
interesting and exciting part and requires great data, on the motion of planets around the sun,
motion of the moon around the earth, pendulums,
ingenuity and persistence of effort. bodies falling towards the earth etc. Each of these
What lies behind the phenomenal progress required a separate explanation, which was more
of physics in the last few centuries? Great or less qualitative. What the universal law of
progress usually accompanies changes in our gravitation says is that, if we assume that any two
basic perceptions. First, it was realised that for bodies in the universe attract each other with a
scientific progress, only qualitative thinking, force proportional to the product of their masses
though no doubt important, is not enough. and inversely proportional to the square of the
Quantitative measurement is central to the distance between them, then we can explain all
growth of science, especially physics, because these observations in one stroke. It not only explains
the laws of nature happen to be expressible in these phenomena, it also allows us to predict the
precise mathematical equations. The second results of future experiments.
most important insight was that the basic laws A hypothesis is a supposition without assuming
of physics are universal — the same laws apply that it is true. It would not be fair to ask anybody
in widely different contexts. Lastly, the strategy to prove the universal law of gravitation, because
it cannot be proved. It can be verified and
of approximation turned out to be very substantiated by experiments and observations.
successful. Most observed phenomena in daily
An axiom is a self-evident truth while a model
life are rather complicated manifestations of the is a theory proposed to explain observed
basic laws. Scientists recognised the importance
phenomena. But you need not worry at this stage
of extracting the essential features of a about the nuances in using these words. For
phenomenon from its less significant aspects. example, next year you will learn about Bohr’s model
It is not practical to take into account all the of hydrogen atom, in which Bohr assumed that an
complexities of a phenomenon in one go. A good electron in the hydrogen atom follows certain rules
strategy is to focus first on the essential features, (postutates). Why did he do that? There was a large
discover the basic principles and then introduce amount of spectroscopic data before him which no
corrections to build a more refined theory of the other theory could explain. So Bohr said that if we
phenomenon. For example, a stone and a feather assume that an atom behaves in such a manner,
dropped from the same height do not reach the we can explain all these things at once.
ground at the same time. The reason is that the Einstein’s special theory of relativity is also
based on two postulates, the constancy of the speed
essential aspect of the phenomenon, namely free
fall under gravity, is complicated by the of electromagnetic radiation and the validity of
physical laws in all inertial frame of reference. It
presence of air resistance. To get the law of free would not be wise to ask somebody to prove that
fall under gravity, it is better to create a the speed of light in vacuum is constant,
situation wherein the air resistance is independent of the source or observer.
negligible. We can, for example, let the stone and In mathematics too, we need axioms and
the feather fall through a long evacuated tube. hypotheses at every stage. Euclid’s statement that
In that case, the two objects will fall almost at parallel lines never meet, is a hypothesis. This means
the same rate, giving the basic law that that if we assume this statement, we can explain
acceleration due to gravity is independent of the several properties of straight lines and two or three
mass of the object. With the basic law thus dimensional figures made out of them. But if you
found, we can go back to the feather, introduce don’t assume it, you are free to use a different axiom
corrections due to air resistance, modify the and get a new geometry, as has indeed happened in
the past few centuries and decades.
existing theory and try to build a more realistic
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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
The connection between physics, technology urgent need to discover new and affordable
and society can be seen in many examples. The sources of energy. Considerable progress has
discipline of thermodynamics arose from the already been made in this direction (for
need to understand and improve the working of example, in conversion of solar energy,
heat engines. The steam engine, as we know, geothermal energy, etc., into electricity), but
is inseparable from the Industrial Revolution in much more is still to be accomplished.
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 Physics is the study of nature and natural
basis of nuclear power reactors and nuclear phenomena. Physicists try to discover the rules
weapons. Yet another important example of that are operating in nature, on the basis of
physics giving rise to technology is the silicon observations, experimentation and analysis.
‘chip’ that triggered the computer revolution in Physics deals with certain basic rules/laws
the last three decades of the twentieth century. 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 Principle of buoyancy; Principle of the lever Greece
Galileo Galilei Law of inertia Italy
Christiaan Huygens Wave theory of light Holland
Isaac Newton Universal law of gravitation; Laws of motion; U.K.
Reflecting telescope
Michael Faraday Laws of electromagnetic induction U.K.
James Clerk Maxwell Electromagnetic theory; Light-an U.K.
electromagnetic wave
Heinrich Rudolf Hertz Generation of electromagnetic waves Germany
J.C. Bose Ultra short radio waves India
W.K. Roentgen X-rays Germany
J.J. Thomson Electron U.K.
Marie Sklodowska Curie Discovery of radium and polonium; Studies on Poland
natural radioactivity
Albert Einstein Explanation of photoelectric effect; Germany
Theory of relativity
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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.
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Table 1.2 Link between technology and physics
Technology Scientific principle(s)
Steam engine Laws of thermodynamics
Nuclear reactor Controlled nuclear fission
Radio and Television Generation, propagation and detection
of electromagnetic waves
Computers Digital logic
Lasers Light amplification by stimulated emission of
radiation
Production of ultra high magnetic Superconductivity
fields
Rocket propulsion Newton’s laws of motion
Electric generator Faraday’s laws of electromagnetic induction
Hydroelectric power Conversion of gravitational potential energy into
electrical energy
Aeroplane Bernoulli’s principle in fluid dynamics
Particle accelerators Motion of charged particles in electromagnetic
fields
Sonar Reflection of ultrasonic waves
Optical fibres Total internal reflection of light
Non-reflecting coatings Thin film optical interference
Electron microscope Wave nature of electrons
Photocell Photoelectric effect
Fusion test reactor (Tokamak) Magnetic confinement of plasma
Giant Metrewave Radio Detection of cosmic radio waves
Telescope (GMRT)
Bose-Einstein condensate Trapping and cooling of atoms by laser beams and
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 In principle, this means that the laws for
shall get familiar with some of these forces in later ‘derived’ forces (such as spring force, friction)
parts of this course. 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 At the present stage of our understanding,
small number of fundamental forces in nature. we know of four fundamental forces in nature,
For example, the elastic spring force arises due which are described in brief here :
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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
2
equation E = mc . 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,
The gravitational force is the force of mutual is 10 36 times the gravitational force between
attraction between any two objects by virtue of them, for any fixed distance.
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
1.4.2 Electromagnetic Force like ‘tension’, ‘friction’, ‘normal force’, ‘spring
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
Coulomb’s law : attractive for unlike charges and repulsive. Another way of putting it is that mass
comes only in one variety (there is no negative
repulsive for like charges. Charges in motion
produce magnetic effects and a magnetic field mass), but charge comes in two varieties :
gives rise to a force on a moving charge. Electric positive and negative charge. This is what
and magnetic effects are, in general, makes all the difference. Matter is mostly
inseparable – hence the name electromagnetic electrically neutral (net charge is zero). Thus,
force. Like the gravitational force, electric force is largely zero and gravitational
electromagnetic force acts over large distances force dominates terrestrial phenomena. Electric
and does not need any intervening medium. It force manifests itself in atmosphere where the
is enormously strong compared to gravity. The atoms are ionised and that leads to lightning.
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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 –15 m). 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 Recent developments have, however,
the book, at the surface in contact. If indicated that protons and neutrons are built
electromagnetic force were not intrinsically so out of still more elementary constituents called
much stronger than gravity, the hand of the quarks.
strongest man would crumble under the
1.4.4 Weak Nuclear Force
weight of a feather ! Indeed, to be consistent,
in that circumstance, we ourselves would The weak nuclear force appears only in certain
crumble under our own weight ! nuclear processes such as the β-decay of a
nucleus. In β-decay, the nucleus emits an
1.4.3 Strong Nuclear Force
electron and an uncharged particle called
The strong nuclear force binds protons and neutrino. The weak nuclear force is not as weak
neutrons in a nucleus. It is evident that without as the gravitational force, but much weaker
some attractive force, a nucleus will be than the strong nuclear and electromagnetic
unstable due to the electric repulsion between forces. The range of weak nuclear force is
its protons. This attractive force cannot be exceedingly small, of the order of 10 –16 m.
gravitational since force of gravity is negligible
compared to the electric force. A new basic force 1.4.5 Towards Unification of Forces
must, therefore, be invoked. The strong nuclear We remarked in section 1.1 that unification is a
force is the strongest of all fundamental forces, basic quest in physics. Great advances in
about 100 times the electromagnetic force in physics often amount to unification of different
2018-19
10 PHYSICS
Table 1.3 Fundamental forces of nature
Name Relative Range Operates among
strength
Gravitational force 10 –39 Infinite All objects in the universe
Weak nuclear force 10 –13 Very short, Sub-nuclear Some elementary particles,
size (∼10 –16 m) particularly electron and
neutrino
Electromagnetic force 10 –2 Infinite Charged particles
Strong nuclear force 1 Short, nuclear Nucleons, heavier
size (∼10 –15 m) elementary particles
theories and domains. Newton unified terrestrial 1.5 NATURE OF PHYSICAL LAWS
and celestial domains under a common law of
Physicists explore the universe. Their
gravitation. The experimental discoveries of
investigations, based on scientific processes,
Oersted and Faraday showed that electric and
range from particles that are smaller than
magnetic phenomena are in general
atoms in size to stars that are very far away. In
inseparable. Maxwell unified electromagnetism addition to finding the facts by observation and
and optics with the discovery that light is an experimentation, physicists attempt to discover
electromagnetic wave. Einstein attempted to the laws that summarise (often as mathematical
unify gravity and electromagnetism but could equations) these facts.
not succeed in this venture. But this did not In any physical phenomenon governed by
deter physicists from zealously pursuing the different forces, several quantities may change
goal of unification of forces. with time. A remarkable fact is that some special
Recent decades have seen much progress on physical quantities, however, remain constant
this front. The electromagnetic and the weak in time. They are the conserved quantities of
nuclear force have now been unified and are nature. Understanding these conservation
seen as aspects of a single ‘electro-weak’ force. principles is very important to describe the
What this unification actually means cannot observed phenomena quantitatively.
be explained here. Attempts have been (and are For motion under an external conservative
being) made to unify the electro-weak and the force, the total mechanical energy i.e. the sum
strong force and even to unify the gravitational of kinetic and potential energy of a body is a
force with the rest of the fundamental forces. constant. The familiar example is the free fall of
Many of these ideas are still speculative and an object under gravity. Both the kinetic energy
inconclusive. Table 1.4 summarises some of the of the object and its potential energy change
milestones in the progress towards unification continuously with time, but the sum remains
of forces in nature. fixed. If the object is released from rest, the initial
Table 1.4 Progress in unification of different forces/domains in nature
2018-19
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 Until the advent of Einstein’s theory of
Thermodynamics). 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 According to Einstein’s theory, mass m is
surroundings) remains unchanged. equivalent to energy E given by the relation
2
The law of conservation of energy is thought E = mc , where c is speed of light in vacuum.
to be valid across all domains of nature, from In a nuclear process mass gets converted to
the microscopic to the macroscopic. It is energy (or vice-versa). This is the energy which
routinely applied in the analysis of atomic, is released in a nuclear power generation and
nuclear and elementary particle processes. At 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.
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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
fundamental laws in physics. At this moment,
are also conserved quantities. These laws can
there are many such conservation laws. Apart from
be derived from Newton’s laws of motion in the above four, there are others which mostly deal
mechanics. But their validity goes beyond with quantities which have been introduced in
mechanics. They are the basic conservation nuclear and particle physics. Some of the
laws of nature in all domains, even in those conserved quantities are called spin, baryon
number, strangeness, hypercharge, etc, but you
where Newton’s laws may not be valid.
need not worry about them.
Besides their great simplicity and generality,
A conservation law is a hypothesis, based on
the conservation laws of nature are very useful observations and experiments. It is important to
in practice too. It often happens that we cannot remember that a conservation law cannot be
solve the full dynamics of a complex problem proved. It can be verified, or disproved, by
involving different particles and forces. The experiments. An experiment whose result is in
conformity with the law verifies or substantiates
conservation laws can still provide useful
the law; it does not prove the law. On the other
results. For example, we may not know the hand, a single experiment whose result goes
complicated forces that act during a collision against the law is enough to disprove it.
of two automobiles; yet momentum It would be wrong to ask somebody to prove
conservation law enables us to bypass the the law of conservation of energy. This law is an
complications and predict or rule out possible outcome of our experience over several centuries,
and it has been found to be valid in all
outcomes of the collision. In nuclear and
experiments, in mechanics, thermodynamics,
elementary particle phenomena also, the
electromagnetism, optics, atomic and nuclear
conservation laws are important tools of physics, or any other area.
analysis. Indeed, using the conservation laws Some students feel that they can prove the
of energy and momentum for β-decay, Wolfgang conservation of mechanical energy from a body
Pauli (1900-1958) correctly predicted in 1931 falling under gravity, by adding the kinetic and
potential energies at a point and showing that it
the existence of a new particle (now called
turns out to be constant. As pointed out above,
neutrino) emitted in β-decay along with the this is only a verification of the law, not its proof.
electron.
Conservation laws have a deep connection
with symmetries of nature that you will explore because of differing conditions at different
in more advanced courses in physics. For locations. For example, the acceleration due to
example, an important observation is that the gravity at the moon is one-sixth that at the earth,
laws of nature do not change with time! If you but the law of gravitation is the same both on
perform an experiment in your laboratory today the moon and the earth.) This symmetry of the
and repeat the same experiment (on the same laws of nature with respect to translation in
objects under identical conditions) after a year, space gives rise to conservation of linear
the results are bound to be the same. It turns momentum. In the same way isotropy of space
out that this symmetry of nature with respect to (no intrinsically preferred direction in space)
translation (i.e. displacement) in time is underlies the law of conservation of angular
equivalent to the law of conservation of energy. momentum*. The conservation laws of charge and
Likewise, space is homogeneous and there is no other attributes of elementary particles can also
(intrinsically) preferred location in the universe. be related to certain abstract symmetries.
To put it more clearly, the laws of nature are the Symmetries of space and time and other abstract
same everywhere in the universe. (Caution : the symmetries play a central role in modern theories
phenomena may differ from place to place of fundamental forces in nature.
* See Chapter 7
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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.
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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
2018-19
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.
2018-19
CHAPTER TWO
UNITS AND MEASUREMENT
2.1 INTRODUCTION
Measurement of any physical quantity involves comparison
with a certain basic, arbitrarily chosen, internationally
accepted reference standard called unit. The result of a
2.1 Introduction
measurement of a physical quantity is expressed by a
2.2 The international system of number (or numerical measure) accompanied by a unit.
units
Although the number of physical quantities appears to be
2.3 Measurement of length very large, we need only a limited number of units for
2.4 Measurement of mass expressing all the physical quantities, since they are inter-
2.5 Measurement of time 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
2.7 Significant figures derived quantities are called derived units. A complete set
of these units, both the base units and derived units, is
2.8 Dimensions of physical
quantities known as the system of units.
2.9 Dimensional formulae and 2.2 THE INTERNATIONAL SYSTEM OF UNITS
dimensional equations In earlier time scientists of different countries were using
2.10 Dimensional analysis and its different systems of units for measurement. Three such
applications systems, the CGS, the FPS (or British) system and the MKS
Summary system were in use extensively till recently.
Exercises The base units for length, mass and time in these systems
Additional exercises were as follows :
• In CGS system they were centimetre, gram and second
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, was developed and recommended
by General Conference on Weights and Measures in 1971 for
2018-19
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 (a)
Table 2.1. Besides the seven base units, there
are two more 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 as the ratio of
the intercepted area dA of the spherical surface,
described about the apex O as the centre, to
the square of its radius r, as shown in Fig. 2.1(a)
and (b) respectively. The unit for plane angle is
radian with the symbol rad and the unit for the (b)
solid angle is steradian with the symbol sr. Both Fig. 2.1 Description of (a) plane angle dθ and
these are dimensionless quantities. (b) solid angle dΩ .
Table 2.1 SI Base Quantities and Units*
Base SI Units
quantity Name Symbol Definition
Length metre m The metre is the length of the path travelled by light in vacuum
during a time interval of 1/299,792,458 of a second. (1983)
Mass kilogram kg The kilogram is equal to the mass of the international prototype
of the kilogram (a platinum-iridium alloy cylinder) kept at
international Bureau of Weights and Measures, at Sevres, near
Paris, France. (1889)
Time second s The second is the duration of 9,192,631,770 periods of the
radiation corresponding to the transition between the two
hyperfine levels of the ground state of the cesium-133 atom.
(1967)
Electric ampere A The ampere is that constant current which, if maintained in
current two straight parallel conductors of infinite length, of negligible
circular cross-section, and placed 1 metre apart in vacuum,
would produce between these conductors a force equal to 2×10 –7
newton per metre of length. (1948)
Thermo kelvin K The kelvin, is the fraction 1/273.16 of the thermodynamic
dynamic temperature of the triple point of water. (1967)
Temperature
Amount of mole mol The mole is the amount of substance of a system, which contains
substance as many elementary entities as there are atoms in 0.012
kilogram of carbon - 12. (1971)
Luminous candela cd The candela is the luminous intensity, in a given
intensity direction, of a source that emits monochromatic radiation of
12
frequency 540×10 hertz and that has a radiant intensity in
that direction of 1/683 watt per steradian. (1979)
* 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.
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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 Large distances such as the distance of a planet
may be atoms, molecules, ions, electrons, or a star from the earth cannot be measured
other particles or specified groups of such directly with a metre scale. An important method
particles.
We employ units for some physical quantities in such cases is the parallax method.
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
Common SI prefixes and symbols for multiples the distance between the eyes.
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
2.3 MEASUREMENT OF LENGTH along which the planet is viewed at these two
You are already familiar with some direct methods points. The ∠ASB in Fig. 2.2 represented by
for the measurement of length. For example, a symbol θ is called the parallax angle or
–3
metre scale is used for lengths from 10 m to 10 2 parallactic angle.
m. A vernier callipers is used for lengths to an b
accuracy of 10 –4 m. A screw gauge and a As the planet is very far away, << 1, and
spherometer can be used to measure lengths as D
less as to 10 m. To measure lengths beyond these therefore, θ is very small. Then we
–5
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
2018-19
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
= 40 (θ is known as ‘parallax’) estimate
0
b the distance of the tower C from his original
D = (2.1)
θ position A.
Fig. 2.2 Parallax method.
Fig. 2.3
Having determined D, we can employ a similar Answer We have, parallax angle θ = 40 0
method to determine the size or angular diameter From Fig. 2.3, AB = AC tan θ
of the planet. If d is the diameter of the planet AC = AB/tanθ = 100 m/tan 40 0
and α the angular size of the planet (the angle = 100 m/0.8391 = 119 m t
subtended by d at the earth), we have
α = d/D (2.2) t Example 2.3 The moon is observed from
The angle α can be measured from the same two diametrically opposite points A and B
location on the earth. It is the angle between on Earth. The angle θ subtended at the
the two directions when two diametrically moon by the two directions of observation
o
opposite points of the planet are viewed through is 1 54′. Given the diameter of the Earth to
×
the telescope. Since D is known, the diameter d be about 1.276 × ×× × 10 7 m, compute the
of the planet can be determined using Eq. (2.2). distance of the moon from the Earth.
′
Example 2.1 Calculate the angle of Answer We have θ = 1° 54 = 114 ′
t
(a) 1 (degree) (b) 1′ (minute of arc or arcmin) ′′ -6
0
and (c) 1″(second of arc or arc second) in = (114 60× ) ( 4.85 10× × ) rad
radians. Use 360 =2π rad, 1 =60′ and −2
0
0
1′ = 60 ″ = 3.32 ×10 rad,
−
6
×
"
Answer (a) We have 360 = 2π rad since 1 = 4.85 10 rad .
0
1 = (π /180) rad = 1.745×10 rad Also b = AB =1.276 ×10 7 m
–2
0
(b) 1 = 60′ = 1.745×10 rad Hence from Eq. (2.1), we have the earth-moon
0
–2
1′ = 2.908×10 –4 rad 2.91×10 –4 rad distance,
(c) 1′ = 60″ = 2.908×10 rad D = /θ
–4
b
1″ = 4.847×10 –4 rad 4.85×10 –6 rad t 1.276 10 7
×
Example 2.2 A man wishes to estimate = 3.32 10 -2
×
t
the distance of a nearby tower from him.
8
×
He stands at a point A in front of the tower = 3.84 10 m t
C and spots a very distant object O in line
with AC. He then walks perpendicular to t Example 2.4 The Sun’s angular diameter
AC up to B, a distance of 100 m, and looks is measured to be 1920′′. The distance D of
at O and C again. Since O is very distant, the Sun from the Earth is 1.496 × 10 11 m.
the direction BO is practically the same as What is the diameter of the Sun ?
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20 PHYSICS
Answer Sun’s angular diameter α of this solution and dilute it to 20 cm , using
3
= 1920" alcohol. So, the concentration of the solution is
= 1920 4.85 10 −6 rad equal to 1 cm of oleic acid/cm 3 of
3
×
×
= 9.31 ×10 rad 20 × 20
−3
Sun’s diameter solution. Next we lightly sprinkle some
d = α D lycopodium powder on the surface of water in a
large trough and we put one drop of this solution
×
×
= 9.31 10 − 3 1.496 10 11 m
×
in the water. The oleic acid drop spreads into a
thin, large and roughly circular film of molecular
=1.39 10 m× 9 t thickness on water surface. Then, we quickly
measure the diameter of the thin film to get its
2.3.2 Estimation of Very Small Distances: area A. Suppose we have dropped n drops in
Size of a Molecule the water. Initially, we determine the
3
To measure a very small size, like that of a approximate volume of each drop (V cm ).
–8
molecule (10 m to 10 –10 m), we have to adopt Volume of n drops of solution
special methods. We cannot use a screw gauge = nV cm 3
or similar instruments. Even a microscope has
certain limitations. An optical microscope uses Amount of oleic acid in this solution
visible light to ‘look’ at the system under
investigation. As light has wave like features, nV 1 cm 3
the resolution to which an optical microscope = 20 20
×
can be used is the wavelength of light (A detailed This solution of oleic acid spreads very fast
explanation can be found in the Class XII on the surface of water and forms a very thin
Physics textbook). For visible light the range of layer of thickness t. If this spreads to form a
wavelengths is from about 4000 Å to 7000 Å cm , then the thickness of the
2
(1 angstrom = 1 Å = 10 m). Hence an optical film of area A
-10
film
microscope cannot resolve particles with sizes
smaller than this. Instead of visible light, we can t = Volume of the film
use an electron beam. Electron beams can be Area of the film
focussed by properly designed electric and
magnetic fields. The resolution of such an or, t = nV cm (2.3)
×
electron microscope is limited finally by the fact 20 20 A
that electrons can also behave as waves ! (You
will learn more about this in class XII). The If we assume that the film has mono-molecular
wavelength of an electron can be as small as a thickness, then this becomes the size or diameter
fraction of an angstrom. Such electron of a molecule of oleic acid. The value of this
–9
microscopes with a resolution of 0.6 Å have been thickness comes out to be of the order of 10 m.
built. They can almost resolve atoms and
molecules in a material. In recent times, t Example 2.5 If the size of a nucleus (in
tunnelling microscopy has been developed in the range of 10 –15 to 10 –14 m) is scaled up
which again the limit of resolution is better than to the tip of a sharp pin, what roughly is
an angstrom. It is possible to estimate the sizes the size of an atom ? Assume tip of the pin
–4
–5
of molecules. to be in the range 10 m to 10 m.
A simple method for estimating the molecular
size of oleic acid is given below. Oleic acid is a Answer The size of a nucleus is in the range of
m and 10
m.
The tip of a sharp pin is
10
–15
–14
soapy liquid with large molecular size of the taken to be in the range of 10 m and 10 m.
–5
–4
order of 10 m. Thus we are scaling up by a factor of 10 . An
–9
10
The idea is to first form mono-molecular layer atom roughly of size 10 m will be scaled up to a
–10
of oleic acid on water surface. size of 1 m. Thus a nucleus in an atom is as small
We dissolve 1 cm of oleic acid in alcohol to in size as the tip of a sharp pin placed at the centre
3
make a solution of 20 cm . Then we take 1 cm 3 of a sphere of radius about a metre long. t
3
2018-19
UNITS AND MEASUREMENT 21
2.3.3 Range of Lengths While dealing with atoms and molecules, the
The sizes of the objects we come across in the kilogram is an inconvenient unit. In this case,
universe vary over a very wide range. These may there is an important standard unit of mass,
vary from the size of the order of 10 –14 m of the called the unified atomic mass unit (u), which
tiny nucleus of an atom to the size of the order has been established for expressing the mass
of 10 m of the extent of the observable universe. of atoms as
26
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
1 astronomical unit = 1 AU (average distance determined by a common balance like the one
of the Sun from the Earth) used in a grocery shop. Large masses in the
= 1.496 × 10 m universe like planets, stars, etc., based on
11
1 light year = 1 ly= 9.46 × 10 m (distance Newton’s law of gravitation can be measured by
15
that light travels with velocity of using gravitational method (See Chapter 8). For
3 × 10 m s in 1 year) measurement of small masses of atomic/sub-
–1
8
16
1 parsec = 3.08 × 10 m (Parsec is the atomic particles etc., we make use of mass
distance at which average radius of earth’s orbit spectrograph in which radius of the trajectory
subtends an angle of 1 arc second) is proportional to the mass of a charged particle
moving in uniform electric and magnetic field.
2.4 MEASUREMENT OF MASS
Mass is a basic property of matter. It does not 2.4.1 Range of Masses
depend on the temperature, pressure or location The masses of the objects, we come across in
of the object in space. The SI unit of mass is
kilogram (kg). The prototypes of the International the universe, vary over a very wide range. These
-30
standard kilogram supplied by the International may vary from tiny mass of the order of 10 kg
55
Bureau of Weights and Measures (BIPM) are of an electron to the huge mass of about 10 kg
available in many other laboratories of different of the known universe. Table 2.4 gives the range
countries. In India, this is available at the and order of the typical masses of various
National Physical Laboratory (NPL), New Delhi. objects.
Table 2.3 Range and order of lengths
2018-19
22 PHYSICS
13
Table 2.4 Range and order of masses ± 1 × 10 –13 , i.e. 1 part in 10 . This implies that
the uncertainty gained over time by such a
device is less than 1 part in 10 ; they lose or
13
gain no more than 3 µs in one year. In view of
the tremendous accuracy in time measurement,
the SI unit of length has been expressed in terms
the path length light travels in certain interval
of time (1/299, 792, 458 of a second) (Table 2.1).
The time interval of events that we come
across in the universe vary over a very wide
range. Table 2.5 gives the range and order of
some typical time intervals.
You may notice that there is an interesting
coincidence between the numbers appearing
in Tables 2.3 and 2.5. Note that the ratio of the
longest and shortest lengths of objects in our
universe is about 10 . Interestingly enough,
41
the ratio of the longest and shortest time
intervals associated with the events and objects
2.5 MEASUREMENT OF TIME in our universe is also about 10 . This number,
41
To measure any time interval we need a clock. 10 comes up again in Table 2.4, which lists
41
We now use an atomic standard of time, which typical masses of objects. The ratio of the
is based on the periodic vibrations produced in largest and smallest masses of the objects in
a cesium atom. This is the basis of the cesium our universe is about (10 ) . Is this a curious
41 2
clock, sometimes called atomic clock, used in coincidence between these large numbers
the national standards. Such standards are purely accidental ?
available in many laboratories. In the cesium
atomic clock, the second is taken as the time 2.6 ACCURACY, PRECISION OF INSTRUMENTS
needed for 9,192,631,770 vibrations of the AND ERRORS IN MEASUREMENT
radiation corresponding to the transition
between the two hyperfine levels of the ground Measurement is the foundation of all
state of cesium-133 atom. The vibrations of the experimental science and technology. The result
cesium atom regulate the rate of this cesium of every measurement by any measuring
atomic clock just as the vibrations of a balance instrument contains some uncertainty. This
wheel regulate an ordinary wristwatch or the uncertainty is called error. Every calculated
vibrations of a small quartz crystal regulate a quantity which is based on measured values,
quartz wristwatch. also has an error. We shall distinguish between
The cesium atomic clocks are very accurate. two terms: accuracy and precision. The
In principle they provide portable standard. The accuracy of a measurement is a measure of how
national standard of time interval ‘second’ as close the measured value is to the true value of
the quantity. Precision tells us to what resolution
well as the frequency is maintained through four
cesium atomic clocks. A cesium atomic clock is or limit the quantity is measured.
used at the National Physical Laboratory (NPL), The accuracy in measurement may depend on
New Delhi to maintain the Indian standard of several factors, including the limit or the resolution
time. of the measuring instrument. For example, suppose
In our country, the NPL has the responsibility the true value of a certain length is near 3.678 cm.
of maintenance and improvement of physical In one experiment, using a measuring instrument
standards, including that of time, frequency, etc. of resolution 0.1 cm, the measured value is found to
Note that the Indian Standard Time (IST) is be 3.5 cm, while in another experiment using a
linked to this set of atomic clocks. The efficient measuring device of greater resolution, say 0.01 cm,
cesium atomic clocks are so accurate that they the length is determined to be 3.38 cm. The first
impart the uncertainty in time realisation as measurement has more accuracy (because it is
2018-19
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 systematic errors are those errors that the apparatus or individual’s carelessness
tend to be in one direction, either positive or in taking observations without observing
negative. Some of the sources of systematic proper precautions, etc. For example, if you,
errors are : by habit, always hold your head a bit too far
(a) Instrumental errors that arise from the to the right while reading the position of a
errors due to imperfect design or calibration needle on the scale, you will introduce an
of the measuring instrument, zero error in error due to parallax.
the instrument, etc. For example, the Systematic errors can be minimised by
temperature graduations of a thermometer improving experimental techniques, selecting
may be inadequately calibrated (it may read better instruments and removing personal bias
104 °C at the boiling point of water at STP as far as possible. For a given set-up, these
whereas it should read 100 °C); in a vernier errors may be estimated to a certain extent and
callipers the zero mark of vernier scale may the necessary corrections may be applied to the
not coincide with the zero mark of the main readings.
scale, or simply an ordinary metre scale may
be worn off at one end. Random errors
(b) Imperfection in experimental technique The random errors are those errors, which occur
or procedure To determine the temperature irregularly and hence are random with respect
2018-19
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 The magnitude of the difference
temperature, voltage supply, mechanical between the individual measurement and
vibrations of experimental set-ups, etc), personal the true value of the quantity is called the
(unbiased) errors by the observer taking absolute error of the measurement. This
readings, etc. For example, when the same is denoted by |∆a |. In absence of any other
person repeats the same observation, it is very method of knowing true value, we considered
likely that he may get different readings arithmatic mean as the true value. Then the
everytime.
errors in the individual measurement values
Least count error from the true value, are
The smallest value that can be measured by the ∆a = a – a mean ,
1
1
measuring instrument is called its least count. ∆a = a – a ,
2
mean
2
All the readings or measured values are good only .... .... ....
up to this value. .... .... ....
The least count error is the error ∆a = a – a
associated with the resolution of the instrument. n n mean
For example, a vernier callipers has the least The ∆a calculated above may be positive in
count as 0.01cm; a spherometer may have a certain cases and negative in some other
least count of 0.001 cm. Least count error cases. But absolute error |∆a| will always
belongs to the category of random errors but be positive.
within a limited size; it occurs with both (b) The arithmetic mean of all the absolute errors
systematic and random errors. If we use a metre is taken as the final or mean absolute error
scale for measurement of length, it may have of the value of the physical quantity a. It is
graduations at 1 mm division scale spacing or .
interval. represented by ∆a mean
Using instruments of higher precision, Thus,
improving experimental techniques, etc., we can
reduce the least count error. Repeating the ∆a mean = (|∆a |+|∆a |+|∆a |+...+ |∆a |)/n
2
1
n
3
observations several times and taking the (2.6)
arithmetic mean of all the observations, the
mean value would be very close to the true value n
of the measured quantity. = ∑ |∆a |/n (2.7)
i
= i 1
2.6.1 Absolute Error, Relative Error and If we do a single measurement, the value we
Percentage Error
get may be in the range a ± ∆a
(a) Suppose the values obtained in several mean mean
measurements are a , a , a ...., a . The i.e. a = a ± ∆a
2
3
n
1
arithmetic mean of these values is taken as mean mean
the best possible value of the quantity under or,
the given conditions of measurement as : a mean – ∆a mean ≤ a ≤ a mean + ∆a mean
(2.8)
a = (a +a +a +...+a ) / n (2.4)
mean 1 2 3 n This implies that any measurement of the
or, physical quantity a is likely to lie between
n (a mean + ∆a mean ) and (a mean − ∆a mean ).
mean ∑ (c) Instead of the absolute error, we often use
=
a a / n (2.5)
i
=
i 1 the relative error or the percentage error
This is because, as explained earlier, it is (δa). The relative error is the ratio of the
∆
reasonable to suppose that individual mean absolute error ∆ ∆∆ ∆a mean to the mean
measurements are as likely to overestimate value a mean of the quantity measured.
2018-19
UNITS AND MEASUREMENT 25
Relative error = ∆a /a (2.9) = 2.624 s
mean mean
= 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
is proper to put this mean period also to the
Thus, Percentage error
second decimal.
δa = (∆a /a ) × 100% (2.10)
mean mean The errors in the measurements are
Let us now consider an example. 2.63 s – 2.62 s = 0.01 s
2.56 s – 2.62 s = – 0.06 s
Example 2.6 Two clocks are being tested 2.42 s – 2.62 s = – 0.20 s
t
against a standard clock located in a 2.71 s – 2.62 s = 0.09 s
national laboratory. At 12:00:00 noon by 2.80 s – 2.62 s = 0.18 s
the standard clock, the readings of the two Note that the errors have the same units as the
clocks are : quantity to be measured.
Clock 1 Clock 2 The arithmetic mean of all the absolute errors
(for arithmetic mean, we take only the
Monday 12:00:05 10:15:06 magnitudes) is
Tuesday 12:01:15 10:14:59
Wednesday 11:59:08 10:15:18 ∆Τ = [(0.01+ 0.06+0.20+0.09+0.18)s]/5
mean
Thursday 12:01:50 10:15:07 = 0.54 s/5
Friday 11:59:15 10:14:53 = 0.11 s
Saturday 12:01:30 10:15:24
Sunday 12:01:19 10:15:11 That means, the period of oscillation of the
simple pendulum is (2.62 ± 0.11) s i.e. it lies
If you are doing an experiment that requires
precision time interval measurements, which between (2.62 + 0.11) s and (2.62 – 0.11) s or
of the two clocks will you prefer ? between 2.73 s and 2.51 s. As the arithmetic
mean of all the absolute errors is 0.11 s, there
is already an error in the tenth of a second.
Answer The range of variation over the seven Hence there is no point in giving the period to a
days of observations is 162 s for clock 1, and hundredth. A more correct way will be to write
31 s for clock 2. The average reading of clock 1
is much closer to the standard time than the T = 2.6 ± 0.1 s
average reading of clock 2. The important point Note that the last numeral 6 is unreliable, since
is that a clock’s zero error is not as significant it may be anything between 5 and 7. We indicate
for precision work as its variation, because a this by saying that the measurement has two
‘zero-error’ can always be easily corrected. significant figures. In this case, the two
Hence clock 2 is to be preferred to clock 1. t
significant figures are 2, which is reliable and
6, which has an error associated with it. You
Example 2.7 We measure the period of will learn more about the significant figures in
t
oscillation of a simple pendulum. In section 2.7.
successive measurements, the readings For this example, the relative error or the
turn out to be 2.63 s, 2.56 s, 2.42 s, 2.71s percentage error is
and 2.80 s. Calculate the absolute errors,
relative error or percentage error.
.
01
δa = ×100 = 4 % t
Answer The mean period of oscillation of the 2.6
pendulum
+
+
(2.63 2.56 2.42 + 2.71 2.80 )s
+
T = 2.6.2 Combination of Errors
5 If we do an experiment involving several
13.12 measurements, we must know how the errors
= s
5 in all the measurements combine. For example,
2018-19
26 PHYSICS
(a) Error of a sum or a difference
How will you measure the length of a line?
Suppose two physical quantities A and B have
What a naïve question, at this stage, you might measured values A ± ∆A, B ± ∆B respectively
say! But what if it is not a straight line? Draw where ∆A and ∆B are their absolute errors. We
a zigzag line in your copy, or on the blackboard. wish to find the error ∆Z in the sum
Well, not too difficult again. You might take a Z = A + B.
thread, place it along the line, open up the We have by addition, Z ± ∆Z
thread, and measure its length. = (A ± ∆A) + (B ± ∆B).
Now imagine that you want to measure the
length of a national highway, a river, the railway The maximum possible error in Z
track between two stations, or the boundary ∆Z = ∆A + ∆B
between two states or two nations. If you take For the difference Z = A – B, we have
a string of length 1 metre or 100 metre, keep it Z ± ∆ Z = (A ± ∆A) – (B ± ∆B)
along the line, shift its position every time, the = (A – B) ± ∆A ± ∆B
arithmetic of man-hours of labour and expenses
on the project is not commensurate with the or, ± ∆Z = ± ∆A ± ∆B
outcome. Moreover, errors are bound to occur The maximum value of the error ∆Z is again
in this enormous task. There is an interesting ∆A + ∆B.
fact about this. France and Belgium share a
common international boundary, whose length Hence the rule : When two quantities are
mentioned in the official documents of the two added or subtracted, the absolute error in the
countries differs substantially! final result is the sum of the absolute errors
Go one step beyond and imagine the in the individual quantities.
coastline where land meets sea. Roads and rivers
have fairly mild bends as compared to a The temperatures of two
coastline. Even so, all documents, including our t Example 2.8
school books, contain information on the length bodies measured by a thermometer are 0
0
0
0
of the coastline of Gujarat or Andhra Pradesh, t = 20 C ± 0.5 C and t = 50 C ± 0.5 C.
1
2
or the common boundary between two states, Calculate the temperature difference and
etc. Railway tickets come with the distance the error theirin.
between stations printed on them. We have
‘milestones’ all along the roads indicating the Answer t′ = t –t = (50 C±0.5 C)– (20 C±0.5 C)
0
0
0
0
distances to various towns. So, how is it done? 2 1
0
0
One has to decide how much error one can t′ = 30 C ± 1 C t
tolerate and optimise cost-effectiveness. If you (b) Error of a product or a quotient
want smaller errors, it will involve high
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 Z ± ∆Z = (A ± ∆A) (B ± ∆B)
belongs to the areas of fractals, which has lately
become popular in theoretical physics. Even = AB ± B ∆A ± A ∆B ± ∆A ∆B.
then one doesn’t know how much to rely on Dividing LHS by Z and RHS by AB we have,
the figure that props up, as is clear from the 1±(∆Z/Z) = 1 ± (∆A/A) ± (∆B/B) ± (∆A/A)(∆B/B).
story of France and Belgium. Incidentally, this
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
mass density is obtained by deviding mass by ∆Z/ Z = (∆A/A) + (∆B/B).
the volume of the substance. If we have errors You can easily verify that this is true for division
in the measurement of mass and of the sizes or also.
dimensions, we must know what the error will
be in the density of the substance. To make such Hence the rule : When two quantities are
estimates, we should learn how errors combine multiplied or divided, the relative error in the
in various mathematical operations. For this, result is the sum of the relative errors in the
we use the following procedure. multipliers.
2018-19
UNITS AND MEASUREMENT 27
Then,
Example 2.9 The resistance R = V/I where ∆Z/Z = (∆A/A) + (∆A/A) = 2 (∆A/A).
t
V = (100 ± 5)V and I = (10 ± 0.2)A. Find the Hence, the relative error in A is two times the
2
percentage error in R.
error in A.
Answer The percentage error in V is 5% and in In general, if Z = A B /C r
q
p
I it is 2%. The total error in R would therefore Then,
be 5% + 2% = 7%. t ∆Z/Z = p (∆A/A) + q (∆B/B) + r (∆C/C).
Example 2.10 Two resistors of resistances Hence the rule : The relative error in a
t
R = 100 ±3 ohm and R = 200 ± 4 ohm are physical quantity raised to the power k is the
2
1
connected (a) in series, (b) in parallel. Find k times the relative error in the individual
the equivalent resistance of the (a) series quantity.
combination, (b) parallel combination. Use
for (a) the relation R = R + R and for (b) t Example 2.11 Find the relative error in
1 2,
Z, if Z = A B 1/3 /CD 3/2 .
4
and
Answer The relative error in Z is ∆Z/Z =
Answer (a) The equivalent resistance of series 4(∆A/A) +(1/3) (∆B/B) + (∆C/C) + (3/2) (∆D/D).
combination t
R = R + R = (100 ± 3) ohm + (200 ± 4) ohm
1 2 t Example 2.12 The period of oscillation of
= 300 ± 7 ohm. a simple pendulum is T = 2π L/g.
(b) The equivalent resistance of parallel Measured value of L is 20.0 cm known to 1
combination mm accuracy and time for 100 oscillations
R R 200 of the pendulum is found to be 90 s using
R′= 1 2 = = 66.7 ohm a wrist watch of 1 s resolution. What is the
R + R 3
1 2 accuracy in the determination of g ?
1 1 1 2 2
Then, from R′ = R 1 + R 2 Answer g = 4π L/T t ∆ ∆ t ∆
t
we get, Here, T = n and T∆ = n . Therefore, T T = t .
∆ R′ = ∆ R 1 + ∆ R 2 The errors in both L and t are the least count
R′ 2 R 2 R 2 errors. Therefore,
2
1
(∆g/g) = (∆L/L) + 2(∆T/T )
)
)
.
∆ R′ = ( R′ 2 ∆ R 1 + ( R′ 2 ∆ R 2 0 1 1
.
R 1 2 R 2 2 = 20 0 + 2 90 = 0 027
.
66.7 2 66.7 2 Thus, the percentage error in g is
=
3 + 4 100 (∆g/g) = 100(∆L/L) + 2 × 100 (∆T/T )
100 200
= 3% t
= 1.8
2.7 SIGNIFICANT FIGURES
±
Then, R′ = 66.7 1.8 ohm
(Here, ∆R is expresed as 1.8 instead of 2 to As discussed above, every measurement
keep in confirmity with the rules of significant involves errors. Thus, the result of
figures.) t measurement should be reported in a way that
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
Suppose Z = A , digit that is uncertain. The reliable digits plus
2
2018-19
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 decimal point are significant.
say the period of oscillation of a simple [The numbers 3.500 or 0.06900 have four
pendulum is 1.62 s, the digits 1 and 6 are significant figures each.]
reliable and certain, while the digit 2 is
uncertain. Thus, the measured value has three (2) There can be some confusion regarding the
significant figures. The length of an object trailing zero(s). Suppose a length is reported to
reported after measurement to be 287.5 cm has be 4.700 m. It is evident that the zeroes here
four significant figures, the digits 2, 8, 7 are are meant to convey the precision of
certain while the digit 5 is uncertain. Clearly, measurement and are, therefore, significant. [If
reporting the result of measurement that these were not, it would be superfluous to write
them explicitly, the reported measurement
includes more digits than the significant digits would have been simply 4.7 m]. Now suppose
is superfluous and also misleading since it would we change units, then
give a wrong idea about the precision of
measurement. 4.700 m = 470.0 cm = 4700 mm = 0.004700 km
The rules for determining the number of Since the last number has trailing zero(s) in a
significant figures can be understood from the number with no decimal, we would conclude
following examples. Significant figures indicate, erroneously from observation (1) above that the
as already mentioned, the precision of number has two significant figures, while in
measurement which depends on the least count fact, it has four significant figures and a mere
of the measuring instrument. A choice of change of units cannot change the number of
change of different units does not change the significant figures.
number of significant digits or figures in a (3) To remove such ambiguities in
measurement. This important remark makes determining the number of significant
most of the following observations clear: figures, the best way is to report every
(1) For example, the length 2.308 cm has four measurement in scientific notation (in the
significant figures. But in different units, the power of 10). In this notation, every number is
same value can be written as 0.02308 m or 23.08 b
expressed as a × 10 , where a is a number
mm or 23080 µm. between 1 and 10, and b is any positive or
All these numbers have the same number of negative exponent (or power) of 10. In order to
significant figures (digits 2, 3, 0, 8), namely four. get an approximate idea of the number, we may
This shows that the location of decimal point is round off the number a to 1 (for a ≤ 5) and to 10
of no consequence in determining the number (for 5<a ≤ 10). Then the number can be
of significant figures. expressed approximately as 10 b in which the
The example gives the following rules : exponent (or power) b of 10 is called order of
• All the non-zero digits are significant. magnitude of the physical quantity. When only
• All the zeros between two non-zero digits an estimate is required, the quantity is of the
b
are significant, no matter where the order of 10 . For example, the diameter of the
7
7
decimal point is, if at all. earth (1.28×10 m) is of the order of 10 m with
• If the number is less than 1, the zero(s) the order of magnitude 7. The diameter of
on the right of decimal point but to the hydrogen atom (1.06 ×10 –10 m) is of the order of
–10
left of the first non-zero digit are not 10 m, with the order of magnitude
significant. [In 0.00 2308, the underlined –10. Thus, the diameter of the earth is 17 orders
zeroes are not significant]. of magnitude larger than the hydrogen atom.
• The terminal or trailing zero(s) in a It is often customary to write the decimal after
number without a decimal point are not the first digit. Now the confusion mentioned in
(a) above disappears :
significant. 2
[Thus 123 m = 12300 cm = 123000 mm has 4.700 m = 4.700 × 10 cm –3
3
three significant figures, the trailing zero(s) = 4.700 × 10 mm = 4.700 × 10 km
being not significant.] However, you can also The power of 10 is irrelevant to the
see the next observation. determination of significant figures. However, all
2018-19
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
Thus, in the scientific notation, no confusion consistent with the precision of the input
arises about the trailing zero(s) in the base measured values :
number a. They are always significant. (1) In multiplication or division, the final
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
with the least significant figures.
the rules adopted in the preceding example : Thus, in the example above, density should
• For a number greater than 1, without any be reported to three significant figures.
decimal, the trailing zero(s) are not 4.237g
significant. Density = = 1.69 g cm -3
2.51 cm 3
• For a number with a decimal, the trailing
zero(s) are significant. Similarly, if the speed of light is given as
3 × 10 8 m s -1 (one significant figure) and one
(5) The digit 0 conventionally put on the left of a 7
decimal for a number less than 1 (like 0.1250) year (1y = 365.25 d) has 3.1557 × 10 s (five
15
significant figures), the light year is 9.47 × 10 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
should retain as many decimal places as are
(6) The multiplying or dividing factors which are
neither rounded numbers nor numbers there in the number with the least decimal
representing measured values are exact and places.
For example, the sum of the numbers
have infinite number of significant digits. For
436.32 g, 227.2 g and 0.301 g by mere arithmetic
d addition, is 663.821 g. But the least precise
example in r = or s = 2πr, the factor 2 is an
2 measurement (227.2 g) is correct to only one
exact number and it can be written as 2.0, 2.00 decimal place. The final result should, therefore,
be rounded off to 663.8 g.
t Similarly, the difference in length can be
or 2.0000 as required. Similarly, in T = , n is
n expressed as :
an exact number.
–3
0.307 m – 0.304 m = 0.003 m = 3 × 10 m.
2.7.1 Rules for Arithmetic Operations with Note that we should not use the rule (1)
Significant Figures applicable for multiplication and division and
The result of a calculation involving approximate write 664 g as the result in the example of
–3
measured values of quantities (i.e. values with addition and 3.00 × 10 m in the example of
limited number of significant figures) must reflect subtraction. They do not convey the precision
the uncertainties in the original measured values. of measurement properly. For addition and
It cannot be more accurate than the original subtraction, the rule is in terms of decimal
measured values themselves on which the result places.
is based. In general, the final result should not
have more significant figures than the original 2.7.2 Rounding off the Uncertain Digits
data from which it was obtained. Thus, if mass of The result of computation with approximate
an object is measured to be, say, 4.237 g (four numbers, which contain more than one
significant figures) and its volume is measured to uncertain digit, should be rounded off. The rules
be 2.51 cm , then its density, by mere arithmetic for rounding off numbers to the appropriate
3
division, is 1.68804780876 g/cm upto 11 decimal significant figures are obvious in most cases. A
3
places. It would be clearly absurd and irrelevant number 2.746 rounded off to three significant
to record the calculated value of density to such a figures is 2.75, while the number 2.743 would
precision when the measurements on which the be 2.74. The rule by convention is that the
value is based, have much less precision. The preceding digit is raised by 1 if the
2018-19
30 PHYSICS
insignificant digit to be dropped (the
Example 2.14 5.74 g of a substance
underlined digit in this case) is more than
t
occupies 1.2 cm . Express its density by
3
5, and is left unchanged if the latter is less keeping the significant figures in view.
than 5. But what if the number is 2.745 in
which the insignificant digit is 5. Here, the Answer There are 3 significant figures in the
convention is that if the preceding digit is measured mass whereas there are only 2
even, the insignificant digit is simply significant figures in the measured volume.
dropped and, if it is odd, the preceding digit Hence the density should be expressed to only
is raised by 1. Then, the number 2.745 rounded 2 significant figures.
off to three significant figures becomes 2.74. On 5.74 −3
the other hand, the number 2.735 rounded off Density = 1.2 g cm
to three significant figures becomes 2.74 since = 4.8 g cm --3 . t
the preceding digit is odd.
In any involved or complex multi-step 2.7.3 Rules for Determining the Uncertainty
calculation, you should retain, in intermediate in the Results of Arithmatic
steps, one digit more than the significant digits Calculations
and round off to proper significant figures at the
end of the calculation. Similarly, a number The rules for determining the uncertainty or
known to be within many significant figures, error in the number/measured quantity in
arithmetic operations can be understood from
such as in 2.99792458 × 10 m/s for the speed the following examples.
8
of light in vacuum, is rounded off to an (1) If the length and breadth of a thin
approximate value 3 × 10 m/s , which is often rectangular sheet are measured, using a metre
8
employed in computations. Finally, remember scale as 16.2 cm and, 10.1 cm respectively, there
that exact numbers that appear in formulae like are three significant figures in each
measurement. It means that the length l may
L
2 π in T = 2π , have a large (infinite) number be written as
g
l = 16.2 ± 0.1 cm
of significant figures. The value of π = = 16.2 cm ± 0.6 %.
3.1415926.... is known to a large number of
significant figures. You may take the value as Similarly, the breadth b may be written as
3.142 or 3.14 for π, with limited number of b = 10.1 ± 0.1 cm
significant figures as required in specific = 10.1 cm ± 1 %
cases.
Then, the error of the product of two (or more)
Example 2.13 Each side of a cube is experimental values, using the combination of
t
measured to be 7.203 m. What are the errors rule, will be
total surface area and the volume of the 2
cube to appropriate significant figures? l b = 163.62 cm + 1.6%
= 163.62 + 2.6 cm 2
Answer The number of significant figures in
the measured length is 4. The calculated area This leads us to quote the final result as
and the volume should therefore be rounded off l b = 164 + 3 cm 2
to 4 significant figures.
Here 3 cm 2 is the uncertainty or error in the
Surface area of the cube = 6(7.203) m 2 estimation of area of rectangular sheet.
2
= 311.299254 m 2 (2) If a set of experimental data is specified
= 311.3 m 2 to n significant figures, a result obtained by
combining the data will also be valid to n
Volume of the cube = (7.203) m 3
3
significant figures.
= 373.714754 m 3
However, if data are subtracted, the number of
= 373.7 m 3 t significant figures can be reduced.
2018-19
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 The dimensions of a physical quantity are the
than the number of significant figures in any of powers (or exponents) to which the base
the number added or subtracted). quantities are raised to represent that
quantity. Note that using the square brackets
(3) The relative error of a value of number
specified to significant figures depends not [ ] round a quantity means that we are dealing
only on n but also on the number itself. with ‘the dimensions of’ the quantity.
In mechanics, all the physical quantities can
For example, the accuracy in measurement of be written in terms of the dimensions [L], [M]
mass 1.02 g is ± 0.01 g whereas another and [T]. For example, the volume occupied by
measurement 9.89 g is also accurate to ± 0.01 g.
The relative error in 1.02 g is an object is expressed as the product of length,
= (± 0.01/1.02) × 100 % breadth and height, or three lengths. Hence the
3
3
= ± 1% dimensions of volume are [L] × [L] × [L] = [L] = [L ].
Similarly, the relative error in 9.89 g is As the volume is independent of mass and time,
= (± 0.01/9.89) × 100 % it is said to possess zero dimension in mass [M°],
= ± 0.1 % zero dimension in time [T°] and three
Finally, remember that intermediate results in dimensions in length.
Similarly, force, as the product of mass and
a multi-step computation should be acceleration, can be expressed as
calculated to one more significant figure in Force = mass × acceleration
every measurement than the number of = mass × (length)/(time) 2
digits in the least precise measurement.
2
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 ]. 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 Note that in this type of representation, the
0.104 calculated to three significant figures is magnitudes are not considered. It is the quality
9.62. However, if we had written 1/9.58 = 0.1044 of the type of the physical quantity that enters.
Thus, a change in velocity, initial velocity,
and then taken the reciprocal to three significant average velocity, final velocity, and speed are
figures, we would have retrieved the original all equivalent in this context. Since all these
value of 9.58. quantities can be expressed as length/time,
This example justifies the idea to retain one their dimensions are [L]/[T] or [L T ].
–1
more extra digit (than the number of digits in
the least precise measurement) in intermediate 2.9 DIMENSIONAL FORMULAE AND
steps of the complex multi-step calculations in DIMENSIONAL EQUATIONS
order to avoid additional errors in the process The expression which shows how and which of
of rounding off the numbers. the base quantities represent the dimensions
of a physical quantity is called the dimensional
2.8 DIMENSIONS OF PHYSICAL QUANTITIES formula of the given physical quantity. For
example, the dimensional formula of the volume
The nature of a physical quantity is described is [M° L T°], and that of speed or velocity is
3
by its dimensions. All the physical quantities [M° L T ]. Similarly, [M° L T ] is the dimensional
–2
-1
represented by derived units can be expressed formula of acceleration and [M L –3 T°] that of
in terms of some combination of seven mass density.
fundamental or base quantities. We shall call An equation obtained by equating a physical
these base quantities as the seven dimensions quantity with its dimensional formula is called
of the physical world, which are denoted with the dimensional equation of the physical
2018-19
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
mass density [ρ] may be expressed as are not same, the equation is wrong. Hence, if
we derive an expression for the length (or
[V] = [M L T ] distance) of an object, regardless of the symbols
0
0
3
0
[v] = [M L T ]
–1
[F] = [M L T ] appearing in the original mathematical relation,
–2
[ρ] = [M L T ] when all the individual dimensions are
0
–3
simplified, the remaining dimension must be
The dimensional equation can be obtained that of length. Similarly, if we derive an equation
from the equation representing the relations of speed, the dimensions on both the sides of
between the physical quantities. The equation, when simplified, must be of length/
dimensional formulae of a large number and time, or [L T ].
–1
wide variety of physical quantities, derived from Dimensions are customarily used as a
the equations representing the relationships preliminary test of the consistency of an
among other physical quantities and expressed equation, when there is some doubt about the
in terms of base quantities are given in
Appendix 9 for your guidance and ready correctness of the equation. However, the
reference. dimensional consistency does not guarantee
correct equations. It is uncertain to the extent
2.10 DIMENSIONAL ANALYSIS AND ITS of dimensionless quantities or functions. The
APPLICATIONS arguments of special functions, such as the
trigonometric, logarithmic and exponential
The recognition of concepts of dimensions, which
guide the description of physical behaviour is functions must be dimensionless. A pure
of basic importance as only those physical number, ratio of similar physical quantities,
quantities can be added or subtracted which such as angle as the ratio (length/length),
have the same dimensions. A thorough refractive index as the ratio (speed of light in
understanding of dimensional analysis helps us vacuum/speed of light in medium) etc., has no
in deducing certain relations among different dimensions.
physical quantities and checking the derivation, Now we can test the dimensional consistency
accuracy and dimensional consistency or or homogeneity of the equation
homogeneity of various mathematical 2
expressions. When magnitudes of two or more x = x + v t + (1/2 ) a t
0
0
physical quantities are multiplied, their units for the distance x travelled by a particle or body
should be treated in the same manner as in time t which starts from the position x with
0
ordinary algebraic symbols. We can cancel an initial velocity v at time t = 0 and has uniform
0
identical units in the numerator and acceleration a along the direction of motion.
denominator. The same is true for dimensions The dimensions of each term may be written as
of a physical quantity. Similarly, physical [x] = [L]
quantities represented by symbols on both sides [x ] = [L]
of a mathematical equation must have the same [v t] = [L T ] [T]
0
–1
dimensions. = [L]
0
[(1/2) a t ] = [L T ] [T ]
–2
2
2
2.10.1 Checking the Dimensional = [L]
Consistency of Equations
As each term on the right hand side of this
The magnitudes of physical quantities may be equation has the same dimension, namely that
added together or subtracted from one another of length, which is same as the dimension of
only if they have the same dimensions. In other left hand side of the equation, hence this
words, we can add or subtract similar physical equation is a dimensionally correct equation.
quantities. Thus, velocity cannot be added to It may be noted that a test of consistency of
force, or an electric current cannot be subtracted dimensions tells us no more and no less than a
2018-19
UNITS AND MEASUREMENT 33
test of consistency of units, but has the (b) and (d); [M L T ] for (c). The quantity on the
–2
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 L T ], formulas (a), (c) and (e)
2
–2
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). t
wrong.
2.10.2 Deducing Relation among the
Example 2.15 Let us consider an equation Physical Quantities
The method of dimensions can sometimes be
t
1
2
m v = m g h used to deduce relation among the physical
2 quantities. For this we should know the
where m is the mass of the body, v its dependence of the physical quantity on other
velocity, g is the acceleration due to quantities (upto three physical quantities or
gravity and h is the height. Check linearly independent variables) and consider it
whether this equation is dimensionally as a product type of the dependence. Let us take
correct. an example.
Answer The dimensions of LHS are t Example 2.17 Consider a simple
–1
–2
2
[M] [L T ] = [M] [ L T ] pendulum, having a bob attached to a
2
= [M L T ] string, that oscillates under the action of
–2
2
The dimensions of RHS are the force of gravity. Suppose that the period
[M][L T ] [L] = [M][L T ] of oscillation of the simple pendulum
2
–2
–2
= [M L T ] depends on its length (l), mass of the bob
–2
2
The dimensions of LHS and RHS are the same and (m) and acceleration due to gravity (g).
hence the equation is dimensionally correct. t Derive the expression for its time period
using method of dimensions.
Example 2.16 The SI unit of energy is
t
J = kg m s ; that of speed v is m s and Answer The dependence of time period T on
–1
2 –2
–2
of acceleration a is m s . Which of the the quantities l, g and m as a product may be
formulae for kinetic energy (K) given below written as :
y
x
can you rule out on the basis of T = k l g m z
dimensional arguments (m stands for the where k is dimensionless constant and x, y
mass of the body) : and z are the exponents.
2
(a) K = m v 3 By considering dimensions on both sides, we
(b) K = (1/2)mv 2 have
(c) K = ma [L M T ]=[L ] [L T ] [M ]
1
o
o
1
–2 y
1 z
1 x
(d) K = (3/16)mv 2
= L x+y T –2y M z
2
(e) K = (1/2)mv + ma
On equating the dimensions on both sides,
Answer Every correct formula or equation must we have
have the same dimensions on both sides of the x + y = 0; –2y = 1; and z = 0
equation. Also, only quantities with the same 1 1
physical dimensions can be added or So that x = ,y = – ,z = 0
subtracted. The dimensions of the quantity on 2 2
3
–3
2
–2
the right side are [M L T ] for (a); [M L T ] for Then, T = k l g –½
2
½
2018-19
34 PHYSICS
Dimensional analysis is very useful in deducing
l relations among the interdependent physical
or, T = k quantities. However, dimensionless constants
g
cannot be obtained by this method. The method
Note that value of constant k can not be obtained of dimensions can only test the dimensional
by the method of dimensions. Here it does not validity, but not the exact relationship between
matter if some number multiplies the right side physical quantities in any equation. It does not
of this formula, because that does not affect its distinguish between the physical quantities
dimensions. having same dimensions.
A number of exercises at the end of this
l chapter will help you develop skill in
Actually, k = 2π so that T = 2π t
g 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.
2018-19
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 .....m 3
(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 covers....m in 1 s
–1
(d) The relative density of lead is 11.3. Its density is ....g cm or ....kg m .
–3
–3
2.2 Fill in the blanks by suitable conversion of units
(a) 1 kg m s = ....g cm s
2 –2
2
–2
(b) 1 m = ..... ly
(c) 3.0 m s = .... km h –2
–2
3
–2
2
–1
(d) G = 6.67 × 10 –11 N m (kg) = .... (cm) s g .
–2
2.3 A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where 1J =
1 kg m s . Suppose we employ a system of units in which the unit of mass equals α
2
–2
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 β γ in terms of the new units.
–1
–2
2
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
2.9 The photograph of a house occupies an area of 1.75 cm on a 35 mm slide. The slide
2
is projected on to a screen, and the area of the house on the screen is 1.55 m . What
is the linear magnification of the projector-screen arrangement.
2.10 State the number of significant figures in the following :
(a) 0.007 m 2
(b) 2.64 × 10 kg
24
(c) 0.2370 g cm –3
2018-19
36 PHYSICS
(d) 6.320 J
(e) 6.032 N m –2
(f) 0.0006032 m 2
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 2.300 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 :
3
2
P = a b / ( 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 2π t / T + cos 2π t / )
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’ m o 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 :
m
m = 0 1/2
( 1 v− 2 ) .
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 m of a mole of hydrogen atoms ?
3
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
11
of the Earth’s orbit ≈ 3 × 10 m. 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 ?
2018-19
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