physics xi part 1 - ncert

8/20/2019 Physics XI Part 1 - Ncert

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CONTENTS

F OREWORD iii

PREFACE v

A NOTE FOR THE TEACHER x

C H A P T E R 1

PHYSICAL W ORLD

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 3MOTION IN A S TRAIGHT L INE

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 654.2 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

L AWS 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 1005.10 Circular motion 104

5.11 Solving problems in mechanics 105

C H A P T E R 6

W ORK, 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 1216.9 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

S YSTEM OF P ARTICLES AND ROTATIONAL MOTION


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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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


8.2 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 1968.12 Weightlessness 197

APPENDICES 203

ANSWERS 219

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1.1 WHAT IS PHYSICS ?

Humans have always been curious about the world aroundthem. The night sky with its bright celestial objects hasfascinated humans since time immemorial. The regular repetitions of the day and night, the annual cycle of seasons,the eclipses, the tides, the volcanoes, the rainbow have always been a source of wonder. The world has an astonishing variety of materials and a bewildering diversity of life and behaviour. The inquiring and imaginative human mind has respondedto the wonder and awe of nature in different ways. One kindof response from the earliest times has been to observe thephysical environment carefully, look for any meaningfulpatterns and relations in natural phenomena, and build anduse new tools to interact with nature. This human endeavour led, in course of time, to modern science and technology.

The word Science originates from the Latin verb Scientia meaning ‘ to know ’ . The Sanskrit word Vijnan and the Arabic word Ilm convey similar meaning, namely ‘knowledge’.Science, in a broad sense, is as old as human species. Theearly civilisations of Egypt, India, China, Greece, Mesopotamia and many others made vital contributions to its progress.From the sixteenth century onwards, great strides were madein science in Europe. By the middle of the twentieth century,science had become a truly international enterprise, withmany 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 understandnatural phenomena in as much detail and depth as possible,and use the knowledge so gained to predict, modify andcontrol phenomena. Science is exploring, experimenting andpredicting from what we see around us. The curiosity to learnabout the world, unravelling the secrets of nature is the first step towards the discovery of science. The scientific methodinvolves several interconnected steps : Systematicobservations, controlled experiments, qualitative and

CHAPTER ONE

PHYSICAL W ORLD

1.1 What is physics ?

1.2 Scope and excitement of physics

1.3 Physics, technology andsociety

1.4 Fundamental forces innature

1.5 Nature of physical laws

Summary

Exercises


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quantitative reasoning, mathematicalmodelling, prediction and verification or falsification of theories. Speculation andconjecture also have a place in science; but ultimately, a scientific theory, to be acceptable,must be verified by relevant observations or experiments. There is much philosophicaldebate about the nature and method of sciencethat we need not discuss here.

The interplay of theory and observation (or experiment) is basic to the progress of science.Science is ever dynamic. There is no ‘final’theory in science and no unquestionedauthority among scientists. As observationsimprove in detail and precision or experiments yield new results, theories must account for them, if necessary, by introducing modifications.

Sometimes the modifications may not be drasticand may lie within the framework of existingtheory. For example, when Johannes Kepler (1571-1630) examined the extensive data onplanetary motion collected by Tycho Brahe(1546-1601), the planetary circular orbits inheliocentric theory (sun at the centre of thesolar system) imagined by Nicolas Copernicus(1473–1543) had to be replaced by ellipticalorbits to fit the data better. Occasionally,however, the existing theory is simply unableto explain new observations. This causes a major upheaval in science. In the beginning of the twentieth century, it was realised that Newtonian mechanics, till then a very successful theory, could not explain some of themost basic features of atomic phenomena.Similarly, the then accepted wave picture of light failed to explain the photoelectric effect properly. This led to the development of a radically new theory (Quantum Mechanics ) to deal with atomicand molecular phenomena.

Just as a new experiment may suggest analternative theoretical model, a theoreticaladvance may suggest what to look for in some

experiments. The result of experiment of scattering of alpha particles by gold foil, in 1911 by Ernest Rutherford (1871–1937) establishedthe nuclear model of the atom, which then became the basis of the quantum theory of hydrogen atom given in 1913 by Niels Bohr (1885–1962). On the other hand, the concept of antiparticle was first introduced theoretically by Paul Dirac (1902–1984) in 1930 and confirmedtwo years later by the experimental discovery of positron (antielectron) by Carl Anderson.

Physics is a basic discipline in the category of Natural Sciences, which also includes other disciplines like Chemistry and Biology. The wordPhysics comes from a Greek word meaning

nature. Its Sanskrit equivalent is Bhautiki that is used to refer to the study of the physical world. A precise definition of this discipline is neither possible nor necessary. We can broadly describephysics as a study of the basic laws of natureand their manifestation in different naturalphenomena. The scope of physics is described briefly in the next section. Here we remark ontwo principal thrusts in physics : unificationand reduction.

In Physics, we attempt to explain diversephysical phenomena in terms of a few concepts and laws . The effort is to see the physical worldas manifestation of some universal laws indifferent domains and conditions. For example,the same law of gravitation (given by Newton)describes the fall of an apple to the ground, themotion of the moon around the earth and themotion of planets around the sun. Similarly, the basic laws of electromagnetism (Maxwell ’sequations) govern all electric and magneticphenomena. The attempts to unify fundamentalforces of nature (section 1.4) reflect this samequest for unification.

A related effort is to derive the properties of a

bigger, more complex, system from the propertiesand interactions of its constituent simpler parts. This approach is called reductionism and isat the heart of physics. For example, the subject of thermodynamics, developed in the nineteenthcentury, deals with bulk systems in terms of macroscopic quantities such as temperature,internal energy, entropy, etc. Subsequently, thesubjects of kinetic theory and statisticalmechanics interpreted these quantities in termsof the properties of the molecular constituentsof the bulk system. In particular, the

temperature was seen to be related to the averagekinetic energy of molecules of the system.

1.2 SCOPE AND EXCITEMENT OF PHYSICS

We can get some idea of the scope of physics by looking at its various sub-disciplines. Basically,there are two domains of interest : macroscopicand microscopic. The macroscopic domainincludes phenomena at the laboratory, terrestrialand astronomical scales. The microscopic domainincludes atomic, molecular and nuclear


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Ampere and Faraday, and encapsulated by Maxwell in his famous set of equations. Themotion of a current-carrying conductor in a magnetic field, the response of a circuit to an ac voltage (signal), the working of an antenna, thepropagation of radio waves in the ionosphere, etc.,are problems of electrodynamics. Optics deals with the phenomena involving light. The workingof telescopes and microscopes, colours exhibited by th in films , etc. , are topics in optics .

Thermodynamics, in contrast to mechanics, doesnot deal with the motion of bodies as a whole.Rather, it deals with systems in macroscopicequilibrium and is concerned with changes ininternal energy, temperature, entropy, etc., of thesystem through external work and transfer of heat. The efficiency of heat engines andrefrigerators, the direction of a physical or

You can now see that the scope of physics is

truly vast. It covers a tremendous range of

magnitude of physical quantities like length,

mass, time, energy, etc. At one end, it studies

phenomena at the very small scale of length

(10-14

m or even less) involving electrons, protons,

etc.; at the other end, it deals with astronomical

phenomena at the scale of galaxies or even the

entire universe whose extent is of the order of

1026

m. The two length scales differ by a factor of

1040 or even more. The range of time scales can be obtained by dividing the length scales by the

speed of light : 10 –22

s to 1018 s. The range of

masses goes from, say, 10 –30 kg (mass of an

electron) to 1055

kg (mass of known observable

universe). Terrestrial phenomena lie somewhere

in the middle of this range.

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.

* 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.

phenomena *. Classical Physics deals mainly with macroscopic phenomena and includessubjects like Mechanics, Electrodynamics,Optics and Thermodynamics. Mechanics

founded on Newton’s laws of motion and the law of gravitation is concerned with the motion (or equilibrium) of particles, rigid and deformable bodies, and general systems of particles. Thepropulsion of a rocket by a jet of ejecting gases,propagation of water waves or sound waves inair, the equilibrium of a bent rod under a load,etc., are problems of mechanics. Electrodynamicsdeals with electric and magnetic phenomena associated with charged and magnetic bodies.Its basic laws were given by Coulomb, Oersted,

chemical process, etc., are problems of interest

in thermodynamics.

The microscopic domain of physics deals with

the constitution and structure of matter at the

minute scales of atoms and nuclei (and evenlower scales of length) and their interaction with

different probes such as electrons, photons and

other elementary particles. Classical physics is

inadequate to handle this domain and Quantum

Theory is currently accepted as the proper

framework for explaining microscopic

phenomena. Overall, the edifice of physics is

beautiful and imposing and you will appreciate

it more as you pursue the subject.


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Physics is exciting in many ways. To some peoplethe excitement comes from the elegance anduniversality of its basic theories, from the fact that a few basic concepts and laws can explainphenomena covering a large range of magnitudeof physical quantities. To some others, the challengein carrying out imaginative new experiments tounlock the secrets of nature, to verify or refutetheories, is thrilling. Applied physics is equally demanding. Application and exploitation of physical laws to make useful devices is the most interesting and exciting part and requires great ingenuity and persistence of effort.

What lies behind the phenomenal progressof physics in the last few centuries? Great progress usually accompanies changes in our basic perceptions. First, it was realised that for

scientific progress, only qualitative thinking,though no doubt important, is not enough.Quantitative measurement is central to thegrowth of science, especially physics, becausethe laws of nature happen to be expressible inprecise mathematical equations. The secondmost important insight was that the basic lawsof physics are universal — the same laws apply in widely different contexts. Lastly, the strategy of approximation turned out to be very successful. Most observed phenomena in daily life are rather complicated manifestations of the basic laws. Scientists recognised the importance

of extracting the essential features of a phenomenon from its less significant aspects.It is not practical to take into account all thecomplexities of a phenomenon in one go. A goodstrategy is to focus first on the essential features,discover the basic principles and then introducecorrections to build a more refined theory of thephenomenon. For example, a stone and a feather dropped from the same height do not reach theground at the same time. The reason is that theessential aspect of the phenomenon, namely freefall under gravity, is complicated by thepresence of air resistance. To get the law of free

fall under gravity, it is better to create a situation wherein the air resistance isnegligible. We can, for example, let the stone andthe feather fall through a long evacuated tube.In that case, the two objects will fall almost at the same rate, giving the basic law that acceleration due to gravity is independent of themass of the object. With the basic law thusfound, we can go back to the feather, introducecorrections due to air resistance, modify theexisting theory and try to build a more realistic

Hypothesis, axioms and models

One should not think that everything can be proved with physics and mathematics. All physics, and also

mathematics, is based on assumptions, each of which is variously called a hypothesis or axiom or postulate, etc.

For example, the universal law of gravitationproposed by Newton is an assumption or hypothesis, which he proposed out of his ingenuity. Before him,there were several observations, experiments anddata, on the motion of planets around the sun,motion of the moon around the earth, pendulums, bodies falling towards the earth etc. Each of theserequired a separate explanation, which was moreor less qualitative. What the universal law of gravitation says is that, if we assume that any two bodies in the universe attract each other with a

force proportional to the product of their massesand inversely proportional to the square of thedistance between them, then we can explain allthese observations in one stroke. It not only explainsthese phenomena, it also allows us to predict theresults of future experiments.

A hypothesis is a supposition without assumingthat it is true. It would not be fair to ask anybody to prove the universal law of gravitation, becauseit cannot be proved. It can be verified andsubstantiated by experiments and observations.

An axiom is a self-evident truth while a modelis a theory proposed to explain observedphenomena. But you need not worry at this stage

about the nuances in using these words. For example, next year you will learn about Bohr’s modelof hydrogen atom, in which Bohr assumed that anelectron in the hydrogen atom follows certain rules(postutates). Why did he do that? There was a largeamount of spectroscopic data before him which noother theory could explain. So Bohr said that if weassume that an atom behaves in such a manner, we can explain all these things at once.

Einstein’s special theory of relativity is also based on two postulates, the constancy of the speedof electromagnetic radiation and the validity of physical laws in all inertial frame of reference. It would not be wise to ask somebody to prove that

the speed of light in vacuum is constant,independent of the source or observer.

In mathematics too, we need axioms andhypotheses at every stage. Euclid’s statement that parallel lines never meet, is a hypothesis. This meansthat if we assume this statement, we can explainseveral properties of straight lines and two or threedimensional figures made out of them. But if youdon’t assume it, you are free to use a different axiomand get a new geometry, as has indeed happened inthe past few centuries and decades.


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Table 1.1 Some physicists from different countries of the world and their major contributions

theory of objects falling to the earth under gravity.

1.3 PHYSICS, TECHNOLOGY AND SOCIETY

The connection between physics, technology and society can be seen in many examples. Thediscipline of thermodynamics arose from theneed to understand and improve the working of heat engines. The steam engine, as we know,is inseparable from the Industrial Revolution inEngland in the eighteenth century, which hadgreat impact on the course of humancivilisation. Sometimes technology gives rise tonew physics; at other times physics generatesnew technology. An example of the latter is the wireless communication technology that followedthe discovery of the basic laws of electricity andmagnetism in the nineteenth century. Theapplications of physics are not always easy toforesee. As late as 1933, the great physicist Ernest Rutherford had dismissed the possibility of tapping energy from atoms. But only a few years la ter, in 1938, Hahn and Me itner discovered the phenomenon of neutron-inducedfission of uranium, which would serve as the basis of nuclear power reactors and nuclear weapons. Yet another important example of physics giving rise to technology is the silicon‘chip’ that triggered the computer revolution inthe last three decades of the twentieth century.

A most significant area to which physics hasand will contribute is the development of alternative energy resources. The fossil fuels of the planet are dwindling fast and there is anurgent need to discover new and affordablesources of energy. Considerable progress hasalready been made in this direction (for example, in conversion of solar energy,geothermal energy, etc., into electricity), but much more is still to be accomplished.

Table1.1 lists some of the great physicists,their major contribution and the country of origin. You will appreciate from this table themulti-cultural, international character of thescientific endeavour. Table 1.2 lists someimportant technologies and the principles of physics they are based on. Obviously, thesetables are not exhaustive. We urge you to try toadd many names and items to these tables withthe help of your teachers, good books and websites on science. You will find that thisexercise is very educative and also great fun. And, assuredly, it will never end. The progressof science is unstoppable!

Physics is the study of nature and naturalphenomena. Physicists try to discover the rulesthat are operating in nature, on the basis of observations, experimentation and analysis.Physics deals with certain basic rules/lawsgoverning the natural world. What is the nature

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 Polandnatural radioactivity

Albert Einstein Explanation of photoelectric effect; Germany

Theory of relativity


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* 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.

of physical laws? We shall now discuss thenature of fundamental forces and the laws that govern the diverse phenomena of the physical world.

1.4 FUNDAMENTAL FORCES IN NATURE*

We all have an intuitive notion of force. In our experience, force is needed to push, carry or

throw objects, deform or break them. We alsoexperience the impact of forces on us, like whena moving object hits us or we are in a merry-go-round. Going from this intuitive notion to theproper scientific concept of force is not a trivialmatter. Early thinkers like Aristotle had wrong

ideas about it. The correct notion of force wasarrived at by Isaac Newton in his famous laws of motion. He also gave an explicit form for the forcefor gravitational attraction between two bodies. We shall learn these matters in subsequent chapters.

In the macroscopic world, besides thegravitational force, we encounter several kinds

of forces: muscular force, contact forces between bodies, friction (which is also a contact forceparallel to the surfaces in contact), the forcesexerted by compressed or elongated springs andtaut strings and ropes (tension), the force of buoyancy and viscous force when solids are in

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 statisticsEdwin 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 Pakistaninteractions

Name Major contribution/discovery Country of Origin


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contact with fluids, the force due to pressure of a fluid, the force due to surface tension of a liquid,and so on. There are also forces involving chargedand magnetic bodies. In the microscopic domain

again, we have electric and magnetic forces,nuclear forces involving protons and neutrons,interatomic and intermolecular forces, etc. Weshall get familiar with some of these forces in later parts of this course.

A great insight of the twentieth century physics is that these different forces occurringin different contexts actually arise from only a small number of fundamental forces in nature.For example, the elastic spring force arises due

to the net attraction/repulsion between theneighbouring atoms of the spring when thespring is elongated/compressed. This net attraction/repulsion can be traced to the

(unbalanced) sum of electric forces between thecharged constituents of the atoms.

In principle, this means that the laws for ‘derived’ forces (such as spring force, friction)are not independent of the laws of fundamentalforces in nature. The origin of these derivedforces is, however, very complex.

At the present stage of our understanding, we know of four fundamental forces in nature, which are described in brief here :

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 electromagneticfields

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 andmagnetic fields.


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1.4.1 Gravitational Force

The gravitational force is the force of mutualattraction between any two objects by virtue of their masses. It is a universal force. Every object experiences this force due to every other object in the universe. All objects on the earth, for example, experience the force of gravity due to

the earth. In particular, gravity governs themotion of the moon and artificial satellites aroundthe earth, motion of the earth and planetsaround the sun, and, of course, the motion of bodies falling to the earth. It plays a key role inthe large-scale phenomena of the universe, suchas formation and evolution of stars, galaxies andgalactic clusters.

1.4.2 Electromagnetic Force

Electromagnetic force is the force betweencharged particles. In the simpler case whencharges are at rest, the force is given by Coulomb’s law : attractive for unlike charges andrepulsive for like charges. Charges in motionproduce magnetic effects and a magnetic fieldgives rise to a force on a moving charge. Electricand magnetic effects are, in general,inseparable – hence the name electromagneticforce. Like the gravitational force,electromagnetic force acts over large distancesand does not need any intervening medium. It is enormously strong compared to gravity. The

electric force between two protons, for example,

is 1036

times the gravitational force between

them, for any fixed distance.

Matter, as we know, consists of elementary

charged constituents like electrons and

protons. Since the electromagnetic force is so

much stronger than the gravitational force, it

dominates all phenomena at atomic andmolecular scales. (The other two forces, as we

shall see, operate only at nuclear scales.) Thus

it is mainly the electromagnetic force that

governs the structure of atoms and molecules,

the dynamics of chemical reactions and the

mechanical, thermal and other properties of

materials. It underlies the macroscopic forces

like ‘tension’, ‘friction’, ‘normal force’, ‘spring

force’, etc.

Gravity is always attractive, while

electromagnetic force can be attractive or

repulsive. Another way of putting it is that masscomes only in one variety (there is no negative

mass), but charge comes in two varieties :

positive and negative charge. This is what

makes all the difference. Matter is mostly

electrically neutral (net charge is zero). Thus,

electric force is largely zero and gravitational

force dominates terrestrial phenomena. Electric

force manifests itself in atmosphere where the

atoms are ionised and that leads to lightning.

Albert Einstein (1879-1955)

Albert Einstein, born in Ulm, Germany in 1879, is universally regarded asone of the greatest physicists of all time. His astonishing scientific career

began with the publication of three path-breaking papers in 1905. In thefirst 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, hedeveloped 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 hisnew theory which included, among other things, the mass-energy equivalence enshrined in his famousequation E = mc2. He also created the general version of relativity (The General Theory of Relativity), which is the modern theory of gravitation. Some of Einstein’s most significant later contributions are:the notion of stimulated emission introduced in an alternative derivation of Planck’s blackbody radiationlaw, 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 tophysics, in year 1905, describing revolutionary scientific ideas that have since influenced all of modernphysics.


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If we reflect a little, the enormous strengthof the electromagnetic force compared togravity is evident in our daily life. When wehold a book in our hand, we are balancing thegravitational force on the book due to the huge

mass of the earth by the ‘normal force’provided by our hand. The latter is nothing but the net electromagnetic force between thecharged constituents of our hand andthe book, at the surface in contact. If electromagnetic force were not intrinsically somuch stronger than gravity, the hand of thestrongest man would crumble under the weight of a feather ! Indeed, to be consistent,in that circumstance, we ourselves wouldcrumble under our own weight !

1.4.3 Strong Nuclear Force

The strong nuclear force binds protons andneutrons in a nucleus. It is evident that without some attractive force, a nucleus will beunstable due to the electric repulsion betweenits protons. This attractive force cannot begravitational since force of gravity is negligiblecompared to the electric force. A new basic forcemust, therefore, be invoked. The strong nuclear force is the strongest of all fundamental forces,about 100 times the electromagnetic force in

strength. It is charge-independent and actsequally between a proton and a proton, a neutron and a neutron, and a proton and a neutron. Its range is, however, extremely small,of about nuclear dimensions (10

–15m). It is

responsible for the stability of nuclei. Theelectron, it must be noted, does not experiencethis force.

Recent developments have, however,indicated that protons and neutrons are built out of still more elementary constituents calledquarks.

1.4.4 Weak Nuclear Force

The weak nuclear force appears only in certain

nuclear processes such as the β-decay of a nucleus. In β-decay, the nucleus emits anelectron and an uncharged particle called

neutrino. The weak nuclear force is not as weak as the gravitational force, but much weaker than the strong nuclear and electromagneticforces. The range of weak nuclear force isexceedingly small, of the order of 10-16 m.

1.4.5 Towards Unification of Forces

We remarked in section 1.1 that unification is a basic quest in physics. Great advances inphysics often amount to unification of different

Satyendranath Bose (1894-1974)

Satyendranath Bose, born in Calcutta in 1894, is among the great Indianphysicists who made a fundamental contribution to the advance of science

in the twentieth century. An outstanding student throughout, Bose startedhis career in 1916 as a lecturer in physics in Calcutta University; five yearslater 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 whoimmediately recognised its great significance, translated it in German andforwarded 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 toparticles 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 thesame lowest energy state. Some seventy years were to pass before the pioneering ideas of Bose, developedfurther by Einstein, were dramatically confirmed in the observation of a new state of matter in a dilutegas of ultra cold alkali atoms - the Bose-Eintein condensate.


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theories and domains. Newton unified terrestrialand celestial domains under a common law of gravitation. The experimental discoveries of Oersted and Faraday showed that electric and

magnetic phenomena are in generalinseparable. Maxwell unified electromagnetismand optics with the discovery that light is anelectromagnetic wave. Einstein attempted tounify gravity and electromagnetism but couldnot succeed in this venture. But this did not deter physicists from zealously pursuing thegoal of unification of forces.

Recent decades have seen much progress onthis front. The electromagnetic and the weak nuclear force have now been unified and areseen as aspects of a single ‘electro-weak’ force.

What this unification actually means cannot be explained here. Attempts have been (and are being) made to unify the electro-weak and thestrong force and even to unify the gravitationalforce with the rest of the fundamental forces.Many of these ideas are still speculative andinconclusive. Table 1.4 summarises some of themilestones in the progress towards unificationof forces in nature.

1.5 NATURE OF PHYSICAL LAWS

Physicists explore the universe. Their investigations, based on scientific processes,range from particles that are smaller thanatoms in size to stars that are very far away. Inaddition to finding the facts by observation andexperimentation, physicists attempt to discover the laws that summarise (often as mathematicalequations) these facts.

In any physical phenomenon governed by different forces, several quantities may change with time. A remarkable fact is that some specialphysical quantities, however, remain constant in time. They are the conserved quantities of nature. Understanding these conservationprinciples is very important to describe the

observed phenomena quantitatively.For motion under an external conservative

force, the total mechanical energy i.e. the sumof kinetic and potential energy of a body is a constant. The familiar example is the free fall of an object under gravity. Both the kinetic energy of the object and its potential energy changecontinuously with time, but the sum remainsfixed. If the object is released from rest, the initial

Table 1.4 Progress in unification of different forces/domains in nature

Table 1.3 Fundamental forces of nature

Name Relative Range Operates amongstrength

Gravitational force 10 –39 Infinite All objects in the universe

Weak nuclear force 10 –13 Very short, Sub-nuclear Some elementary particles,size (∼10 –16m) particularly electron and

neutrino

Electromagnetic force 10 –2 Infinite Charged particles

Strong nuclear force 1 Short, nuclear Nucleons, heavier size (∼10 –15m) elementary particles


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PHYSICAL WORLD 11

potential energy is completely converted into thekinetic energy of the object just before it hitsthe ground. This law restricted for a conservativeforce should not be confused with the generallaw of conservation of energy of an isolatedsystem (which is the basis of the First Law of Thermodynamics).

The concept of energy is central to physicsand the expressions for energy can be writtenfor every physical system. When all forms of energy e.g., heat, mechanical energy, electricalenergy etc., are counted, it turns out that energy is conserved. The general law of conservation of energy is true for all forces and for any kind of transformation between different forms of energy. In the falling object example, if youinclude the effect of air resistance during the

fall and see the situation after the object hitsthe ground and stays there, the totalmechanical energy is obviously not conserved. The general law of energy conservation, however,is still applicable. The initial potential energy of the stone gets transformed into other formsof energy : heat and sound. (Ultimately, soundafter it is absorbed becomes heat.) The totalenergy of the system (stone plus thesurroundings) remains unchanged.

The law of conservation of energy is thought to be valid across all domains of nature, fromthe microscopic to the macroscopic. It isroutinely applied in the analysis of atomic,nuclear and elementary particle processes. At

the other end, all kinds of violent phenomena occur in the universe all the time. Yet the totalenergy of the universe (the most ideal isolatedsystem possible !) is believed to remainunchanged.

Until the advent of Einstein’s theory of relativity, the law of conservation of mass wasregarded as another basic conservation law of nature, since matter was thought to beindestructible. It was (and still is) an important principle used, for example, in the analysis of chemical reactions. A chemical reaction is basically a rearrangement of atoms amongdifferent molecules. If the total binding energy of the reacting molecules is less than the total binding energy of the product molecules, thedifference appears as heat and the reaction is

exothermic. The opposite is true for energy absorbing (endothermic) reactions. However,since the atoms are merely rearranged but not destroyed, the total mass of the reactants is thesame as the total mass of the products in a chemical reaction. The changes in the bindingenergy are too small to be measured as changesin mass.

According to Einstein’s theory, mass m isequivalent to energy E given by the relationE= mc

2, where c is speed of light in vacuum.

In a nuclear process mass gets converted toenergy (or vice-versa). This is the energy whichis released in a nuclear power generation andnuclear 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 theIndian 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 hisevening 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 waselected ‘Fellow’ of the Royal Society of London and received Nobel prize in Physicsin 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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PHYSICS12

Energy is a scalar quantity. But all conservedquantities are not necessarily scalars. The totallinear momentum and the total angular

momentum (both vectors) of an isolated system

are also conserved quantities. These laws can be derived from Newton’s laws of motion inmechanics. But their validity goes beyond

mechanics. They are the basic conservationlaws of nature in all domains, even in those where Newton’s laws may not be valid.

Besides their great simplicity and generality,

the conservation laws of nature are very usefulin practice too. It often happens that we cannot solve the full dynamics of a complex probleminvolving different particles and forces. The

conservation laws can still provide useful

results. For example, we may not know thecomplicated forces that act during a collisionof two automobiles; yet momentum

conservation law enables us to bypass thecomplications and predict or rule out possibleoutcomes of the collision. In nuclear andelementary particle phenomena also, the

conservation laws are important tools of analysis. Indeed, using the conservation lawsof energy and momentum for β-decay, WolfgangPauli (1900-1958) correctly predicted in 1931

the existence of a new particle (now called

neutrino) emitted in β-decay along with theelectron.Conservation laws have a deep connection

with symmetries of nature that you will explorein more advanced courses in physics. For example, an important observation is that thelaws of nature do not change with time! If you

perform an experiment in your laboratory today and repeat the same experiment (on the sameobjects under identical conditions) after a year,the results are bound to be the same. It turns

out that this symmetry of nature with respect totranslation (i.e. displacement) in time isequivalent to the law of conservation of energy.Likewise, space is homogeneous and there is no

(intrinsically) preferred location in the universe. To put it more clearly, the laws of nature are thesame everywhere in the universe. (Caution : the

phenomena may differ from place to place

because of differing conditions at different locations. For example, the acceleration due to

gravity at the moon is one-sixth that at the earth,

but the law of gravitation is the same both onthe moon and the earth.) This symmetry of the

laws of nature with respect to translation in

space gives rise to conservation of linear momentum. In the same way isotropy of space

(no intrinsically preferred direction in space)

underlies the law of conservation of angular momentum*. The conservation laws of charge andother attributes of elementary particles can also

be related to certain abstract symmetries.Symmetries of space and time and other abstract

symmetries play a central role in modern theories

of fundamental forces in nature.

* See Chapter 7

Conservation laws in physics

Conservation of energy, momentum, angular momentum, charge, etc are considered to befundamental laws in physics. At this moment,

there are many such conservation laws. Apart fromthe above four, there are others which mostly deal with quantities which have been introduced innuclear and particle physics. Some of theconserved quantities are called spin, baryonnumber, strangeness, hypercharge, etc, but youneed not worry about them.

A conservation law is a hypothesis, based onobservations and experiments. It is important toremember that a conservation law cannot beproved. It can be verified, or disproved, by experiments. An experiment whose result is inconformity with the law verifies or substantiatesthe law; it does not prove the law. On the other

hand, a single experiment whose result goesagainst the law is enough to disprove it.

It would be wrong to ask somebody to provethe law of conservation of energy. This law is anoutcome of our experience over several centuries,and it has been found to be valid in allexperiments, in mechanics, thermodynamics,electromagnetism, optics, atomic and nuclear physics, or any other area.

Some students feel that they can prove theconservation of mechanical energy from a body falling under gravity, by adding the kinetic andpotential energies at a point and showing that it turns out to be constant. As pointed out above,

this is only a verification of the law, not its proof.


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PHYSICAL WORLD 13

SUMMARY

1. Physics deals with the study of the basic laws of nature and their manifestation indifferent 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 physicalquantities.

3. Physics and technology are related to each other. Sometimes technology gives rise tonew 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 themacroscopic and the microscopic world. These are the ‘gravitational force’, the‘electromagnetic force’, the ‘strong nuclear force’, and the ‘weak nuclear force’. Unificationof different forces/domains in nature is a basic quest in physics.

5. The physical quantities that remain unchanged in a process are called conservedquantities. 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 surroundingscience, technology and society and to encourage you to think and formulate your viewsabout 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 didEinstein mean when he said : “The most incomprehensible thing about the world isthat it is comprehensible”?

1.2 “Every great physical theory starts as a heresy and ends as a dogma”. Give someexamples 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 thisobserved fact. Which of these strikes you as a scientific explanation ?

(a) A tragic sea accident several centuries ago drowned a young Samurai. As a tributeto his bravery, nature through its inscrutable ways immortalised his face by imprinting it on the crab shells in that area.


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PHYSICS14

(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 shaperesembling the face of a Samurai. Consequently, the particular shape of thecrab 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’ highlightsthe 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 centuriesago was triggered by some key scientific and technological advances. What were theseadvances ?

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 areasof 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 scienceand 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 havenothing but dangerous consequences for the human society. How, if at all, will youresolve 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 superstitiousand 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 themagree 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 whichstrike you as beautiful.

1.15 Though the statement quoted above may be disputed, most physicists do have a feelingthat the great laws of physics are at once simple and beautiful. Some of the notablephysicists, besides Dirac, who have articulated this feeling, are : Einstein, Bohr,Heisenberg, Chandrasekhar and Feynman. You are urged to make special efforts to get


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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 laughor 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.


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CHAPTER T WO

UNITS AND MEASUREMENT

2.1 INTRODUCTIONMeasurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit. The result of a measurement of a physical quantity is expressed by a number (or numerical measure) accompanied by a unit. Although the number of physical quantities appears to be very large, we need only a limited number of units for expressing all the physical quantities, since they are inter-related with one another. The units for the fundamental or base quantities are called fundamental or base units. Theunits of all other physical quantities can be expressed as

combinations of the base units. Such units obtained for thederived quantities are called derived units. A complete set of these units, both the base units and derived units, isknown as the system of units.

2.2 THE INTERNATIONAL SYSTEM OF UNITS

In earlier time scientists of different countries were usingdifferent systems of units for measurement. Three suchsystems, the CGS, the FPS (or British) system and the MKSsystem were in use extensively till recently.

The base units for length, mass and time in these systems 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

2.1 Introduction

2.2 The international system of units

2.3 Measurement of length

2.4 Measurement of mass

2.5 Measurement of time

2.6 Accuracy, precision of instruments and errors in

measurement 2.7 Significant figures

2.8 Dimensions of physicalquantities

2.9 Dimensional formulae anddimensional equations

2.10 Dimensional analysis and itsapplications

Summary Exercises

Additional exercises


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international usage in scientific, technical,industrial and commercial work. Because SIunits used decimal system, conversions within

the system are quite simple and convenient. Weshall follow the SI units in this book.

In SI, there are seven base units as given in Table 2.1. Besides the seven base units, thereare two more units that are defined for (a) planeangle dθ as the ratio of length of arc ds to theradius r and (b) solid angle dΩ as the ratio of the intercepted area d A of the spherical surface,described about the apex O as the centre, tothe square of its radius r , as shown in Fig. 2.1(a)and (b) respectively. The unit for plane angle isradian with the symbol rad and the unit for the

solid angle is steradian with the symbol sr. Boththese are dimensionless quantities.

Table 2.1 SI Base Quantities and Units*

UNITS AND MEASUREMENT 17

* 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.

(a)

(b)

Fig. 2.1 Description of (a) plane angle d θ and (b) solid angle d Ω .

Base SI Units

quantity Name Symbol Definition

Length metre m The metre is the length of the path travelled by light in vacuumduring 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 prototypeof 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 theradiation corresponding to the transition between the twohyperfine levels of the ground state of the cesium-133 atom.(1967)

Electric ampere A The ampere is that constant current which, if maintained incurrent 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 thermodynamicdynamic 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 0.012kilogram of carbon - 12. (1971)

Luminous candela cd The candela is the luminous intensity, in a givenintensity direction, of a source that emits monochromatic radiation of

frequency 540×1012 hertz and that has a radiant intensity inthat direction of 1/683 watt per steradian. (1979)


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PHYSICS18

Table 2.2 Some units retained for general use (Though outside SI)

Note that when mole is used, the elementary entities must be specified. These entitiesmay be atoms, molecules, ions, electrons,other particles or specified groups of suchparticles.

We employ units for some physical quantitiesthat can be derived from the seven base units(Appendix A 6). Some derived units in terms of the SI base units are given in (Appendix A 6.1).

Some SI derived units are given special names(Appendix A 6.2 ) and some derived SI units makeuse of these units with special names and theseven base units (Appendix A 6.3). These aregiven in Appendix A 6.2 and A 6.3 for your ready reference. Other units retained for general useare given in Table 2.2.

Common SI prefixes and symbols for multiplesand sub-multiples are given in Appendix A2.General guidelines for using symbols for physicalquantities, chemical elements and nuclides aregiven in Appendix A7 and those for SI units andsome other units are given in Appendix A8 for your guidance and ready reference.

2.3 MEASUREMENT OF LENGTH

You are already familiar with some direct methodsfor the measurement of length. For example, a metre scale is used for lengths from 10 –3 m to 102

m. A vernier callipers is used for lengths to anaccuracy of 10 –4 m. A screw gauge and a spherometer can be used to measure lengths asless as to 10 –5 m. To measure lengths beyond theseranges, we make use of some special indirect methods.

2.3.1 Measurement of Large Distances

Large distances such as the distance of a planet or a star from the earth cannot be measureddirectly with a metre scale. An important methodin such cases is the parallax method.

When you hold a pencil in front of you against some specific point on the background (a wall)and look at the pencil first through your left eye

A (closing the right eye) and then look at thepencil through your right eye B (closing the left eye), you would notice that the position of thepencil seems to change with respect to the point on the wall. This is called parallax. Thedistance between the two points of observationis called the basis. In this example, the basis isthe distance between the eyes.

To measure the distance D of a far away planet S by the parallax method, we observe it from two different positions (observatories) A andB on the Earth, separated by distance AB = b

at the same time as shown in Fig. 2.2. Wemeasure the angle between the two directionsalong which the planet is viewed at these twopoints. The ∠ ASB in Fig. 2.2 represented by symbol θ is called the parallax angle or parallactic angle.

As the planet is very far away, 1,b

D


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the radius AS = BS so that AB = b = D θ whereθ is in radians.

D =b

(2.1)

Having determined D , we can employ a similar method to determine the size or angular diameter of the planet. If d is the diameter of the planet and α the angular size of the planet (the anglesubtended by d at the earth), we have

α = d/D (2.2) The angleα can be measured from the same

location on the earth. It is the angle betweenthe two directions when two diametrically opposite points of the planet are viewed throughthe telescope. Since D is known, the diameter d of the planet can be determined using Eq. (2.2).

Example 2.1 Calculate the angle of (a) 10 (degree) (b) 1′ (minute of arc or arcmin)and (c) 1″ (second of arc or arc second) inradians. Use 3600=2π rad, 10=60′ and1′ = 60 ″

Answer (a) We have 3600 = 2π rad10 = (π /180) rad = 1.745×10 –2 rad

(b) 10 = 60′ = 1.745×10 –2 rad1′ = 2.908×10 –4 rad ; 2.91×10 –4 rad(c) 1′ = 60″ = 2.908×10 –4 rad

1″ = 4.847×10 –4 rad ; 4.85×10 –6 rad Example 2.2 A man wishes to estimatethe distance of a nearby tower from him.He stands at a point A in front of the tower C and spots a very distant object O in line with AC. He then walks perpendicular to AC up to B, a distance of 100 m, and looksat O and C again. Since O is very distant,the direction BO is practically the same as

AO; but he finds the line of sight of C shiftedfrom the original line of sight by an angle θ = 400 (θ is known as ‘parallax’) estimate

the distance of the tower C from his originalposition A.

Fig. 2.3

Answer We have, parallax angle θ = 400

From Fig. 2.3, AB = AC tan θ AC = AB/tanθ = 100 m/tan 400

= 100 m/0.8391 = 119 m

Example 2.3 The moon is observed fromtwo diametrically opposite points A and Bon Earth. The angle θ subtended at the

moon by the two directions of observationis 1o 54′ . Given the diameter of the Earth to be about 1.276 ××××× 107 m, compute thedistance of the moon from the Earth.

Answer We have θ = 1° 54′ = 114′

( ) ( )-6114 60 4.85 10= ′′× × × rad

= 3.32 × −10 2 rad,

since 61 4.85 10 .=" rad −×

Also b = AB =1.276 m×107

Hence from Eq. (2.1), we have the earth-moondistance,

D b = /θ

=1.276 103.32 10

7

-2

××

83.84 10 m= ×

Example 2.4 The Sun’s angular diameter is measured to be 1920′ ′. The distanceD of the Sun from the Earth is 1.496 × 1011 m. What is the diameter of the Sun ?

Fig. 2.2 Parallax method.


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PHYSICS20

Answer Sun’s angular diameter α = 1920"

= × ×

−

1920 4.85 10 rad

6

= × −9.31 10 rad3Sun’s diameter d D =

3 119.31 10 1.496 10 m

−= × × ×

9=1.39 10 m×

2.3.2 Estimation of Very Small Distances:Size of a Molecule

To measure a very small size like that of a

molecule (10

–8

m to 10

–10

m), we have to adopt special methods. We cannot use a screw gaugeor similar instruments. Even a microscope hascertain limitations. An optical microscope uses visible light to ‘look’ at the system under investigation. As light has wave like features,the resolution to which an optical microscopecan be used is the wavelength of light (A detailedexplanation can be found in the Class XIIPhysics textbook). For visible light the range of wavelengths is from about 4000 Å to 7000 Å (1 angstrom = 1 Å = 10-10 m). Hence an opticalmicroscope cannot resolve particles with sizes

smaller than this. Instead of visible light, we canuse an electron beam. Electron beams can befocussed by properly designed electric andmagnetic fields. The resolution of such anelectron microscope is limited finally by the fact that electrons can also behave as waves ! (You will learn more about this in class XII). The wavelength of an electron can be as small as a fraction of an angstrom. Such electronmicroscopes with a resolution of 0.6 Å have been built . They can almost resolve atoms andmolecules in a material. In recent times,

tunnelling microscopy has been developed in which again the limit of resolution is better thanan angstrom. It is possible to estimate the sizesof molecules.

A simple method for estimating the molecular size of oleic acid is given below. Oleic acid is a soapy liquid with large molecular size of theorder of 10 –9 m.

The idea is to first form mono-molecular layer of oleic acid on water surface.

We dissolve 1 cm3 of oleic acid in alcohol tomake a solution of 20 cm3. Then we take 1 cm3

of this solution and dilute it to 20 cm3, usingalcohol. So, the concentration of the solution is

equal to31

cm20 20

×

of oleic acid/cm3

of solution. Next we lightly sprinkle somelycopodium powder on the surface of water in a large trough and we put one drop of this solutionin the water. The oleic acid drop spreads into a thin, large and roughly circular film of molecular thickness on water surface. Then, we quickly measure the diameter of the thin film to get itsarea A . Suppose we have dropped n drops inthe water. Initially, we determine theapproximate volume of each drop (V cm3).

Volume of n drops of solution = nV cm3

Amount of oleic acid in this solution

= nV1

20 20×

cm

3

This solution of oleic acid spreads very fast on the surface of water and forms a very thinlayer of thickness t . If this spreads to form a film of area A cm2, then the thickness of thefilm

t = Volume of the film Area of the film

or, cm20 20nV

t A

=× (2.3)

If we assume that the film has mono-molecular thickness, then this becomes the size or diameter of a molecule of oleic acid. The value of thisthickness comes out to be of the order of 10 –9 m.

Example 2.5 If the size of a nucleus (inthe range of 10 –15 to 10 –14 m) is scaled upto the tip of a sharp pin, what roughly isthe size of an atom ? Assume tip of the pinto be in the range 10 –5m to 10 –4m.

Answer The size of a nucleus is in the range of 10 –15 m and 10 –14 m. The tip of a sharp pin istaken to be in the range of 10 –5 m and 10 –4 m. Thus we are scaling up by a factor of 1010. Anatom roughly of size 10 –10 m will be scaled up to a size of 1 m. Thus a nucleus in an atom is as smallin size as the tip of a sharp pin placed at the centreof a sphere of radius about a metre long.


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2.3.3 Range of Lengths

The sizes of the objects we come across in theuniverse vary over a very wide range. These may vary from the size of the order of 10 –14 m of thetiny nucleus of an atom to the size of the order of 1026 m of the extent of the observable universe. Table 2.6 gives the range and order of lengthsand sizes of some of these objects.

We also use certain special length units for short and large lengths. These are1 fermi = 1 f = 10 –15 m1 angstrom = 1 Å = 10 –10 m1 astronomical unit = 1 AU (average distance

of the Sun from the Earth)

= 1.496 × 1011

m1 light year = 1 ly= 9.46 × 1015 m (distance that light travels with velocity of

3 × 108 m s –1 in 1 year)1 parsec = 3.08 × 1016 m (Parsec is thedistance at which average radius of earth’s orbit subtends an angle of 1 arc second)

2.4 MEASUREMENT OF MASS

Mass is a basic property of matter. It does not depend on the temperature, pressure or locationof the object in space. The SI unit of mass is

kilogram (kg). The prototypes of the Internationalstandard kilogram supplied by the InternationalBureau of Weights and Measures (BIPM) areavailable in many other laboratories of different countries. In India, this is available at theNational Physical Laboratory (NPL), New Delhi.

While dealing with atoms and molecules, thekilogram is an inconvenient unit. In this case,there is an important standard unit of mass,

called the unified atomic mass unit (u), whichhas been established for expressing the massof atoms as

1 unified atomic mass unit = 1u = (1/12) of the mass of an atom of carbon-12isotope ( )6

12 C including the mass of electrons = 1.66 × 10 –27 kg

Mass of commonly available objects can bedetermined by a common balance like the oneused in a grocery shop. Large masses in the

universe like planets, stars, etc., based onNewton’s law of gravitation can be measured by using gravitational method (See Chapter 8). For measurement of small masses of atomic/sub-atomic particles etc., we make use of massspectrograph in which radius of the trajectory is proportional to the mass of a charged particlemoving in uniform electric and magnetic field.

2.4.1 Range of Masses

The masses of the objects, we come across inthe universe, vary over a very wide range. Thesemay vary from tiny mass of the order of 10-30 kgof an electron to the huge mass of about 1055 kgof the known universe. Table 2.4 gives the rangeand order of the typical masses of variousobjects.

Table 2.3 Range and order of lengths


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PHYSICS22

Table 2.4 Range and order of masses

2.5 MEASUREMENT OF TIME To measure any time interval we need a clock. We now use an atomic standard of time, whichis based on the periodic vibrations produced ina cesium atom. This is the basis of the cesiumclock, sometimes called atomic clock, used inthe national standards. Such standards areavailable in many laboratories. In the cesiumatomic clock, the second is taken as the timeneeded for 9,192,631,770 vibrations of theradiation corresponding to the transition between the two hyperfine levels of the groundstate of cesium-133 atom. The vibrations of thecesium atom regulate the rate of this cesiumatomic clock just as the vibrations of a balance wheel regulate an ordinary wristwatch or the vibrations of a small quartz crystal regulate a quartz wristwatch.

The cesium atomic clocks are very accurate.In principle they provide portable standard. Thenational standard of time interval ‘second’ as

well as the frequency is maintained through four cesium atomic clocks. A cesium atomic clock isused at the National Physical Laboratory (NPL),New Delhi to maintain the Indian standard of time.

In our country, the NPL has the responsibility of maintenance and improvement of physicalstandards, including that of time, frequency, etc.Note that the Indian Standard Time (IST) islinked to this set of atomic clocks. The efficient cesium atomic clocks are so accurate that they impart the uncertainty in time realisation as

± 1 × 10 –13, i.e. 1 part in 1013. This implies that the uncertainty gained over time by such a device is less than 1 part in 1013; they lose or

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 termsthe path length light travels in certain intervalof time (1/299, 792, 458 of a second) (Table 2.1).

The time interval of events that we comeacross in the universe vary over a very widerange. Table 2.5 gives the range and order of some typical time intervals.

You may notice that there is an interestingcoincidence between the numbers appearingin Tables 2.3 and 2.5. Note that the ratio of the

longest and shortest lengths of objects in our universe is about 1041. Interestingly enough,the ratio of the longest and shortest timeintervals associated with the events and objectsin our universe is also about 1041. This number,1041 comes up again in Table 2.4, which liststypical masses of objects. The ratio of thelargest and smallest masses of the objects inour universe is about (1041)2. Is this a curiouscoincidence between these large numberspurely accidental ?

2.6 ACCURACY, PRECISION OF INSTRUMENTS AND ERRORS IN MEASUREMENT

Measurement is the foundation of allexperimental science and technology. The result of every measurement by any measuringinstrument contains some uncertainty. Thisuncertainty is called error. Every calculatedquantity which is based on measured values,also has an error. We shall distinguish betweentwo terms: accuracy and precision. Theaccuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. Precision tells us to what resolutionor limit the quantity is measured.

The accuracy in measurement may depend onseveral factors, including the limit or the resolutionof the measuring instrument. For example, supposethe true value of a certain length is near 3.678 cm.In one experiment, using a measuring instrument of resolution 0.1 cm, the measured value is found to be 3.5 cm, while in another experiment using a measuring device of greater resolution, say 0.01 cm,the length is determined to be 3.38 cm. The first measurement has more accuracy (because it is


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closer to the true value) but less precision (itsresolution is only 0.1 cm), while thesecond measurement is less accurate but more precise. Thus every measurement is

approximate due to errors in measurement. Ingeneral, the errors in measurement can be broadly classified as (a) systematic errors and(b) random errors.

Systematic errors

The systematic errors are those errors that tend to be in one direction, either positive or negative. Some of the sources of systematicerrors are :

(a) Instrumental errors that arise from theerrors due to imperfect design or calibration

of the measuring instrument, zero error inthe instrument, etc. For example, thetemperature graduations of a thermometer may be inadequately calibrated (it may read104 °C at the boiling point of water at STP whereas it should read 100 °C); in a vernier callipers the zero mark of vernier scale may not coincide with the zero mark of the mainscale, or simply an ordinary metre scale may be worn off at one end.

(b) Imperfection in experimental techniqueor procedure To determine the temperature

of a human body, a thermometer placedunder the armpit will always give a temperature lower than the actual value of the body temperature. Other externalconditions (such as changes in temperature,humidity, wind velocity, etc.) during theexperiment may systematically affect themeasurement.

(c) Personal errors that arise due to anindividual’s bias, lack of proper setting of the apparatus or individual’s carelessnessin taking observations without observingproper precautions, etc. For example, if you, by habit, always hold your head a bit too far to the right while reading the position of a needle on the scale, you will introduce anerror due to parallax.

Systematic errors can be minimised by improving experimental techniques, selecting better instruments and removing personal biasas far as possible. For a given set-up, theseerrors may be estimated to a certain extent andthe necessary corrections may be applied to thereadings.

Random errors

The random errors are those errors, which occur irregularly and hence are random with respect

Table 2.5 Range and order of time intervals


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PHYSICS24

to sign and size. These can arise due to randomand unpredictable fluctuations in experimentalconditions (e.g. unpredictable fluctuations in

temperature, voltage supply, mechanical vibrations of experimental set-ups, etc), personal(unbiased) errors by the observer takingreadings, etc. For example, when the sameperson repeats the same observation, it is very likely that he may get different readingseverytime.

Least count error

The smallest value that can be measured by themeasuring instrument is called its least count. All the readings or measured values are good only up to this value.

The least count error is the error associated with the resolution of the instrument.For example, a vernier callipers has the least count as 0.01cm; a spherometer may have a least count of 0.001 cm. Least count error belongs to the category of random errors but within a limited size; it occurs with bothsystematic and random errors. If we use a metrescale for measurement of length, it may havegraduations at 1 mm division scale spacing or interval.

Using instruments of higher precision,

improving experimental techniques, etc., we canreduce the least count error. Repeating theobservations several times and taking thearithmetic mean of all the observations, themean value would be very close to the true valueof the measured quantity.

2.6.1 Absolute Error, Relative Error andPercentage Error

(a) Suppose the values obtained in severalmeasurements are a

1, a

2 , a

3...., a

n . The

arithmetic mean of these values is taken as

the best possible value of the quantity under the given conditions of measurement as :

a mean

= (a 1+a

2 +a

3+...+a

n ) / n (2.4)

or,

a a / n mean i i 1

n

==∑ (2.5)

This is because, as explained earlier, it isreasonable to suppose that individualmeasurements are as likely to overestimate

as to underestimate the true value of thequantity.

The magni tude of the di ffe rence

between the true value of the quantityand the individual measurement value is

called the absolute error of the

measurement. This is denoted by | ∆a | . Inabsence of any other method of knowing true value, we considered arithmatic mean as thetrue value. Then the errors in the individualmeasurement values are

∆a 1 = a

mean – a

1,

∆a 2 = a

mean – a

2 ,

.... .... ....

.... .... ....∆a

n = a

mean – a

n

The ∆a calculated above may be positive incertain cases and negative in some other cases. But absolute error |∆a | will always be positive.

(b) The arithmetic mean of all the absolute errors is taken as the final or mean absolute error of the value of the physical quantity a . It isrepresented by ∆a

mean .

Thus,

∆a mean

= (|∆a 1|+|∆a

2 |+|∆a

3|+...+ |∆a

n |)/n

(2.6)

==∑i 1

n

|∆a i |/n (2.7)

If we do a single measurement, the value weget may be in the range a

mean ± ∆a

mean

i.e. a = a mean

± ∆a mean

or, a mean – ∆a mean ≤a ≤a mean + ∆a mean

(2.8)

This implies that any measurement of thephysical quantity a is likely to lie between

(a mean

+ ∆a mean

) and (a mean

− ∆a mean

).(c) Instead of the absolute error, we often use

the relative error or the percentage error(δa ). T he relative error is the ratio of themean absolute error ∆∆∆∆∆a

mean to the mean

value a mean

of the quantity measured.


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Relative error = ∆a mean

/a mean

(2.9)

When the relative error is expressed in per

cent, it is called the percentage error (δa ).

Thus, Percentage error

δa = (∆a mean

/a mean

) × 100% (2.10)

Let us now consider an example.

Example 2.6 Two clocks are being testedagainst a standard clock located in a national laboratory. At 12:00:00 noon by the standard clock, the readings of the twoclocks are :

Clock 1 Clock 2

Monday 12:00:05 10:15:06 Tuesday 12:01:15 10:14:59 Wednesday 11:59:08 10:15:18 Thursday 12:01:50 10:15:07Friday 11:59:15 10:14:53Saturday 12:01:30 10:15:24Sunday 12:01:19 10:15:11

If you are doing an experiment that requiresprecision time interval measurements, whichof the two clocks will you prefer ?

Answer The range of variation over the sevendays of observations is 162 s for clock 1, and31 s for clock 2. The average reading of clock 1is much closer to the standard time than theaverage reading of clock 2. The important point is that a clock’s zero error is not as significant for precision work as its variation, because a ‘zero-error’ can always be easily corrected.Hence clock 2 is to be preferred to clock 1.

Example 2.7 We measure the period of oscillation of a simple pendulum. In

successive measurements, the readingsturn out to be 2.63 s, 2.56 s, 2.42 s, 2.71sand 2.80 s. Calculate the absolute errors,relative error or percentage error.

Answer The mean period of oscillation of thependulum

( )T =

+ + + +2.63 2.56 2.42 2.71 2.80 s

5

=13.12

5 s

= 2.624 s= 2.62 s

physics xi part 1 - ncert

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