Class XI Physics Lab Manual

Laboratory ManualLaboratory ManualLaboratory ManualLaboratory ManualLaboratory Manual

PHYSICSClass XI

The National Council of Educational Research and Training (NCERT) is the apexbody concerning all aspects of refinement of School Education. It has recentlydeveloped textual material in Physics for Higher Secondary stage which is basedon the National Curriculum Framework (NCF)–2005. NCF recommends thatchildren’s experience in school education must be linked to the life outside schoolso that learning experience is joyful and fills the gap between the experience athome and in community. It recommends to diffuse the sharp boundaries betweendifferent subjects and discourages rote learning. The recent development of syllabiand textual material is an attempt to implement this basic idea. The presentLaboratory Manual will be complementary to the textbook of Physics for ClassXI. It is in continuation to the NCERT’s efforts to improve upon comprehensionof concepts and practical skills among students. The purpose of this manual isnot only to convey the approach and philosophy of the practical course to studentsand teachers but to provide them appropriate guidance for carrying outexperiments in the laboratory. The manual is supposed to encourage children toreflect on their own learning and to pursue further activities and questions. Ofcourse, the success of this effort also depends on the initiatives to be taken bythe principals and teachers to encourage children to carry out experiments inthe laboratory and develop their thinking and nurture creativity.The methods adopted for performing the practicals and their evaluation willdetermine how effective this practical book will prove to make the children’s lifeat school a happy experience, rather than a source of stress and boredom. Thepractical book attempts to provide space to opportunities for contemplation andwondering, discussion in small groups, and activities requiring hands-onexperience. It is hoped that the material provided in this manual will help studentsin carrying out laboratory work effectively and will encourage teachers tointroduce some open-ended experiments at the school level.

PROFESSOR YASH PAL

Chairperson

National Steering Committee

National Council of Educational

Research and Training

FOREWORD

The development of the present laboratory manual is in continuation to theNCERT’s efforts to support comprehension of concepts of science and alsofacilitate inculcation of process skills of science. This manual is complementaryto the Physics Textbook for Class XI published by NCER T in 2006 following theguidelines enumerated in National Curriculum Framework (NCF)-2005. One of thebasic criteria for validating a science curriculum recommended in NCF–2005, isthat ‘it should engage the learner in acquiring the methods and processes thatlead to the generation and validation of scientific knowledge and nurture thenatural curiosity and creativity of the child in science’. The broad objective ofthis laboratory manual is to help the students in performing laboratory basedexercises in an appropriate manner so as to develop a spirit of enquiry in them.It is envisaged that students would be given all possible opportunities to raisequestions and seek their answers from various sources.The physics practical work in this manual has been presented under foursections (i) experiments (ii) activities (iii) projects and (iv) demonstrations. Awrite-up on major skills to be developed through practical work in physics hasbeen given in the beginning which includes discussion on objectives of practicalwork, experimental errors, logarithm, plotting of graphs and general instructionsfor recording experiments.Experiments and activities prescribed in the NCERT syllabus (covering CBSEsyllabus also) of Class XI are discussed in detail. Guidelines for conductingeach experiment has been presented under the headings (i) apparatus andmaterial required (ii) principle (iii) procedure (iv) observations (v) calculations(vi) result (vii) precautions (viii) sources of error. Some important experimentalaspects that may lead to better understanding of result are also highlighted inthe discussion. Some questions related to the concepts involved have been raisedso as to help the learners in self assessment. Additional experiments/activitiesrelated to a given experiment are put forth under suggested additionalexperiments/activities at the end.A number of project ideas, including guidelines are suggested so as to cover alltypes of topics that may interest young learners at higher secondary level.A large number of demonstration experiments have also been suggested for theteachers to help them in classroom transaction. Teachers should encourageparticipation of the students in setting up and improvising apparatus, indiscussions and give them opportunity to analyse the experimental data to arriveat conclusions.

PREFACE

Appendices have been included with a view to try some innovative experimentsusing improvised apparatus. Data section at the end of the book enlists a numberof useful Tables of physical constants.Each experiment, activity, project and demonstration suggested in this manualhave been tried out by the experts and teachers before incorporating them. Wesincerely hope that students and teachers will get motivated to perform theseexperiments supporting various concepts of physics thereby enriching teachinglearning process and experiences.It may be recalled that NCER T brought out laboratory manual in physics forsenior secondary classes earlier in 1989. The write-ups on activities, projects,demonstrations and appendices included in physics manual published byNCERT in 1989 have been extensively used in the development of the presentmanual.We are grateful to the teachers and subject experts who participated in theworkshops organised for the review and refinement of the manuscript of thislaboratory manual.I acknowledge the valuable contributions of Prof. B.K. Sharma and other teammembers who contributed and helped in finalising this manuscript. I alsoacknowledge with thanks the dedicated efforts of Sri R. Joshi who looked afterthe coordinatorship after superannuation of Professor B.K. Sharma in June,2008.We warmly welcome comments and suggestions from our valued readers forfurther improvement of this manual.

HUKUM SINGH

Professor and Head

Department of Education inScience and Mathematics

vi

MEMBERS

B.K. Sharma, Professor, DESM, NCERT, New Delhi

Gagan Gupta, Reader, DESM, NCER T, New Delhi

R. Joshi, Lecturer (S.G.), DESM, NCERT, New Delhi

S.K. Dash, Reader, DESM, NCERT, New Delhi

Shashi Prabha, Senior Lecturer, DESM, NCERT, New Delhi

V.P. Srivastava, Reader, DESM, NCERT, New Delhi

MEMBER-COORDINATORS

B.K. Sharma, Professor, DESM, NCERT, New Delhi

R. Joshi, Lecturer (S.G.), DESM, NCERT, New Delhi

DEVELOPMENT TEAM

ACKNOWLEDGEMENT

The National Council of Educational Research and Training (NCERT)

acknowledges the valuable contributions of the individuals and theorganisations involved in the development of Laboratory Manual of Physics

for Class XI. The Council also acknowledges the valuable contributions of thefollowing academics for the reviewing, refining and editing the manuscript of

this manual : A.K. Das, PGT, St. Xavier’s Senior Secondary School, Raj NiwasMarg, Delhi; A.K. Ghatak, Professor (Retired), IIT, New Delhi; A.W. Joshi,

Hon. Visiting Scientist, NCRA, Pune; Anil Kumar, Principal, R.P.V.V., BT -Block, Shalimar Bagh, New Delhi; Anuradha Mathur, PGT, Modern School

Vasant Vihar, New Delhi; Bharthi Kukkal, PGT, Kendriya Vidyalaya, PushpVihar, New Delhi; C.B. Verma, Principal (Retired), D.C. Arya Senior Secondary

School, Lodhi Road, New Delhi; Chitra Goel, PGT, R.P.V.V., Tyagraj Nagar, NewDelhi; Daljeet Kaur Bhandari, Vice Principal, G.H.P.S., Vasant Vihar, New

Delhi; Girija Shankar, PGT, R.P.V.V., Surajmal Vihar, New Delhi; H.C. Jain,Principal (Retired), Regional Institute of Education (NCERT), Ajmer; K.S.

Upadhyay, Principal, Jawahar Navodaya Vidyalaya, Farrukhabad, U.P.; M.N.Bapat, Reader, Regional Institute of Education (NCERT), Bhopal; Maneesha

Pachori, Maharaja Agrasen College, University of Delhi, New Delhi; P.C. Agarwal,Reader, Regional Institute of Education (NCERT), Ajmer; P.C. Jain, Professor

(Retired), University of Delhi, Delhi; P.K. Chadha, Principal, St. Soldier PublicSchool, Paschim Vihar, New Delhi; Pragya Nopany, PGT, Birla Vidya Niketan,

Pushp Vihar-IV, New Delhi; Pushpa Tyagi, PGT, Sanskriti School,Chanakyapuri, New Delhi; R.P. Sharma, Education Officer (Science),

CBSE, New Delhi; R.S. Dass, Vice Principal (Retired), Balwant Ray MehtaVidya Bhawan, Lajpat Nagar, New Delhi; Rabinder Nath Kakarya, PGT, Darbari

Lal, DAVMS, Pitampura, New Delhi; Rachna Garg, Lecturer (Senior Scale),CIET, NCERT; Rajesh Kumar, Principal, District Institute of Educational

Research and Training, Pitampura, New Delhi; Rajeshwari Prasad Mathur,Professor, Aligarh Muslim University, Aligarh; Rakesh Bhardwaj, PGT, Maharaja

Agrasen Model School, CD-Block, Pitampura, New Delhi; Ramneek Kapoor,PGT, Jaspal Kaur Public School, Shalimar Bagh, New Delhi; Rashmi Bargoti,

PGT, S.L.S. D.A.V. Public School, Mausam Vihar, New Delhi; S.N. Prabhakara,PGT, Demonstration Multipurpose School, Mysore; S.R. Choudhury, Raja

Ramanna Fellow, Centre for Theoretical Physics, Jamia Millia Islamia, NewDelhi; S.S. Islam, Professor, Jamia Millia Islamia, New Delhi; Sher Singh, PGT,

Navyug School, Lodhi Road, New Delhi; Shirish R. Pathare, Scientific Officer;

Homi Bhabha Centre for Science Education (TIFR), Mumbai; SubhashChandra Samanta, Reader (Retired), Midnapur College, Midnapur (W.B.);

Sucharita Basu Kasturi, PGT, Sardar Patel Vidyalaya, New Delhi; SurajitChakrabarti, Reader, Maharaja Manindra Chandra College, Kolkata; Suresh

Kumar, PGT, Delhi Public School, Dwarka, New Delhi; V.K. Gautam, Education

Officer (Science), Kendriya Vidyalaya Sangathan, Shaheed Jeet Singh Marg,

New Delhi; Ved Ratna, Professor (Retired), DESM, NCERT, New Delhi; VijayH. Raybagkar, Reader, N. Wadia College, Pune; Vishwajeet D. Kulkarni,

Smt. Parvatibai Chowgule College, Margo, Goa; Y.K. Vijay, CDPE Universityof Rajasthan, Jaipur, Rajasthan; Yashu Kumar, PGT, Kulachi Hansraj Model

School, New Delhi. We are thankful to all of them. Special thanks are due toHukum Singh, Professor and Head, DESM, NCERT for providing all academic

and administrative support.The Council also acknowledges the support provided by the APC Office and

administrative staff of DESM, Deepak Kapoor, Incharge, Computer Station;Bipin Srivastva, Rohit Verma and Mohammad Jabir Hussain, DTP Operators

for typing the manuscript, preparing CRC and refining and drawing some ofthe illustrations; Dr. K. T. Chitralekha, Copy Editor; Abhimanu Mohanty,

Proof Reader. The efforts of the Publication Department are also highlyappreciated.

ix

xi

CONTENTS

iiiv

1

2

4

5

10

14

14

19

20

23

33

42

48

55

60

68

FOREWORD

PREFACE

Major Skills in Physics Practical Work

I 1.1 Introduction

I 1.2 Objectives of practical work

I 1.3 Specific objectives of laboratory work

I 1.4 Experimental errors

I 1.5 Logarithms

I 1.6 Natural sine/cosine table

I 1.7 Plotting of graphs

I 1.8 General instructions for performing experiments

I 1.9 General instructions for recording experiments

EXPERIMENTS

E1 Use of Vernier Callipers to

(i) measure diameter of a small spherical/cylindrical body,(ii) measure the dimensions of a given regular body of known mass

and hence to determine its density and

(iii) measure the internal diameter and depth of a given cylindrical objectlike beaker/glass/calorimeter and hence to calculate its volume

E2 Use of screw gauge to(a) measure diameter of a given wire,

(b) measure thickness of a given sheet and

(c) determine volume of an irregular lamina

E3 To determine the radius of curvature of a given spherical surface bya spherometer

E4 To determine mass of two different objects using a beam balance

E5 Measurement of the weight of a given body (a wooden block) usingthe parallelogram law of vector addition

E6 Using a simple pendulum plot L – T and L – T2 graphs, hence findthe effective length of second's pendulum using appropriate graph

E7 To study the relation between force of limiting friction and normalreaction and to find the coefficient of friction between surface of amoving block and that of a horizontal surface

xii

E8 To find the downward force, along an inclined plane, acting on aroller due to gravity and study its relationship with the angle ofinclination by plotting graph between force and sin θ

E9 To determine Young's modulus of the material of a given wire byusing Searle's apparatus

E10 To find the force constant and effective mass of a helical spring byplotting T 2 - m graph using method of oscillation

E11 To study the variation in volume (V ) with pressure (P ) for a sampleof air at constant temperature by plotting graphs between P and V,

and between P and 1V

E12 To determine the surface tension of water by capillary rise method

E13 To determine the coefficient of viscosity of a given liquid by measuringthe terminal velocity of a spherical body

E14 To study the relationship between the temperature of a hot bodyand time by plotting a cooling curve

E15 (i) To study the relation between frequency and length of a givenwire under constant tension using a sonometer(ii) To study the relation between the length of a given wire and tensionfor constant frequency using a sonometer

E16 To determine the velocity of sound in air at room temperature usinga resonance tube

E17 To determine the specific heat capacity of a given (i) solid and (ii) a liquidby the method of mixtures

ACTIVITIESA1 To make a paper scale of given least count: (a) 0.2 cm and (b) 0.5 cm

A2 To determine the mass of a given body using a metre scale by theprinciple of moments

A3 To plot a graph for a given set of data choosing proper scale andshow error bars due to the precision of the instruments

A4 To measure the force of limiting rolling friction for a roller (woodenblock) on a horizontal plane

A5 To study the variation in the range of a jet of water with the changein the angle of projection

A6 To study the conservation of energy of a ball rolling down an inclinedplane (using a double inclined plane)

A7 To study dissipation of energy of a simple pendulum with time

A8 To observe the change of state and plot a cooling curve for molten wax

A9 To observe and explain the effect of heating on a bi-metallic strip

74

78

83

89

95

99

104

109

114

119

125

128

132

137

140

144

148

152

155

xiii

A10 To study the effect of heating on the level of a liquid in a containerand to interpret the observations

A11 To study the effect of detergent on surface tension of water byobserving capillary rise

A12 To study the factors affecting the rate of loss of heat of a liquid

A13 To study the effect of load on depression of a suitably clampedmetre scale loaded (i) at its end and (ii) in the middle

PROJECTS

P1 To investigate whether the energy of a simple pendulum is conserved

P2 To determine the radius of gyration about the centre of mass of ametre scale used as a bar pendulum

P3 To investigate changes in the velocity of a body under the actionof a constant force and to determine its acceleration

P4 To compare the effectiveness of different materials asinsulator of heat

P5 To compare the effectiveness of different materials as absorbersof sound

P6 To compare the Young’s modules of elasticity of differentspecimen of rubber and compare them by drawing their elastichysteresis curve

P7 To study the collision of two balls in two-dimensions

P8 To study Fortin’s Barometer and use it to measure theatmospheric pressure

P9 To study of the spring constant of a helical spring from itsload-extension graph

P10 To study the effect of nature of surface on emission and absorptionof radiation

P11 To study the conservation of energy with a 0.2 pendulum

DEMONSTRATIONS

D1 To demonstrate uniform motion in a straight line

D2 To demonstrate the nature of motion of a ball on aninclined track

D3 To demonstrate that a centripetal force is necessary for moving abody with a uniform speed along a circle, and that magnitude ofthis force increases with angular speed

D4 To demonstrate the principle of centrifuge

D5 To demonstrate interconversion of potential and kinetic energy

D6 To demonstrate conservation of momentum

D7 To demonstrate the effect of angle of launch on range of a projectile

158

160

163

167

173

181

186

190

193

197

200204

208

213

216

219

223

224

226

227

228

229

xiv

D8 To demonstrate that the moment of inertia of a rod changes with thechange of position of a pair of equal weights attached to the rod

D9 To demonstrate the shape of capillary rise in a wedge-shaped gapbetween two glass sheets

D10 To demonstrate affect of atmospheric pressure by making partialvacuum by condensing steam

D11 To study variation of volume of a gas with its pressure at constanttemperature with a doctors’ syringe

D12 To demonstrate Bernoulli’s theorem with simple illustrations

D13 To demonstrate the expansion of a metal wire on heating

D14 To demonstrate that heat capacities of equal masses of aluminium,iron, copper and lead are different

D15 To demonstrate free oscillations of different vibrating systems

D16 To demonstrate resonance with a set of coupled pendulums

D17 To demonstrate damping of a pendulum due to resistance ofthe medium

D18 To demonstrate longitudinal and transverse waves

D19 To demonstrate reflection and transmission of waves at theboundary of two media

D20 To demonstrate the phenomenon of beats due to superpositionof waves produced by two tuning forks of slightly differentfrequencies

D21 To demonstrate standing waves with a spring

Appendices (A-1 to A-14)

Bibliography

Data Section

230

232

233

235

237

240241

243247

248

249251

253

254

256–263

264–265

266–275

I: MAJOR SKILLS IN: MAJOR SKILLS IN: MAJOR SKILLS IN: MAJOR SKILLS IN: MAJOR SKILLS INPHYSICS PRACTICALPHYSICS PRACTICALPHYSICS PRACTICALPHYSICS PRACTICALPHYSICS PRACTICAL

WORKWORKWORKWORKWORK

I 1.1 INTRODUCTION

The higher secondary stage is the most crucial and challenging stageof school education because at this stage the general undifferentiatedcurriculum changes into a discipline-based, content area-orientedcourse. At this stage, students take up physics as a discipline, withthe aim of pursuing their future careers either in basic sciences or inscience-based professional courses like engineering, medicine,information technology etc.

Physics deals with the study of matter and energy associated with theinanimate as well as the animate world. Although all branches ofscience require experimentation, controlled laboratory experimentsare of central importance in physics. The basic purpose of laboratoryexperiments in physics, in general, is to verify and validate the concepts,principles and hypotheses related to the physical phenomena. Onlydoing this does not help the learners become independent thinkers orinvestigate on their own. In view of this, laboratory work is very muchrequired and encouraged in different ways.These may include notonly doing experiments but investigate different facets involved in doingexperiments. Many activities as well as project work will thereforeensure that the learners are able to construct and reconstruct theirideas on the basis of first hand experiences through investigation in thelaboratory. Besides, learners will be able to integrate experimental workwith theory which they are studying at higher secondary stage throughtheir environment.

The history of science reveals that many significant discoveries have beenmade while carrying out experiments. In the growth of physics,experimental work is as important as the theoretical understanding of aphenomenon. Performing experiments by one’s own hands in a laboratoryis important as it generates a feeling of direct involvement in the processof generating knowledge. Carrying out experiments in a laboratorypersonally and analysis of the data obtained also help in inculcatingscientific temper, logical thinking, rational outlook, sense of self-confidence,ability to take initiative, objectivity, cooperative attitude, patience, self-reliance, perseverance, etc. Carrying out experiments also developmanipulative, observational and reporting skills.

2

LABORATORY MANUALLABORATORY MANUAL

The ‘National Curriculum Framework’ (NCF-2005) and the Syllabusfor Secondary and Higher Secondary stages (NCERT, 2006) havetherefore, laid considerable emphasis on laboratory work as an integralpart of the teaching-learning process.

NCERT has already published Physics Textbook for Classes XII,based on the new syllabus. In order to supplement the conceptualunderstanding and to integrate the laboratory work in physics andcontents of the physics course, this laboratory manual has beendeveloped. The basic purpose of a laboratory manual in physics is tomotivate the students towards practical work by involving them in“process-oriented performance” learning (as opposed to ‘product-orresult-oriented performance’) and to infuse life into the sagging practicalwork in schools. In view of the alarming situation with regard to theconduct of laboratory work in schools, it is hoped that this laboratorymanual will prove to be of considerable help and value.

I 1.2 OBJECTIVES OF PRACTICAL WORK

Physics deals with the understanding of natural phenomena andapplying this understanding to use the phenomena fordevelopment of technology and for the betterment of society.Physics practical work involves ‘learning by doing’. It clarifiesconcepts and lays the seed for enquiry.

Careful and stepwise observation of sequences during an experimentor activity facilitate personal investigation as well as small group orteam learning.

A practical physics course should enable students to do experimentson the fundamental laws and principles, and gain experience of usinga variety of measuring instruments. Practical work enhances basiclearning skills. Main skills developed by practical work in physics arediscussed below.

I 1.2.1 MANIPULATIVE SKILLS

The learner develops manipulative skills in practical work if she/heis able to

(i) comprehend the theory and objectives of the experiment,(ii) conceive the procedure to perform the experiment,(iii) set-up the apparatus in proper order,(iv) check the suitability of the equipment, apparatus, tool

regarding their working and functioning,(v) know the limitations of measuring device and find its least

count, error etc.,(vi) handle the apparatus carefully and cautiously to avoid any

damage to the instrument as well as any personal harm,

3

UNIT NAMEMAJOR SKILLS...

(vii) perform the experiment systematically.

(viii) make precise observations,

(ix) make proper substitution of data in formula, keeping properunits (SI) in mind,

(x) calculate the result accurately and express the same withappropriate significant figures, justified by the degree ofaccuracy of the instrument,

(xi) interpret the results, verify principles and draw conclusions; and

(xii) improvise simple apparatus for further investigations byselecting appropriate equipment, apparatus, tools, materials.

I 1.2.2 OBSERVATIONAL SKILLS

The learner develops observational skills in practical work if she/heis able to

(i) read about instruments and measure physical quantities,keeping least count in mind,

(ii) follow the correct sequence while making observations,

(iii) take observations carefully in a systematic manner; and

(iv) minimise some errors in measurement by repeating everyobservation independently a number of times.

I 1.2.3 DRAWING SKILLSThe learner develops drawing skills for recording observed data ifshe/he is able to

(i) make schematic diagram of the apparatus,

(ii) draw ray diagrams, circuit diagrams correctly and label them,

(iii) depict the direction of force, tension, current, ray of light etc,by suitable lines and arrows; and

(iv) plot the graphs correctly and neatly by choosing appropriatescale and using appropriate scale.

I 1.2.4 REPORTING SKILLS

The learner develops reporting skills for presentation of observationdata in practical work if she/he is able to

(i) make a proper presentation of aim, apparatus, formula used,principle, observation table, calculations and result for theexperiment,

4


(ii) support the presentation with labelled diagram usingappropriate symbols for components,

(iii) record observations systematically and with appropriate unitsin a tabular form wherever desirable,

(iv) follow sign conventions while recording measurements inexperiments on ray optics,

(v) present the calculations/results for a given experimentalongwith proper significant figures, using appropriate symbols,units, degree of accuracy,

(vi) calculate error in the result,

(vii) state limitations of the apparatus/devices,

(viii) summarise the findings to reject or accept a hypothesis,

(ix) interpret recorded data, observations or graphs to drawconclusion; and

(x) explore the scope of further investigation in the work performed.

However, the most valued skills perhaps are those that pertain to therealm of creativity and investigation.

I 1.3 SPECIFIC OBJECTIVES OF LABORATORY WORK

Specific objectives of laboratory may be classified as process-orientedperformance skills and product-oriented performance skills.

I 1.3.1 PROCESS - ORIENTED PERFORMANCE SKILLS

The learner develops process-oriented performance skills in practicalwork if she/he is able to

(i) select appropriate tools, instruments, materials, apparatus andchemicals and handle them appropriately,

(ii) check for the working of apparatus beforehand,

(iii) detect and rectify instrumental errors and their limitations,

(iv) state the principle/formula used in the experiment,

(v) prepare a systematic plan for taking observations,

(vi) draw neat and labelled diagram of given apparatus/raydiagram/circuit diagram wherever needed,

(vii) set up apparatus for performing the experiment,

5


(viii) handle the instruments, chemicals and materials carefully,

(ix) identify the factors that will influence the observations andtake appropriate measures to minimise their effects,

(x) perform experiment within stipulated time with reasonablespeed, accuracy and precision,

(xi) represent the collected data graphically and neatly by choosingappropriate scale and neatly, using proper scale,

(xii) interpret recorded data, observations, calculation or graphsto draw conclusion,

(xiii) report the principle involved, procedure and precautionsfollowed in performing the experiment,

(xiv) dismantle and reassemble the apparatus; and

(xv) follow the standard guidelines of working in a laboratory.

I 1.3.2 PRODUCT - ORIENTED PERFORMANCE SKILLS

The learner develops product-oriented performance skills in practicalwork if she/he is able to

(i) identify various parts of the apparatus and materials used inthe experiment,

(ii) set-up the apparatus according to the plan of the experiment,

(iii) take observations and record data systematically so as tofacilitate graphical or numerical analysis,

(iv) present the observations systematically using graphs,calculations etc. and draw inferences from recordedobservations,

(v) analyse and interpret the recorded observations to finalise theresults; and

(vi) accept or reject a hypothesis based on the experimentalfindings.

I 1.4 EXPERIMENTAL ERRORSThe ultimate aim of every experiment is to measure directly orindirectly the value of some physical quantity. The very process ofmeasurement brings in some uncertainties in the measured value.THERE IS NO MEASUREMENT WITHOUT ERRORS. As such the valueof a physical quantity obtained from some experiments may bedifferent from its standard or true value. Let ‘a’ be the experimentally

6


observed value of some physical quantity, the ‘true’ value of which is‘a

0’. The difference (a – a

0) = e is called the error in the measurement.

Since a0, the true value, is mostly not known and hence it is notpossible to determine the error e in absolute terms. However, it ispossible to estimate the likely magnitude of e. The estimated value oferror is termed as experimental error. The error can be due to leastcount of the measuring instrument or a mathematical relation involvingleast count as well as the variable. The quality of an experiment isdetermined from the experimental uncertainty of the result. Smallerthe magnitude of uncertainty, closer is the experimentally measuredvalue to the true value. Accuracy is a measure of closeness of themeasured value to the true value. On the other hand, if a physicalquantity is measured repeatedly during the same experiment againand again, the values so obtained may be different from each other.This dispersion or spread of the experimental data is a measure of theprecision of the experiment/instrument. A smaller spread in theexperimental value means a more precise experiment. Thus, accuracyand precision are two different concepts. Accuracy is a measureof the nearness to truth, while precision is a measure of thedispersion in experimental data. It is quite possible that a highprecision experimental data may be quite inaccurate (if there are largesystematic errors present). A rough estimate of the maximum spreadis related to the least count of the measuring instrument.

Experimental errors may be categorised into two types:(a) systematic, and (b) random. Systematic errors may arise becauseof (i) faulty instruments (like zero error in vernier callipers),(ii) incorrect method of doing the experiment, and (iii) due to theindividual who is conducting the experiment. Systematic errors arethose errors for which corrections can be applied and in principlethey can be removed. Some common systematic errors: (i) Zero errorin micrometer screw and vernier callipers readings. (ii) The ‘backlash’error. When the readings on a scale of microscope are taken by rotatingthe screw first in one direction and then in the reverse direction, thereading is less than the actual distance through which the screw ismoved. To avoid this error all the readings must be taken while rotatingthe screw in the same direction. (iii) The ‘bench error’ or ‘indexcorrection’. When distances measured on the scale of an optical benchdo not correspond to the actual distances between the optical devices,addition or substraction of the difference is necessary to obtain correctvalues. (iv) If the relation is linear, and if the systematic error is constant,the straight-line graph will get shifted keeping the slope unchanged,but the intercept will include the systematic error.

In order to find out if the result of some experiments containssystematic errors or not, the same quantity should be measured by adifferent method. If the values of the same physical quantity obtainedby two different methods differ from each other by a large amount,then there is a possibility of systematic error. The experimental value,

7


after corrections for systematic errors still contain errors. All suchresidual errors whose origin cannot be traced are called random errors.Random errors cannot be avoided and there is no way to find theexact value of random errors. However, their magnitude may bereduced by measuring the same physical quantity again and againby the same method and then taking the mean of the measured values(For details, see Physics Textbook for Class XI, Part I, Chapter 2;NCERT, 2006).

While doing an experiment in the laboratory, we measure differentquantities using different instruments having different values of theirleast counts. It is reasonable to assume that the maximum error inthe measured value is not more than the least count of the instrumentwith which the measurement has been made. As such in the case ofsimple quantities measured directly by an instrument, the least countof the instrument is generally taken as the maximum error in themeasured value. If a quantity having a true value A0 is measured as Awith the instrument of least count a, then

( )0A A a= ±

( )0 01 /A a A= ±

( )0 1 aA f= ±

where af is called the maximum fractional error of A. Similarly, for

another measured quantity B, we have

( )bB B f= ±0 1

Now some quantity, say Z, is calculated from the measured value of Aand B, using the formula

Z = A.B

We now wish to calculate the expected total uncertainty (or the likelymaximum error) in the calculated value of Z. We may write

Z = A.B

( ) ( )a bA f .B f= ± ±0 01 1

( )a b a bA B f f f f= ± ± ±0 0 1

( )a bA B f f± +⎡ ⎤⎣ ⎦0 0 1 , [If fa and f

b are very small quantities, their

product fa

fb can be neglected]

or [ ]zZ Z f≈ ±0 1

8


where the fractional error fz in the value of Z may have the largest

value of a bf f+ .

On the other hand, if the quantity Y to be calculated is given as

Y = A/B ( ) ( )a bA f B f= ± ±0 01 / 1

( )( )= ± ±–1

0Y 1 1a bf f ; A

B

⎡ ⎤=⎢ ⎥

⎣ ⎦0

00

Y

( )( )= ± ± + 20Y 1 1a b bf f f

( ) ( )= ± ±0Y 1 1a bf f

~ ( )⎡ ⎤± +⎣ ⎦0Y 1 a bf f

or Y = Y0 ( )1 yf± , with fy = fa + fb, where the maximum fractional

uncertainty fy in the calculated value of Y is again

a bf + f . Note that

the maximum fractional uncertainty is always additive.

Taking a more general case, where a quantity P is calculated fromseveral measured quantities x, y, z etc., using the formula P = xa yb zc,it may be shown that the maximum fractional error f

p in the calculated

value of P is given as

p x y zf a f b f c f= + +

It may be observed that the value of the overall fractional error fp in

the quantity P depends on the fractional errors fx, f

y, f

z etc. of each

measured quantity, as well as on the power a, b, c etc., of thesequantities which appear in the formula. As such, the quantity whichhas the highest power in the formula, should be measured with theleast possible fractional error, so that the contribution of

x y za f b f c f+ + to the overall fraction error fp are of the same order

of magnitude.

Let us calculate the expected uncertainty (or experimental error) in aquantity that has been determined using a formula which involvesseveral measured physical parameters.

A quantity Y, Young’s Modulus of elasticity is calculated using the formula

MgLY =

bd δ

3

34

9


where M is the mass, g is the acceleration due to gravity, L is thelength of a metallic bar of rectangular cross-section, with breadth b,and thickness d, and δ is the depression (or sagging) from the horizontalin the bar when a mass M is suspended from the middle point of thebar, supported at its two ends (Fig. I 1.1).

Now in an actual experiment, mass M may be taken as 1 kg. Normallythe uncertainty in mass is not more than 1 g. It means that the leastcount of the ordinary balance used for measuring mass is 1 g. Assuch, the fractional error fM is 1g/1kg or fM = 1 × 10–3.

Let us assume that the value of acceleration due to gravity g is 9.8 m/s2 and it does not contain any significant error. Hence there will be nofractional error in g, i.e., fg = 0. Further the length L of the bar is, say,1 m and is measured by an ordinary metre scale of least count of 1mm = 0.001 m. The fractional error fL in the length L is therefore,

fL = 0.001 m / 1m = 1 × 10–3.

Next the breadth b of the bar which is, say, 5 cm is measured by avernier callipers of least count 0.01 cm. The fractional error fb is then,

fb = 0.01 cm / 5 cm = 0.002 = 2 × 10 –3.

Similarly, for the thickness d of the bar, a screw gauge of least count0.001 cm is used. If, a bar of thickness, say, 0.2 cm is taken so that

fd = 0.001 cm / 0.2 cm = 0.005 = 5 × 10–3.

Finally, the depression δ which is measured by a spherometer of leastcount 0.001 cm, is about 5 mm, so that

fδ = 0.001 cm / 0.5 cm = 0.002 = 2 × 10–3.

Having calculated the fractional errors in each quantity, let uscalculate the fractional error in Y as

fY = (1) fM + (1) fg + (3) fL + (1) fb + (3) fd + (1) f δ

= 1 × (1 × 10–3 ) + 1 × 0 + 3 × (1 × 10–3) + 1 × (2 × 10–3) + 3 × (5 × 10–3) + 1 × (2 × 10–3)

= 1 × 10–3 + 3 × 10–3 + 2 × 10–3 + 15 × 10–3 + 2 × 10–3

or, fY = 22 × 10–3 = 0.022.

Hence the possible fractional error (or uncertainty) is fy × 100 = 0.022

× 100 = 2.2%. It may be noted that, for a good experiment, thecontribution to the maximum fractional error f

y in the calculated value

of Y contributed by various terms, i.e., fM, 3fL, fb, 3fd, and fδ should beof the same order of magnitude. It should not happen that one ofthese quantities becomes so large that the value of f

y is determined by

that factor only. If this happens, then the measurement of otherquantities will become insignificant. It is for this reason that the lengthL is measured by a metre scale which has a large least count (0.1 cm)while smaller quantities d and δ are measured by screw gauge and

10


spherometer, respectively, which have smaller leastcount (0.001 cm). Also those quantities which havehigher power in the formula, like d and L should bemeasured more carefully with an instrument ofsmaller least count.

The end product of most of the experiments is themeasured value of some physical quantity. Thismeasured value is generally called the result of theexperiment. In order to report the result, three mainthings are required. These are – the measured value,the expected uncertainty in the result (orexperimental error) and the unit in which thequantity is expressed. Thus the measured value isexpressed alongwith the error and proper unit as thevalue ± error (units).

Suppose a result is quoted as A ± a (unit).

This implies that the value of A is estimated to an accuracy of 1 partin A/a, both A and a being numbers. It is a general practice to includeall digits in these numbers that are reliably known plus the first digitthat is uncertain. Thus, all reliable digits plus the first uncertain digittogether are called SIGNIFICANT FIGURES. The significant figures ofthe measured value should match with that of the errors. In the presentexample assuming Young Modulus of elasticity, Y = 18.2 × 1010 N/m2; (please check this value by calculating Y from the given data) and

error, Δ

= y

Yf

Y

ΔY = fy.Y

= 0.022 × 18.2 × 1010 N/m2

= 0.39 × 1010 N/m2, where ΔY is experimental error.

So the quoted value of Y should be (18.2 ± 0.4) × 1010 N/m2.

I 1.5 LOGARITHMSThe logarithm of a number to a given base is the index of the power towhich the base must be raised to equal that number.

If ax = N then x is called logarithm of N to the base a, and is denoted byloga N [read as log N to the base a]. For example, 24 = 16. The log of16 to the base 2 is equal to 4 or, log2 16 = 4.

In general, for a number we use logarithm to the base 10. Here log 10= 1, log 100 = log 102 and so on. Logarithm to base 10 is usuallywritten as log.

Fig. 1.1: A mass M is suspended from themetallic bar supported at its twoends

11


(i) COMMON LOGARITHMLogarithm of a number consists of two parts:

(i) Characteristic — It is the integral part [whole of naturalnumber]

(ii) Mantissa — It is the fractional part, generally expressed indecimal form (mantissa is always positive).

(ii) HOW TO FIND THE CHARACTERISTIC OF A NUMBER?

The characteristic depends on the magnitude of the number and isdetermined by the position of the decimal point. For a number greaterthan 1, the characteristic is positive and is less than the number ofdigits to the left of the decimal point.

For a number smaller than one (i.e., decimal fraction), the characteristicis negative and one more than the number of zeros between the decimalpoint and the first digit. For example, characteristic of the number

430700 is 5; 4307 is 3; 43.07 is 1;

4.307 is 0; 0.4307 is –1; 0.04307 is –2;

0.0004307 is –4 0.00004307 is –5.

The negative characteristic is usually written as 1,2,4,5 etc and readas bar 1, bar 2, etc.

I 1.5.1 HOW TO FIND THE MANTISSA OF A NUMBER?

The value of mantissa depends on the digits and their order and isindependent of the position of the decimal point. As long as the digitsand their order is the same, the mantissa is the same, whatever be theposition of the decimal point.

The logarithm Tables 1 and 2, on pages 266–269, give the mantissaonly. They are usually meant for numbers containing four digits, andif a number consists of more than four figures, it is rounded off to fourfigures after determining the characteristic. To find mantissa, the tablesare used in the following manner :

(i) The first two significant figures of the number are found at theextreme left vertical column of the table wherein the numberlying between 10 and 99 are given. The mantissa of the figureswhich are less than 10 can be determined by multiplying thefigures by 10.

(ii) Along the horizontal line in the topmost column the figures

12


are given. These correspond to the third significant figure ofthe given number.

(iii) Further right column under the figures (digits) corresponds tothe fourth significant figures.

Example 1 : Find the logarithm of 278.6.

Answer : The number has 3 figures to the left of the decimal point.Hence, its characteristic is 2. To find the mantissa, ignore thedecimal point and look for 27 in the first vertical column. For 8,look in the central topmost column. Proceed from 27 along ahorizontal line towards the right and from 8 vertically downwards.The two lines meet at a point where the number 4440 is written.This is the mantissa of 278. Proceed further along the horizontalline and look vertically below the figure 6 in difference column.You will find the figure 9. Therefore, the mantissa of 2786 is 4440+ 9 = 4449.

Hence, the logarithm of 278.6 is 2.4449 ( or log 278.6 = 2.4449).

Example 2 : Find the logarithm of 278600.

Answer : The characteristic of this number is 5 and the mantissa isthe same as in Example 1, above. We can find the mantissa of onlyfour significant figures. Hence, we neglect the last 2 zero.

∴ log 278600 = 5.4449

Example 3 : Find the logarithm of 0.00278633.

Answer : The characteristic of this number is 3 , as there are twozeros following the decimal point. We can find the mantissa of onlyfour significant figures. Hence, we neglect the last 2 figures (33) andfind the mantissa of 2786 which is 4449.

∴ log 0.00278633 = 3 .4449

When the last figure of a number consisting of more than 4 significantfigures is equal to or more than 5, the figure next to the left of it israised by one and so on till we have only four significant figures and ifthe last figure is less than 5, it is neglected as in the above example.

If we have the number 2786.58, the last figure is 8. Therefore, weshall raise the next in left figure to 6 and since 6 is greater than 5, weshall raise the next figure 6 to 7 and find the logarithmof 2787.

0 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

13


I 1.5.2 ANTILOGARITHMS

The number whose logarithm is x is called antilogarithm and is denotedby antilog x.

Thus, since log 2 = 0.3010, then antilog 0.3010 = 2.

Example 1 : Find the number whose logarithm is 1.8088.

Answer : For this purpose, we use antilogarithms table which is usedfor fractional part.

(i) In Example 1, fractional part is 0.8088. The first two figuresfrom the left are 0.80, the third figure is 8 and the fourth figureis also 8.

(ii) In the table of the antilogarithms, first look in the verticalcolumn for 0.80. In this horizontal row under the columnheaded by 8, we find the number 6427 at the intersection. Itmeans the number for mantissa 0.808 is 6427.

(iii) In continuation of this horizontal row and under the meandifference column on the right under 8, we find the number12 at the intersection. Adding 12 to 6427 we get 6439. Now6439 is the figure of which .8088 is the mantissa.

(iv) The characteristic is 1. This is one more than the number ofdigits in the integral part of the required number. Hence, thenumber of digits in the integral part of the requirednumber = 1 + 1 =2. The required number is 64.39 i.e., antilog1.8088 = 64.39.

Example 2 : Find the antilog of 2 .8088.

Answer : As the characteristic is 2 , there should be one zero on the

right of decimal in the number, hence antilog 2 .8088 = 0.06439.

Properties of logarithms:

(i) loga mn = log

am + log

an

(ii) loga m/n = logam – logan

(iii) loga mn = n log

am

The definition of logarithm:

loga 1 = 0 [since a0 = 1]

The log of 1 to any base is zero,

and loga a = 1 [since the logarithm of the base to itself is 1, a1 = a]

14


I 1.6 NATURAL SINE / COSINE TABLETo find the sine or cosine of some angles we need to refer to Tables oftrignometric functions. Natural sine and cosine tables are given in theDATA SECTION (Tables 3 and 4, Pages 270–273). Angles are givenusually in degrees and minutes, for example : 35°6′ or 35.1°.

I 1.6.1 READING OF NATURAL SINE TABLE

Suppose we wish to know the value of sin 35°10′. You may proceed as follows:

(i) Open the Table of natural sines.

(ii) Look in the first column and locate 35°. Scan horizontally,move from value 0.5736 rightward and stop under the columnwhere 6′ is marked. You will stop at 0.5750.

(iii) But it is required to find for 10′.

The difference between 10′ and 6′ is 4′. So we look into the column ofmean difference under 4′ and the corresponding value is 10. Add 10to the last digits of 0.5750 and we get 0.5760.

Thus, sin (35°10′) is 0.5760.

I 1.6.2 READING OF NATURAL COSINE TABLE

Natural cosine tables are read in the same manner. However, becauseof the fact that value of cos θ decrease as θ increases, the meandifference is to be subtracted. For example, cos 25° = 0.9063. To readthe value of cosine angle 25°40′, i.e., cos 25°40′, we read for cos 25°36′= 0.9018. Mean difference for 4′ is 5 which is to be subtracted fromthe last digits of 0.9018 to get 0.9013. Thus, cos 25°40′ = 0.9013.

I 1.6.3 READING OF NATURAL TANGENTS TABLENatural Tangents table are read the same way as the natural sinetable.

I 1.7 PLOTTING OF GRAPHSA graph pictorially represents the relation between two variablequantities. It also helps us to visualise experimental data at a glanceand shows the relation between the two quantities. If two physicalquantities a and b are such that a change made by us in a results ina change in b, then a is called independent variable and b is calleddependent variable. For example, when you change the length of thependulum, its time period changes. Here length is independent variablewhile time period is dependent variable.

15


A graph not only shows the relation between two variable quantitiesin pictorial form, it also enables verification of certain laws (such asBoyle’s law) to find the mean value from a large number ofobservations, to extrapolate/interpolate the value of certain quantitiesbeyond the limit of observation of the experiment, to calibrate orgraduate a given instrument for measurement and to find themaximum and minimum values of the dependent variable.

Graphs are usually plotted on a graph paper sheet ruled inmillimetre/centimetre squares. For plotting a graph, the following stepsare observed:

(i) Identify the independent variable and dependent variable.Represent the independent variable along the x-axis and thedependent variable along the y-axis.

(ii) Determine the range of each of the variables and count thenumber of big squares available to represent each, along therespective axis.

(iii) Choice of scale is critical for plotting of a graph. Ideally, thesmallest division on the graph paper should be equal to theleast count of measurement or the accuracy to which theparticular parameter is known. Many times, for clarity of thegraph, a suitable fraction of the least count is taken as equalto the smallest division on the graph paper.

(iv) Choice of origin is another point which has to be donejudiciously. Generally, taking (0,0) as the origin serves thepurpose. But such a choice is to be adopted generally whenthe relation between variables begins from zero or it is desiredto find the zero position of one of the variables, if its actualdetermination is not possible. However, in all other cases theorigin need not correspond to zero value of the variable. It is,however, convenient to represent a round number nearest tobut less than the smallest value of the corresponding variable.On each axis mark only the values of the variable in roundnumbers.

(v) The scale markings on x-and y-axis should not be crowded.Write the numbers at every fifth cm of the axis. Write also theunits of the quantity plotted. Use scientific representations ofthe numbers, i.e., write the number with decimal point followingthe first digit and multiply the number by appropriate powerof ten. The scale conversion may also be written at the right orleft corner at the top of the graph paper.

(vi) Write a suitable caption below the plotted graph mentioningthe names or symbols of the physical quantities involved.Also indicate the scales taken along both the axes on thegraph paper.

16


(vii) When the graph is expected to be a straight line, generally 6 to7 readings are enough. Time should not be wasted in taking avery large number of observations. The observations must becovering all available range evenly.

(viii) If the graph is a curve, first explore the range by covering theentire range of the independent variable in 6 to 7 steps. Thentry to guess where there will be sharp changes in the curvatureof the curve. Take more readings in those regions. For example,when there is either a maximum or minimum, more readingsare needed to locate the exact point of extremum, as in thedetermination of angle of minimum deviation (δm) you mayneed to take more observations near about δm.

(ix) Representation of “data” points also has a meaning. The sizeof the spread of plotted point must be in accordance with theaccuracy of the data. Let us take an example in which theplotted point is represented as , a point with a circle aroundit. The central dot is the value of measured data. The radius ofcircle of ‘x’ or ‘y’ side gives the size of uncertainty. If the circleradius is large, it will mean as if uncertainty in data is more.Further such a representation tells that accuracy along x- andy-axis are the same. Some other representations used whichgive the same meaning as above are , , , , ×, etc.

In case, uncertainty along the x-axis and y-axis are different,some of the notations used are (accuracy along x-axis ismore than that on y-axis); (accuracy along x-axis is less

than that on y-axis). , , , , are some of such other

symbols. You can design many more on your own.

(x) After all the data points are plotted, it is customary to fit asmooth curve judiciously by hand so that the maximumnumber of points lie on or near it and the rest are evenlydistributed on either side of it. Now a days computers are alsoused for plotting graphs of a given data.

I 1.7.1 SLOPE OF A STRAIGHT LINE

The slope m of a straight line graph AB is defined asy

mx

Δ=Δ

where Δy is the change in the value of the quantity plotted on they-axis, corresponding to the change Δx in the value of the quantityplotted on the x-axis. It may be noted that the sign of m will be positivewhen both Δx and Δy are of the same sign, as shown in Fig. I 1.2. Onthe other hand, if Δy is of opposite sign (i.e., y decreases when xincreases) than that of Δx, the value of the slope will be negative. Thisis indicated in Fig. I 1.3.

17


Further, the slope of a given straight line has the same value, for allpoints on the line. It is because the value of y changes by the sameamount for a given change in the value of x, at every point of thestraight line, as shown in Fig. I 1.4. Thus, for a given straight line, theslope is fixed.

Fig. 1.2 Value of slope is positive Fig. 1.3 Value of slope is negative

Fig. 1.4: Slope is fixed for a given straight line

While calculating the slope, always choose the x-segment of sufficientlength and see that it represents a round number of the variable. Thecorresponding interval of the variable on y-segment is then measuredand the slope is calculated. Generally, the slope should not have morethan two significant digits. The values of the slope and the intercepts,if there are any, should be written on the graph paper.

18


Do not show slope as tanθ. Only when scales along both the axes areidentical slope is equal to tanθ. Also keep in mind that slope of a graphhas physical significance, not geometrical.

Often straight-line graphs expected to pass through the origin arefound to give some intercepts. Hence, whenever a linear relationshipis expected, the slope should be used in the formula instead of themean of the ratios of the two quantities.

I 1.7.2 SLOPE OF A CURVE AT A GIVEN POINT ON ITAs has been indicated, the slope of a straight line has the same valueat each point. However, it is not true for a curve. As shown in Fig. I 1.5, theslope of the curve CD may have different values of slope at points A′,A, A′′, etc.

Fig. I 1.5: Tangent at a point A

Therefore, in case of a non-straight line curve, we talk of the slope at

a particular point. The slope of the curve at a particular point, saypoint A in Fig. I 1.5, is the value of the slope of the line EF which is the

tangent to the curve at point A. As such, in order to find the slope of acurve at a given point, one must draw a tangent to the curve at the

desired point.

In order to draw the tangent to a given curve at a given point, one may

use a plane mirror strip attached to a wooden block, so that it standsperpendicular to the paper on which the curve is to be drawn. This is

illustrated in Fig. I 1.6 (a) and Fig. I 1.6 (b). The plane mirror stripMM′ is placed at the desired point A such that the image D′ A of the

part DA of the curve appears in the mirror strip as continuation of

19


DA. In general, the image D′ A will not appear to be smoothlyjoined with the part of the curve DA as shown in Fig. I 1.6 (a).

Next rotate the mirror strip MM′, keeping its position at point A fixed.The image D′Α in the mirror will also rotate. Now adjust the position

of MM′ such that DAD′ appears as a continuous, smooth curve asshown in Fig. I 1.6 (b). Draw the line MAM′ along the edge of the mirror

for this setting. Next using a protractor, draw a perpendicular GH tothe line MAM′ at point A.

GAH is the line, which is the required tangent to the curve DAC at

point A. The slope of the tangent GAH (i.e., Δy /Δx) is the slope of thecurve CAD at point A. The above procedure may be followed for finding

the slope of any curve at any given point.

I 1.8 GENERAL INSTRUCTIONS FOR PERFORMING EXPERIMENTS

1. The students should thoroughly understand the principle of theexperiment. The objective of the experiment and procedure to befollowed should be clear before actually performing the experiment.

2. The apparatus should be arranged in proper order. To avoidany damage, all apparatus should be handled carefully andcautiously. Any accidental damage or breakage of theapparatus should be immediately brought to the notice of theconcerned teacher.

(a), (b)

Fig. 1.6 (a), (b): Drawing tangent at point A using a plane mirror

20


3. Precautions meant for each experiment should be observed strictlywhile performing it.

4. Repeat every observation, a number of times, even if measuredvalue is found to be the same. The student must bear in mind theproper plan for recording the observations. Recording in tabularform is essential in most of the experiments.

5. Calculations should be neatly shown (using log tables whereverdesired). The degree of accuracy of the measurement of eachquantity should always be kept in mind, so that final result doesnot reflect any fictitious accuracy. The result obtained should besuitably rounded off.

6. Wherever possible, the observations should be represented withthe help of a graph.

7. Always mention the result in proper SI unit, if any, along withexperimental error.

I1.9 GENERAL INSTRUCTIONS FOR RECORDING EXPERIMENTS

A neat and systematic recording of the experiment in the practical fileis very important for proper communication of the outcome of theexperimental investigations. The following heads may usually befollowed for preparing the report:

DATE:-------- EXPERIMENT NO:---------- PAGE NO.-------

AIMState clearly and precisely the objective(s) of the experiment to beperformed.

APPARATUS AND MATERIAL REQUIREDMention the apparatus and material used for performing the experiment.

DESCRIPTION OF APPARATUS INCLUDING MEASURING DEVICES (OPTIONAL)

Describe the apparatus and various measuring devices used inthe experiment.

TERMS AND DEFINITIONS OR CONCEPTS (OPTIONAL)

Various important terms and definitions or concepts used in theexperiment are stated clearly.

21


PRINCIPLE / THEORYMention the principle underlying the experiment. Also, write theformula used, explaining clearly the symbols involved (derivation notrequired). Draw a circuit diagram neatly for experiments/activitiesrelated to electricity and ray diagrams for light.

PROCEDURE (WITH IN-BUILT PRECAUTIONS)Mention various steps followed with in-built precautions actuallyobserved in setting the apparatus and taking measurements in asequential manner.

OBSERVATIONS

Record the observations in tabular form as far as possible, neatlyand without any overwriting. Mention clearly, on the top of theobservation table, the least counts and the range of each measuringinstrument used.

However, if the result of the experiment depends upon certainconditions like temperature, pressure etc., then mention the valuesof these factors.

CALCULATIONS AND PLOTTING GRAPHSubstitute the observed values of various quantities in the formulaand do the computations systematically and neatly with the help oflogarithm tables. Calculate experimental error.

Wherever possible, use the graphical method for obtainingthe result.

RESULTState the conclusions drawn from the experimental observations.[Express the result of the physical quality in proper significant figuresof numerical value along with appropriate SI units and probable error].Also mention the physical conditions like temperature, pressure etc.,if the result happens to depend upon them.

PRECAUTIONSMention the precautions actually observed during the course of theexperiment/activity.

22


SOURCES OF ERROR

Mention the possible sources of error that are beyond the control ofthe individual while performing the experiment and are liable to affectthe result.

DISCUSSION

The special reasons for the set up etc., of the experiment are to bementioned under this heading. Also mention any special inferenceswhich you can draw from your observations or special difficulties facedduring the experimentation. These may also include points for makingthe experiment more accurate for observing precautions and, ingeneral, for critically relating theory to the experiment for betterunderstanding of the basic principle involved.

EXPERIMENTEXPERIMENTEXPERIMENTEXPERIMENTEXPERIMENT

AIM

Use of Vernier Callipers to

(i) measure diameter of a small spherical/cylindrical body,

(ii) measure the dimensions of a given regular body of known massand hence to determine its density; and

(iii) measure the internal diameter and depth of a given cylindrical objectlike beaker/glass/calorimeter and hence to calculate its volume.

APPARATUS AND MATERIAL REQUIREDVernier Callipers, Spherical body, such as a pendulum bob or a glassmarble, rectangular block of known mass and cylindrical object likea beaker/glass/calorimeter

DESCRIPTION OF THE MEASURING DEVICE

1. A Vernier Calliper has two scales–one main scale and a Vernierscale, which slides along the main scale. The main scale and Vernierscale are divided into small divisions though of differentmagnitudes.

The main scale is graduated in cm and mm. It has two fixed jaws, Aand C, projected at right angles to the scale. The sliding Vernier scalehas jaws (B, D) projecting at right angles to it and also the main scaleand a metallic strip (N). The zero ofmain scale and Vernier scale coincidewhen the jaws are made to touch eachother. The jaws and metallic strip aredesigned to measure the distance/diameter of objects. Knob P is used toslide the vernier scale on the mainscale. Screw S is used to fix the vernierscale at a desired position.

2. The least count of a common scaleis 1mm. It is difficult to furthersubdivide it to improve the leastcount of the scale. A vernier scaleenables this to be achieved.

11111

Fig. E 1.1 Vernier Calliper

EXPERIMENTS

LABORATORY MANUAL

24

LABORATORY MANUAL

PRINCIPLE

The difference in the magnitude of one main scale division (M.S.D.)and one vernier scale division (V.S.D.) is called the least count of theinstrument, as it is the smallest distance that can be measured usingthe instrument.

n V.S.D. = (n – 1) M.S.D.

Formulas Used

(a) Least count of vernier callipers

the magnitude of the smallest division on the main scale

the total number of small divisions on the vernier scale=

(b) Density of a rectangular body = mass m m

volume V l.b.h= = where m is

its mass, l its length, b its breadth and h the height.

(c) The volume of a cylindrical (hollow) object V = πr2h' = π ′2D

. h'4

where h' is its internal depth, D' is its internal diameter and r isits internal radius.

PROCEDURE(a) Measuring the diameter of a small spherical or cylindrical

body.

1. Keep the jaws of Vernier Callipers closed. Observe the zero mark ofthe main scale. It must perfectly coincide with that of the vernierscale. If this is not so, account for the zero error for all observations tobe made while using the instrument as explained on pages 26-27.

2. Look for the division on the vernier scale that coincides with adivision of main scale. Use a magnifying glass, if available andnote the number of division on the Vernier scale that coincideswith the one on the main scale. Position your eye directly over thedivision mark so as to avoid any parallax error.

3. Gently loosen the screw to release the movable jaw. Slide it enoughto hold the sphere/cylindrical body gently (without any unduepressure) in between the lower jaws AB. The jaws should be perfectlyperpendicular to the diameter of the body. Now, gently tighten thescrew so as to clamp the instrument in this position to the body.

4. Carefully note the position of the zero mark of the vernier scaleagainst the main scale. Usually, it will not perfectly coincide with

UNIT NAME

25

EXPERIMENT

any of the small divisions on the main scale. Record the main scaledivision just to the left of the zero mark of the vernier scale.

5. Start looking for exact coincidence of a vernier scale division withthat of a main scale division in the vernier window from left end(zero) to the right. Note its number (say) N, carefully.

6. Multiply 'N' by least count of the instrument and add the productto the main scale reading noted in step 4. Ensure that the productis converted into proper units (usually cm) for addition to be valid.

7. Repeat steps 3-6 to obtain the diameter of the body at differentpositions on its curved surface. Take three sets of reading ineach case.

8. Record the observations in the tabular form [Table E 1.1(a)] withproper units. Apply zero correction, if need be.

9. Find the arithmetic mean of the corrected readings of the diameterof the body. Express the results in suitable units with appropriatenumber of significant figures.

(b) Measuring the dimensions of a regular rectangular body todetermine its density.

1. Measure the length of the rectangular block (if beyond the limitsof the extended jaws of Vernier Callipers) using a suitable ruler.Otherwise repeat steps 3-6 described in (a) after holding the blocklengthwise between the jaws of the Vernier Callipers.

2. Repeat steps 3-6 stated in (a) to determine the other dimensions(breadth b and height h) by holding the rectangular block in properpositions.

3. Record the observations for length, breadth and height of therectangular block in tabular form [Table E 1.1 (b)] with properunits and significant figures. Apply zero corrections wherevernecessary.

4. Find out the arithmetic mean of readings taken for length, breadthand height separately.

[c] Measuring the internal diameter and depth of the given beaker(or similar cylindrical object) to find its internal volume.

1. Adjust the upper jaws CD of the Vernier Callipers so as to touchthe wall of the beaker from inside without exerting undue pressureon it. Tighten the screw gently to keep the Vernier Callipers in thisposition.

2. Repeat the steps 3-6 as in (a) to obtain the value of internal diameterof the beaker/calorimeter. Do this for two different (angular)positions of the beaker.

1

LABORATORY MANUAL

26

LABORATORY MANUAL

3. Keep the edge of the main scale of Vernier Callipers, to determinethe depth of the beaker, on its peripheral edge. This should bedone in such a way that the tip of the strip is able to go freelyinside the beaker along its depth.

4. Keep sliding the moving jaw of the Vernier Callipers until the stripjust touches the bottom of the beaker. Take care that it does sowhile being perfectly perpendicular to the bottom surface. Nowtighten the screw of the Vernier Callipers.

5. Repeat steps 4 to 6 of part (a) of the experiment to obtain depth ofthe given beaker. Take the readings for depth at different positionsof the breaker.

6. Record the observations in tabular form [Table E 1.1 (c)] withproper units and significant figures. Apply zero corrections, ifrequired.

7. Find out the mean of the corrected readings of the internal diameterand depth of the given beaker. Express the result in suitable unitsand proper significant figures.

OBSERVATIONS

(i) Least count of Vernier Callipers (Vernier Constant)

1 main scale division (MSD) = 1 mm = 0.1 cm

Number of vernier scale divisions, N = 10

10 vernier scale divisions = 9 main scale divisions

1 vernier scale division = 0.9 main scale division

Vernier constant = 1 main scale division – 1 vernier scale division

= (1– 0.9) main scale divisions

= 0.1 main scale division

Vernier constant (VC) = 0.1 mm = 0.01 cm

Alternatively,

1MSDVernier constant =

N1 mm

=10

Vernier constant (VC) = 0.1 mm = 0.01 cm

(ii) Zero error and its correction

When the jaws A and B touch each other, the zero of the Verniershould coincide with the zero of the main scale. If it is not so, theinstrument is said to possess zero error (e). Zero error may be

UNIT NAME

27

EXPERIMENT

positive or negative, depending upon whether the zero of vernierscale lies to the right or to the left of the zero of the main scale. Thisis shown by the Fig. E1.2 (ii) and (iii). In this situation, a correctionis required to the observed readings.

(iii) Positive zero error

Fig E 1.2 (ii) shows an example of positive zero error. From thefigure, one can see that when both jaws are touching each other,zero of the vernier scale is shifted to the right of zero of the mainscale (This might have happened due to manufacturing defect ordue to rough handling). This situation makes it obvious that whiletaking measurements, the reading taken will be more than theactual reading. Hence, a correction needs to be applied which isproportional to the right shift of zero of vernier scale.

In ideal case, zero of vernier scale should coincide with zero ofmain scale. But in Fig. E 1.2 (ii), 5th vernier division is coincidingwith a main scale reading.

∴ Zero Error = + 5 × Least Count = + 0.05 cm

Hence, the zero error is positive in this case. For any measurementsdone, the zero error (+ 0.05 cm in this example) should be‘subtracted’ from the observed reading.

∴ True Reading = Observed reading – (+ Zero error)

(iv) Negative zero error

Fig. E 1.2 (iii) shows an example of negative zero error. From thisfigure, one can see that when both the jaws are touching eachother, zero of the vernier scale is shifted to the left of zero of themain scale. This situation makes it obvious that while takingmeasurements, the reading taken will be less than the actualreading. Hence, a correction needs to be applied which isproportional to the left shift of zero of vernier scale.

In Fig. E 1.2 (iii), 5th vernier scale division is coinciding with amain scale reading.

∴ Zero Error = – 5 × Least Count

= – 0.05 cm

1

Fig. E 1.2: Zero error (i) no zero error (ii) positive zero error(iii) negative zero error

LABORATORY MANUAL

28

LABORATORY MANUAL

Note that the zero error in this case is considered to be negative.For any measurements done, the negative zero error, ( –0.05 cm inthis example) is also substracted ‘from the observed reading’,though it gets added to the observed value.

∴ True Reading = Observed Reading – (– Zero error)

Table E 1.1 (a): Measuring the diameter of a small spherical/cylindrical body

Zero error, e = ± ... cm

Mean observed diameter = ... cm

Corrected diameter = Mean observed diameter – Zero Error

Table E 1.1 (b) : Measuring dimensions of a given regular body(rectangular block)

S.No.

Main Scalereading, M(cm/mm)

Number ofcoinciding

vernierdivision, N

Vernier scale reading,V = N × VC (cm/mm)

Measureddimension

M + V (cm/mm)

123

Dimension

Length (l)

Breadth (b)

Height (h)

123

123

S.No.


Number ofcoinciding

vernierdivision, N

Vernier scalereading, V = N × VC

(cm/mm)

Measured

diameter, M + V(cm/mm)

1

2

3

4

Zero error = ± ... mm/cm

Mean observed length = ... cm, Mean observed breadth = ... cm

Mean observed height = ... cm

Corrected length = ... cm; Corrected breath = ... cm;

Corrected height = ...cm

UNIT NAME

29

EXPERIMENT

Table E 1.1 (c) : Measuring internal diameter and depth of agiven beaker/ calorimeter/ cylindrical glass

Mean diameter = ... cm

Mean depth= ... cm

Corrected diameter = ... cm

Corrected depth = ... cm

CALCULATION(a) Measurement of diameter of the sphere/ cylindrical body

Mean measured diameter, 1 2 6o

D D ... DD

6

+ + += cm

Do = ... cm = ... × 10–2 m

Corrected diameter of the given body, D = Do – ( ± e ) = ... × 10–2 m

(b) Measurement of length, breadth and height of the rectangularblock

Mean measured length, 1 2 3o

l l ll

3

+ += cm

lo = ... cm = ... × 10–2 m

Corrected length of the block, l = lo – ( ± e ) = ... cm

Mean observed breadth, 1 2 3o

b b bb

3

+ +=

Mean measured breadth of the block, b0 = ... cm = ... × 10–2 m

Corrected breadth of the block,

b = bO – ( ± e ) cm = ... × 10–2 m

S.No.


Number ofcoinciding

vernierdivision, N

Vernier scale reading,V = N × VC (cm/mm)

Measureddiameter depth,M + V (cm/mm)

123

Dimension

Internaldiameter(D′)

Depth (h′)123

1

LABORATORY MANUAL

30

LABORATORY MANUAL

Mean measured height of block 1 2 3o

h h hh

+ +=

3

Corrected height of block h = ho – ( ± e ) = ... cm

Volume of the rectangular block,

V = lbh = ... cm3 = ... × 10–6 m3

Density ρ of the block,

m= =...

Vρ –3kgm

(c) Measurement of internal diameter of the beaker/glass

Mean measured internal diameter, 1 2 3o

D D DD

+ +=

3

Do = ... cm = ... × 10–2 m

Corrected internal diameter,

D = Do – ( ± e ) = ... cm = ... × 10–2 m

Mean measured depth of the beaker, 1 2 3o

h h hh

+ +=

3

= ... cm = ... × 10–2 m

Corrected measured depth of the beaker

h = ho – ( ± e ) ... cm = ... × 10–2 m

Internal volume of the beaker

πV

2–6 3D h

= =...×10 m4

RESULT(a) Diameter of the spherical/ cylindrical body,

D = ... × 10–2m

(b) Density of the given rectangular block,

ρ = ... kgm–3

(c) Internal volume of the given beaker

V'= ... m3

UNIT NAME

31

EXPERIMENT

PRECAUTIONS1. If the vernier scale is not sliding smoothly over the main scale,

apply machine oil/grease.

2. Screw the vernier tightly without exerting undue pressure to avoidany damage to the threads of the screw.

3. Keep the eye directly over the division mark to avoid any errordue to parallax.

4. Note down each observation with correct significant figuresand units.

SOURCES OF ERROR

Any measurement made using Vernier Callipers is likely to beincorrect if-

(i) the zero error in the instrument placed is not accounted for; and

(ii) the Vernier Callipers is not in a proper position with respect to thebody, avoiding gaps or undue pressure or both.

DISCUSSION

1. A Vernier Callipers is necessary and suitable only for certaintypes of measurement where the required dimension of the objectis freely accessible. It cannot be used in many situations. e.g.suppose a hole of diameter 'd' is to be drilled into a metal block.If the diameter d is small - say 2 mm, neither the diameter northe depth of the hole can be measured with a Vernier Callipers.

2. It is also important to realise that use of Vernier Callipers formeasuring length/width/thickness etc. is essential only whenthe desired degree of precision in the result (say determinationof the volume of a wire) is high. It is meaningless to use it whereprecision in measurement is not going to affect the result much.For example, in a simple pendulum experiment, to measurethe diameter of the bob, since L >> d.

SELF ASSESSMENT

1. One can undertake an exercise to know the level of skills developedin making measurements using Vernier Callipers. Objects, suchas bangles/kangan, marbles whose dimensions can be measuredindirectly using a thread can be used to judge the skill acquiredthrough comparison of results obtained using both the methods.

2. How does a vernier decrease the least count of a scale.

1

SUGGESTED ADDITIONAL EXPERIMENTS/ACTIVITIES

1. Determine the density of glass/metal of a (given) cylindrical vessel.

2. Measure thickness of doors and boards.

3. Measure outer diameter of a water pipe.

ADDITIONAL EXERCISE

1. In the vernier scale normally used in a Fortin's barometer, 20 VSDcoincide with 19 MSD (each division of length 1 mm). Find the leastcount of the vernier.

2. In vernier scale (angular) normally provided in spectrometers/sextant,60 VSD coincide with 59 MSD (each division of angle 1°). Find the leastcount of the vernier.

3. How would the precision of the measurement by Vernier Callipers beaffected by increasing the number of divisions on its vernier scale?

4. How can you find the value of π using a given cylinder and a pair ofVernier Callipers?

[Hint : Using the Vernier Callipers, - Measure the diameter D and findthe circumference of the cylinder using a thread. Ratio of circumferenceto the diameter (D) gives π.]

5. How can you find the thickness of the sheet used for making of a steeltumbler using Vernier Callipers?

[Hint: Measure the internal diameter (Di) and external diameter (D

o) of

the tumbler. Then, thickness of the sheet Dt = (D

o – D

i)/2.]


32

AIM

Use of screw gauge to

(a) measure diameter of a given wire,

(b) measure thickness of a given sheet; and

(c) determine volume of an irregular lamina.

APPARATUS AND MATERIAL REQUIRED

Wire, metallic sheet, irregular lamina, millimetre graph paper, penciland screw gauge.

DESCRIPTION OF APPARATUS

With Vernier Callipers. you are usually able to measure lengthaccurately up to 0.1 mm. More accurate measurement of length, upto 0.01 mm or 0.005 mm, may be made by using a screw gauge. Assuch a Screw Gauge is aninstrument of higher precision thana Vernier Callipers. You might haveobserved an ordinary screw [Fig E2.1(a)]. There are threads on a screw. Theseparation between any twoconsecutive threads is the same. Thescrew can be moved backward orforward in its nut by rotating it anti-clockwise or clockwise [Fig E2.1(b)].

The distance advanced by the screwwhen it makes its one completerotation is the separation betweentwo consecutive threads. Thisdistance is called the Pitch of thescrew. Fig. E 2.1(a) shows the pitch(p) of the screw. It is usually 1 mmor 0.5 mm. Fig. E 2.2 shows ascrew gauge. It has a screw ’S’which advances forward orbackward as one rotates the headC through rachet R. There is a linear

EXPERIMENTEXPERIMENTEXPERIMENTEXPERIMENTEXPERIMENT22222

Fig.E 2.1A screw (a) without nut (b) with nut

Fig.E 2.2: View of a screw gauge

LABORATORY MANUAL

34

LABORATORY MANUAL

scale ‘LS’ attached to limb D of the U frame. The smallest division onthe linear scale is 1 mm (in one type of screw gauge). There is a circularscale CS on the head, which can be rotated. There are 100 divisionson the circular scale. When the end B of the screw touches the surfaceA of the stud ST, the zero marks on the main scale and the circularscale should coincide with each other.

ZERO ERROR

When the end of the screw and the surface of thestud are in contact with each other, the linear scaleand the circular scale reading should be zero. Incase this is not so, the screw gauge is said to havean error called zero error.

Fig. E 2.3 shows an enlarged view of a screw gaugewith its faces A and B in contact. Here, the zeromark of the LS and the CS are coinciding with eachother.

When the reading on the circular scale across thelinear scale is more than zero (or positive), theinstrument has Positive zero error as shown inFig. E 2.4 (a). When the reading of the circular scaleacross the linear scale is less than zero (or negative),the instrument is said to have negative zero erroras shown in Fig. E 2.4 (b).

Fig.E 2.4 (a): Showing a positive zero error

Fig.E 2.4 (b): Showing a negative zero error

TAKING THE LINEAR SCALE READING

The mark on the linear scale which lies close to theleft edge of the circular scale is the linear scalereading. For example, the linear scale reading asshown in Fig. E 2.5, is 0.5 cm.

TAKING CIRCULAR SCALE READING

The division of circular scale which coincides withthe main scale line is the reading of circular scale.For example, in the Fig. E 2.5, the circular scalereading is 2.

Fig.E 2.5: Measuring thickness with a screwguage

Fig.E 2.3: A screw gauge with no zero error

UNIT NAME

35

EXPERIMENT

TOTAL READING

Total reading

= linear scale reading + circular scale reading × least count

= 0.5 + 2 × 0.001

= 0.502 cm

PRINCIPLE

The linear distance moved by the screw is directly proportional to therotation given to it. The linear distance moved by the screw when it isrotated by one division of the circular scale, is the least distance thatcan be measured accurately by the instrument. It is called the leastcount of the instrument.

pitchLeast count =

No. of divisions on circular scale

For example for a screw gauge with a pitch of 1mm and 100 divisionson the circular scale. The least count is

1 mm/100 = 0.01 mm

This is the smallest length one can measure with this screw gauge.

In another type of screw gauge, pitch is 0.5 mm and there are 50divisions on the circular scale. The least count of this screw gaugeis 0.5 mm/50 = 0.01 mm. Note that here two rotations of thecircular scale make the screw to advance through a distance of 1mm. Some screw gauge have a least count of 0.001 mm (i.e. 10–6

m) and therefore are called micrometer screw.

(a) Measurement of Diameter of a Given Wire

PROCEDURE

1. Take the screw gauge and make sure that the rachet R on thehead of the screw functions properly.

2. Rotate the screw through, say, ten complete rotations and observethe distance through which it has receded. This distance is thereading on the linear scale marked by the edge of the circularscale. Then, find the pitch of the screw, i.e., the distance moved bythe screw in one complete rotation. If there are n divisions on thecircular scale, then distance moved by the screw when it is rotatedthrough one division on the circular scale is called the least countof the screw gauge, that is,

pitchLeast count =

n

2

LABORATORY MANUAL

36

LABORATORY MANUAL

3. Insert the given wire between the screw and the stud of the screwgauge. Move the screw forward by rotating the rachet till the wireis gently gripped between the screw and the stud as shown inFig. E 2.5. Stop rotating the rachet the moment you hear a clicksound.

4. Take the readings on the linear scale and the circular scale.

5. From these two readings, obtain the diameter of the wire.

6. The wire may not have an exactlycircular cross-section. Therefore. it isnecessary to measure the diameter of thewire for two positions at right angles toeach other. For this, first record thereading of diameter d1 [Fig. E 2.6 (a)]and then rotate the wire through 90° atthe same cross-sectional position.Record the reading for diameter d

2 in this

position [Fig. E 2.6 (b)].

7. The wire may not be truly cylindrical.Therefore, it is necessary to measure thediameter at several different places andobtain the average value of diameter. Forthis, repeat the steps (3) to (6) for threemore positions of the wire.

8. Take the mean of the different values of diameter so obtained.

9. Substract zero error, if any, with proper sign to get the correctedvalue for the diameter of the wire.

OBSERVATIONS AND CALCULATIONThe length of the smallest division on the linear scale = ... mm

Distance moved by the screw when it is rotatedthrough x complete rotations, y = ... mm

Pitch of the screw = y

x= ... mm

Number of divisions on the circular scale n = ...

Least Count (L.C.) of screw guage

=pitch

No. of divisions on the circular scale= ... mm

Zero error with sign (No. of div. × L. C.) = ... mm

Fig.E 2.6 (a): Two magnified views (a) and (b) of a wireshowing its perpendicular diameters d1

and d2. d2 is obtained after the rotatingthe wire in the clockwise directionthrough 90°.

UNIT NAME

37

EXPERIMENT

Table E 2.1: Measurement of the diameter of the wire

Mean diameter = ... mm

Mean corrected value of diameter

= measured diameter – (zero error with sign) = ... mm

RESULT

The diameter of the given wire as measured by screw gauge is ... m.

PRECAUTIONS1. Rachet arrangement in screw gauge must be utilised to avoid undue

pressure on the wire as this may change the diameter.

2. Move the screw in one direction else the screw may develop “play”.

3. Screw should move freely without friction.

4. Reading should be taken atleast at four different points along thelength of the wire.

5. View all the reading keeping the eye perpendicular to the scale toavoid error due to parallax.

SOURCES OF ERROR

1. The wire may not be of uniform cross-section.

2. Error due to backlash though can be minimised but cannot becompletely eliminated.

S.No.

LinearscalereadingM (mm)

Reading alongone direction

(d1)

Readingalong

perpendiculardirection (d2)

Measured

diameter

1 2d dd

+=

2

Circularscalereading(n)

Diameter

d1 = M + n × L.C.

(mm)

LinearscalereadingM (mm)

Circularscalereading(n)

Diameter

d2 = M + n × L.C.

(mm)

1

2

3

4

2

LABORATORY MANUAL

38

LABORATORY MANUAL


1. Think of a method to find the ‘pitch’ of bottle caps.

2. Compare the ‘pitch’ of an ordinary screw with that of a screw guage.In what ways are the two different?

3. Measure the diameters of petioles (stem which holds the leaf) ofdifferent leaf and check if it has any relation with the mass or surfacearea of the leaf. Let the petiole dry before measuring its diameter byscrew gauge.

BACKLASH ERROR

In a good instrument (either screw gauge or a spherometer) thethread on the screw and that on the nut (in which the screw moves),should tightly fit with each other. However, with repeated use,the threads of both the screw and the nut may get worn out. Asa result a gap develops between these two threads, which is called“play”. The play in the threads may introduce an error inmeasurement in devices like screw gauge. This error is calledbacklash error. In instruments having backlash error, the screwslips a small linear distance without rotation. To prevent this, itis advised that the screw should be moved in only one directionwhile taking measurements.

3. The divisions on the linear scale and the circular scale may not beevenly spaced.

DISCUSSION1. Try to assess if the value of diameter obtained by you is realistic

or not. There may be an error by a factor of 10 or 100 . You canobtain a very rough estimation of the diameter of the wire bymeasuring its thickness with an ordinary metre scale.

2. Why does a screw gauge develop backlash error with use?

SELF ASSESSMENT1. Is the screw gauge with smaller least count always better? If you

are given two screw gauges, one with 100 divisions on circularscale and another with 200 divisions, which one would you preferand why?

2. Is there a situation in which the linear distance moved by the screwis not proportional to the rotation given to it?

3. Is it possible that the zero of circular scale lies above the zero lineof main scale, yet the error is positive zero error?

4. For measurement of small lengths, why do we prefer screw gaugeover Vernier Callipers?

UNIT NAME

39

EXPERIMENT

(b) Measurement of Thickness of a Given Sheet

PROCEDURE1. Insert the given sheet between the studs of the screw gauge and

determine the thickness at five different positions.2. Find the average thickness and calculate the correct thickness by

applying zero error following the steps followed earlier.

OBSERVATIONS AND CALCULATIONLeast count of screw gauge = ... mm

Zero error of screw gauge = ... mm

Table E 2.2 Measurement of thickness of sheet

Mean thickness of the given sheet = ... mm

Mean corrected thickness of the given sheet

= observed mean thickness – (zero error with sign) = ... mm

RESULT

The thickness of the given sheet is ... m.

4. Measure the thickness of the sheet of stainless steel glasses ofvarious make and relate it to their price structure.

5. Measure the pitch of the ‘screw’ end of different types of hooks andcheck if it has any relation with the weight each one of these hooksare expected to hold.

6. Measure the thickness of different glass bangles available in theMarket. Are they made as per some standard?

7. Collect from the market, wires of different gauge numbers, measuretheir diameters and relate the two. Find out various uses of wires ofeach gauge number.

S.No.

Linear scalereading M

(mm)

Circularscale reading

n

Thickness

t = M + n × L.C.(mm)

1

2

3

4

5

2

LABORATORY MANUAL

40

LABORATORY MANUAL

SOURCES OF ERROR1. The sheet may not be of uniform thickness.

2. Error due to backlash though can be minimised but cannot beeliminated completely.

DISCUSSION

1. Assess whether the thickness of sheet measured by you is realisticor not. You may take a pile of say 20 sheets, and find its thicknessusing a metre scale and then calculate the thickness of one sheet.

2. What are the limitations of the screw gauge if it is used to measurethe thickness of a thick cardboard sheet?


1. Find out the thickness of different wood ply boards available in themarket and verify them with the specifications provided by thesupplier.

2. Measure the thickness of the steel sheets used in steel almirahsmanufactured by different suppliers and compare their prices. Is itbetter to pay for a steel almirah by mass or by the guage of steelsheets used?

3. Design a cardboard box for packing 144 sheets of paper and giveits dimensions.

4. Hold 30 pages of your practical notebook between the screw andthe stud and measure its thickness to find the thickness of onesheet.

5. Find the thickness of plastic ruler/metal sheet of the geometry box.

(C) Determination of Volume of the Given Irregular Lamina

PROCEDURE1. Find the thickness of lamina as in Experiment E 2(b).

2. Place the irregular lamina on a sheet of paper with mm graph.Draw the outline of the lamina using a sharp pencil. Count thetotal number of squares and also more than half squares withinthe boundary of the lamina and determine the area of the lamina.

3. Obtain the volume of the lamina using the relationmean thickness × area of lamina.

OBSERVATIONS AND CALCULATION

Same as in Experiment E 2(b). The first section of the table is now forreadings of thickness at five different places along the edge of the

UNIT NAME

41

EXPERIMENT

lamina. Calculate the mean thickness and make correction for zeroerror, if any.

From the outline drawn on the graph paper:-

Total number of complete squares = ... mm2 = ... cm2

Volume of the lamina = ... mm3 = ... cm3

RESULT

Volume of the given lamina = ... cm3


1. Find the density of cardboard.

2. Find the volume of a leaf (neem, bryophyte).

3. Find the volume of a cylindrical pencil.

2

LABORATORY MANUAL

42

LABORATORY MANUAL


Fig. E 3.1: A spherometer

AIM

To determine the radius of curvature of a given spherical surface by aspherometer.


A spherometer, a spherical surface such as a watch glass or a convexmirror and a plane glass plate of about 6 cm × 6 cm size.


A spherometer consists of a metallic triangular frame F supported onthree legs of equal length A, B and C (Fig. E 3.1). The lower tips of the

legs form three corners of an equilateral triangle ABCand lie on the periphery of a base circle of known radius,r. The spherometer also consists of a central leg OS (anaccurately cut screw), which can be raised or loweredthrough a threaded hole V (nut) at the centre of the frameF. The lower tip of the central screw, when lowered tothe plane (formed by the tips of legs A, B and C) touchesthe centre of triangle ABC. The central screw also carriesa circular disc D at its top having a circular scale dividedinto 100 or 200 equal parts. A small vertical scale Pmarked in millimetres or half-millimetres, called mainscale is also fixed parallel to the central screw, at one endof the frame F. This scale P is kept very close to the rim ofdisc D but it does not touch the disc D. This scale readsthe vertical distance which the central leg moves throughthe hole V. This scale is also known as pitch scale.

TERMS AND DEFINITIONSPitch: It is the vertical distance moved by the central screw in onecomplete rotation of the circular disc scale.

Commonly used spherometers in school laboratories havegraduations in millimetres on pitch scale and may have100 equaldivisions on circular disc scale. In one rotation of the circular scale,the central screw advances or recedes by 1 mm. Thus, the pitch ofthe screw is 1 mm.

UNIT NAME

43

EXPERIMENT

Least Count: Least count of a spherometer is the distance moved bythe spherometer screw when it is turned through one division on thecircular scale, i.e.,

Least count of the spherometer =Pitchof thespherometerscrew

Numberof divisions on the circular scale

The least count of commonly used spherometers is 0.01 mm.However, some spherometers have least count as small as0.005 mm or 0.001 mm.

PRINCIPLEFORMULA FOR THE RADIUS OF CURVATURE OF A SPHERICALSURFACE

Let the circle AOBXZY (Fig. E 3.2) represent the vertical section ofsphere of radius R with E as its centre (The given spherical surface isa part of this sphere). Length OZ is the diameter (= 2R ) of this verticalsection, which bisects the chord AB. Points A and B are the positionsof the two spherometer legs on the given spherical surface. The positionof the third spherometer leg is not shown in Fig. E 3.2. The point O isthe point of contact of the tip of central screw with the spherical surface.

Fig. E 3.3 showsthe base circle andequilateral triangleABC formed by thetips of the threespherometer legs.From this figure, itcan be noted that thepoint M is not onlythe mid point of lineAB but it is thecentre of base circleand centre of theequilateral triangleABC formed by thelower tips of the legs ofthe spherometer (Fig.E 3.1).

In Fig. E 3.2 thedistance OM is theheight of centralscrew above the planeof the circular sectionABC when its lower

Fig. E 3.2: Measurement of radiusof curvature of a spheri-cal surface

Fig. E 3.3: The base circle ofthe spherometer

3

LABORATORY MANUAL

44

LABORATORY MANUAL

tip just touches the spherical surface. This distance OM is also calledsagitta. Let this be h. It is known that if two chords of a circle, suchas AB and OZ, intersect at a point M then the areas of the rectanglesdescribed by the two parts of chords are equal. Then

AM.MB = OM.MZ

(AM)2 = OM (OZ – OM) as AM = MB

Let EZ (= OZ/2) = R, the radius of curvature of the given sphericalsurface and AM = r, the radius of base circle of the spherometer.

r2 = h (2R – h)

Thus, =2r h

R +h2 2

Now, let l be the distance between any two legs of the spherometer orthe side of the equilateral triangle ABC (Fig. E 3.3), then from geometrywe have

Thus, l

r =3

, the radius of curvature (R) of the given spherical surface

can be given by

l hR = +

h

2

6 2

PROCEDURE

1. Note the value of one division on pitch scale of the givenspherometer.

2. Note the number of divisions on circular scale.

3. Determine the pitch and least count (L.C.) of the spherometer. Placethe given flat glass plate on a horizontal plane and keep thespherometer on it so that its three legs rest on the plate.

4. Place the spherometer on a sheet of paper (or on a page in practicalnote book) and press it lightly and take the impressions of the tipsof its three legs. Join the three impressions to make an equilateraltriangle ABC and measure all the sides of ΔABC. Calculate themean distance between two spherometer legs, l.

In the determination of radius of curvature R of the given sphericalsurface, the term l 2 is used (see formula used). Therefore, greatcare must be taken in the measurement of length, l.

UNIT NAME

45

EXPERIMENT

5. Place the given spherical surface on the plane glass plate and thenplace the spherometer on it by raising or lowering the central screwsufficiently upwards or downwards so that the three spherometerlegs may rest on the spherical surface (Fig. E 3.4).

6. Rotate the central screw till it gently touches the spherical surface.To be sure that the screw touches the surface one can observe itsimage formed due toreflection from the surfacebeneath it.

7. Take the spherometerreading h

1 by taking the

reading of the pitch scale.Also read the divisions ofthe circular scale that is inline with the pitch scale.Record the readings inTable E 3.1.

8. Remove the sphericalsurface and place the spherometer on plane glass plate. Turn thecentral screw till its tip gently touches the glass plate. Take thespherometer reading h

2 and record it in Table E 3.1. The difference

between h1 and h

2 is equal to the value of sagitta (h).

9. Repeat steps (5) to (8) three more times by rotating the sphericalsurface leaving its centre undisturbed. Find the mean value of h.

OBSERVATIONS

A. Pitch of the screw:

(i) Value of smallest division on the vertical pitch scale = ... mm

(ii) Distance q moved by the screw for p complete rotations of thecircular disc = ... mm

(iii) Pitch of the screw ( = q / p ) = ... mm

B. Least Count (L.C.) of the spherometer:

(i) Total no. of divisions on the circular scale (N ) = ...

(ii) Least count (L.C.) of the spherometer

=Pitchof the spherometerscrew

Numberof divisionson the circularscale

L.C.= Pitchof the screw

N = ... cm

Fig.E 3.4: Measurement of sagitta h

3

LABORATORY MANUAL

46

LABORATORY MANUAL

C. Determination of length l (from equilateral triangle ABC)

(i) Distance AB = ... cm

(ii) Distance BC = ... cm

(iii) Distance CA = ... cm

Mean l = AB+ BC +CA

3= ... cm

Table E 3.1 Measurement of sagitta h

Mean h = ... cm

CALCULATIONS

A. Using the values of l and h, calculate the radius of curvature Rfrom the formula:

R =l h

+h

2

6 2;

the term h/2 may safely be dropped in case of surfaces of large radii

of curvature (In this situation error in ⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

l

h

2

6 is of the order of h/2).

RESULT

The radius of curvature R of the given spherical surface is ... cm.

S.No.

Spherometer readings

Spherometerreading withsphericalsurfaceh

1 = x + z

(cm)

PitchScalereadingx (cm)

(h1 – h2)

With Spherical Surface

Circularscaledivisioncoincidingwith pitchscale y

Circularscalereadingz =y × L.C.(cm)

Horizontal Plane Surface

Spherometerreading withsphericalsurfaceh

2=x′ + z′

(cm)

PitchScalereadingx1 (cm)

Circularscaledivisioncoincidingwith pitchscale y ′

Circularscalereading

z′=y × L.C.(cm)

UNIT NAME

47

EXPERIMENT

PRECAUTIONS

1. The screw may have friction.

2. Spherometer may have backlash error.

SOURCES OF ERROR1. Parallax error while reading the pitch scale corresponding to the

level of the circular scale.

2. Backlash error of the spherometer.

3. Non-uniformity of the divisions in the circular scale.

4. While setting the spherometer, screw may or may not be touchingthe horizontal plane surface or the spherical surface.

DISCUSSIONDoes a given object, say concave mirror or a convex mirror, have thesame radius of curvature for its two surfaces? [Hint: Does the thicknessof the material of object make any difference?]

3


1. Determine the focal length of a convex/concave spherical mirrorusing a spherometer.

2. (a) Using spherometer measure the thickness of a small piece ofthin sheet of metal/glass.

(b) Which instrument would be precise for measuring thickness ofa card sheet – a screw gauge or a spherometer?

LABORATORY MANUAL

48

LABORATORY MANUAL


Fig. E 4.1: A beam balance and set of weights

AIMTo determine mass of two different objects using a beam balance.


Physical balance, weight box with a set of milligram masses andforceps, spirit level and two objects whose masses are to be determined.

DESCRIPTION OF PHYSICAL BALANCE

A physical balance is a device that measures the weight (or gravitationalmass) of an object by comparing it with a standard weight (or standardgravitational mass).

The most commonly used two-pan beambalance is an application of a lever. It consistsof a rigid uniform bar (beam), two panssuspended from each end, and a pivotal pointin the centre of the bar (Fig. E 4.1). At this pivotalpoint, a support (called fulcrum ) is set at rightangles to the beam. This beam balance workson the principle of moments.

For high precision measurements, a physicalbalance (Fig. E 4.2) is often used in laboratories.Like a common beam balance, a physicalbalance too consists of a pair of scale pans P1

and P2, one at each end of a rigid beam B. The

pans P1 and P2 are suspended through stirrupsS1 and S2 respectively, on inverted knife-edgesE

1 and E

2, respectively, provided symmetrically

near the end of the beam B. The beam is alsoprovided with a hard material (like agate) knife-edge (E) fixed at the centre pointing downwards

and is supported on a vertical pillar (V) fixed on a wooden baseboard(W). The baseboard is provided with three levelling screws W

1, W

2 and

W3. In most balances, screws W1 and W2 are of adjustable heights andthrough these the baseboard W is levelled horizontally. The third screwW3, not visible in Fig. E 4.2, is not of adjustable height and is fixed inthe middle at the back of board W. When the balance is in use, the

UNIT NAME

49

EXPERIMENT

knife-edge E rests on aplane horizontal platefixed at the top of pillarV. Thus, the centraledge E acts as a pivotor fulcrum for the beamB. When the balance isnot in use, the beamrests on the supportsX

1 and X

2, These

supports, X1 and X2,are fixed to anotherhorizontal bar attachedwith the central pillar V.Also, the pans P

1 and

P2 rest on supports A1

and A2, respectively,

fixed on the woodenbaseboard. In somebalances, supports A

l and A

2 are not fixed and in that case the pans

rest on board W, when the balance is not in use.

At the centre of beam B, a pointer P is also fixed at right angles to it. Aknob K, connected by a horizontal rod to the vertical pillar V, is alsoattached from outside with the board W. With the help of this knob,the vertical pillar V and supports A

1 and A

2 can be raised or lowered

simultaneously. Thus, at the 'ON' position of the knob K, the beamB also gets raised and is then suspended only by the knife-edge Eand oscillates freely. Along with the beam, the pans P

1 and P

2 also

begin to swing up and down. This oscillatory motion of the beamcan be observed by the motion of the pointer P with reference to ascale (G) provided at the base of the pillar V. When the knob K isturned back to 'OFF' position, the beam rests on supports X1 and X2

keeping the knife-edge E and plate T slightly separated; and thepans P1 and P2 rest on supports A1 and A2 respectively. In the 'OFF'position of the knob K, the entire balance is said to be arrested.Such an arresting arrangement protects the knife-edges from unduewear and tear and injury during transfer of masses (unknown andstandards) from the pan.

On turning the knob K slowly to its ‘ON’ position, when there are nomasses in the two pans, the oscillatory motion (or swing) of thepointer P with reference to the scale G must be same on either sideof the zero mark on G. And the pointer must stop its oscillatorymotion at the zero mark. It represents the vertical position of thepointer P and horizontal position of the beam B. However, if theswing is not the same on either side of the zero mark, the twobalancing screws B

1 and B

2 at the two ends of the beam are adjusted.

The baseboard W is levelled horizontal1y to make the pillar V vertical.

Fig. E 4.2: A physical balance and a weight box

4

LABORATORY MANUAL

50

LABORATORY MANUAL

This setting is checked with the help of plumb line (R) suspended bythe side of pillar V. The appartus is placed in a glass case with twodoors.

For measuring the gravitational mass of an object using a physicalbalance, it is compared with a standard mass. A set of standardmasses (100 g, 50 g, 20 g, 10g, 5 g, 2 g, and 1 g) along with a pair offorceps is contained in a wooden box called Weight Box. The massesare arranged in circular grooves as shown in Fig. E 4.2. A set ofmilligram masses (500 mg, 200 mg, 100 mg, 50 mg, 20 mg 10 mg,5 mg, 2 mg, and 1 mg) is also kept separately in the weight box. Aphysical balance is usually designed to measure masses of bodiesup to 250 g.

PRINCIPLEThe working of a physical balance is based on the principle ofmoments. In a balance, the two arms are of equal length and the twopans are also of equal masses. When the pans are empty, the beamremains horizontal on raising the beam base by using the lower knob.When an object to be weighed is placed in the left pan, the beamturns in the anticlockwise direction. Equilibrium can be obtainedby placing suitable known standard weights on the right hand pan.Since, the force arms are equal, the weight (i.e., forces) on the twopans have to be equal.

A physical balance compares forces. The forces are the weights (mass× acceleration due to gravity) of the objects placed in the two pans ofthe physical balance. Since the weights are directly proportional tothe masses if weighed at the same place, therefore, a physical balanceis used for the comparison of gravitational masses. Thus, if an objectO having gravitational mass m is placed in one pan of the physicalbalance and a standard mass O′ of known gravitational mass m

s is

put in the other pan to keep the beam the horizontal, then

Weight of body O in one pan = Weight of body O′ in other pan

Or, mg = msg

where g is the acceleration due to gravity, which is constant. Thus,

m = ms

That is,

the mass of object O in one pan = standard mass in the other pan

PROCEDURE

1. Examine the physical balance and recognise all of its parts. Checkthat every part is at its proper place.

UNIT NAME

51

EXPERIMENT

2. Check that set of the weight, both in gram and milligram, in theweight box are complete.

3. Ensure that the pans are clean and dry.

4. Check the functioning of arresting mechanism of the beam B bymeans of the knob K.

5. Level the wooden baseboard W of the physical balancehorizontally with the help of the levelling screws W

1 and W

2. In

levelled position, the lower tip of the plumb line R should beexactly above the fixed needle point N. Use a spirit level for thispurpose.

6. Close the shutters of the glass case provided for covering thebalance and slowly raise the beam B using the knob K.

7. Observe the oscillatory motion of the pointer P with reference tothe small scale G fixed at the foot of the vertical pillar V. In case,the pointer does not start swinging, give a small gentle jerk toone of the pans. Fix your eye perpendicular to the scale to avoidparallax. Caution: Do not touch the pointer.

8. See the position of the pointer P. Check that it either stops at thecentral zero mark or moves equally on both sides of the centralzero mark on scale G. If not, adjust the two balancing screws B1

and B2 placed at the two ends of the beam B so that the pointer

swings equally on either side of the central zero mark or stops atthe central zero mark. Caution: Arrest the balance beforeadjusting the balancing screws.

9. Open the shutter of the glass case of the balance. Put theobject whose mass (M) is to be measured in the left handpan and add a suitable standard mass say M1, (which maybe more than the rough estimate of the mass of the object)in the right hand pan of the balance in its rest (or arrested)position, i.e., when the beam B is lowered and allowed torest on stoppers X

l and X

2. Always use forceps for taking

out the standard mass from the weight box and for puttingthem back.

The choice of putting object on left hand pan and standardmasses on right hand pan is arbitrary and chosen due to theease in handling the standard masses. A left handed person mayprefer to keep the object on right hand pan and standard masseson left hand pan. It is also advised to keep the weight box nearthe end of board W on the side of the pan being used for puttingthe standard masses.

10. Using the knob K, gently raise the beam (now the beam’s knifeedge E will rest on plate T fixed on the top of the pillar V) andobserve the motion of the pointer P. It might rest on one side of

4

LABORATORY MANUAL

52

LABORATORY MANUAL

the scale or might oscillate more in one direction with referenceto the central zero mark on the scale G.

Note: Pans should not swing while taking the observations. Theswinging of pans may be stopped by carefully touching the panwith the finger in the arresting position of the balance.

11. Check whether M1 is more than M or less. For this purpose, thebeam need to be raised to the full extent.

12. Arrest the physical balance. Using forceps, replace the standardmasses kept in the right pan by another mass (say M2). It shouldbe lighter if M

1 is more than the mass M and vice versa.

13. Raise the beam and observe the motion of the pointer P and checkwhether the standard mass kept on right hand pan is still heavier(or lighter) than the mass M so that the pointer oscillates more inone direction. If so, repeat step 12 using standard masses in gramtill the pointer swings nearly equal on both sides of the centralzero mark on scale G. Make the standard masses kept on righthand pan to be slightly lesser than the mass of object. This wouldresult in the measurement of mass M of object with a precision of1 g. Lower the beam B.

14. For fine measurement of mass add extra milligram massesin the right hand pan in descending order until the pointerswings nearly equal number of divisions on either side of thecentral zero mark on scale G (use forceps to pick the milligramor fractional masses by their turned-up edge). In theequilibrium position (i.e., when the masses kept on both thepans are equal), the pointer will rest at the centre zero mark.Close the door of the glass cover to prevent disturbances dueto air draughts.

Note: The beam B of the balance should not be raised to the fullextent until milligram masses are being added or removed.Pointer’s position can be seen by lifting the beam very gently andfor a short moment.

15. Arrest the balance and take out masses from the right hand panone by one and note total mass in notebook. Replace them intheir proper slot in the weight box. Also remove the object fromthe left hand pan.

16. Repeat the step 9 to step 15 two more times for the same object.

17. Repeat steps 9 to 15 and determine the mass of the second givenobject.

Record the observations for the second object in the table similar toTable E 4.1.

UNIT NAME

53

EXPERIMENT

OBSERVATIONS

TABLE E 4.1: Mass of First Object

Mean mass of the first object = ... g

TABLE E 4.2: Mass of Second Object

1

2

3

S.No.

Standard mass Mass of the object (x + y)

Gram weights, x Milligram weights, y

(mg) (g)(g)

Mean mass of the second object = ... g

RESULT

The mass of the first given object is ... g and that of the secondobject is ... g.

PRECAUTIONS

1. The correctness of mass determined by a physical balance dependson minimising the errors, which may arise due to the friction betweenthe knife-edge E and plate T. Friction cannot be removed completely.However, it can be minimised when the knife-edge is sharp andplate is smooth. The friction between other parts of the balance maybe minimised by keeping all the parts of balance dry and clean.

2. Masses should always be added in the descending order ofmagnitude. Masses should be placed in the centre of the pan.

3. The balance should not be loaded with masses more thancapacity. Usually a physical balance is designed to measuremasses upto 250 g.

1

2

3

S.No.

Standard mass Mass of the object (x + y)

Gram weights, x Milligram weights, y

(mg) (g)(g)

4

LABORATORY MANUAL

54

LABORATORY MANUAL

4. Weighing of hot and coId bodies using a physical balance shouldbe avoided. Similarly, active substances like chemicals, liquidsand powders should not be kept directly on the pan.

SOURCES OF ERROR1. There is always some error due to friction at various parts of the

balance.

2. The accuracy of the physical balance is 1 mg. This limits thepossible instrumental error.

DISCUSSION

The deviation of experimental value from the given value may be dueto many factors.

1. The forceps used to load/unload the weights might contain dustparticles sticking to it which may get transferred to the weight.

2. Often there is a general tendancy to avoid use of levelling andbalancing screws to level the beam/physical balance just beforeusing it.

SELF ASSESSMENT

1. Why is it necessary to close the shutters of the glass case for anaccurate measurement?

2. There are two physical balances: one with equal arms and otherwith unequal arms. Which one should be preferred? Whatadditional steps do you need to take to use a physical balancewith unequal arms.

3. The minimum mass that can be used from the weight box is 10 g.Find the possible instrumental error.

4. Instead of placing the mass (say a steel block) on the pan, supposeit is hanged from the same hook S

1 on which the pan P

1 is hanging.

Will the value of measured mass be same or different?


1. Determination of density of material of a non-porous block andverification of Archimedes principle:

Hint: First hang the small block (say steel block) from hook S1 and

determine its mass in air. Now put the hanging block in a half water-filled measuring cylinder. Measure the mass of block in water. Will itbe same, more or less? Also detemine the volume of steel block.Find the density of the material of the block. From the measuredmasses of the steel block in air and water, verify Archimedes principle.

UNIT NAME

55

EXPERIMENT

Fig. E 5.1(a): Gravesand's apparatus Fig. E 5.1(b): Marking forces to scale


AIMMeasurement of the weight of a given body (a wooden block) usingthe parallelogram law of vector addition.

APPARATUS AND MATERIAL REQUIREDThe given body with hook, the parallelogram law of vector apparatus(Gravesand's apparatus), strong thread, slotted weights (two sets),white paper, thin mirror strip, sharp pencil.

DESCRIPTION OF MATERIAL

Gravesand's apparatus: It consists of a wooden board fixedvertically on two wooden pillars as shown in Fig. E 5.1 (a). Two pulleysP

1 and P

2 are provided on its two sides near the upper edge of the

board. A thread carrying hangers for addition of slotted weights ismade to pass over the pulleys so that two forces P and Q can be appliedby adding weights in the hangers. By suspending the given object,whose weight is to be determined, in the middle of the thread, a thirdforce X is applied.

55555

LABORATORY MANUAL

56

LABORATORY MANUAL

PRINCIPLE

Working of this apparatus is based on the parallelogram law ofvector addition. The law states that "when two forces actsimultaneously at a point and are represented in magnitude anddirection by the two adjacent sides of a parallelogram, then theresultant of forces can be represented both in magnitude anddirection by the diagonal of the parallelogram passing through thepoint of application of the two forces.

Let P and Q be the magnitudes of the two forces and θ the angle be-tween them. Then the resultant R of P and Q is given by

θR P PQ2 2 = + Q + 2 cos

If two known forces P and Q and a third unknown force due to theweight of the given body are made to act at a point O [Fig. 5.1 (a)]such that they are in equilibrium, the unknown force is equal tothe resultant of the two forces. Thus, the weight of a given bodycan be found.

PROCEDURE

1. Set the board of Gravesand's apparatus in vertical position byusing a plumb-line. Ensure that the pulleys are movingsmoothly. Fix a sheet of white paper on the wooden board withdrawing pins.

2. Take a sufficiently long piece of string and tie the two hangers atits ends. Tie another shorter string in the middle of the first stringto make a knot at 'O'. Tie the body of unknown weight at theother end of the string. Arrange them on the pulley as shown inFig. E 5.1 (a) with slotted weights on the hangers.

3. Add weights in the hangers such that the junction of the threadsis in equilibrium in the lower half of the paper. Make sure thatneither the weights nor the threads touch the board orthe table.

4. Bring the knot of the three threads to position of no-friction. Forthis, first bring the knot to a point rather wide off its position ofno-friction. On leaving there, it moves towards the position ofno-friction because it is not in equilibrium. While it so moves,tap the board gently. The point where the knot thus come to restis taken as the position of no-friction, mark this point. Repeatthe procedure several times. Each time let the knot approach theposition of no-friction from a different direction and mark thepoint where it comes to rest. Find by judgement the centre ofthose points which are close together. Mark this centre as O.

UNIT NAME

57

EXPERIMENT

5. To mark the direction of the force acting along a string, place amirror strip below the string on the paper . Adjust the position ofthe eye such that there is no parallax between the string and itsimage. Mark the two points A1 and A2 at the edges of the mirrorwhere the image of the string leaves the mirror [Fig E 5.1 (b)].

Similarly, mark the directions of other two forces by points B1 andB2 and by points X1 and X2 along the strings OB and OXrespectively.

6. Remove the hangers and note the weight of each hanger and slottedweights on them.

7. Place the board flat on the table with paper on it. Join the threepairs of points marked on the paper and extend these lines tomeet at O. These three lines represent the directions of the threeforces.

8. Choose a suitable scale, say 0.5 N (50 g wt) = 1cm and cut offlength OA and OB to represent forces P and Q respectively actingat point O. With OA and OB as adjacent sides, complete theparallelogram OACB. Ensure that the scale chosen is such thatthe parallelogram covers the maximum area of the sheet.

9. Join points O and C. The length of OC will measure the weight ofthe given body. See whether OC is along the straight line XO. Ifnot, let it meet BC at some point C′. Measure the angle COC′.

10.Repeat the steps 1 to 9 by suspending two different sets of weightsand calculate the mean value of the unknown weight.

OBSERVATIONSWeight of each hanger = ... N

Scale, 1cm = ... N

Table E 5.1: Measurement of weight of given body

5

Force P = wtof (hanger +slottedweight)

Force Q = wtof (hanger +slotted weight

Length OC= L

Unknownweight X =

L × s

Angle COC′S.No.

P

(N)

OA

(cm)

Q

(N)

OB

(cm)

(cm) (N)

1

2

3

LABORATORY MANUAL

58

LABORATORY MANUAL

RESULT

The weight of the given body is found to be ... N.

PRECAUTIONS

1. Board of Gravesand's apparatus is perpendicular to table on whichit is placed, by its construction. Check up by plumb line that it isvertical. If it is not, make table top horizontal by putting packingbelow appropriate legs of table.

2. Take care that pulleys are free to rotate, i.e., have little frictionbetween pulley and its axle.

SOURCES OF ERROR

1. Friction at the pulleys may persist even after oiling.

2. Slotted weights may not be accurate.

3. Slight inaccuracy may creep in while marking the position ofthread.

DISCUSSION

1. The Gravesand's apparatus can also be used to verify theparallelogram law of vector addition for forces as well astriangle law of vector addition. This can be done by using thesame procedure by replacing the unknown weight by astandard weight.

2. The method described above to find the point of no-friction for thejunction of three threads is quite good experimentally. If you liketo check up by an alternative method, move the junction toextreme left, extreme right, upper most and lower most positionswhere it can stay and friction is maximum. The centre of thesefour positions is the point of no-friction.

3. What is the effect of not locating the point of no-friction accurately?In addition to the three forces due to weight, there is a fourthforce due to friction. These four are in equilibrium. Thus, theresultant of P and Q may not be vertically upwards, i.e., exactlyopposite to the direction of X.

4. It is advised that values of P and Q may be checked by springbalance as slotted weights may have large error in their markedvalue. Also check up the result for X by spring balance.

UNIT NAME

59

EXPERIMENT


1. Interchange position of the body of unknown weight with either ofthe forces and then find out the weight of that body.

2. Keeping the two forces same and by varying the unknown weight,study the angle between the two forces.

3. Suggest suitable method to estimate the density of material of agiven cylinder using parallelogram law of vectors.

4. Implement parallelogram law of vectors in the following situations:-

(a) Catapult (b) Bow and arrow (c) Hand gliding(d) Kite (e) Cycle pedalling

SELF ASSESSMENT

1. State parallelogram law of vector addition.

2. Given two forces, what could be the

(a) Maximum magnitude of resultant force.

(b) Minimum magnitude of resultant force.

3. In which situation this parallelogram can be a rhombus.

4. If all the three forces are equal in magnitude, how will theparallelogram modify?

5. When the knot is in equilibrium position, is any force acting onthe pulleys?

5

60



AIM

Using a Simple Pendulum plot L – T and L – T2 graphs, hence find theeffective length of second's pendulum using appropriate graph.


Clamp stand; a split cork; a heavy metallic (brass/iron) spherical bobwith a hook; a long, fine, strong cotton thread/string (about 2.0 m);stop-watch; metre scale, graph paper, pencil, eraser.

DESCRIPTION OF TIME MEASURING DEVICES IN A SCHOOL LABORATORYThe most common device used for measuring time in a schoollaboratory is a stop-watch or a stop-clock (analog). As the namessuggest, these have the provision to start or stop their working asdesired by the experimenter.

(a) Stop-Watch

Analog

A stop-watch is a special kind of watch. It has a multipurpose knobor button (B) for start/stop/back to zero position [Fig. E 6.1(b)]. It has

two circular dials, the bigger one for a longer second’s handand the other smaller one for a shorter minute’s hand. Thesecond’s dial has 30 equal divisions, each division repre-senting 0.1 second. Before using a stop-watch you shouldfind its least count. In one rotation, the seconds hand covers30 seconds (marked by black colour) then in the secondrotation another 30 seconds are covered (marked by redcolour), therefore, the least count is 0.1 second.

(b) Stop-Clock

The least count of a stop-watch is generally about 0.1s [Fig.E 6.1(b)] while that of a stop-clock is 1s, so for more accuratemeasurement of time intervals in a school laboratory, astop-watch is preferred. Digital stop-watches are alsoavailable now. These watches may be started by pressingthe button and can be stopped by pressing the same button

66666

Fig.E 6.1(a): Stop - Watch

61

UNIT NAMEEXPERIMENT

once again. The lapsed time interval is directlydisplayed by the watch.

TERMS AND DEFINITIONS

1. Second's pendulum: It is a pendulum whichtakes precisely one second to move from oneextreme position to other. Thus, its times periodis precisely 2 seconds.

2. Simple pendulum: A point mass suspended byan inextensible, mass less string from a rigidpoint support. In practice a small heavyspherical bob of high density material of radiusr, much smaller than the length of the suspension, is suspendedby a light, flexible and strong string/thread supported at the otherend firmly with a clamp stand. Fig. E 6.2 is a good approximationto an ideal simple pendulum.

3. Effective length of the pendulum: The distance L between thepoint of suspension and the centre of spherical bob (centre ofgravity), L = l + r + e, is also called the effective length where l is thelength of the string from the top of the bob to the hook, e, thelength of the hook and r the radius of the bob.

PRINCIPLEThe simple pendulum executes Simple Harmonic Motion (SHM)as the acceleration of the pendulum bob is directly proportional toits displacement from the mean position and is always directedtowards it.

The time period (T) of a simple pendulum for oscillations of smallamplitude, is given by the relation

T L / g= π2

where L is the length of the pendulum, and g is the acceleration dueto gravity at the place of experiment.

Eq. (6.1) may be rewritten as

LT

g

π=

22 4

PROCEDURE

1. Place the clamp stand on the table. Tie the hook, attached tothe pendulum bob, to one end of the string of about 150 cm inlength. Pass the other end of the string through two half-piecesof a split cork.

Fig.E 6.1(b): Stop - Clock

(E 6.1)

(E 6.2)

6

62


2. Clamp the split cork firmly in the clamp stand such that the line ofseparation of the two pieces of the split cork is at right angles tothe line OA along which the pendulum oscillates [Fig. E 6.2(a)].Mark, with a piece of chalk or ink, on the edge of the table a verticalline parallel to and just behind the vertical thread OA, the positionof the bob at rest. Take care that the bob hangs vertically (about2 cm above the floor) beyond the edge of the table so that it is freeto oscillate.

3. Measure the effective length of simple pendulum as shownin Fig. E 6.2(b).

Fig.E 6.2 (a): A simple pendulum; B and C showthe extreme positions

4. Displace the bob to one side, not more than 15 degrees angulardisplacement, from the vertical position OA and then release it gently.In case you find that the stand is shaky, put some heavy object onits base. Make sure that the bob starts oscillating in a vertical planeabout its rest (or mean) position OA and does not (i) spin about itsown axis, or (ii) move up and down while oscillating, or (iii) revolvein an elliptic path around its mean position.

5. Keep the pendulum oscillating for some time. After completion ofa few oscillations, start the stop-watch/clock as the thread attachedto the pendulum bob just crosses its mean position (say, from leftto right). Count it as zero oscillation.

6. Keep on counting oscillations 1,2,3,…, n, everytime the bob crossesthe mean position OA in the same direction (from left to right).

Fig.E 6.2 (b): Effective length of asimple pendulum

63

UNIT NAMEEXPERIMENT

Stop the stop-watch/clock, at the count n (say, 20 or 25) ofoscillations, i.e., just when n oscillations are complete. For betterresults, n should be chosen such that the time taken for noscillations is 50 s or more. Read, the total time (t) taken by thebob for n oscillations. Repeat this observation a few times by notingthe time for same number (n) of oscillations. Take the mean ofthese readings. Compute the time for one oscillation, i.e., the timeperiod T ( = t/n) of the pendulum.

7. Change the length of the pendulum, by about 10 cm. Repeat thestep 6 again for finding the time (t) for about 20 oscillations ormore for the new length and find the mean time period. Take 5 or6 more observations for different lengths of penduLum and findmean time period in each case.

8. Record observations in the tabular form with proper units andsignificant figures.

9. Take effective length L along x-axis and T 2 (or T) along y-axis,using the observed values from Table E 6.1. Choose suitable scaleson these axes to represent L and T 2 (or T ). Plot a graph betweenL and T 2 (as shown in Fig. E 6.4) and also between L and T (asshown in Fig. E 6.3). What are the shapes of L –T 2 graph and L –Tgraph? Identify these shapes.

OBSERVATIONS

(i) Radius (r) of the pendulum bob (given) = ... cm

Length of the hook (given) (e) = ... cm

Least count of the metre scale = ... mm = ... cm

Least count of the stop-watch/clock = ... s

Table E 6.1: Measuring the time period T and effective lengthL of the simple pendulum

S.No.

Length of thestring from the

top of the bob tothe point ofsuspension l

Effectivelength, L =

(l+r+e )

Number ofoscillationscounted, n

Time for n oscillationst (s)

Timeperiod T(= t/n)

(cm) m (i)... (ii) (iii) Meant (s)

s

6

64


PLOTTING GRAPH(i) L vs T graphs

Plot a graph between L versus T from observations recorded inTable E 6.1, taking L along x-axis and T along y-axis. You willfind that this graph is a curve, which is part of a parabola asshown in Fig. E 6.3.

(ii) L vs T 2 graph

Plot a graph between L versus T 2 from observations recorded inTable E 6.1, taking L along x-axis and T 2 along y-axis. You willfind that the graph is a straight line passing through origin as shownin Fig. E. 6.4.

(iii) From the T 2 versus L graph locate the effective length of second'spendulum for T 2 = 4s2.

Fig. E 6.3: Graph of L vs T Fig. E 6.4: Graph L vs T 2

RESULT

1. The graph L versus T is curved, convex upwards.

2. The graph L versus T 2 is a straight line.

3. The effective length of second's pendulum from L versus T 2 graphis ... cm.

Note : The radius of bob may be found from its measureddiameter with the help of callipers by placing the pendulum bobbetween the two jaws of (a) ordinary callipers, or (b) Ver nierCallipers, as described in Experiment E 1.1 (a). It can also befound by placing the spherical bob between two parallel cardboards and measuring the spacing (diameter) or distance betweenthem with a metre scale.

65

UNIT NAMEEXPERIMENT

DISCUSSION

1. The accuracy of the result for the length of second's pendulumdepends mainly on the accuracy in measurement of effective length(using metre scale) and the time period T of the pendulum (usingstop-watch). As the time period appears as T 2 in Eq. E 6.2, a smalluncertainty in the measurement of T would result in appreciableerror in T 2, thereby significantly affecting the result. A stop-watchwith accuracy of 0.1s may be preferred over a less accuratestop-watch/clock.

2. Some personal error is always likely to be involved due to stop-watchnot being started or stopped exactly at the instant the bob crossesthe mean position. Take special care that you start and stop thestop-watch at the instant when pendulum bob just crosses themean position in the same direction.

3. Sometimes air currents may not be completely eliminated. Thismay result in conical motion of the bob, instead of its motion invertical plane. The spin or conical motion of the bob may cause atwist in the thread, thereby affecting the time period. Take specialcare that the bob, when it is taken to one side of the rest position,is released very gently.

4. To suspend the bob from the rigid support, use a thin, light, strong,unspun cotton thread instead of nylon string. Elasticity of thestring is likely to cause some error in the effective length of thependulum.

5. The simple pendulum swings to and fro in SHM about the mean,equilibrium position. Eq. (E 6.1) that expresses the relation

between T and L as T = L gπ2 / , holds strictly true for small

amplitude or swing θ of the pendulum.

Remember that this relation is based on the assumption that sinθ ≈ θ, (expressed in radian) holds only for small angulardisplacement θ.

6. Buoyancy of air and viscous drag due to air slightly increase thetime period of the pendulum. The effect can be greatly reduced toa large extent by taking a small, heavy bob of high density material(such as iron/ steel/brass).

SELF ASSESSMENT

1. Interpret the graphs between L and T 2, and also between L and Tthat you have drawn for a simple pendulum.

2. Examine, using Table E 6.1, how the time period T changes as the

6

66



1. To determine 'g', the acceleration due to gravity, at a given place,from the L – T 2 graph, for a simple pendulum.

2. Studying the effect of size of the bob on the time period of thesimple pendulum.

[Hint: With the same experimental set-up, take a few sphericalbobs of same material (density) but of different sizes (diameters).Keep the length of the pendulum the same for each case. Clampthe bobs one by one, and starting from a small angular displacementof about 10o, each time measure the time for 50 oscillations. Findout the time period of the pendulum using bobs of different sizes.Compensate for difference in diameter of the bob by adjusting thelength of the thread.

Does the time period depend on the size of the pendulum bob? Ifyes, see the order in which the change occurs.]

3. Studying the effect of material (density) of the bob on the timeperiod of the simple pendulum.

[Hint: With the same experimental set-up, take a few sphericalbobs (balls) of different materials, but of same size. Keep thelength of the pendulum the same for each case. Find out, in eachcase starting from a small angular displacement of about 10°, thetime period of the pendulum using bobs of different materials,

Does the time period depend on the material (density) of thependulum bob? If yes, see the order in which the change occurs.If not, then do you see an additional reason to use the pendulumfor time measurement.]

4. Studying the effect of mass of the bob on the time period of thesimple pendulum.

[Hint: With the same experimental set-up, take a few bobs ofdifferent materials (different masses) but of same size. Keep thelength of the pendulum same for each case. Starting from a smallangular displacement of about 10° find out, in each case, the timeperiod of the pendulum, using bobs of different masses.

Does the time period depend on the mass of the pendulum bob? Ifyes, then see the order in which the change occurs. If not, thendo you see an additional reason to use the pendulum as a timemeasuring device.]

5. Studying the effect of amplitude of oscillation on the time period ofthe simple pendulum.

[Hint: With the same experimental set-up, keep the mass of thebob and length of the pendulum fixed. For measuring the angularamplitude, make a large protractor on the cardboard and have ascale marked on an arc from 0° to 90° in units of 5°. Fix it on theedge of a table by two drawing pins such that its 0°- line coincides

effective length L of a simple pendulum; becomes 2-fold, 4-fold,and so on.

3. How can you determine the value of 'g', acceleration due to gravity,from the T 2 vs L graph?

67

UNIT NAMEEXPERIMENT

with the suspension thread of the pendulum at rest. Start thependulum oscillating with a very large angular amplitude (say 70°)and find the time period T of the pendulum. Change the amplitudeof oscillation of the bob in small steps of 5° or 10° and determinethe time period in each case till the amplitude becomes small (say5°). Draw a graph between angular amplitude and T. How doesthe time period of the pendulum change with the amplitude ofoscillation?

How much does the value of T for A = 10° differ from that for A=50° from the graph you have drawn?

Find at what amplitude of oscillation , the time period begins to vary?

Determine the limit for the pendulum when it ceases to be a simplependulum.]

6. Studying the effect on time period of a pendulum having a bob ofvarying mass (e.g. by filling the hollow bob with sand, sand beingdrained out in steps)

[Hint: The change in T, if any, in this experiment will be so smallthat it will not be possible to measure it due to the following reasons:

The centre of gravity (CG) of a hollow sphere is at the centre of thesphere. The length of this simple pendulum will be same as that ofa solid sphere (same size) or that of the hollow sphere filledcompletely with sand (solid sphere).

Drain out some sand from the sphere. The situation is as shown inFig. E. 6.5. The CG of bob now goes down to point say A. Theeffective length of the pendulum increases and therefore the T

A

increases (TA > TO), some more sand is drained out, the CG goesdown further to a point B. The effective length further increases,increasing T .

The process continues and L and T change in the same direction(increasing), until finally the entire sand is drained out. The bob isnow a hollow sphere with CG shifting back to centre C. The timeperiod will now become T

0 again.]

Fig. E 6.5: Variation of centre of gravity of sand filledhollow bob on time period of the pendulum;sand being drained out of the bob in steps.

SandA

B

C

6

68



AIM

To study the relation between force of limiting friction and normalreaction and to find the coefficient of friction between surface of amoving block and that of a horizontal surface.

APPARATUS AND MATERIAL REQUIREDA wooden block with a hook, a horizontal plane with a glass orlaminated table top (the table top itself may be used as a horizontalplane), a frictionless pulley which can be fixed at the edge of thehorizontal table/plane, spirit level, a scale, pan, thread or string, springbalance, weight box and five masses of 100 g each.


Friction: The tendency to oppose the relative motion between twosurfaces in contact is called friction.

Static Friction: It is the frictional force acting between two solidsurfaces in contact at rest but having a tendency to move (slide) withrespect to each other.

Limiting Friction: It is the maximum value of force of static frictionwhen one body is at the verge of sliding with respect to the other bodyin contact.

Kinetic (or Dynamic) Friction: It is the frictional force acting betweentwo solid surfaces in contact when they are in relative motion.

PRINCIPLE

The maximum force ofstatic friction, i.e., limitingfriction, FL, between twodry, clean and unlubricatedsolid surfaces is found toobey the following empiricallaws:

(i) The limiting friction isFig. E 7.1: The body is at rest due to

static friction

69

UNIT NAMEEXPERIMENT

directly proportional to the normal reaction, R, which is given bythe total weight W of the body (Fig. E 7.1). The line of action issame for both W and R for horizontal surface,

FL ∝ R ⇒ FL = μLR

i.e. μL = LF

R

Thus, the ratio of the magnitude of the limiting friction, FL, to the

magnitude of the normal force, R, is a constant known as thecoefficient of limiting friction (μ

L) for the given pair of surfaces in

contact.

(ii) The limiting friction depends upon the nature of surfaces in contactand is nearly independent of the surface area of contact over widelimits so long as normal reaction remains constant.

Note that FL= μ

LR is

an equation of astraight line passingthrough the origin.Thus, the slope of thestraight-line graphbetween F

l (along Y-

axis) and R (along X-axis) will give thevalue of coefficient oflimiting friction μ

L.

In this experiment,the relationshipbetween the limitingfriction and normalreaction is studied for a wooden block. The wooden block is madeto slide over a horizontal surface (say glass or a laminated surface)(Fig. E 7.2).

PROCEDURE

1. Find the range and least count of the spring balance.

2. Measure the mass (M) of the given wooden block with hooks on itssides and the scale pan (m) with the help of the spring balance.

3. Place the glass (or a laminated sheet) on a table and make ithorizontal, if required, by inserting a few sheets of paper orcardboard below it. To ensure that the table-top surface ishorizontal use a spirit level. Take care that the top surface mustbe clean and dry.

Fig. E 7.2: Experimental set up to study limiting friction

( )m+q g

q

Pulley

Clear glass (orwood mica) top

Pan

R M p g= ( + )

F

( )M+p g

7

70


4. Fix a frictionless pulley on one edge of table-top as shown in Fig.E 7.2. Lubricate the pulley if need be.

5. Tie one end of a string of suitable length (in accordance with thesize and the height of the table) to a scale pan and tie its other endto the hook of the wooden block.

6. Place the wooden block on the horizontal plane and pass the stringover the pulley (Fig. E7.2). Ensure that the portion of the stringbetween pulley and the wooden block is horizontal. This can bedone by adjusting the height of the pulley to the level of hook ofblock.

7. Put some mass (q) on the scale pan. Tap the table-top gently withyour finger. Check whether the wooden block starts moving.

8. Keep on increasing the mass (q) on the scale pan till the woodenblock just starts moving on gently tapping the glass top. Recordthe total mass kept on the scale pan in Table E 7.1.

9. Place some known mass (say p ) on the top of wooden block andadjust the mass (q′) on the scale pan so that the wooden blockalongwith mass p just begins to slide on gently tapping the tabletop. Record the values of p′ and q′ in Table E 7.1.

10.Repeat step 9 for three or four more values of p and record thecorresponding values of q in Table E 7.1. A minimum of fiveobservations may be required for plotting a graph between FL andR.

OBSERVATIONS

1. Range of spring balance = ... to ... g

2. Least count of spring balance = ... g

3. Mass of the scale pan, (m) = ... g

4. Mass of the wooden block (M) = ... g

5. Acceleration due to gravity (g) at the place of experiment= ... m/s2

Table E 7.1: Variation of Limiting Friction with NormalS.No.

Mass on thewooden block

(p) (g)

Normalforce R due

to mass(M+p)

Mass onthe pan

(q) g

Force oflimitingfriction FL

Coefficientof friction

LL

F

Rμ =

(g) (kg)

Mean μL

N (g) (kg) (N)

12345

71

UNIT NAMEEXPERIMENT

Reaction

GRAPHPlot a graph between the limiting friction (F

L) and

normal force (R) between the wooden block and thehorizontal surface, taking the limiting friction F

L

along the y-axis and normal force R along the x-axis. Draw a line to join all the points marked on it(Fig. E 7.3). Some points may not lie on the straight-line graph and may be on either side of it. Extendthe straight line backwards to check whether thegraph passes through the origin. The slope of thisstraight-line graph gives the coefficient of limitingfriction (μ

L) between the wooden block and the

horizontal surface. To find the slope of straight line,choose two points A and B that are far apart fromeach other on the straight line as shown in Fig. E7.3. Draw a line parallel to x-axis through point Aand another line parallel to y-axis through point B.Let point Z be the point of intersection of these two lines. Then, theslope μ

L of straight line graph AB would be

LL

F BZ= =

R AZμ

RESULT

The value of coefficient of limiting friction μL between surface of woodenblock and the table-top (laminated sheet/glass) is:

(i) From calculation (Table E 7.1) = ...

(ii) From graph = ...

PRECAUTIONS

1. Surface of the table should be horizontal and dust free.

2. Thread connecting wooden block and pulley should be horizontal.

3. Friction of the pulley should be reduced by proper oiling.

4. Table top should always be tapped gently.

SOURCES OF ERROR

1. Always put the mass at the centre of wooden block.

Fig. E 7.3: Graph between force oflimiting friction FL andnormal reaction, R

7

72


2. Surface must be dust free and dry.

3. The thread must be unstretchable and unspun.

DISCUSSION

1. The friction depends on the roughness of the surfaces in contact.If the surfaces in contact are ideally (perfactly) smooth, therewould be no friction between the two surfaces. However, therecannot be an ideally smooth surface as the distribution of atomsor molecules on solid surface results in an inherent roughness.

2. In this experimental set up and calculations, friction at the pulleyhas been neglected, therefore, as far as possible, the pulley, shouldhave minimum friction as it cannot be frictionless.

3. The presence of dust particles between the wooden block andhorizontal plane surface may affect friction and therefore leadto errors in observations. Therefore, the surface of thehorizontal plane and wooden block in contact must be cleanand dust free.

4. The presence of water or moisture between the wooden blockand the plane horizontal surface would change the nature ofthe surface. Thus, while studying the friction between the surfaceof the moving body and horizontal plane these must be keptdry.

5. Elasticity of the string may cause some error in the observation.Therefore, a thin, light, strong and unspun cotton thread mustbe used as a string to join the scale pan and the moving block.

6. The portion of string between the pulley and wooden block mustbe horizontal otherwise only a component of tension in the stringwould act as the force to move the block.

7. It is important to make a judicious choice of the size of the blockand set of masses for this experiment. If the block is too light, itsforce of limiting friction may be even less than the weight of emptypan and in this situation, the observation cannot be taken withthe block alone. Similarly, the maximum mass on the block, whichcan be obtained by putting separate masses on it, should not bevery large otherwise it would require a large force to make theblock move.

8. The additional mass, p, should always be put at the centre ofwooden block.

73

UNIT NAMEEXPERIMENT


1. To study the effect of the nature of sliding surface. [Hint: Repeatthe same experiment for different types of surfaces say, plywood,carpet etc. Or repeat the experiment after putting oil or powder onthe surface.]

2. To study the effect of changing the area of the surfaces in contact.[Hint: Place the wooden block vertically and repeat the experiment.Discuss whether the readings and result of the experiment are same.]

3. To find the coefficient of limiting friction by sliding the block on aninclined plane.

9. The permissible error in measurements of coefficient of friction

= L

L

F R...

F R

Δ Δ+ =

SELF ASSESSMENT1. On the basis of your observations, find the relation between

limiting friction and the mass of sliding body.

2. Why do we not choose a spherical body to study the limitingfriction between the two surfaces?

3. Why should the horizontal surfaces be clean and dry?

4. Why should the portion of thread between the moving body andpulley be horizontal?

5. Why is it essential in this experiment to ensure that the surfaceon which the block moves should be horizontal?

6. Comment on the statement: “The friction between two surfacescan never be zero”.

7. In this experiment, usually unpolished surfaces are preferred,why?

8. What do you understand by self-adjusting nature of force offriction?

9. In an experiment to study the relation between force of limitingfriction and normal reaction, a body just starts sliding on applyinga force of 3 N. What will be the magnitude of force of frictionacting on the body when the applied forces on it are 0.5 N, 1.0 N,2.5 N, 3.5 N, respectively.

7

74



AIM

To find the downward force, along an inclined plane, acting on a rollerdue to gravity and study its relationship with the angle of inclinationby plotting graph between force and sin θ.

APPARATUS AND MATERIAL REQUIREDInclined plane with protractor and pulley, roller, weight box, springbalance, spirit level, pan and thread.

88888

Fig. E 8.1: Experimental set up to find the downwardforce along an inclined plane

Pulley

Pan, M2Mass, M3

W M M= ( + )2 3

M1Roller

Constant v

Protractor

v

Fig. E 8.2: Free body diagram

PRINCIPLEConsider the set up shown in Fig. E 8.1. Here a roller ofmass M

1 has been placed on an inclined plane making

an angle θ with the horizontal. An upward force, alongthe inclined plane, could be applied on the mass M

1 by

adjusting the weights on the pan suspended with a stringwhile its other end is attached to the mass through apulley fixed at the top of the inclined plane. The force onthe the mass M1 when it is moving with a constant velocityv will be

W = M1g sin θ – f

r

where fr is the force of friction due to rolling, M1 ismass of roller and W is the total tension in the string

75

UNIT NAMEEXPERIMENT

(W = weight suspended). Assuming there is no friction betweenthe pulley and the string.

PROCEDURE1. Arrange the inclined plane, roller and the masses in the pan as

shown in Fig. E. 8.1. Ensure that the pulley is frictionless. Lubricateit using machine oil, if necessary.

2. To start with, let the value of W be adjusted so as to permit theroller to stay at the top of the inclined plane at rest.

3. Start decreasing the masses in small steps in the pan until theroller just starts moving down the plane with a constant velocity.Note W and also the angle θ. Fig. E 8.2 shows the free body diagramfor the situation when the roller just begins to move downwards.

4. Repeat steps 2 and 3 for different values of θ. Tabulate yourobservations.

OBSERVATIONSAcceleration due to gravity, g = ... N/m2

Mass of roller, m = (M1) gMass of the pan = (M

2) g

Table E 8.1

PLOTTING GRAPHPlot graph between sin θ andthe force W (Fig. E 8.3). Itshould be a straight line.

Fig. E 8.3: Graph between Wand sin θ

8

1

2

3

S. No. θ ° sin θ Mass added to panM3

ForceW = (M2 + M3 ) g (N)

76


RESULTTherefore, within experimental error, downward force along inclinedplane is directly proportional to sin θ, where θ is the angle of inclinationof the plane.

PRECAUTIONS

1. Ensure that the inclined plane is placed on a horizontal surfaceusing the spirit level.

2. Pulley must be frictionless.

3. The weight should suspend freely without touching the table orother objects.

4. Roller should roll smoothly, that is, without slipping.

5. Weight, W should be decreased in small steps.

SOURCES OF ERROR

1. Error may creep in due to poor judgement of constant velocity.

2. Pulley may not be frictionless.

3. It may be difficult to determine the exact point when the rollerbegins to slide with constant velocity.

4. The inclined surface may not be of uniform smoothness/roughness.

5. Weights in the weight box may not be standardised.

DISCUSSION

As the inclination of the plane is increased, starting from zero,the value of mg sin θ increases and frictional force also increasesaccordingly. Therefore, till limiting friction W = 0, we need not applyany tension in the string.

When we increase the angle still further, net tension in the string isrequired to balance (mg sin θ – f

r ) or otherwise the roller will accelerate

downwards.

It is difficult to determine exact value of W. What we can do is we findtension W

1 (< W) at which the roller is just at the verge of rolling

down and W2 (> W) at which the roller is just at the verge of movingup. Then we can take

( )=

W + WW 1 2

2

77

UNIT NAMEEXPERIMENT


1. From the graph, find the intercept and the slope. Interpret themusing the given equation.

2. Allow the roller to move up the inclined plane by adjusting the massin the pan. Interpret the graph between W ′ and sin θ where W ′ isthe mass in pan added to the mass of the pan required to allow theroller to move upward with constant velocity.

SELF ASSESSMENT

1. Give an example where the force of friction is in the same directionas the direction of motion.

2. How will you use the graph to find the co-efficient of rolling frictionbetween the roller and the inclined plane?

3. What is the relation between downward force and angle ofinclination of the plane?

4. How will you ensure that the roller moves upward/downward withconstant velocity?

8

78



Fig. E 9.1: Searle's apparatus fordetermination of Y

AIMTo determine Young's modulus of the material of a given wire by usingSearle's apparatus.

APPARATUS AND MATERIAL REQUIREDSearle's apparatus, slotted weights, experimentalwire, screw gauge and spirit level.

SEARLE'S APPARATUS

It consists of two metal frames P and Q hingedtogether such that they can move relative to eachother in vertical direction (Fig. E9.1).

A spirit level is supported on a rigid crossbarframe which rests on the tip of a micrometerscrew C at one end and a fixed knife edge K atthe other. Screw C can be moved vertically. Themicrometer screw has a disc having 100 equaldivisions along its circumference. On the side ofit is a linear scale S, attached vertically. If thereis any relative displacement between the twoframes, P and Q, the spirit level no longer remainshorizontal and the bubble of the spirit level isdisplaced from its centre. The crossbar can againbe set horizontal with the help of micrometerscrew and the spirit level. The distance throughwhich the screw has to be moved gives therelative displacement between the two frames.

The frames are suspended by two identical longwires of the same material, from the same rigidhorizontal support. Wire B is called theexperimental wire and wire A acts as a referencewire. The frames, P and Q, are provided withhooks H

1 and H

2 at their lower ends from which

weights are suspended. The hook H1 attached

to the reference wire carries a constant weightW to keep the wire taut.

79

UNIT NAMEEXPERIMENT

To the hook H2 is attached a hanger on which slotted weights canbe placed to apply force on the experimental wire.

PRINCIPLEThe apparatus works on the principle of Hookes’ Law. If l is theextension in a wire of length L and radius r due to force F (=Mg), theYoung's modulus of the material of the given wire, Y, is

=πMgL

Yr l2

PROCEDURE

1. Suspend weights from both the hooks so that the two wires arestretched and become free from any kinks. Attach only the constantweight W on the reference wire to keep it taut.

2. Measure the length of the experimental wire from the point of itssupport to the point where it is attached to the frame.

3. Find the least count of the screw gauge. Determine the diameter ofthe experimental wire at about 5 places and at each place in twomutually perpendicular directions. Find the mean diameter andhence the radius of the wire.

4. Find the pitch and the least count of the miocrometer screwattached to the frame. Adjust it such that the bubble in thespirit level is exactly in the centre. Take the reading of themicrometer.

5. Place a load on the hanger attached to the experimental wire andincrease it in steps of 0.5 kg. For each load, bring the bubble of thespirit level to the centre by adjusting the micrometer screw andthen note its reading. Take precautions to avoid backlash error.

6. Take about 8 observations for increasing load.

7. Decrease the load in steps of 0.5 kg and each time take reading onmicrometer screw as in step 5.

OBSERVATIONSLength of the wire (L) = ...

Pitch of the screw gauge = ...

No. of divisions on the circular scale of the screw gauge = ...

Least count (L.C.) of screw gauge = ...

Zero error of screw gauge = ...

9

80


Table E 9.1: Measurement of diameter of wire

Mean diameter (corrected for zero error) = ...

Mean radius = ...

MEASUREMENT OF EXTENSION l

Pitch of the micrometer screw = ...

No. of divisions on the circular scale = ...

Least count (L.C.) of the micrometer screw = ...

Acceleration due to gravity, g = ...

Table E 9.2: Measurement of extension with load

1

2

3

4

5

6

7

8

S.No.

Load on experimentalwire

M

Mean reading+⎛ ⎞

⎜ ⎟⎝ ⎠x y

2(cm)

Micrometer reading

(kg) Load increasingx

(cm)

Load decreasingy

(cm)

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

a

b

c

d

e

f

g

h

1

2

3

4

5

S.No.

Reading along anydirection

Mean diameter

+=

d dd 1 2

2 (cm)

Reading along perpen-dicular direction

Mainscaleread-ing S(cm)

Circu-lar

scaleread-ing n

Diameterd1 =

S + n × L.C.

Mainscaleread-ing S (cm)

Circu-lar

scaleread-ing n

Diameterd2 =

S + n × L.C.(cm)

81

UNIT NAMEEXPERIMENT

CALCULATIONObservations recorded in Table E 9.2 can be utilised to find extensionof experimental wire for a given load, as shown in Table E 9.3.

Table E 9.3: Calculating extension for a given load

(a – d) +(b – c)+ (c – f)l =∴ Mean

3

= ... cm for 1.5 kg

Young’s modulus, Y, of experimental wire 2 N m2MgLY ... /

r l= =π

GRAPHThe value of Y can also be found by plotting a graph between land Mg. Draw a graph with load on the x-axis and extension on

the y-axis. It should be a straight line. Find the slope Δ

=Δ

l

M of the

line. Using this value, find the value of Y.

RESULTThe Young's modulus Y of the material of the wire

(using half table method) = Y ± ΔY N/m2

(using graph) = Y ± ΔY N/m2

ERRORUncertainty, ΔM, in the measurement of M can be determined by abeam/physical balance using standard weight box/or by using waterbottles of fixed capacity.

S.No.

Mean extension(cm)

Extension l′ for1.5 kg

Load(kg)

Mean extension

0.5 2.0 d – a

1.0 2.5 e – b

1.5 3.0 f – c

9

82


Find the variation in M for each slotted weight of equal mass sayΔM

1 and ΔM

2. Find the mean of these ΔM. This is the uncertainity

(ΔM) in M.

ΔL – the least count of the scale used for measuring L.

Δr – the least count of the micrometer screw gauge used for measuring r.

Δl – least count of the device used for measuring extension.

PRECAUTION

1. Measure the diameter of the wire at different positions, check forits uniformity.

2. Adjust the spirit level only after sufficient time gap following eachloading/unloading.

SOURCES OF ERROR

1. The diameter of the wire may alter while loading.

2. Backlash error of the device used for measuring extension.

3. The nonuniformity in thickness of the wire.

DISCUSSION

Which of the quantities measured in the experiment is likely to havemaximum affect on the accuracy in measurement of Y (Young'smodulus).

SELF ASSESSMENT1. If the length of the wire used is reduced what will be its effect on

(a) extension on the wire and (b) stress on the wire.

2. Use wire of different radii (r1, r2, r3) but of same material in theabove experimental set up. Is there any change in the value ofYoung’s modulus of elasticity of the material? Discuss your result.


1. Repeat the experiment with wires of different materials, if available.

2. Change the length of the experimental wire, of same material andstudy its ef fect on the Young’s modulus of elasticity of the material.


AIMTo find the force constant and effective mass of a helical spring byplotting T 2 - m graph using method of oscillation.

APPARATUS AND MATERIAL REQUIREDLight weight helical spring with a pointer attached at the lowerend and a hook/ring for suspending it from a hanger, (diameterof the spring may be about 1-1.5 cm inside or same as that in aspring balance of 100 g); a rigid support, hanger and five slottedweights of 10 g each (in case the spring constant is of high valueone may use slotted weight of 20 g), clamp stand, a balance, ameasuring scale (15-30 cm) and a stop-watch (with least countof 0.1s).

PRINCIPLE

Spring constant (or force constant) of a spring is given by

Spring constant, K = Restoring Force

Extension

Thus, spring constant is the restoring force per unit extension in thespring. Its value is determined by the elastic properties of the spring.A given object is attached to the free end of a spring which is suspendedfrom a rigid point support (a nail, fixed to a wall). If the object ispulled down and then released, it executes simple harmonicoscillations.

The time period (T ) of oscillations of a helical spring of spring constantK is given by the relation T,

= πm

TK

2 where m is the load that is the mass of the object. If the

spring has a large mass of its own, the expression changes to

+π om m

T =K

2

1010101010

(E 10.1)

(E 10.2)

84


1. Suspend the helical spring SA (having pointerP and the hanger H at its free end A), from arigid support, as shown in Fig. E 10.1.

2. Set the measuring scale, close to the springvertically. Take care that the pointer P movesfreely over the scale without touching it.

3. Find out the least count of the measuring scale(It is usually 1mm or 0.1 cm).

4. Familiarise yourself with the working of thestop-watch and find its least count.

5. Suspend the load or slotted weight with massm

1 on the hanger gently. Wait till the pointer

comes to rest. This is the equilibrium positionfor the given load. Pull the load slightlydownwards and then release it gently so thatit is set into oscillations in a vertical plane

where mo and m define the effective mass of the spring system (thespring along with the pointer and the hanger) and the suspendedobject (load) respectively. The time period of a stiff spring (havinglarge spring constant) is small.

One can easily eliminate the term mo of the spring system appearing

in Eq. (E 10.2) by suspending two different objects (loads) of massesm1 and m2 and measuring their respective periods of oscillations T1

and T2. Then,

10

1 2m m

T =K

+π

and 0 22 2

m mT

K

+= π

Eliminating mo from Eqs. (E 2.3) and (E 2.4), we get

21 2

2 21 2

4 ( )( )

m – mK

T – T

π=

Using Eq. (E 10.5), and knowing the values of m1, m2, T1 and T2, thespring constant K of the spring system can be determined.

PROCEDURE

Fig. E 10.1: Experimental arrangement forstudying spring constant of ahelical spring

(E 10.3)

(E 10.4)

(E 10.5)

85

UNIT NAMEEXPERIMENT 10

about its rest (or equilibrium) position. The rest position (x) ofthe pointer P on the scale is the reference or mean position forthe given load. Start the stop-watch as the pointer P just crossesits mean position (say, from upwards to downwards) andsimultaneously begin to count the oscillations.

6. Keep on counting the oscillations as the pointer crosses themean position (x) in the same direction. Stop the watch aftern (say, 5 to 10) oscillations are complete. Note the time (t)taken by the oscillating load for n oscillations.

7. Repeat this observation alteast thrice and in each occasionnote the time taken for the same number (n) of oscillations.Find the mean time (t

1), for n oscillations and compute the

time for one oscillation, i.e., the time period T1 (= t1/n) ofoscillating helical spring with a load m1.

8. Repeat steps 5 and 6 for two more slotted weights.

9. Calculate time period of oscillation t

Tn

= for each weight and

tabulate your observations.

10.Compute the value of spring constant (K1, K2, K3) for each loadand find out the mean value of spring constant K of the givenhelical spring.

11.The value of K can also be determined by plotting a graph of T 2 vsm with T 2 on y-axis and m on x-axis.

[Note: The number of oscillations, n, should be large enough tokeep the error minimum in measurement of time. One convenientmethod to decide on the number n is based on the least count ofthe stop-watch. If the least count of the stop-watch is 0.1 s. Thento have 1% error in measurement, the minimum time measuredshould be 10.0 s. Hence, the number n for oscillations should beso chosen that oscillating mass takes more than 10.0 s tocomplete them.]

OBSERVATIONSLeast count of the measuring scale = ... mm = ... cm

Least count of the stop-watch = ... s

Mass of load 1, m1 = ... g = ... kg



86


Table E 10.1: Measuring the time period T of oscillations ofhelical spring with load

CALCULATIONSubstitute the values of m

1, m

2, m

3 and T

1, T

2,

T3, in Eq. (E 10.5):

K1 = 4π2 (m

1 – m

2)/(T

12 – T

22);

K2 = 4π2 (m

2 – m

3)/(T

22 – T

32);

K3 = 4π2 (m1 – m3) / (T12 – T3

2)

Compute the values of K1, K2 and K3 and findthe mean value of spring constant K of the givenhelical spring. Express the result in proper SIunits and significant figures.

Alternately one can also find the spring constantand effective mass of the spring from the graphbetween T 2 and m, which is expected to be astraight line as shown in Fig. E 10.2.

The value of spring constant K ( = 4π2/m′) of thehelical spring can be calculated from the slope

m′ of the straight line graph.

From the knowledge of intercept c on y-axis and the slope m, thevalue of effective mass mo (= c/m′ ) of the helical spring can becomputed. Alternatively, the effective mass mo (= –c′ ) of the helicalspring can be directly computed from the knowledge of theintercept c ′ made by the straight line on x-axis.

RESULTSpring constant of the given helical spring = ... N/m-1

Effective mass of helical spring = ... g = ... kg

Error in K, can be calculated from the error in slope

K slope

K slope

Δ Δ=

Fig. E 10.2: Expected graph between T 2 and mfor a helical spring

1

2

1 2 3 Meant (s)

S.No.

Mass ofthe load,

m (kg)

Meanposition ofpointer, x

(cm)

No. of oscilla-tions, (n)

Time for (n)oscillations,

t (s)

T ime period,T = t/n (s)

87

UNIT NAMEEXPERIMENT 10

The error in effective mass m0 will be equal to the error in interceptand the error in slope. Once the error is calculated the result may bestated indicating the error.

DISCUSSION1. The accuracy in determination of the spring constant depends

mainly on the accuracy in measurement of the time period Tof oscillation of the spring. As the time period appears as T 2 inEq. (E 10.5), a small uncertainty in the measurement of T wouldresult in appreciable error in T 2, thereby significantly affectingthe result. A stop-watch with accuracy of 0.1s may bepreferred.

2. Some personal error is always likely to occur in measurement oftime due to delay in starting or stopping the watch.

3. Sometimes air currents may affect the oscillations thereby affectingthe time period. The time period of oscillation may also get affectedif the load is released with a jerk. Take special care that the loadwhile being taken to one side (upwards or downwards) of the rest(or mean) position, is released very gently.

4. The load attached to the spring executes to and fro motion (inSHM) about the mean, equilibrium position. Eqs. (E 10.1)and (E 10.2) hold true for small amplitude of oscillations orsmall extensions of the spring within the elastic limit (Hook’slaw). Take care that initially the load is pulled only through asmall distance before being released gently to let it oscillatevertically.

5. Oscillations of the helical spring are not likely to be absolutelyundamped. Buoyancy of air and viscous drag due to it may slightlyincrease the time period of the oscillations. The effect can be greatlyreduced by taking a small and stiff spring of high density material(such as steel/brass).

6. A rigid support is required for suspending the helical spring. Theslotted weights may not have exactly the same mass as engravedon them. Some error in the time period of its oscillation is likely tocreep in due to yielding (sometimes) of the support and inaccuracyin the accepted value of mass of load.

SELF-ASSESSMENT

1. Two springs A (soft) and B (stiff), loaded with the same mass ontheir hangers, are suspended one by one from the same rigidsupport. They are set into vertical oscillations at different times,and the time period of their oscillations are noted. In which springwill the oscillations be slower?

88


Fig. E 10.3: Springs joined in series Fig. E 10.4: Springs joined in parallel


1. Take three springs with different spring constants K1, K

2, K

3 and join them in series

as shown in Fig. E. 10.3. Determine the time period of oscillation of combined springand check the relation between individual spring constant and combined system.

2. Repeat the above activity with the set up shown in Fig. E. 10.4 and find outwhether there is any difference in the time period and spring constant betweenthe two set ups?

3. What is the physical significance of spring constant 20.5 Nm–1?

4. If possible, measure the mass of the spring. Is this related to the effective mass mo?

2. You are given six known masses (m1, m2 ..., m6), a helicalspring and a stop-watch. You are asked to measure timeperiods (T1, T2, ..., T6) of oscillations corresponding to eachmass when it is suspended from the given helical spring.

(a) What is the shape of the curve you would expect by usingEq. (E 10.2) and plotting a graph between load of mass malong x-axis and T 2 on y-axis?

(b) Interpret the slope, the x-and y-intercepts of the above graph,and hence find (i) spring constant K of the helical spring, and(ii) its effective mass m

o.

[Hint: (a) Eq. (E 10.2), rewritten as: T 2 =(4π2/K) m + (4π2/K) mo, issimilar to the equation of a straight line: y = mx + c, with m as theslope of the straight line and c the intercept on y–axis. The graphbetween m and T 2 is expected to be a straight line AB, as shown inFig. E 10.2. From the above equations given in (a):

Intercept on y–axis (OD), c = (4π2/K) mo; (x = 0, y = c)

Intercept on x-axis (OE), c′=-c/m′ = –mo; (y = 0, x = –c/m′)

Slope, m′ = tan θ = OD/OE = c/c′ = –c/mo = (4π2/K)]

89

UNIT NAMEEXPERIMENT


AIM

To study the variation in volume (V) with pressure (P) for a sample ofair at constant temperature by plotting graphs between P and V, and

between P and 1V

.


Boyle’s law apparatus, Fortin’s Barometer, Vernier Callipers,thermometer, set square and spirit level.

DESCRIPTION AND APPARATUSThe Boyle’s law apparatus consistsof two glass tubes about 25 cm longand 0.5 cm in diameter (Fig. E11.1).One tube AB is closed at one endwhile the other CD is open. The twotubes are drawn into a fine openingat the other end (B and D). The endsB and D are connected by a thickwalled rubber tubing. The glasstube AB is fixed vertically along themetre scale. The other tube CD canbe moved vertically along a verticalrod and may be fixed to it at anyheight with the help of screw S.

The tube CD, AB and rubber tubingare filled with mercury. The closedtube AB traps some air in it. Thevolume of air is proportional to thelength of air column as it is ofuniform cross section.

The apparatus is fixed on ahorizontal platform with a verticalstand. The unit is provided withlevelling screws.

1111111111

Fig. E11.1: Boyle’s law apparatus

90


PROCEDURE(a) Measurement of Pressure:

The pressure of the enclosed air in tube AB is measured by noting thedifference (h) in the mercury levels (X and Y) in the two tubes AB andCD (Fig. E11.2). Since liquid in interconnected vessels have the samepressure at any horizontal level,

P (Pressure of enclosed air) = H ± h

where H is the atmospheric pressure.

Fig. E 11.2 : Pressure of air in tube AB = H + h Fig. E 11.3 : Volume of trapped air in tube AB

(E 11.1)

(b) Measurement of volume of trapped air

In case the closed tube is not graduated.

Volume of air in tube

= Volume of air in length PR – Volume of air in curved portion PQ

Let r be the radius of the tube

Volume of curved portion = volume of the hemisphere of radius r

r r= × π = π 3 31 4 22 3 3

Volume of PQ = πr2 × r = πr3

error in volume r r r= π π = π 3 3 32 1–

3 3

resulting error in length r / r r= π π =3 21 13 3

91

UNIT NAMEEXPERIMENT

correction in length 1 13 3

– r – PQ= =

This should be subtracted from the measured length l .

Boyle’s law: At a constant temperature, the pressure exerted by anenclosed mass of gas is inversely proportional to its volume.

1P

Vα

or PV = constant

Hence the P – V graph is a curve while that of 1–P

V is a straight line.

(c) Measurement of volume of air for a given pressure.

1. Note the temperature of the room with a thermometer.

2. Note the atmospheric pressure using Fortin’s Barometer(Project P-9).

3. Set the apparatus vertically using the levelling screws andspirit level.

4. Slide the tube CD to adjust the mercury level at the same level asin AB. Use set square to read the upper convex meniscus ofmercury.

5. Note the reading of the metre scale corresponding to the top endof the closed tube P and that of level Q where its curvature just

ends. Calculate 1

3 PQ and note it.

6. Raise CD such that the mercury level in tubes AB and CD isdifferent. Use the set square to carefully read the meniscus Xand Y of mercury in tube AB and CD. Note the difference h in themercury level.

7. Repeat the adjustment of CD for 5 more values of ‘h’. This shouldbe done slowly and without jerk. Changing the position of CDwith respect to AB slowly ensures that there is no change intemperature, otherwise the Boyle’s law will not be valid.

8. Use the Vernier Callipers to determine the diameter of the closed

tube AB and hence find ‘r’, its radius 13

PQ = 13

r.

9. Record your observations in the Table E 11.1.

10. Plot graphs (i) P versus V and (ii) P versus 1V

, interpret the graphs.

(E 11.2)

(E 11.3)

11

92


OBSERVATIONS AND CALCULATIONS

1. Room temperature = ... °C.

2. Atmospheric pressure as observed from the Fortins Barometer= ... cm of Hg.

3. For correction in level l due to curved portion of tube AB

(a) Reading for the top of the closed tube AB (P) = ... cm.

Reading where the uniform portion of the tube AB begins (or thecurved portion ends) (Q) = ... cm.

Difference (P – Q) = r = ... cm.

Correction = 1

3r = ...

OR

(b) Diameter of tube AB = d = ... cm.

radius r = 1

2d = ... cm.

correction for level l = 1

3r

RESULT1. Within experimental limits, the graph between P and V is a curve.

2. Within experimental limits, the product PV is a constant (from thecalculation).

Table E 11.1 : Measurement of Pressure and Volume of enclosed air

1

2

3

4

Level ofmercuryin closedtube ABX (cm ofHG)

S.

No.Level ofmercuryin opentubeCDY (cmof Hg)

Pressuredifferenceh = X–Y(cm ofHg)

Pressure ofair in

AB = H ± h(cm of Hg)

Volumeof airXA

1

3–l r

⎛ ⎞⎜ ⎟⎝ ⎠

PV orP × l

1/Vor

1

l

Note: H ± h must be considered according to the levels X and Y takinginto account whether the pressure of air in AB will be more thanatmospheric pressure or less.

93

UNIT NAMEEXPERIMENT

Fig. E11.4 : Graph between Volume, Vand pressure, P

Fig. E11.5 : Graph between 1

V and pressure P

Note that Fig. E 11.4 shows that the graph between P and V is a curve

and that between P and 1

V is a straight line (Fig. E 11.5).

3. The graph P and 1

V is a straight line showing that the pressure of

a given mass of enclosed gas is inversely proportional to its volumeat constant temperature.

PRECAUTIONS

1. The apparatus should be kept covered when not in use.

2. The apparatus should not be shifted in between observations.

3. While measuring the volume of the air, correction for the curvedportion of the closed tube should be taken into account.

4. Mercury used should be clean and not leave any trace on the glass.The open tube should be plugged with cotton wool when not inuse.

5. The set square should be placed tangential to the upper meniscusof the mercury for determining its level.

SOURCES OF ERROR1. The enclosed air may not be dry.

2. Atmospheric pressure and temperature of the laboratory maychange during the course of the experiment.

11

94


3. The closed end of the tube AB may not be hemispherical.

4. The mercury may be oxidised due to exposure to atmosphere.

DISCUSSION

1. The apparatus should be vertical to ensure that the difference inlevel (h) is accurate.

2. The diameter of the two glass tubes may or may not be the samebut the apparatus should be vertical.

3. The open tube CD should be raised or lowered gradually to ensurethat the temperature of the enclosed air remains the same.

4. The readings should be taken in order (above and below theatmospheric pressure). This ensures wider range of consideration,also if they are taken slowly the atmospheric pressure andtemperature over the duration of observation remain the same. Sotime should not be wasted.

5. Why should the upper meniscus of mercury in the two tubesrecorded carefully using a set square?

SELF ASSESSMENT

1. Plot 1

V versus ‘h’ graph and determine the value of

1

V when h = 0.

Compare this to the value of atmospheric pressure. Give a suitableexplanation for your result.

2. Comment on the two methods used for estimation of the volumeof the curved portion of the closed tube. What are the assumptionsmade for the two methods?

3. If the diameter of tube AB is large, why would the estimation ofthe curved portion be unreliable?

4. The apparatus when not in use should be kept covered to avoidcontamination of mercury in the open tube. How will oxidation ofmercury affect the experiment?


1. Tilt apparatus slightly and note the value of ‘h’ for two or threevalues of X and Y.

2. Take a glass U tube. Fill it with water. Pour oil in one arm. Note thedifference in level of water, level of oil and water in the two arms.Deduce the density of oil. What role does atmospheric pressureplay in this experiment?

95

UNIT NAMEEXPERIMENT


Fig.E 12.1: Rise of liquid in a capil-lary tube

AIMTo determine the surface tension of water by capillary rise method.

APPARATUS AND MATERIAL REQUIREDA glass/plastic capillary tube, travelling microscope, beaker, cork withpin, clamps and stand, thermometer, dilute nitric acid solution, dilutecaustic soda solution, water, plumb line.

PRINCIPLE

When a liquid rises in a capillary tube[Fig. E 12.1], the weight of the column of theliquid of density ρ below the meniscus, issupported by the upward force of surfacetension acting around the circumference of thepoints of contact. Therefore

22 rT r h gρπ = π (approx) for water

or2

h grT

ρ=

where T = surface tension of the liquid,

h = height of the liquid column and

r = inner radius of the capillary tube

PROCEDURE

1. Do the experiment in a well-lit place for example, near a windowor use an incandescent bulb.

2. Clean the capillary tube and beaker successively in caustic sodaand nitric acid and finally rinse thoroughly with water.

3. Fill the beaker with water and measure its temperature.

4. Clamp the capillary tube near its upper end, keeping it above thebeaker. Set it vertical with the help of a plumbline held near it.

96


Move down the tube so that its lower end dips into the water inthe beaker.

5. Push a pin P through a cork C, and fix it on another clamp suchthat the tip of the pin is just above the water surface as shown inFig. E 12.1. Ensure that the pin does not touch the capillarytube. Slowly lower the pin till its tip just touches the watersurface. This can be done by coinciding the tip of the pin with itsimage in water.

6. Now focus the travelling microscope M on the meniscus ofthe water in capillary A, and move the microscope until thehorizontal crosswire is tangential to the lowest point of themeniscus, which is seen inverted in M. If there is anydifficulty in focussing the meniscus, hold a piece of paperat the lowest point of the meniscus outside the capillary tubeand focus it first, as a guide. Note the reading of travellingmicroscope.

7. Mark the position of the meniscus on the capillary with a pen.Now carefully remove the capillary tube from the beaker, and thenthe beaker without disturbing the pin.

8. Focus the microscope on the tip of the pin and note the microscopereading.

9. Cut the capillary tube carefully at the point marked on it. Fix thecapillary tube horizontally on a stand. Focus the microscope onthe transverse cross section of the tube and take readings tomeasure the internal diameter of the tube in two mutuallyperpendicular directions.

OBSERVATIONSDetermination of h

Least count (L.C.) of the microscope = ... mm

Table E 12.1 : Measurement of capillary rise

1

2

3

S.

No.

Reading of meniscus h1

(cm)Reading of tip of pin touching

surface of water h2 (cm)

h = h1 – h2

M.S.R.S (cm)

V.S.R.n

h1 = (S +

n × L.C.)

M.S.R.S ′ (cm)

V.S.R.n ′

h2 = (S ′+n′×L.C.)(cm)

97

UNIT NAMEEXPERIMENT

Table E 12.2 : Measurement of diameter of the capillary tube

1

2

3

S.

No.

Reading along adiameter (cm)

Diameterd1 (x2– x1)

Meandiameter

d

Reading alongperpendicular

diameter

Diameterd2 (y2– y1)

Oneend

otherend

Oneend

otherend

(cm) (cm) 1 2+

=2

d d

x1 x2 y1 y2

Mean radius r = ... cm; Temperature of water θ = ... °C;

Density of water at 0° C = ... g cm–3

CALCULATIONSubstitute the value of h and r and ρ g in the formula for T and calculatethe surface tension.

RESULTThe surface tension of water at ... °C = ... ± ... Nm–1

PRECAUTIONS1. To make capillary tube free of contamination, it must be rinsed

first in a solution of caustic soda then with dilute nitric acid andfinally cleaned with water thoroughly.

2. The capillary tube must be kept vertical while dipping it in water.

3. To ensure that capillary tube is sufficiently wet, raise and lowerwater level in container by lifting or lowering the beaker. It shouldhave no effect on height of liquid level in the capillary tube.

4. Water level in the capillary tube should be slightly above the edgeof the beaker/dish so that the edge does not obstruct observations.

5. Temperature should be recorded before and after the experiment.

6. Height of liquid column should be measured from lowest point ofconcave meniscus.

SOURCES OF ERROR1. Inserting dry capillary tube in the liquid can cause gross error in

the measurement of surface tension as liquid level in capillary tubemay not fall back when the level in container is lowered.

12

98



1. Experiment can be performed at different temperatures and effectof temperature on surface tension can be studied.

2. Experiment can be performed by adding some impurities and effectof change in impurity concentration (like adding NaCl or sugar) onsurface tension can be studied.

3. Study the effect of inclination of capillary tube on height of liquidrise in the capillary tube.

2. Surface tension changes with impurities and temperature of theliquid.

3. Non-vertical placement of the capillary tube may introduce errorin the measurement of height of the liquid column in the tube.

4. Improper focussing of meniscus in microscope could cause anerror in measurement of the height of liquid column in thecapillary tube.

DISCUSSION

1. In a fine capillary tube, the meniscus surface may be consideredto be semispherical and the weight of the liquid above the lowest

point of the meniscus as r gρ π313 . Taking this into account, the

formula for surface tension is modified to r

T gr hρ ⎛ ⎞= +⎜ ⎟⎝ ⎠12 3

. More

precise calculation of surface tension can be done using thisformula.

2. If the capillary is dry from inside the water that rises to a certainheight in it will not fall back, so the capillary should be wet frominside. To wet the inside of the capillary tube thoroughly, it is firstdipped well down in the water and raised and clamped.Alternatively, the beaker may be lifted up and placed down.

SELF ASSESSMENT

1. Suppose the length of capillary tube taken is less than the heightupto which liquid could rise. What do you expect if such a tube isinserted inside the liquid? Explain your answer.

2. Two match sticks are floating parallel and quite close to each other.What would happen if a drop of soap solution or a drop of hotwater falls between the two sticks? Explain your answer.

99

UNIT NAMEEXPERIMENT


v

6 rv

g4/3 r3

g4/3 r3 (E 13.2)

(E 13.1)

AIMTo determine the coefficient of viscosity of a given liquid by measuringthe terminal velocity of a spherical body.


A wide bore tube of transparent glass/acrylic (approximately 1.25 mlong and 4 cm diameter), a short inlet tube of about 10 cm length and1 cm diameter (or a funnel with an opening of 1 cm), steel balls of knowndiameters between 1.0 mm to 3 mm, transparent viscous liquid (castoroil/glycerine), laboratory stand, forceps, rubber bands, two rubberstoppers (one with a hole), a thermometer (0-50 °C), and metre scale.

PRINCIPLE

When a spherical body of radius r and density σ falls freely througha viscous liquid of density ρ and viscosity η, with terminal velocity v,then the sum of the upward buoyant force and viscous drag, force F,is balanced by the downward weight of the ball (Fig. E13.1).

= Buoyant force on the ball + viscous force

r g r g rvσ ρ ηπ = π + π3 34 46

3 3

or( ) ( )r g r g

r q

σ ρ σ ρυ

η η

π − −= =

π

32

423

6

where v is the terminal velocity, the constantvelocity acquired by a body while movingthrough viscous fluid under application ofconstant force.

The terminal velocity depends directly on thesquare of the size (diameter) of the sphericalball. Therefore, if several spherical balls ofdifferent radii are made to fall freely throughthe viscous liquid then a plot of v vs r2 wouldbe a straight line as illustrated in Fig. E 13.2.

Fig.E 13.1: Forces acting on aspherical body fallingthrough a viscousliquid with terminalvelocity

100


The shape of this line will give an average value of υ 2r which may be

used to find the coefficient of viscosity η of the given liquid. Thus

( ) ( )( )

22 29 9 slope of line

grg .

v

σ ρη σ ρ

−= − =

= ... Nsm–2 (poise)

The relation given by Eq. (E 13.3) holds good ifthe liquid through which the spherical body fallsfreely is in a cylindrical vessel of radius R >> rand the height of the cylinder is sufficient enoughto let the ball attain terminal velocity. At the sametime the ball should not come in contact withthe walls of the vessel.

PROCEDURE

1.Find the least count of the stop-watch.

2.Note the room temperature, using a thermometer.

3.Take a wide bore tube of transparent glass/acrylic (of diameterabout 4 cm and of length approximately1.25 m). Fit a rubberstopper at one end of the wide tube and ensure that it is airtight.Fill it with the given transparent viscous liquid (say glycerine).Fix the tube vertically in the clamp stand as shown in Fig. E13.3. Ensure that there is no air bubble inside the viscous liquidin the wide bore tube.

4.Put three rubber bands A, B, and C around the wide bore tubedividing it into four portions (Fig. E 13.3), such that AB = BC,each about 30 cm. The rubber band A should be around 40 cmbelow the mouth of the wide bore tube (length sufficient to allowthe ball to attain terminal velocity).

5.Separate a set of clean and dry steel balls of different radii. Theset should include four or five identical steel balls of same knownradii (r

1). Rinse these balls thoroughly with the experimental

viscous liquid (glycerine) in a petridish or a watch glass. Otherwise

Fig.E 13.2: Graph between terminal velocity v, andsquare of radius of ball, r2

(E 13.3)

101

UNIT NAMEEXPERIMENT

these balls may develop air bubble(s) ontheir surfaces as they enter the liquidcolumn.

6.Fix a short inlet tube vertically at theopen end of the wide tube through arubber stopper fixed to it. Alternately onecan also use a glass funnel instead of aninlet tube as shown in Fig. E 13.3. Withthe help of forceps hold one of the ballsof radius r

1 near the top of tube. Allow

the ball to fall freely. The ball, afterpassing through the inlet tube, will fallalong the axis of the liquid column.

7.Take two stop watches and start both ofthem simultaneously as the sphericalball passes through the rubber band A.Stop one the watches as the ball passesthrough the band B. Allow the secondstop-watch to continue and stop it whenthe ball crosses the band C.

8.Note the times t1 and t2 as indicated bythe two stop watches, t1 is then the timetaken by the falling ball to travel from Ato B and t2 is the time taken by it in falling from A to C. If terminalvelocity had been attained before the ball crosses A, then t2 = 2 t1.If it is not so, repeat the experiment with steel ball of same radiiafter adjusting the positions of rubber bands.

9.Repeat the experiment for other balls of different diameters.

10.Obtain terminal velocity for each ball.

11.Plot a graph between terminal velocity, v and square of the radiusof spherical ball, r2. It should be a straight line. Find the slope ofthe line and hence determine the coefficient of viscosity of theliquid using the relation given by Eq. (E 13.3).

OBSERVATIONS

1. Temperature of experimental liquid (glycerine) θ = ...°C.

2. Density of material of steel balls σ = ... kg m-3

3. Density of the viscous liquid used in the tube = ... kgm–3

4. Density of experimental viscous liquid ρ = ... kg m-3

Fig.E 13.3: Steel ball falling along theaxis of the tube filled witha viscous liquid.

13

102


1

2

3

S.

No.

Diameter ofspherical

balls

Time taken for covering distanceh = ... cm between rubber bands

Square ofthe

radius ofthe balls

r 2

(m2)

TerminalVelocity

hv

t=

(m–1)d r =d/2

(cm) (m)

A and B

t1

A and C

t2

B and C

t3= t2– t1

Mean time

1 3

2

t tt

+=

(s) (s) (s) (s)

5. Internal diameter of the wide bore tube =... cm = ... m

6. Length of wide bore tube = ... cm = ... m

7. Distance between A and B = ... cm = ... m

8. Distance between B and C = ... cm = ... m

Average distance h between two consecutive rubber bands= ... cm = ... m

9. Acceleration due to gravity at the place of experiment, = ... gms–2

10.Least count of stop-watch = ... s

Table E 13.1: Measurement of time of fall of steel balls

GRAPHPlot a graph between r2 and v taking r2 along x -axis and v alongy-axis. This graph will be similar to that shown in Fig. E 13.2.

Slope of line 2

RTST

v

r=

So( )

( )2

29 slope of line

r gσ ρη

−=

Error2 slope

sloper

r

ηη

Δ Δ Δ= +

Standard value of η = ... Nsm–2

% error in η = ... %

RESULT

The coefficient of viscosity of the given viscous liquid at temperatureθ °C = ... ± ... Nsm–2

103

UNIT NAMEEXPERIMENT

PRECAUTIONS AND SOURCES OF ERROR

1. In order to minimise the effects, although small, on the value ofterminal velocity (more precisely on the value of viscous drag, forceF), the radius of the wide bore tube containing the experimentalviscous liquid should be much larger than the radius of the fallingspherical balls.

2. The steel balls should fall without touching the sides of the tube.

3. The ball should be dropped gently in the tube containing viscous/liquid.

DISCUSSION

1. Ensure that the ball is spherical. Otherwise formula used forterminal velocity will not be valid.

2. Motion of falling ball must be translational.

3. Diameter of the wide bore tube should be much larger than thatof the spherical ball.

SELF ASSESSMENT1. Do all the raindrops strike the ground with the same velocity

irrespective of their size?

2. Is Stokes’ law applicable to body of shapes other than spherical?

3. What is the effect of temperature on coefficient of viscosity ofa liquid?


1. Value of η can be calculated for steel balls of dif ferent radiiand compared with that obtained from the experiment.

2. To find viscosity of mustard oil [Hint: Set up the apparatusand use mustard oil instead of glycerine in the wide boretube].

3. To check purity of milk [Hint: Use mustard oil in the talltube. Take an eye dropper, fill milk in it. Drop one drop ofmilk in the oil at the top of the wide bore tube and find itsterminal velocity. Use the knowledge of coefficient of viscosityof mustard oil to calculate the density of milk].

4. Study the effect of viscosity of water on the time of rise of airbubble [Hint: Use the bubble maker used in an aquarium.Place it in the wide bore tube. Find the terminal velocity ofrising air bubble].

13

104



Stirrer

Lid

Double walledcontainer

Calorimeter

-10

010

2030

4050

6070

8090

100

-10

010

2030

4050

6070

8090

100 T1T2

Fig.E 14.1: Newton's law of cooling apparatus

AIMTo study the relationship between the temperature of a hot body andtime by plotting a cooling curve.

APPARATUS AND MATERIAL REQUIREDNewton’s law of cooling apparatus that includes a copper calorimeterwith a wooden lid having two holes for inserting a thermometer and astirrer and an open double – walled vessel, two celsius thermometers(each with least count 0.5 oC or 0.1 oC), a stop clock/watch, a heater/burner, liquid (water), a clamp stand, two rubber stoppers with holes,strong cotton thread and a beaker.


As shown in Fig. E 14.1, the law of coolingapparatus has a double walled container, whichcan be closed by an insulating lid. Water filledbetween double walls ensures that the temperatureof the environment surrounding the calorimeterremains constant. Temperature of the liquid andthe calorimeter also remains constant for a fairlylong period of time so that temperaturemeasurement is feasible. Temperature of water incalorimeter and that of water between double wallsof container is recorded by two thermometers.

THEORYThe rate at which a hot body loses heat is directlyproportional to the difference between thetemperature of the hot body and that of itssurroundings and depends on the nature ofmaterial and the surface area of the body. This is

Newton’s law of cooling.

For a body of mass m and specific heat s, at its initial temperature θhigher than its surrounding’s temperature θ

o, the rate of loss of heat

105

UNIT NAMEEXPERIMENT

is ddQ

t, where dQ is the amount of heat lost by the hot body to its

surroundings in a small interval of time.

Following Newton’s law of cooling we have

Rate of loss of heat, ddQ

t = – k (θ – θo)

AlsoddQ

t= ms

ddt

θ⎛ ⎞⎜ ⎟⎝ ⎠

Using Eqs. (E 14.1) and (E 14.2), the rate of fall of temperature is given by

ddt

θ = –

k

ms (θ – θo)

where k is the constant of proportionality and k ′ = k/ms is anotherconstant (The term ms also includes the water equivalent of thecalorimeter with which the experiment is performed). Negative signappears in Eqs. (E 14.2) and (E 14.3) because loss of heat impliestemperature decrease. Eq. (E 14.3) may be re written as

dθ = - k′ (θ – θo) dt

On integrating, we get

d

o

k ' dt–

θθ θ

= −∫ ∫

or ln (θ – θo) = log

e (θ – θ

o) = – k′t + c

or ln (θ – θo) = 2.303 log

10 (θ – θ

o) = – k′t + c

where c is the constant of integration.

Eq. (E 14.4) shows that the shape of a plot between log10 (θ – θo) and twill be a straight line.

PROCEDURE1. Find the least counts of thermometers T1 and T2. Take some water

in a beaker and measure its temperature (at room temperature θo)

with one (say T1) of the thermometers.

2. Examine the working of the stop-watch/clock and find its least count.

3. Pour water into the double- walled container (enclosure) at roomtemperature. Insert the other thermometer T

2 in water contained

in it, with the help of the clamp stand.

4. Heat some water separately to a temperature of about 40 oC abovethe room temperature θ

o. Pour hot water in calorimeter up to its top.

(E 14.1)

(E 14.2)

(E 14.3)

(E 14.4)

14

106


5. Put the calorimeter, with hot water, back in the enclosure andcover it with the lid having holes. Insert the thermometer T

1 and

the stirrer in the calorimeter through the holes provided in thelid, as shown in Fig. E14.1.

6. Note the initial temperature of the water between enclosure ofdouble wall with the thermometer T2, when the difference ofreadings of two thermometers T1 and T2 is about 30 oC. Note theinitial reading of the thermometer T

1.

7. Keep on stirring the water gently and constantly. Note thereading of thermometer T1, first after about every half a minute,then after about one minute and finally after two minutesduration or so.

8. Keep on simultaneously noting the reading of the stop-watch andthat of the thermometer T

1, while stirring water gently and

constantly, till the temperature of water in the calorimeter falls toa temperature of about 5 oC above that of the enclosure. Note thetemperature of the enclosure, by the thermometer T2.

9. Record observations in tabular form. Find the excess oftemperature (θ − θ

ο) and also log

10 (θ − θ

ο) for each reading, using

logarithmic tables. Record these values in the correspondingcolumns in the table.

10.Plot a graph between time t, taken along x-axis and log10

(θ – θo)

taken along y-axis. Interpret the graph.

OBSERVATIONSLeast count of both the identical thermometers = ... °C

Least count of stop-watch/clock = ... s

Initial temperature of water in the enclosure θ1 = ... °C

Final temperature of water in the enclosure θ2 = ... oC

Mean temperature of the water in the enclosure θο = (θ1 + θ2)/2 = ... oC

Table E 14.1: Measuring the change in temperature of water with time

1

2

.

.

20

S.No.

Time (t)(s)

Temperatureof hot waterθ °C

Excess Temperatureof hot water (θ – θ0)°C

log10 (θ – θ0)

107

UNIT NAMEEXPERIMENT

PLOTTING GRAPH(i) Plot a graph between (θ – θ

o) and t as shown in Fig. E 14.2 taking t

along x-axis and (θ – θo) along y-axis. This is called cooling curve.

(ii) Also plot a graph between log10 (θ - θo) and time t, as shown in Fig.E 14.3 taking time t along x-axis and log

10 (θ - θ

o) along y-axis.

Choose suitable scales on these axes. Identify the shape of thecooling curve and the other graph.

Fig.E 14.2: Graph between (θ – θo) and t forcooling

Fig.E 14.3: Graph between log10 (θ – θo) and t

RESULTThe cooling curve is an exponential decay curve (Fig. E 14.2). It isobserved from the graph that the logarithm of the excess of temperatureof hot body over that of its surroundings varies linearly with time asthe body cools.

PRECAUTIONS1. The water in the calorimeter should be gently stirred continuously.

2. Ideally the space between the double walls of the surroundingvessel should be filled with flowing water to make it an enclosurehaving a constant temperature.

3. Make sure that the openings for inserting thermometers are airtight and no heat is lost to the surroundings through these.

4. The starting temperature of water in the calorimeter should beabout 30°C above the room temperature.

SOURCES OF ERROR

1. Some personal error is always likely to be involved due to delay instarting or stopping the stop-watch. Take care in starting andstopping the stop-watch.

14

108



1. Find the slope and intercept on y-axis of the straight line graph (Fig.E 14.2) you have drawn. Determine the value of constant k and theconstant of integration c from this graph.

[Hint: Eq. (E 14.4) is similar to the equation of a straight line: y =m′ x + c′ , with m′ as the slope of the straight line and c′ theintercept on y-axis. It is clear m′ = k′/2.303 and c′ = c' × 2.303.]

2. The cooling experiment is performed with the calorimeter, filled withsame volume of water and turpentine oil successively, by maintainingthe same temperature difference between the calorimeter and thesurrounding enclosure. What ratio of the rates of heat loss wouldyou expect in this case?

2. The accuracy of the result depends mainly on the simultaneousmeasurement of temperature of hot water (decrease intemperature being fast in the beginning and then comparativelyslower afterwards) and the time. Take special care while readingthe stop-watch and the thermometer simultaneously.

3. If the opening for the thermometer is not airtight, some loss of heatcan occur.

4. The temperature of the water in enclosure is not constant.

DISCUSSION

Each body radiates heat and absorbs heat radiated by the other. Thewarmer one (here the calorimeter) radiates more and receives less.Radiation by surface occurs at all temperatures. Higher thetemperature difference with the surroundings, higher is rate of heatradiation. Here the enclosure is at a lower temperature so it radiatesless but receives more from the calorimeter. So, finally the calorimeterdominates in the process.

SELF ASSESSMENT

1. State Newton's law of cooling and express this law mathematically.

2. Does the Newton’s law of cooling hold good for all temperaturedifferences?

3. How is Newton's law of cooling different from Stefan's law of heatradiation?

4. What is the shape of cooling curve?

5. Find the specific heat of a solid/liquid using Newton's law ofcooling apparatus.

109

UNIT NAMEEXPERIMENT


Fig. E 15.1: A Sonometer

(E 15.1)

AIM

(i) To study the relation between frequency and length of a givenwire under constant tension using a sonometer.

(ii) To study the relation between the length of a given wire and tensionfor constant frequency using a sonometer.


Sonometer, six tuning forksof known frequencies, metrescale, rubber pad, paperrider, hanger with half-kilogram weights, woodenbridges.

SONOMETER

It consists of a longsounding board or a hollowwooden box W with a peg Gat one end and a pulley atthe other end as shown in Fig E 15.1. One end of a metal wire S isattached to the peg and the other end passes over the pulley P. Ahanger H is suspended from the free end of the wire. By placing slottedweights on the hanger tension is applied to the wire. By placing twobridges A and B under the wire, the length of the vibrating wire can befixed. Position of one of the bridges, say bridge A is kept fixed so thatby varying the position of other bridge, say bridge B, the vibratinglength can be altered.

PRINCIPLE

The frequency n of the fundamental mode of vibration of a string isgiven by

12

Tn =

l m

where m = mass per unit length of the string

l = length of the string between the wedges

110


T = Tension in the string (including the weight of thehanger) = Mg

M = mass suspended, including the mass of the hanger

(a) For a given m and fixed T,

1n

lα or n l = constant.

(b) If frequency n is constant, for a given wire (m isconstant),

Tl

is constant. That is l 2 ∝ T.

Fig. E 15.2: Variation of resonant lengthwith frequency of tuning fork

(i) Variation of frequency with length

PROCEDURE1. Set up the sonometer on the table and clean the groove on the

pully to ensure that it has minimum friction. Stretch the wire byplacing a suitable load on the hanger.

2. Set a tuning fork of frequency n1 into vibrations by striking itagainst the rubber pad and hold it near one of your ears. Pluckthe sonometer wire and compare the two sounds, one producedby the tuning fork and the other by the plucked wire. Make a noteof difference between the two sounds.

3. Adjust the vibrating length of the wire by sliding the bridge B tillthe two sounds appear alike.

4. For final adjustment, place a small paper rider R inthe middle of wire AB. Sound the tuning fork andplace its shank stem on the bridge A or on thesonometer box. Slowly adjust the position of bridgeB till the paper rider is agitated violently, whichindicates resonance.

The length of the wire between A and B is the resonantlength such that its frequency of vibration of thefundamental mode equals the frequency of the tuningfork. Measure this length with the help of a metre scale.

5. Repeat the above procedures for other five tuningforks keeping the load on the hanger unchanged. Plota graph between n and l (Fig. E 15.2)

6. After calculating frequency, n of each tuning fork, plota graph between n and 1/l where l is the resonatinglength as shown in Fig. E 15.3.

Fig. E 15.3: Variation of 1/l with n

111

UNIT NAMEEXPERIMENT

OBSERVATIONS (A)

Tension (constant) on the wire (weight suspended from the hangerincluding its own weight) T = ... N

Table E 15.1: Variation of frequency with length

CALCULATIONS AND GRAPH

Calculate the product nl for each fork. and, calculate the reciprocals, 1l

of the resonating lengths l. Plot 1l

vs n, taking n along x axis and 1l

along y axis, starting from zero on both axes. See whether a straightline can be drawn from the origin to lie evenly between the plotted points.

RESULT

Check if the product n l is found to be constant and the graph of 1

l vs n

is also a straight line. Therefore, for a given tension, the resonant lengthof a given stretched string varies as reciprocal of the frequency.

DISCUSSION

1. Error may occur in measurement of length l. There is always anuncertainty in setting the bridge in the final adjustment.

2. Some friction might be present at the pulley and hence the tensionmay be less than that actually applied.

3. The wire may not be of uniform cross section.

(ii) Variation of resonant length with tension for constantfrequency

1. Select a tuning fork of a certain frequency (say 256 Hz) and hanga load of 1kg from the hanger. Find the resonant length as before.

n1

Frequency n oftuning fork (Hz)

R e s o n a t i n glength l (cm)

nl (Hz cm)

l–11

(cm )

n2

n3

n4

n5

n6

15

112


2. Increase the load on the hanger in steps of 0.5 kg and eachtime find the resonating length with the same tuning fork.Do it for at least four loads.

3. Record your observations.

4. Plot graph between l 2 and T as shown in Fig. E 15.4.

OBSERVATIONS (B)

Frequency of the tuning fork = ... Hz

Table E 15.2: Variation of resonant length with tensionFig. E 15.4: Graph between l2

and T

CALCULATIONS AND GRAPH

Calculate the value of 2T

l for the tension applied in each case.

Alternatively, plot a graph of l 2 vs T taking l 2 along y-axis and Talong the x-axis.

RESULT

It is found that value of T/l 2 is constant within experimental error.The graph of l 2 vs T is found to be a straight line. This shows that

l 2 α T or l Tα .

Thus, the resonating length varies as square root of tension for agiven frequency of vibration of a stretched string.

PRECAUTIONS

1. Pulley should be frictionless ideally. In practice friction at the pulleyshould be minimised by applying grease or oil on it.

2. Wire should be free from kinks and of uniform cross section,ideally. If there are kinks, they should be removed by stretchingas far as possible.

Tension applied T(including weight ofthe hanger) (N)

Resonating length lof the wire

T/l2 (N cm–2)

l2 (cm2)

113

UNIT NAMEEXPERIMENT

3. Bridges should be perpendicular to the wire, its height should beadjustes so that a node is formed at the bridge.

4. Tuning fork should be vibrated by striking its prongs against asoft rubber pad.

5. Load should be removed after the experiment.

SOURCES OF ERROR1. Pulley may not be frictionless.

2. Wire may not be rigid and of uniform cross section.

3. Bridges may not be sharp.

DISCUSSION1. Error may occur in measurement of length l. There is always an

uncertainty in setting the bridge in the final adjustment.

2. Some friction might be present at the pulley and hence the tensionmay be less than that actually applied.

3. The wire may not be of uniform cross section.

4. Care should be taken to hold the tuning fork by the shank only.

SELF ASSESSMENT1. What is the principle of superposition of waves?

2. What are stationary waves?

3. Under what circumstances are stationary waves formed?

4. Identify the nodes and antinodes in the string of your sonometer.

5. What is the ratio of the first three harmonics produced in a stretchedstring fixed at two ends?

6. Keeping material of wire and tension fixed, how will the resonantlength change if the diameter of the wire is increased?


1. Take wires of the same material but of three different diametersand find the value of l for each of these for a given frequency, nand tension, T .

2. Plot a graph between the value of m and 12l

obtained, in 1 above,

with m along X axis.

3. Pluck the string of an stringed musical instrument like a sitar, voilinor guitar with different lengths of string for same tension or samelength of string with different tension. Observe how the frequencyof the sound changes.

15

114



Fig. E 16.1 : Formation of standingwave in glass tube ABclosed at one end

A A

(a) (b)

B B

(E 16.1)

AIMTo determine the velocity of sound in air at room temperature using aresonance tube.

APPARATUS AND MATERIAL REQUIREDResonance tube apparatus, a tuning fork of known frequency(preferably of 480 Hz or 512 Hz), a rubber pad, a thermometer, spiritlevel, a set-square, beaker and water.

PRINCIPLEWhen a vibrating tuning fork of known frequency ν isheld over the top of an air column in a glass tube AB (Fig.E 16.1), a standing wave pattern could be formed in thetube. Under the right conditions, a superposition betweena forward moving and reflected wave occurs in the tube tocause resonance. This gives a very noticeable rise in theamplitude, or loudness, of the sound. In a closed organpipe like a resonance tube, there is a zero amplitude pointat the closed end (Fig. E 16.2). For resonance to occur, anode must be formed at the closed end and an antinodemust be formed at the open end. Let the first loud soundbe heard at length l

1 of the air column [Fig. E 16.2(a)].

That is, when the natural frequency of the air column oflength l

1 becomes equal to the natural frequency of the

tuning fork, so that the air column vibrates with themaximum amplitude. In fact the length of air columnvibrating is slightly longer than the length of the air columnin tube AB. Thus,

14 = l + e

λ

where e (= 0.6 r, where r = radius of the glass tube) is the end correctionfor the resonance tube and λ is the wave-length of the sound producedby the tuning fork.

Now on further lowering the closed end of the tube AB, let the secondresonance position be heard at length l2 of the air column in the tube

115

UNIT NAMEEXPERIMENT

[Fig. E 16.2(b)]. This length l2 wouldapproximately be equal to three quarters ofthe wavelength. That is,

(E 16.2) 2

34

= l + eλ

Subtracting Eq. (E 16.l) from Eq. (E 16.2)gives

(E. 16.3) λ = 2 (l2–l1 )

Thus, the velocity of sound in air atroom temperature (v = ν λ) would bev = 2 ν ( l2 – l1) .

PROCEDUREADJUSTMENT OF RESONANCE TUBE

The apparatus usually consists of a narrow glass tube about a metrelong and 5 cm in diameter, rigidly fixed in its vertical position with awooden stand. The lower end of this tube is attached to a reservoir bya rubber tube. Using a clamp, the reservoir can be made to slide upor down along a vertical rod. A pinch cock is provided with the rubbertube to keep the water level (or the length of air column) fixed in thetube. A metre scale is also fixed along the tube. The whole apparatusis fixed on a horizontal wooden base that can be levelled using thescrews provided at the bottom. Both the reservoir and tube containwater. When reservoir is raised the length of the air column in thetube goes down, and when it is lowered the length of the air column inthe tube goes up. Now:–

1. Set the resonance tube vertical with the help of a spirit level andlevelling screws provided at the bottom of the wooden base of theapparatus.

2. Note the room temperature with a thermometer.

3. Note the frequency ν of given tuning fork.

4. Fix the reservoir to the highest point of the vertical rod with thehelp of clamp.

De--termination of First Resonance Position

5. Fill the water in the reservoir such that the level of water in thetube reaches up to its open end.

6. Close the pinch cock and lower down the position of reservoir onthe vertical rod.

7. Gently strike the given tuning fork on a rubber pad and putit nearly one cm above the open end of the tube. Keep both the

Fig. E 16.2: Vibrations in a resonance tube

l1

(a)

l2

(b)

16

116


prongs of the tuning fork parallel to the ground and lying oneabove the other so that the prongs vibrate in the vertical plane.Try to listen the sound being produced in the tube. It may notbe audible in this position.

8. Slowly loosen the pinch cock to let the water level fall in thetube very slowly. Keep bringing the tuning fork near the openend of the resonance tube, notice the increasing loudness ofthe sound.

9. Repeat steps 7 and 8 till you get the exact position of waterlevel in the tube for which the intensity of sound being producedin the tube is maximum. This corresponds to the first resonanceposition or fundamental node, if the length of air column isminimum. Close the pinch cock at this position and note theposition of water level or length l1 of air column in the tube[Fig. E 16.2]. This is the determination of first resonanceposition while the level of water is falling in the tube.

10. Repeat steps (5) to (9) to confirm the first resonance position.

11. Next find out the first resonance position by gradually raisingthe level of water in resonance tube, and holding the vibratingtuning fork continuously on top of its open end. Fix the tubeat the position where the sound of maximum intensity is heard.

Determination of Second Resonance Position

12. Lower the position of the water level further in the resonance tubeby sliding down the position of reservoir on the vertical stand andopening the pinch cock till the length of air column in the tubeincreases about three times of the length l1.

13. Find out the second resonance position and determine the lengthof air column l

2 in the tube with the same tuning fork having

frequency ν1 and confirm the length l2 by taking four readings,two when the level of water is falling and the other two when thelevel of water is rising in the tube.

14. Repeat steps (5) to (13) with a second tuning fork having frequencyν

2 and determine the first and second resonance positions.

15. Calculate the velocity of sound in each case.

OBSERVATIONS

1. Temperature of the room θ = ... o C

2. Frequency of first tuning fork , ν1

= ... Hz

3. Frequency of second tuning fork, ν2 = ... Hz

117

UNIT NAMEEXPERIMENT

CALCULATIONS(i) For first tuning fork having frequency ν

1 = ... Hz

Velocity of sound in air v1 = 2 ν1 (l2– l1) = ... ms–1

(ii) For second tuning fork having frequency ν2 = ... Hz

Velocity of sound in air v2 = 2ν2 (l2– l1) = ... ms–1

Obtain the mean velocity v of sound in air.

RESULTThe velocity of sound v in air at room temperature is

11 2 ms2

–v + v= ...

PRECAUTIONS1. The resonance tube should be kept vertical using the levelling

screws.

2. The experiment should be performed in a quiet atmosphere sothat the resonance positions may be identified properly.

3. Striking of tuning fork on rubber pad must be done very gently.

4. The lowering and raising of water level in the resonance tube shouldbe done very slowly.

5. The choice of frequencies of the tuning forks being used should besuch that the two resonance positions may be achieved in the aircolumn of the resonance tube.

6. The vibrating tuning fork must be kept about 1 cm above the topof the resonance tube. In any case it should not touch the walls ofthe resonance tube.

Table E 16.1: Determination of length of the resonant air columns

Frequency oftuning fork

used

S.No.

length l1 for the firstresonance position of

the tube

length l2 for thesecond resonanceposition of the tube

Waterlevel isfalling

Waterlevel isrising

Meanlength,l1 cm

Waterlevel isfalling

Waterlevel isrising

Meanlength,l2 cm

ν1 = ... Hz

ν2 = ... Hz

1212

16

118


7. The prongs of the vibrating tuning fork must be kept parallel tothe ground and keeping one over the other so that the vibrationsreaching the air inside the tube are vertical.

8. Room temperature during the performance of experiment shouldbe measured two to three times and a mean value should be taken.

SOURCES OF ERROR1. The air inside the tube may not be completely dry and the presence

of water vapours in the air column may exhibit a higher value ofvelocity of sound.

2. Resonance tube must be of uniform area of cross-section.

3. There must be no wind blowing in the room.

DISCUSSION1. Loudness of sound in second resonance position is lower than the

loudness in first resonance. We determine two resonance positionsin this experiment to apply end correction. But the experimentcan also be conducted by finding first resonance position onlyand applying end correction in resonating length as e = 0.6 r.

2. For a given tuning fork, change in the resonating length of aircoloumn in 2nd resonance does not change the frequency,wavelength or velocity of sound. Thus, the second resonance isnot the overtone of first resonance.

SELF ASSESSMENT1. Is the velocity of sound temperature dependent? If yes, write the

relation.

2. What would happen if resonance tube is not vertical?

3. Name the phenomenon responsible for the resonance in thisexperiment.

4. Write two other examples of resonance of sound from day today life.


1. Calculate the end correction in the resonance tube.

2. Compare the end correction required for the resonance tubes ofdifferent diameters and study the relation between the end correctionand the diameter of the tube.

3. Perform the same experiment with an open pipe.

119

UNIT NAMEEXPERIMENT


(E 17.1)

AIMTo determine the specific heat capacity of a given (i) and solid(ii) a liquid by the method of mixtures.

APPARATUS AND MATERIAL REQUIREDCopper calorimeter with lid, stirrer and insulating cover (the lidshould have provision to insert thermometer in addition to thestirrer), two thermometers (0 °C to 100 °C or 110 °C with a leastcount of 0.5 °C), a solid, preferably metallic (brass/copper/steel/aluminium) cylinder which is insoluble in given liquid and water,given liquid, two beakers (100 mL and 250 mL), a heating device(heater/hot plate/gas burner); physical balance, spring balance withweight box (including fractional weights), a piece of strong non-flexible thread (25-30 cm long), water, laboratory stand, tripod standand wire gauze.

PRINCIPLE / THEORYFor a body of mass m and specific heat s, the amount of heat Qlost/gained by it when its temperature falls/rises by Δt is given by

ΔQ = ms Δt

Specific heat capacity: It is the amount of heat required to raise thetemperature of unit mass of a substance through 1°C. Its S.I unit isJkg–1 K–1.

Principle of Calorimetry: If bodies of different temperatures arebrought in thermal contact, the amount of heat lost by the body athigher temperature is equal to the amount of heat gained by the bodyat lower temperature, at thermal equilibrium, provided no heat is lostto the surrounding.

(a) Specific heat capacity of given solid by method of mixtures

PROCEDURE1. Set the physical balance and make sure there is no zero error.

2. Weigh the empty calorimeter with stirrer and lid with the physicalbalance/spring balance. Ensure that calorimeter is clean and dry.

120


Note the mass m1 of the calorimeter. Pour the given water in the

calorimeter. Make sure that the quantity of water taken would besufficient to completely submerge the given solid in it. Weigh thecalorimeter with water along with the stirrer and the lid and noteits mass m2. Place the calorimeter in its insulating cover.

3. Dip the solid in water and take it out. Now shake it to removewater sticking to its surface. Weigh the wet solid with the physicalbalance and note down its mass m3.

4. Tie the solid tightly with the thread at its middle. Make sure that itcan be lifted by holding the thread without slipping.

Place a 250 mL beaker on the wire gauze kept on a tripod stand asshown in the Fig. E 17.1(a). Fill the beaker up to the half withwater. Now suspend the solid in the beaker containing water bytying the other end of the thread to a laboratory stand. The solidshould be completely submerged in water and should be atleast0.5 cm below the surface. Now heat the water with the solidsuspended in it [Fig. E 17.1 (a)].

Fig. E 17.1: Experimental setup for determining specific heat of a given solid

5. Note the least count of the thermometer. Measure the temperatureof the water taken in the calorimeter. Record the temperature t1 ofthe water.

6. Let the water in the beaker boil for about 5-10 minutes. Nowmeasure the temperature t2 of the water with the other thermometerand record the same. Holding the solid with the thread tied to it,

121

UNIT NAMEEXPERIMENT

remove it from the boiling water, shake it to remove water stickingon it and quickly put it in the water in the calorimeter and replacethe lid immediately (Fig. E 17.1 (b)). Stir the water with the stirrer.Measure the temperature of the water once equilibrium is attained,that is, temperature of the mixture becomes constant. Record thistemperature as t3.

OBSERVATIONSMass of the empty calorimeter with stirrer (m

1) = ... g

Mass of the calorimeter with water (m2) = ... g

Mass of solid (m3) = ... g

Initial temperature of the water (t1) = ... °C = ... K

Temperature of the solid in boiling water (t2) = ... °C = ... K

Temperature of the mixture (t3) = ... °C

Specific heat capacity of material of calorimeter s1 = ... Jkg–1 °C–1 (Jkg–1 K–1)

Specific heat capacity of water (s) = ... Jkg–1 K–1

CALCULATIONS

1. Mass of the water in calorimeter (m2 – m1) = ... g = ... kg

2. Change in temperature of liquid and calorimeter (t3 – t

1) = ... °C

3. Change in temperature of solid (t2 – t

3) == ... °C

Heat given by solid in cooling from t2 to t

3.

= Heat gained by liquid in raising its temperature from t1 to t3 +heat gained by calorimeter in raising its temperature from t

1 to t

3.

m3s

o (t

2 – t

3) = (m

2 – m

1) s (t

2 – t

1) + m

1s

1 (t

3 – t

1)

( )2 1 2 1 1 1 3 1o

3 2 3

m – m s(t t ) m s (t t )s

m (t t )

− + −=

− = ... J kg–1 °C–1

(b) Specific heat capacity of given liquid by method of mixtures

PROCEDURE1. Set the phyiscal balance and make sure there is no zero error.

2. Weigh the empty calorimeter with stirrer and lid with thephysical balance/spring balance. Ensure that calorimeter isclean and dry. Note the mass m1 of the calorimeter. Pour the

17

122


given liquid in the calorimeter. Make sure that the quantity ofliquid taken would be sufficient to completely submerge thesolid in it. Weigh the calorimeter with liquid along with thestirrer and the lid and note its mass m2. Place the calorimeterin its insulating cover.

3. Take a metallic cylinder whose specific heat capacity is known.Dip it in water in a container and shake it to remove the watersticking to its surface. Weigh the wet solid with the physical balanceand note down its mass m3.

4. Tie the solid tightly with the thread at its middle. Make sure that itcan be lifted by holding the thread without slipping.

Place a 250 mL beaker on the wire gauze kept on a tripod standas shown in Fig. E 17.1(a). Fill the beaker up to half with water.Now suspend the solid in the beaker containing water by tyingthe other end of the thread to a laboratory stand. The solid shouldbe completely submerged in water and should be atleast 0.5 cmbelow the surface. Now heat the water with the solid suspended init [Fig. E 17.1(a)].

5. Note the least count of the thermometer. Measure the temperatureof the water taken in the calorimeter. Record the temperature t1 ofthe water.

6. Let the liquid in the beaker boil for about 5-10 minutes. Nowmeasure the temperature t

2 of the liquid with the other thermometer

and record the same. Holding the solid with the thread tied to itremove it from the boiling water, shake it to remove water stickingon it and quickly put it in the liquid in the calorimeter and replacethe lid immediately [Fig. E 17.1(b)]. Stir it with the stirrer. Measurethe temperature of the liquid once equilibrium is attained, that is,temperature of the mixture becomes constant. Record thistemperature as t

3.

OBSERVATIONSMass of the empty calorimeter with stirrer (m1) = ... g

Mass of the calorimeter with liquid (m2) = ... g

Mass of solid (m3) = ... g

Initial temperature of the liquid (t1) = °C = ... K

Temperature of the solid in boiling water (t2) = °C = ... K

Temperature of the mixture (t3) = °C = ... K

Specific heat capacity of material of calorimeter s1

= ... Jkg–1 °C–1 (Jkg–1 K–1)

Specific heat capacity of solid (s0) = ... Jkg–1 K–1

123

UNIT NAMEEXPERIMENT

CALCULATIONS1. Mass of the liquid in calorimeter (m

2 – m

1) = ... g = ... kg

2. Change in temperature of liquid and calorimeter (t3 – t

1) = ... °C

3. Change in temperature of solid (t2 – t3) = ... °C

Heat given by solid in cooling from t2 to t3.

= Heat gained by liquid in raising its temperature from t1 to t

3 +

heat gained by calorimeter in raising its temperature from t1 to t3.

m3so (t2 – t3) = (m2 – m1) s (t2 – t1) + m1s1 (t3 – t1)

3 0 2 3 1 1 3 1

2 1 2 1

m s (t t ) – m s (t t )s

(m m ) (t t )

− −=

− − = ... J kg–1 °C–1

RESULT

(a) The specific heat of the given solid is ... Jkg–1 K–1 withinexperimental error.

(b) The specific heat of the given liquid is ... Jkg–1 K–1 withinexperimental error.

PRECAUTIONS1. Physical balance should be in proper working condition and ensure

that there is no zero error.

2. The two thermometers used should be of the same range and leastcount.

3. The solid used should not be chemically reactive with the liquidused or water.

4. The calorimeter should always be kept in its insulated cover and ata sufficient distance from the source of heat and should not beexposed to sunlight so that it absorbs no heat from the surrounding.

5. The solid should be transferred quickly so that its temperature issame as recorded when it is dropped in the liquid.

6. Liquid should not be allowed to splash while dropping the solidin it in the calorimeter. It is advised that the solid should be loweredgently into the liquid with the help of the thread tied to it.

7. While measuring the temperature, the thermometers should alwaysbe held in vertical position. The line of sight should beperpendicular to the mercury level while recording the temperature.

17

124


SOURCES OF ERROR1. Radiation losses cannot be completely eliminated.

2. Heat loss that takes place during the short period while transferringhot solid into calorimeter, cannot be accounted for.

3. Though mercury in the thermometer bulb has low specific heat, itabsorbs some heat.

4. There may be some error in measurement of mass and temperature.

DISCUSSION1. There may be some heat loss while transferring the solid, from

boiling water to the liquid kept in the calorimeter. Heat loss mayalso occur due to time lapsed between putting of hot solid incalorimeter and replacing its lid.

2. The insulating cover of the calorimeter may not be a perfectinsulator.

3. Error in measurement of mass of calorimeter, calorimeter with liquidand that of the solid may affect the calculation of specific heatcapacity of the liquid.

4. Calculation of specific heat capacity of the liquid may also beaffected by the error in measurement of temperatures.

5. Even though the metal piece is kept in boiling water, it may nothave exactly the same temperature as that of boiling water.

SELF ASSESSMENT1. What is water equivalent?

2. Why do we generally use a calorimeter made of copper?

3. Why is it important to stir the contents before taking thetemperature of the mixture?

4. Is specific heat a constant quantity?

5. What is thermal equilibrium?


We can verify the principle of calorimetry, if specific heat capacity of thesolid and the liquid are known.

AIM To make a paper scale of given least count: (a) 0.2 cm and (b) 0.5 cm

APPARATUS AND MATERIAL REQUIREDThick ivory/drawing sheet; white paper sheet; pencil; sharpener;eraser; metre scale (ruler); fine tipped black ink or gel pen.

PRINCIPLELeast count of a measuring instrument is the smallest measurementthat can be made accurately with the given measuring instrument.A metre scale normally has graduations at 1 mm (or 0.1 cm) spacing,as the smallest division on its scale. You cannot measure lengthswith this scale with accuracy better than 1mm (or 0.1 cm).

You can make paper scale of least count (a) 0.2 cm (b) 0.5 cm, bydividing one centimetre length into smaller divisions by a simplemethod, without using mm marks.

PROCEDURE(a) Making Paper Scale of Least Count 0.2 cm

1. Fold a white paper sheet in the middle along its length.

2. Using a sharp pencil, draw a line AB, of length 30 cm in eitherhalf of the white paper sheet [Fig. A1.1(a)].

3. Starting with the left end marked A as zero,mark very small dots on the line AB afterevery 1.0 cm and write 0,1,2 ..., 30 atsuccessive dots.

4. Draw thin, sharp straight lines, each 5 cmin length, perpendicular to the line AB atthe position of each dot mark.

5. Draw 5 thin, sharp lines parallel to the lineAB at distances of 1.0 cm, 2.0 cm, 3.0cm, 4.0 cm and 5.0 cm respectively. Letthe line at 5 cm be DC while those at 1 cm,

Fig. A1.1(a): Making a paper scaleof least count 0.2 cm

ACTIVITYACTIVITYACTIVITYACTIVITYACTIVITY 11111ACTIVITIES

126


2 cm, 3 cm, and 4 cm be A1B1, A2B2, A3B3 and A4B4 respectively[Fig A 1.1(a)].

6. Join point D with the dot at 1 cm on line AB. Intersection of thisline with lines parallel to AB at A4, A3, A2 and A1 are respectively0.2 cm, 0.4 cm, 0.6 cm and 0.8 cm in length.

7. Use this arrangement to measure length of a pencil or a knittingneedle with least count of 0.2 cm.

(b) Making Paper Scale of Least Count 0.5 cm

1. Using a sharp pencil, draw a line AB of length 30 cmin the other half of the white paper sheet [Fig. A1.1(b)].

2. Repeat steps 3 to 6 as in the above Activity 1.1(a), butdraw only two lines parallel to AB at distances 1.0 cmand 2.0 cm instead of 5 cm.

3. Join diagonal 1-D by fine tipped black ink pen [Fig. A1.1 (b)].

4. Use this scale to measure length of a pencil/knitting needle withleast count of 0.5 cm. Fractional part of length 0.5 cm is measuredon line A

1B

1.

(c) Measuring the Length of a Pencil Using the Paper Scales A and B

1. Place the pencil PP’ along the length of the paper scale A (leastcount 0.2 cm) such that its end P is on a full mark (say 1.0 cm or2.0 cm etc. mark). The position of the other end P′ is on diagonal1–D. If P′ goes beyond the diagonal, place it on next upper line, inwhich fraction of intersection is 0.2 cm larger, and so on. Thus, inFig. A 1.1 (a), length of the pencil = 3 cm + .2 × 2 cm = 3.4 cm. Takecare that you take the reading with one eye closed and the othereye directly over the required graduation mark. The reading islikely to be incorrect if the eye is inclined to the graduation mark.

2. Repeat preceding step 1, using the paper scale B having leastcount 0.5 cm and record your observation in proper units.

OBSERVATIONSLeast count of the paper scale A = 0.2 cm

Least count of the paper scale B = 0.5 cm

RESULT(i) Scale of least count 0.2 cm and 0.5 cm have been made; and

(ii) Length of pencil as measured by using the scales made above is

(a) ... cm and (b) ... cm.

Fig. A1.1(b):

127

UNIT NAMEACTIVITY

PRECAUTIONS(i) Very sharp pencil should be used.

(ii) Scale should be cut along the boundary by using a sharp papercutter.

(iii) Observation should be recorded showing accuracy of the scale.

(iv) While measuring lengths, full cm mark should be made tocoincide with one end of the object and other end should be readon the scale.

SOURCES OF ERROR

The line showing the graduations may not be as sharp as required.

DISCUSSION1. The accuracy of measurement of length with the scale so formed

depends upon the accuracy of the graduation and thickness ofline drawn.

2. Some personal error is likely to be involved e.g. parallax error.

1

128


ACTIVITYACTIVITYACTIVITYACTIVITYACTIVITY 22222

AIMTo determine the mass of a given body using a metre scale by theprinciple of moments.

APPARATUS AND MATERIAL REQUIREDA wooden metre scale of uniform thickness (a wooden strip of onemetre length having uniform thickness and width can also be used);load of unknown mass, wooden or metal wedge with sharp edge, weightbox, thread (nearly 30 cm long), a spirit level, and a raised platform ofabout 20 cm height (such as a wooden or metal block).

PRINCIPLEFor a body free to rotate about a fixed axis, in equilibrium, the sum ofthe clockwise moments is equal to the sum of the anticlockwisemoments.

If M1 is the known mass, suspended at a distance l

1 on one side from

the centre of gravity of a beam and M2 is the unknown mass, suspendedat a distance l

2 on the other side from the centre of gravity, and the

beam is in equilibrium, then M2 l

2 = M

1 l

1.

PROCEDURE1. Make a raised platform on a table. One can use a wooden or a

metal block to do so. However, the platform should be a sturdy,place a wedge having a sharp edge on it. Alternately one can fixthe wedge to a laboratory stand at about 20 cm above the tabletop. With the help of a spirit level set the level of the wedgehorizontal.

2. Make two loops of thread to be used for suspending the unknownmass and the weights from the metre scale (beam). Insert the loopsat about 10 cm from the edge of the metre scale from both sides.

3. Place the metre scale with thread loops on the wedge and adjustit till it is balanced. Mark two points on the scale above the wedgewhere the scale is balanced. Join these two points with a straightline which would facilitate to pin point the location of balance

129

UNIT NAMEACTIVITY

position even if the scale topples over from the wedge due to somereason. This line is passing through the centre of gravity of scale.

4. Take the unknown mass in one hand. Select a weight from theweight box which feels nearly equal to the unknown mass whenit is kept on the other hand.

5. Suspend the unknown mass fromeither of the two loops of threadattached to the metre scale.Suspend the known weight fromthe other loop (Fig. A 2.1).

6. Adjust the position of the knownweight by moving the loop till themetre scale gets balanced on thesharp wedge. Make sure that inbalanced position the line drawn inStep 3 is exactly above the wedgeand also that the thread of two loops passing over the scale isparallel to this line.

7. Measure the distance of the position of the loops from the linedrawn in Step 3. Record your observations.

8. Repeat the activity atleast two times with a slightly lighter and aheavier weight. Note the distances of unknown mass and weightfrom line drawn in Step 3 in each case.

OBSERVATIONSPosition of centre of gravity = ... cm

Table A 2.1: Determination of mass of unknown object

Fig. A 2.1: Experimental set up for determinationof mass of a given body

S.No.

Mass M1

suspendedfrom thethreadloop tobalancethe metrescale (g)

Distance ofthe massfrom thewedge l1

(cm)

Distance ofsolid ofunknownmass fromthe wedgel2 (cm)

Mass ofunknownload M2 (g)

= 1 1

2

M l

l

Averagemass ofunknownload (g)

1

2

3

4

5

Wedge

W mg=

m

x yUnknownMass, m Known standard

mass

GA B

2

130


CALCULATIONSIn balanced position of the metre scale, moment of the force on oneside of the wedge will be equal to the moment of the force on the otherside.

Moment of the force due to known weight = (M1l1) g

Moment of the force due to unknown weight = (M2l2) g

In balanced position

M1l1 = M2l2

or M2 = 1 1

2

M l

l

Average mass of unknown load = ... g

RESULTMass of given body = ... g (within experimental)

PRECAUTIONS

1. Wedge should be sharp and always perpendicular to the length of thescale.

2. Thread loops should be perpendicular to the length of the scale.

3. Thread used for loops should be thin, light and strong.

4. Air currents should be minimised.

SOURCES OF ERROR

1. Mass per unit length may not be uniform along the length of themetre scale due to variation in its thickness and width.

2. The line marked on the scale may not be exactly over the wedgewhile balancing the weights in subsequent settings.

3. The thread of the loops may not be parallel to the wedge when theweights are balanced, which in turn would introduce some errorin measurement of weight-arm.

4. It may be difficult to adjudge balance position of the scale exactly.A tilt of even of the order of 1° may affect the measurement ofmass of the load.

131

UNIT NAMEACTIVITY

DISCUSSION1. What is the name given to the point on the scale at which it is

balanced horizontally on the wedge?

2. How does the least count of the metre scale limit the accuracy inthe measurement of mass?

3. What is the resultant torque on the metre scale, due to gravitationalforce, when the scale is perfectly horizontal?

4. Explain, how a physical balance works on the principle ofmoments.

5. What problems would air currents cause in this activity?


1. We can determine the accuracy of various weights available inthe laboratory, by finding out their mass by the above methodand comparing with their marked values.

2. Verify the principle of moments using a metre scale. Afterbalancing the metre scale at its centre of gravity, suspendmasses M1 and M2 at distances l1 and l2 respectively, from thecentre of gravity, on either side. Adjust the distances l

1 and l

2so that the metre scale is horizontal. Calculate and compareM

1l1 and M

2l2. Repeat with other combinations of masses M

1and M

2.

2

132



AIMTo plot a graph for a given set of data choosing proper scale and showerror bars due to the precision of the instruments.

APPARATUS AND MATERIAL REQUIREDGraph paper, a pencil, a scale and a set of data

PRINCIPLEGraphical representation of experimentally obtained data helps ininterpreting, communicating and understanding the interdependencebetween the variable parameters of a given phenomena. Measuredvalues of variables have some error or expected uncertainty. For thisreason each data point on the graph cannot have a unique position.That means depending upon the errors, the x-axis coordinate and y-axis coordinate of every point plotted on the graph will lie in a rangeknown as an error bar.

Any measurement using a device has an uncertainty in its valuedepending on the precision of the device used. For example, in themeasurement of diameter of a spherical bob, the correct way is torepresent it d + Δd, where Δd is the uncertainty in measurement of dgiven by the least count of the vernier/screw gauze used.Representation of d + Δd in a graph is shown as a line having a lengthof + Δd about point ‘d’. This is known as the error bar of d.

We take an example where the diameters of objects, circular in shape,are measured using a vernier calipers of least count 0.01 cm. Thesemeasured values are given in Table 1. From the measured values ofdiameters, it is required to calculate the radius of each object and toround off the digits in the radius to the value consistent with the leastcount of the measuring instrument, in this case, the vernier calipers.We also estimate the maximum possible fractional uncertainty (orerror) in the values of radius. Next, the area A of each object is thencalculated using the formula.

Area, 2d

Aπ

=4

where π is the well-known constant.

133

UNIT NAMEACTIVITY

Graphical representation of experimental data provides a convenientway to look for interdependence or patterns between variousparameters associated with a given experiment or phenomenon or anevent. Graphs also provide a useful tool to communicate a given datain pictorial form. We are often required to graphically represent thedata collected during an experiment in the laboratory, to verify a givenrelation or to infer inter-relationships between the variables. It is,therefore, imperative that we must know the method for representinga given set of data on a graph, develop skill to draw most appropriatecurve to represent the plotted data and learn as to how to interpret agiven graph to infer relevant information.

Basic ideas about the steps involved in plotting a line graph for agiven data and finding the slope of the curve have already beendiscussed in Chapter I. The steps involved in plotting a graph includechoice of axes (independent variable versus dependent variable), choiceof scale, marking the points on the graph for each pair of data anddrawing a smooth curve/line by joining maximum number of pointscorresponding to the given data. Interpretation of the graph usuallyinvolves finding the slope of the curve/line, inferring nature ofdependence between variables/parameters, interpolating/extrapolating the graph to find desired value of the dependable variablecorresponding to a given value of independent variable or vice versa.However, so far you have learnt to graphically represent the data forwhich uncertainty or error is either ignored or is presumed not toexist. As you know every data has some uncertainty/error due lackof precision in measurement or some other factors inherent in theprocess/method of data collection. It is possible to plot a graph thatdepicts the extent of uncertainty/error in the given data. Such adepiction in the graph is called an error bar. In general error barsallow us to graphically illustrate actual errors, the statistical probabilityof errors in the measurement or typical data points in comparison tothe rest of the data.

You have learnt to show uncertainty in measurement of a physicalquantity like length, mass, temperature and time on the basis of theleast count of the measuring instruments used. For example, thediameter of a wire measured with a screw gauge having least count0.001cm is expressed as 0.181 cm ± 0.001 cm. The figure ± 0.001 cmin the measurement indicates that the actual value of diameter of thewire may lie between 0.180 and 0.182 cm. However, the error inmeasurement may also be due to many other factors, such as personalerror, experimental error etc. In some cases the error in data may bedue to factors other than those associated with measurement. Forexample, angles of scattering of charge particles in an experiment onscattering of α–particles or opinion collected from a section of apopulation on a social issue. The uncertainty due to such errors isestimated through a variety of statistical methods about which you willlearn in higher classes. Here we shall consider uncertainty in

3

134


measurements only due to the least count of the measuring instrumentso as to learn how uncertainty for a given data is shown in a line graph.

Let us take the example of the graph between time period, T, and thelength, l, of a simple pendulum. The uncertainty in measurement oftime period will depend on the least count of the stop watch/clockwhile that in measurement of length of the pendulum will depend onthe least count of the device(s) used to measure length. Table A 3.1gives the data for the time period of simple pendulum measured in anexperiment along with the uncertainty in measurement of the lengthand time period of the pendulum.

1 80.0 80±0.1 1.8 1.8±0.1 3.24±0.2

2 90.0 90±0.1 1.9 1.9±0.1 3.61±0.2

3 100.0 100±0.1 2.0 2.0±0.1 4.0±0.2

4 110.0 110±0.1 2.1 2.1±0.1 4.41±0.2

5 120.0 120±0.1 2.2 2.2±0.1 4.84±0.2

6 130.0 130±0.1 2.3 2.3±0.1 5.29±0.2

7 140.0 140±0.1 2.4 2.4±0.1 5.76±0.2

8 150.0 150±0.1 2.4 2.4±0.1 5.76±0.2

S.No.

Length of the pendulum Time period

Length asmeasured withmetre scale, L

(cm)

Length withuncertainty in L

(least count of scale0.1 cm)

(cm)

Average timeperiod asmeasuredwith stopwatch, T

(s)

Time periodwith

uncertainty inT (least countof stop watch

0.1 s)(s)

Square oftime period

T 2 withuncertainty

Table A 3.1 Time period of simple pendulums of different lengths

PLOTTING OF A GRAPH WITH ERROR BARSSteps involved in drawing a graph with error bars on it are as follows:

1. Draw x- and y- axes on a graph sheet and select an appropriatescale for plotting of the graph. In order to show uncertainty/errorin given data, it is advisable that the scale chosen should be suchthat the lowest value of uncertainty/error on either axes could beshown by at least the smallest division on the graph sheet.

2. Mark the points on the graph for each pair of data without takinginto account the given uncertainty/error.

135

UNIT NAMEACTIVITY

3. Each point marked on the graph in Step 2 has an uncertainty inthe value shown on either the x-axis or the y-axis or both. Forexample, let us consider thecase for the pointcorresponding to (80, 1.8)marked on the graph. If wetake into account theuncertainty in measurementfor this case, the actual lengthof the pendulum may liebetween 79.9 cm and 80.1cm. This uncertainty in thedata is shown in the graph bya line of length 0.2 cm drawnparallel to x-axis with itsmidpoint at 80.0 cm, inaccordance with the scalechosen. The line of length 0.2cm parallel to x-axis showsthe error bar for the pendulumof length 80.0 cm. One cansimilarly draw error bar foreach length of the pendulum.

4. Repeat the procedure explainedin Step 3 to draw error bars foruncertainty in measurement oftime period. However, the errorbars in this case will be parallelto the y-axis.

5. Once the error bars showingthe uncertainty for data in boththe axes of the graph have beenmarked, each pair of data onthe graph will be marked witha + or <% or <% sign,depending on the extent ofuncertainty and the scalechosen for each axis, insteadof a point usually marked fordrawing line graph (Fig. A 3.1).

6. A smooth curve drawnpassing as close as possiblethrough all the + marksmarked on the graph, insteadof points, gives us the plotbetween the two givenvariables (Fig. A 3.2).

Fig. A 3.1: Error bars corresponding to uncertaintyin time period of the given pendulum(uncertainty in length is not shown due tolimitation of scale)

Fig. A 3.2: Graph showing variation in time periodof a simple pendulum with its lengthalong with error bars

3

136


RESULTA given set of data gives unique points. However, when plotted, a curverepresenting that data may not physically pass through these points.It must, however, pass through the area enclosed by the error barsaround each point.

PRECAUTIONS1. In this particular case the point of intersection of the two x-axis

and y-axis represent the origin of O at (0, 0). However, this is notalways necessary to take the values of physical quantities beingplotted as zero at the intersection of the x-axis and y-axis. For agiven set of data, try to maximize the use of the graph paper area.

2. While deciding on scale for plotting the graph, efforts should bemade to choose a scale which would enable to depict uncertaintyby at least one smallest division on the graph sheet.

3. While joining the data points on the graph sheet, enough careshould be taken to join them smoothly. The curve or line shouldbe thin.

4. Every graph must be given a suitable heading, which should bewritten on top of the graph.

SOURCES OF ERROR1. Improper choice of origin and the scale.

2. Improper marking of observation points.


how error bars in the graphs plotted for the data obtained while doingExperiment Nos. 6, 9, 10, 11, 14 and 15.

Note:

As the aim of the Activity is to choose proper scale while plotting a graphalongwith uncertainty only due to the measuring devices, the calculation inthe activity should be avoided.

Suggested alternate Activity for plotting cooling curve with error bars(Experiment No. 14) where temperature and time are measured using athermometer and a stop-clock (stop-watch) with complete set of data /

observations with LC of the measuring devices and θ

θ

Δ and

T

T

Δ values

be given.

Additionally the same curve along with error bar be asked to be drawn usingtwo different scales and the discussion may be done using them.

137

UNIT NAMEACTIVITY


Fig. A 4.1: Setup to study rolling friction

AIMTo measure the force of limiting rolling friction for a roller (woodenblock) on a horizontal plane.

APPARATUS AND MATERIAL REQUIREDWooden block with a hook on one side, set of weights, horizontal planefitted with a frictionless pulley at one end, pan, spring balance, thread,spirit level, weight box and lead shots (rollers).

PRINCIPLERolling friction is the least force required to make a body start rollingover a surface. Rolling friction is less than the sliding friction.

PROCEDURE1. Check that the pulley is almost frictionless otherwise oil it to

reduce friction.

2. Check the horizontal surface with a spirit level and spread a layerof lead shots on it as shown in Fig. A 4.1.

3. Weigh the wooden block.

4. Find the weight of the pan.Tie one end of the thread tothe pan and let it hang overthe pulley.

5. Now put the block over thelayer of lead shots and tie theother end of the thread to itshook.

6. Put a small weight in the panand observe whether thewooden block kept on rollersbegin to move.

138


S.No.

Mass ofstandard

weights onwoodenblock, W

Total weightbeing pulled= (W + w) × g

= NormalReaction, R

(N)

Mass onpan (p) (kg)

Total weight (force)pulling the block andstandard weights (P+p) g

7. If the block does not start rolling, put some more weights on thepan from the weight box increasing weights in the pan graduallytill the block just starts rolling.

8. Note the total weight put in the pan, including the weight of thepan and record them in the observation table.

9. Put a 100 g weight over the wooden block and repeat Steps (7) to (9).

10. Increase the weights in steps over the wooden block and repeatSteps (7) to (9).

OBSERVATIONSMass of wooden block m = ... g = ... kg

Weight of wooden block, W (mg) = ... N

Weight on the pan

= (Mass of the pan + weight) × acceleration due to gravity (g)

= ... N

Table A 4.1: Table for additional weights

RESULTAs the total weight being pulled increases limiting value of rollingfriction increases/decreases.

PRECAUTIONS1. The pulley should be frictionless. It should be lubricated, if

necessary.

2. The portion of the string between the pulley and the hook shouldbe horizontal.

1

2

3

4

139

UNIT NAMEACTIVITY


1. Find the co-efficient of rollingfriction μ

r by plotting the

graph between rolling friction,F and normal reaction, R.

2. What will be the effect ofgreasing the lead shots, andthe horizontal surface onwhich they are placed.

3. Study the rolling motion ofa roller as shown in Fig. A4.2 and compare it with themotion in the arrangementfor the above Activity.

3. The surfaces of lead shots as well as the plane and the blockshould be clean, dry and smooth.

4. The weights in the pan should be placed carefully and very gently.

SOURCES OF ERROR1. Friction at the pulley tends to give larger value of limiting friction.

2. The plane may not be exactly horizontal.

DISCUSSION1. The two segments of the thread joining the block and the pan

passing over the pulley should lie in mutually perpendicularplanes.

2. The total weight pulling the block (including that of pan) shouldbe such that the system just rolls without acceleration.

3. While negotiating a curve on a road, having sand spread over it, atwo wheeler has to be slowed down to avoid skidding, why?

Fig. A 4.2:

4

140



AIMTo study the variation in the range of a jet of water with the change inthe angle of projection.

APPARATUS AND MATERIAL REQUIREDPVC or rubber pipe, a nozzle, source of water under pressure (i.e., atap connected to an overhead water tank or water supply line), ameasuring tape, large size protractor.

PRINCIPLEThe motion of water particles in a jet of water could be taken as anexample of a projectile motion under acceleration due to gravity 'g'.Its range R is given by

20 sin 2 0v

R = g

θ

where θ0 is the angle of projection and v

0 is the velocity of projection.

PROCEDURE1. Making a large protractor: Take a circular plyboard or thick

circular cardboard sheet of radius about 25 cm. Draw a diameterthrough its centre. Cut it along the diameter to form two dees. Onone of the dees, draw angles at an interval of 15° starting with 0°.

2. Attach one end of pipe to a tap. At the other end of the pipe fix anozzle to obtain a jet of water. Ensure that there is no leakage inthe pipe.

3. Fix the protractor vertically on the ground with its graduated–facetowards yourself, as shown in Fig. A 5.1.

4. Place the jet at the centre O of the protractor and direct the nozzleof the jet along 15° mark on the protractor.

5. Open the tap to obtain a jet of water. The water coming out of the

141

UNIT NAMEACTIVITY

Fig. A 5.2: Variation of range with angleof projection

Fig. A 5.1: Setup for studying the variation inthe range of a jet of water with theangle of projection

30 60x

y

0 15 45 75

v02/2g

R(c

m)

(Degrees)θ

jet would strike the ground after completing its parabolictrajectory. Ask your friend to mark the point (A) where the waterfalls. Close the tap.

6. Measure the distance between point O and A. This gives the rangeR corresponding to the angle of projection, 15°.

7. Now, vary θ0 in steps of 15° upto 75° and measure the

corresponding range for each angle of projection.

8. Plot a graph between the angle of projection θ0 and range R(Fig. A 5.2).

OBSERVATIONSLeast count of measuring tape = ... cm

Table A 5.1: Measurement of range

S.No.

1

2

3

4

5

Angle of projection θ0

(degrees)RangeR (cm)

15°

30°

45°

60°

75°

5

142


GRAPHPlot a graph between angle of projection (on x-axis) and range(on y-axis).

RESULTThe range of jet of water varies with the angle of projection as shownin Fig. A 5. 2.

The range of jet of water is maximum when θ0 = ... °

PRECAUTIONS1. There should not be any leakage in the pipe and the pressure

with which water is released from the jet should not vary duringthe experiment.

2. The jet of water does not strike the ground at a point but getsspread over a small area. The centre of this area should beconsidered for measurement of the range.

3. The nozzle should be small so as to get a thin stream of water.

SOURCES OF ERROR1. The pressure of water and hence the projection velocity of water

may not remain constant, particularly if there is leakage inthe pipe.

2. The markings on the protractor may not be accurateor uniform.

DISCUSSION1. Why do you get same range for angles of projection 15° and 75°?

2. Why has a big protractor been taken? Would a protractor of radiusabout 10 cm be preferable? Why?

SELF ASSESSMENT

1. This Activity requires the pressure of inlet water be kept constantto keep projection velocity of water constant. How can this beachieved?

2. How would the range change if the velocity of projection isincreased or decreased?

143

UNIT NAMEACTIVITY


1. Study the variation in maximum height attained by the water streamfor different angles of projection.

2. Study the variation in range of water stream by varying the height atwhich the water supply tank is kept.

3, Take a toy gun which shoots plastic balls and repeat the Activityusing this gun.

4. Calculate velocity of projection by using maximum value of horizontalrange measured as above.

5

144



Fig.A 6.1: Set up for studying the conservation of energy using double inclined tracks

AIM

To study the conservation of energy of a ball rolling down an inclinedplane (using a double inclined plane).


A double inclined plane having hard surface, (for guided motion ofthe ball on the double inclined plane it is suggested that an aluminiumchannel or rails of two steel wires be used for it), a steel ball of about2.5 cm diameter, two wooden blocks, spirit level, tissue paper or cotton,and a half metre scale.

PRINCIPLEThe law of conservation of energy states that ‘energy can neitherbe created nor destroyed but can only be changed from one formto another’.

For a mechanical system, viz., therolling of a steel ball on a perfactlysmooth inclined plane, the energy ofball remains in the form of its kineticand potential energies and during thecourse of motion, a continuoustransformation between these energiestakes place. The sum of its kinetic andpotential energies remains constantprovided there is no dissipation ofenergy due to air resistance, friction etc.

In this experiment, the law ofconservation of energy is illustrated by the motion of a steel ball rollingon a double inclined plane. A steel ball rolling on a hard surface ofinclined plane is an example of motion with low friction. When the ballis released from point A on inclined plane AO, it will roll down theslope and go up the opposite side on the plane OB to about the sameheight h from which it was released. If the angle of the slope on righthand plane is changed, the ball will still move till it reaches the samevertical height from which it was released.

145

UNIT NAMEACTIVITY

At point of release, A, say on the right hand inclined plane, the steelball possesses only potential energy that is proportional to the verticalheight, h, of the point of release and has a zero kinetic energy. Thispotential energy transfers completely into kinetic energy when thesteel ball rolls down to the lowest point O on the double inclined plane.It then starts rolling up on the second inclined plane during which itskinetic energy changes into potential energy. At point B where it stopson the left hand inclined plane OB, it again has only potential energyand zero kinetic energy. The law of conservation of mechanical energycan be verified by the equality of two vertical heights AA′ and BB′.

PROCEDURE

1. Adjust the experimental table horizontally with the help of spiritlevel.

2. Clean the steel ball and inclined planes with cotton or tissue paper.Even a minute amount of dust or stain on the ball or on theplane can cause much friction.

3. Keep the clean double inclined plane on a horizontal table.

Note: In order to reduce friction and thereby reduce loss of energydue to it one can also design an unbreakable double inclinedtrack apparatus, in which the steel ball rolls on stainless steelwire track. In a try outs with such an inclined plane it has beenobserved that the rolling friction is extremely low and it is verygood for this Activity. It also does not develop a kink in the centre,unlike the apparatus presently in use in many schools.

4. Insert identical wooden blocks W1 and W2 underneath each planeat equal distance from point O. The two planes will be inclinednearly equally, as shown in Fig. A 6.1. The inclined plane shouldbe stable on horizontal table otherwise there would be energylosses due to the movement of inclined plane as well.

5. Release the steel ball from A, on either of the two inclined.

6. Find the vertical height AA′ (x) of the point A from the tableusing a scale.

7. Note the point B up to which the ball reaches the inclined planeon the other side and find the vertical height BB′ (y) (Fig. A. 6.1).Record the observations. While observing the highest position ofthe steel ball on other plane, observer has to be very alert as theball stays at the highest position only for an instant.

8. Shift the wooden block W1 and W2, kept under either of the two planes,towards the centre point O by a small distance. Now the angle of theslope of one of the planes would be larger than that of the other.

9. Release the ball again from point A on one of the two planes andmark the point B on the other plane up to which the steel ballrolls up. Also find the vertical height BB′.

6

146


10. Repeat Steps (8) and (9) for one more angle of the slope of theinclined plane.

11. Repeat the observations for another point of release on the sameinclined plane.

OBSERVATIONSTable A 6.1:

RESULTIt is observed that initial vertical height and final vertical height uptowhich the ball rolls up are approximately same. Thus, the rolling steelball has same initial and final potential energies, though during themotion, the form of energy changes. The total mechanical energy (sumof kinetic and potential energies) remains same. This is the verificationof law of conservation of energy.

PRECAUTIONS

1. Steel balls and inclined planes must be cleaned properly withcotton/tissue paper.

2. Both wings of the inclined plane must lie in the same vertical plane.

3. Both the planes must be stable and should not have any movementdue to rolling of the ball or otherwise.

4. The position of the ball at the highest point while climbing up theplane must be noted quickly and carefully.

S.No.

Reading on inclinedplane from which the

ball is rolled down

Position ofmark A

Position ofmark up towhich theball rolls

up

Verticalheight,

y(cm)

Mean, y(cm)

1

2

3

1

2

3

Reading on the inclinedplane in which the ball

rolls up

Difference(x – y)(cm)

VerticalheightAA′, x(cm)

B =

C =

D =

B =

C =

D =

147

UNIT NAMEACTIVITY

SOURCES OF ERROR

1. Some energy is always lost due to friction.

2. Due to lack of continuity at junction of two inclined planes, rollingball usually suffers a collision with second plane and henceresults in some loss of energy.

DISCUSSION1. The key to the success of this Activity for the verification of law

of conservation of energy is in keeping the rolling friction betweenthe steel ball and inclined plane as low as possible. Therefore,the ball and inclined plane surfaces should be smooth, cleanand dry.

2. The dissipation of energy due to friction can be minimised byminimising the area of contact between the steel ball and inclinedplane. Therefore, it is advised that the inclined planes should bemade of polished aluminium channels having narrow grooves.

3. The surface of inclined planes should be hard and smooth sothat role of friction remains minimum.

4. If the inclination of the planes is large then the dissipation ofenergy will be more (how)? Therefore inclination of the planesshould be kept small.

SELF ASSESSMENT1. Can this Activity be performed successfully with a steel ball of

smaller diameter?

2. If the ball is not reaching exactly up to the same height on theother wing, comment on the observations?


1. Study of the effects of mass and size of the ball on rolling down aninclined plane.

2. Study the effect of inclination of the planes on coefficient of rollingfriction.

6

148



(A 7.1)

(A 7.2)

(A 7.3)

(A 7.4)

AIM

To study dissipation of energy of a simple pendulum with time.


A heavy metallic spherical ball with a hook; a rigid support; a longfine strong cotton thread (1.5 m to 2m); metre scale; weighing balance;sheet of paper; cotton; cellophane sheet.

PRINCIPLEWhen a simple pendulum executes simple harmonic motion, the re-storing force F is given by

F(t) = –kx (t)

Where x (t) is the displacement at time t and k = mg/L, the symbols k,m, g and L have been explained in Experiment E 6. The displacementis given by

x (t) = A0 cos (ωt – θ )

where ω is the (angular) frequency and θ is a constant. A0 is the maxi-mum displacement in each oscillation, which is called the amplititude.The total energy of the pendulum is given as

E° =

12

k 20A

The total energy remains a constant in an ideal pendulum, becauseits amplitude remains constant.

But in a real pendulum, the amplitude never remains constant. Itdecreases with time due to several factors like air drag, some play atthe point of suspension, imperfection in rigidity of the string and sus-pension, etc. Therefore, the amplitude of A

0 falls with time at each

successive oscillation. The amplitude becomes a function of time andis given by

A(t) = A0e –λ t/2

where A0 is the initial amplitude and λ is a contant which depends on

149

UNIT NAMEACTIVITY

damping and the mass of the bob. The total energy of the pendulumat time t is then given by

E (t) = 12

kA2(t)

= E0 e–λ t

Thus, the energy falls with time, because some of the energy is beinglost to the surroundings.

The frequency of a damped oscillator does not depend much on theamplitude. Therefore, instead of measuring the time, we can alsomeasure the number of oscillations n. At the end of n oscillations,t = nT, where T is the time period. Then Eq. (A 7.5) can be written inthe form E

n = E

0 e–α n

where α = λt

and En is the energy of the oscillator at the end of n oscillations.

PROCEDURE

1. Find the mass of the pendulum bob.

2. Repeat Steps 1 to 5 of Experiment E 6.

3. Fix a metre scale just below the pendulum so that it is in theplane of oscillations of the pendulum, and such that the zero markof the scale is just below the bob at rest.

4. When the pendulum oscillates, you have to observe the point onthe scale above which the bob rises at its maximum displace-ment. In doing this, do not worry about millimetre marks. Takeobservations only upto 0.5 cm.

5. Pull the pendulum bob so that it is above the 15 cm mark. Thus,the initial amplitude will be A

0 = 15 cm at n = 0. Leave the bob

gently so that it starts oscillating.

6. Keep counting the number of oscillations when the bob is at itsmaximum displacement on the same side.

7. Record the amplitude An at the end of n oscillations for n = 5, 10,15, ..., that is at the end of every five oscillations. You may evennote An after every ten oscillations.

8. Plot a graph of 2nA versus n and intepret the graph (Fig. A 7.1).

9. Stick a bit of cotton or a small strip of paper to the bob so as toincrease the damping, and repeat the experiment.

(A 7.5)

(A 7.6)

7

150


Fig. A 7.1: Graph between A2 n and nfor a simple pendulum

( )m2

n

A2n

OBSERVATIONS

Least count of the balance = ... g

Least count of the metre scale = ... cm

Mass of the pendulum bob. m = ... g

Radius (r) of the pendulum bob (given) = ... cm

Effective length of the pendulum (from the tip of the bob to the pointof suspension), L = ... cm

Force constant, k = mg/L = ... N m–1

Initial amplitude of oscillation, A0 = ... cm

Initial energy, E0 = 1/2 (k A2)= ... J

Table A 7.1 : Decay of amplitude with time and dissipation ofenergy of simple pendulum

RESULTFrom the graphs, we may conclude that the energy of a simplependulum dissipates with time.

S.No.

Amplitude,An (cm)

1

2

3

4

Energy ofoscillater, En

(J)

Dissipationof energy,(En–Eo) (J)

Number ofoscillations,

n

A2n

151

UNIT NAMEACTIVITY

PRECAUTIONS1. The experiment should be performed in a section of the laboratory

where air flow is minimum.

2. The pendulum must swing for atleast a couple of oscillationsbefore recording its amplitude, this will ensure that the pendulumis moving in the same plane.

SOURCES OF ERROR1. Some movement of air is always there in the laboratory.

2. Accurate measurement of amplitude is difficult.

DISCUSSION

1. Which graph among the A – n and A2 – n graph would you preferfor studying the dissipation of energy of simple pendulum withtime and why?

2. How would the amplitude of oscillation change with time withthe variation in (a) size and (b) mass of the pendulum bob; and(c) length of the pendulum?

SELF ASSESSMENT1. Interpret the graph between A2 and n you have drawn for a sim-

ple pendulum.

2. Examine how the amplitude of oscillations changes with time.

3. What does the decreasing amplitude of oscillation with time indi-cate in terms of variations in energy of simple pendulum with time.

4. In what way does graph between A and n differ from that betweenthe A2 and n graph, you have drawn.

5. Compare the A2 – n plots for

(a) oscillations with small damping and

(b) oscillations with large damping.


Take a plastic ball (5 cm diameter) make two holes in it along itsdiameter. Fill it with sand. Use the sand filled ball to make a pendulumof 100 cm length.

Swing the pendulum allowing the sand to drain out of the hole.Find the rate at which the amplitude of pendulum falls and compareit with the case when mass of the bob is constant.

7

152



Fig. A 8.1: Cooling curve

x

y

0Time (min)

TM

TR

Tem

per

atu

re (°C

)

AIMTo observe the change of state and plot a cooling curve formolten wax.

APPARATUS AND MATERIAL REQUIREDA 500 mL beaker, tripod stand, wire gauge, clamp stand, hard glassboiling tube, celsius thermometer of least count 0.5 °C, a stop-watch/stop-clock, burner, parraffin wax, cork with a hole to fit the boilingtube and hold a thermometer vertically.

PRINCIPLE

Matter exists in three states – solid, liquid and gas.

On heating a solid expands and its temperature increases. If wecontinue to heat the solid, it changes its state.

The process of conversion of solid to a liquid state is called melting.The temperature at which the change takes place is called meltingpoint. Melting does not take place instantaneously throughoutthe bulk of a solid, the temperature of solid-liquid remains

constant till the whole solid changes intoliquid. The time for melting depends upon thenature and mass of solid.

A liquid when cooled freezes to solid state at thesame temperature as its melting point. In this casealso the temperature of liquid-solid remainsconstant till all the liquid solidifies.

Paraffin wax is widely used in daily life. We candetermine the melting point of wax by plottinga cooling curve. The temperature of moltenwax is recorded at equal intervals of time. Firstthe temperature of wax falls with time thenbecomes constant at TM, the melting point,when it solidifies. On further cooling thetemperature of solid wax falls to roomtemperature TR as shown in Fig. A 8.1.

153

UNIT NAMEACTIVITY

PROCEDURE

1. Note the least count and range of the thermometer.

2. Note the least count of the stop-clock.

3. Record the room temperature.

4. Set up the tripod, burner, heatingarrangement as shown in Fig A 8.2.

5. Adjust the boiling tube and the thermometersuch that the graduation marks could beeasily read by you.

6. Heat the water and observe the state of wax.Continue to heat till all the wax melts, notethe approximate melting point.

7. Continue to heat the wax in the water bathtill the temperatue is atleast 20°C above theapproximate melting point as observed in Step 6.

8. Turn off the burner, and carefully raise the clamp to remove theboiling tube from the water bath.

9. Record readings of temperature after every 2 minutes.

10. Plot a graph of temperature of wax versus time, (temperature ony – axis).

11. From the graph

(i) determine the melting point of wax.

(ii) mark the time interval for which the wax is in liquid state/solid state.

OBSERVATIONSLeast count of thermometer = ... °C

Thermometer range ... °C to ... °C

Room temperature = ... °C

Least count of stop clock = ... s

Table A 8.1: Change in temperature of molten wax with time

Fig. A 8.2: Experimental set up

S.No.

times

1234

temperature°C

8

154


RESULT

The cooling curve of molten wax is shown in the graph. From thegraph (i) the melting point of wax is ... °C and (ii) the wax remains inliquid state for ... s and in solid state for ... s.

PRECAUTIONS1. The boiling tube with wax should never be heated directly on

a flame.

2. The stop clock should be placed on the right hand side of theapparatus as it may be easy to see.

3. Wax should not be heated more than 20°C above its melting point.

SOURCES OF ERRORSimultaneous recording of temperature and time may give rise tosome errors.

SELF ASSESSMENT1. Why should we never heat the wax directly over a flame?

2. Why is water bath used to melt the wax and heat it further?

3. What is the maximum temperature to which molten wax can beheated in a water bath?

4. Would this method be suitable to determine the melting point ofplastics? Give reason for your answer.

5. Will the shape of the curve for coding of hot water be differentthan that for wax?


1. Find melting point of ice.

2. Study the effect of addition of colour/fragrance on the melting pointof a given sample of colour less wax. Find the change in meltingpoint of wax by adding colour/fragrance in different proportions.

155

UNIT NAMEACTIVITY

ACTIVITYACTIVITYACTIVITYACTIVITYACTIVITY99999

Fig. A 9.1: A bi-metallic strip in (a) straight, and (b)bent positions

AIMTo observe and explain the effect of heating on a bi-metallic strip.


A iron-brass bi-metallic strip with an insulating (wooden) handle;heater/burner.

DESCRIPTION OF THE DEVICE

A bi-metallic strip is made oftwo bars/strips of differentmetals (materials), but of samedimensions. These metallicbars/strips (A and B) areput together lengthwise andfirmly rivetted. An insulating(wooden) handle is also fixedat one end of the bi-metallicstrip. A bi-metallic strip canbe made by selecting metals(materials) with widelydifferent values of coefficientsof linear thermal expansion.

The bi-metallic strip isstraight at room temperature,as shown in position (a) ofFig. A 9.1. When the bi-metallicstrip is heated, both metallic pieces expand to different extentsbecause of their different linear thermal expansivities, as shownin position (b) of Fig. A 9.1. As a result, the bimetallic stripappears to bend.

PRINCIPLEThe linear thermal expansion is the change in length of a bar onheating. If L1 and L2 are the lengths of rod/bar of a metal attemperatures t

1°C and t

2°C (such that t

2 > t

1), the change in length

156


(L2 – L1) is directly proportional to the original length L1 and the risein temperature (t

2 – t

1).

Then, (L2 – L1) = α L1 (t2 – t1)

or L2 = L1 [1 + α (t2 – t1)]

and α = (L2 – L1)/(t2 – t1)

where α is the coefficient of linear thermal expansion of the materialof the bar/rod.The coefficient of linear thermal expansion (α) is the increase in lengthper unit length for unit degree rise in temperature of the bar. It isexpressed in SI units as K–1.

PROCEDURE

1. Light a burner or switch on the electric heater.

2. Keep the bi-metallic strip in the horizontal position by holding itwith the insulated handle and heat it with the help of burner/heater. Note which side of the bi-metallic strip is in direct contactof heat source.

3. Observe the effect of heating the strip. Note carefully the directionof the bending of the free end of the bi-metallic strip, whether it isupwards or downwards?

4. Identify the metal (A or B) which is on the convex side of thebi-metallic strip and also the one which is on its concave side.Which one of the two metals/materials strips have a largerthermal expansion? (The one on the convex side of the bi-metallic strip will expand more and hence have larger linearthermal expansion).

5. Note down the known values of coefficient of linear thermalexpansion of two metals (A and B) of the bi-metallic strip. Verifywhether the direction of bending (upward or downward) is onthe side of the metal/material having lower coefficient of linearthermal expansion.

6. Take the bi-metallic strip away from the heat source. Allow thestrip to cool to room temperature.

7. Repeat the Steps 1 to 6 to heat the other side of the bi-metallicstrip. Observe the direction of bending of the bi-metallic strip.What change, if any, do you observe in the direction ofbending of the strip in this case relative to that observedearlier in Step 3?

(A 9.1)

(A 9.2)

(A 9.3)

157

UNIT NAMEACTIVITY

RESULTThe bending of a bi-metallic strip on heating is due to difference incoefficient of linear expansion of the two metals of the strip.

PRECAUTIONSThe two bars (strips) should be firmly rivetted near their ends.

DISCUSSION

The direction of bending of the bi-metallic strip is towards the side ofthe metal which has lower value of linear thermal expansion.

SELF ASSESSMENT

1. You have been given bars of identical dimensions of followingmetals/materials along with their α - values, for making a bi-metallic strip:

Aluminium (α = 23 × 10–6 K–1); Nickel (α = 13 × 10–6 K–1)

Copper (α = 17 × 10–6 K–1); Invar (α = 0.9 × 10–6 K–1)

Iron (α = 12 × 10–6 K–1); Brass (α = 18 × 10–6 K–1)

which pair of metals/materials would you select as best choicefor making a bi-metallic strip for pronounced effect of bending?Why?

2. What would be the effect on the bending of the bi-metallic strip ifit is heated to a high temperature?

3. Name a few devices in which bi-metallic strips are generally usedas a thermostat?


Design fire alarm circuit using a bi-metallic strip.

9

158



Fig.A 10.1: Expansion of liquid (glycerine)

AIM

To study the effect of heating on the level of a liquid in a container andto interpret the observations.


A round bottom flask of 500 mL capacity, a narrow tube about 20 cmlong and of internal diameter 2mm, a rubber cork, glycerine, hot water,a stand for holding the flask, a strip of graph paper, a thermometer.

PRINCIPLEA container is required to keep the liquid. When we heat the liquid, thecontainer also gets heated. On being heated, liquid and container bothexpand. Therefore, the observed expansion of liquid is its apparentexpansion, i.e. (the expansion of the liquid) – (the expansion of thecontainer). For finding the real expansion of the liquid, we must takeinto account the expansion of the container. Real expansion = apparentexpansion of the liquid + expansion of the container.

PROCEDURE

1. Fill the flask with glycerine upto the brim. Closeits mouth with a tight fitting cork having a longnarrow tube fixed in it. Glycerine will rise inthe tube; mark the level of the glycerine inthe tube as A. Set the apparatus as shown inFig. A 10.1.

2. Place the flask in the trough filled with hot waterand hold the flask in position with the help of astand as shown.

OBSERVATION

It is observed that as the flask is immersed in hotwater, the level of glycerine in the tube first falls downto a point, say B, and then rises up to a level C.

159

UNIT NAMEACTIVITY

DISCUSSION

The level falls from A to B on account of expansion of the flask oncoming in contact with hot water. This fall is equal to the expansionof the container. After some time glycerine also gets heated andexpands. Finally, the glycerine level attains a stationary level C.Obviously the glycerine has expanded from B to C. B C gives the realexpansion and A C is the apparent expansion.

SELF ASSESSMENTWater in a flask is heated in one case from 25°C to 45°C and in anothercase, from 50°C to 70°C. Will the apparent expansion/real expansionbe the same in the two cases?


Take equal volume of water in a glass tumbler and a steel tumbler havingsimilar shape and size. Cover them both with thermocol sheet and inserta narrow bore tube in each. Heat both from 25°C to 50°C and study theapparent/real expansion in both cases. Are they equal? Give reason foryour answer.

10

160



Fig. A 11.1: Rise of water in capillary tube

AIMTo study the effect of detergent on surface tension of water byobserving capillary rise.

APPARATUS AND MATERIAL REQUIREDA capillary tube, a beaker of 250 mL, small quantity of solid/liquiddetergent, 15/30 cm plastic scale, rubber band, stand with clampand water.

PRINCIPLE

Substances that can be used to separate grease, dust and dirtsticking to a surface are called detergents. When added to waterdetergents lower its surface tension due to additionalintermolecular interactions.

The lowering of surface tension by addition ofdetergent in water can be observed by capillary risemethod.

For a vertically placed capillary tube of radius r ina water - filled shallow vessel, the rise of water incapillary tube h (Fig. A11.1) is given by:

2 cosSh

gr

θρ

=

Or

2cosh gr

Sρ

θ =

where S is the surface tension of the water vapourfilm; θ is the contact angle (Fig. A11.1), ρ is the

density of water and g is the acceleration due to gravity. For pureor distilled water in contact with a clean glass capillary tube θ ≈ 8°or cos θ ≈ 1. Thus,

1=

2 S h grρ

161

UNIT NAMEACTIVITY

Using this result, the surface tension of different detergent solutions(colloidal) in water can be compared. In a detergent solution, thecapillary rise (or the surface tension) would be lower than that forpure and distilled water. And an increase in detergent’s concentrationwould result in a further lowering the rise of solution in the capillary.

A detergent for which the capillary rise is minimum (or the one thatcauses maximum lowering of surface tension), is said to have bettercleansing effect.

PROCEDURE

1. Take a capillary tube of uniform bore. Clean and rinse it withdistilled water. Also clean and rinse the beaker with water. Pourwater to fill the beaker up to half. Make sure that the capillarytube is dry and free from grease, oil etc. Also check that the topof the capillary tube is open and not blocked by anything.

2. Take a plastic scale and mount the capillary tube on it usingrubber bands.

3. Hold the scale with capillary in vertical position with the help of aclamp stand.

4. Place the half filled beaker below the lower end of the scale andgradually lower down the scale till its lower end get immersedbelow the surface of water in the beaker as shown in Fig. A 11.2.

5. Read the position of the water level inside and outside the capillarytube on the scale. Let the positions be h

2 and h

1 respectively. The

rise of water in the capillary is h = h2 – h

1.

6. Rinse the capillary thoroughly in running water and dry it.

7. Take a little quantity of the givendetergent and mix it with water inthe beaker.

8. Repeat the experiment withdetergent solution and findcapillary rise again. Let it be h′.

Note

The concentration of detergent mustnot be made high, otherwise thedensity of solution (colloidal) willchange substantially as comparedto water. Mor eover, the value ofangle of contact between the surfaceof glass and solution may alsochange substantially.

Fig. A 11.2: To study capillary rise in waterand detergent mixed in it

h

Water with

Capillary tube

Rubber

Scale

Rubberband

band

h2

h1

detergent

11

162


OBSERVATIONSThe height to which water rose in the capillary h = ... cm

The height to which the detergent solution rose in the capillaryh′ = ... cm.

RESULTThe capillary rise of detergent solution h′ is less than the capillary riseof water, h.

PRECAUTIONS1. The inner surface of the beaker and the part of capillary tube to

be immersed in water or solution in the beaker should not betouched by hand after cleaning them. This is essential to avoidcontamination by the hand.

2. To wet the inside of the capillary tube freely, it is first dipped welldown in the water and then raised and clamped. Alternatively,the beaker may be lifted up and then put down.

SOURCES OF ERROR

1. Contamination of liquid surface as also of the capillary tube cannotbe completely ruled out.

2. The tube may not be at both ends or its one end may be openblocked.

DISCUSSION

Can you also think of materials, which have a property of increasingthe surface tension of a liquid? If yes, what are these?

[Hint: There are some polymeric materials which can increase thesurface tension of water. Such materials are called hydrophilic.These have immense use in pumping out oil from the ground withless power.]

163

UNIT NAMEACTIVITY


AIM

To study the factors affecting the rate of loss of heat of a liquid.

APPARATUS AND MATERIAL REQUIREDTwo copper calorimeters of different sizes (one small and anotherbig); two copper calorimeters of same size (one painted black andthe other highly polished), two tumblers of same size (one metallicand another plastic); two thermometers having a range of - 10° C to110° C and least count 0.5 °C, stop watch/clock, cardboard lids forcalorimeters, two laboratory stands, a pan to heat water; a measuringcylinder, a plastic mug.

PRINCIPLE

Hot bodies cool whenever placed in a cooler surrounding.

Rate of loss of heat is given by ddQs

Q = mass × specific heat capacity(s) × temperature (θ ) = msθ

ddQ

t= ms

ddt

θ

hence rate of loss of heat is proportional to rate of change oftemperature.

The rate of loss of heat of a body depends upon (a) the difference intemperature of the hot body and its surroundings, (b) area of thesurface losing heat, (c) nature of the surface losing heat and (d) materialof the container.

PROCEDURE

(A). Effect of area of surface on rate of loss of heat.

1. Note the room temperature, least count of the two thermometers(T

A and T

B).

164


Fig. A 12.2: Cooling curve for water cooledin calorimeters A and B.Surface area of water is morefor calorimeter B than for thecalorimeter A

2. Take the big (A) and small (B)calorimeters.

3. Heat water in the pan up tonearly 80°C (no need to boilthe water).

4. Pour 100 mL of hot water incalorimeter (A) and also incalorimeter (B). This should bedone carefully and with least timeloss. One can use a plastic mugto pour 100 mL of hot water in ameasuring cylinder.

5. Insert a thermometer in eachof the two calorimeters. Usestands to keep the thermometersvertical. Also ensure that thethermometer bulb is wellinside the hot water in thecalorimeters (Fig. A 12.1).

6. Note the temperature of thewater in the two calorimetersinitially at an interval of 1minute till the temperature ofwater in the calorimeter is about40–30°C above the roomtemperature and thereafter atintervals of 2 minutes when thetemperature of hot water isabout 20–10°C above roomtemperature.

7. Record your observation inTable A 12.1. Plot graphsbetween θA versus time and θB

versus time for both thecalorimeters on the same graphpaper (Fig. A 12.2).

8. Determine the slope of θ versus tgraph after 5 minute interval.

Fig. A 12.1: Experimental set up for studying theeffect of surface area on cooling

OBSERVATIONS

Least count of thermometer = ... °C

Room temperature = ... °C

165

UNIT NAMEACTIVITY

S.No.

Calorimeter Black (A) Calorimeter White (B)

Time Temp θA S.No.

Time Temp θB

Table A 12.1: Effect of area of surface on rate of cooling

C. Effect of material of container on rate of cooling of a liquid

1. Use the metallic tumbler (A) and the plastic tumbler (B) instead ofcalorimeters.

2. Repeat Steps 3 to 8 as in part A. Record your observations in atable similar to Table A 12.1.

RESULT

From the six graphs plotted on 3 graph sheets complete the following:

1. The rate of cooling is ... °C/min in the larger calorimeter ascompared to the smaller calorimeter.

B. Effect of nature of surface of container on rate of cooling ofa liquid

1. Use the two identical small calorimeters; one with black (A)and the other with highly polished (B) surfaces.

2. Repeat Steps 3 to 8 as in part A.

Table A 12.2: Effect of nature of surface on rate of cooling

S.No.

Calorimeter A (Big) Calorimeter B (Small)

Time Temp θA

S.No.

Time Temp θB

12

166



1. Compare the effectiveness of disposable thermocole tumblers withthat of glass for taking tea.

2. Study the rate of cooling of tea contained in a stainless steel (metallic)teapot and a ceremic teapot.

3. Compare the rate of cooling of tea in a cup and in a saucer.

2. Least rate of cooling is ... °C/min observed in calorimeter ... partA/B/C.

3. Black surfaces radiate ... heat as compared to white or polishedsurface in the same time when heated to the same temperature.

4. Plastic mugs are preferred for drinking tea, as the rate of coolingof a liquid in them is ...

PRECAUTIONS

1. θA, θB and time recordings are to be done simultaneously so a set-up that allows both thermometers could be read quickly and atthe same time should be planned.

2. The lid of the calorimeter should be covered with insultatingmaterial to make sure that the heat is lost (cooling takes place)only from the calorimeter surface.

3. All three activities should be performed under similar conditionsof wind and temperature of the surrounding to reduce their effecton the rate of cooling.

DISCUSSION

1. The rate of cooling in summers is lower than in winters. Give areason for your answer.

2. Surface of metallic kettles are often polished to keep the tea warmfor a long time.

3. Why does the rate of cooling decrease when the temperature ofliquid is closer to the room temperature?

167

UNIT NAMEACTIVITY

ACTIVITYACTIVITYACTIVITYACTIVITYACTIVITY1313131313AIM

To study the effect of load on depression of a suitably clamped metrescale loaded (i) at its end; and (ii) in the middle.

A. Bending of a metre scale loaded at its end


Metre scale (or a thick wooden strip of about 1 m length), thread,slotted weights with hanger (10 g, 20 g, 50 g, 100 g), anothergraduated scale to be used to measure depression, a pin, cellotapeand clamp.

THEORYThe depression 'y' of a cantilever of length 'L' clamped at one end andloaded at the free end with a load M (weight Mg) is given by relation

3

33 12MgL

yY (bd / )

=

where L, b and d are length, width and thickness of the rectangularcantilever respectively and Y is the modulus of elasticity of the materialof the rod.

or 3

3

4MgLy

Y bd=

The readings of depression ‘y’ of the cantilever, in this case withvariation of load suspended at the other end, are taken. The variationof depression with load is expected to be linear.

PROCEDURE1. Clamp the metre scale firmly to the edge of the table. As shown in

Fig. A 13.1 ensure that the length and breadth of the scale are inhorizontal plane and 90 cm of the length of the scale is projectedout. Fix a pin with a tape at the free end of the metre scale alongits length to act as a pointer.

168


2. Fix a graduated scale vertically near thefree end of the clamped metre scale andnote its least count. Ensure that the pointedend of the pin is just above the graduationmarks of the scale but do not touch it.

3. Read the pointer 'p' when metre scalecantilever is without any load.

4. Suspend a hanger of known mass forkeeping slotted weights to depress the freeend of the cantilever.

5. Read the pointer on vertical scale and recordthe observation.

6. Keep on adding 20 g masses to the hangerand record the reading of the pointereverytime when it stops vibrating.

S .No.

Load M(g)

Depressiony = l

m – l

o

l1 (cm) whenload is

increasing

l2 (cm) whenload is

decreasing

Mean

1 2

2m

l ll

+=

(cm)

Reading of free end ofcantilever

1234567

Fig. A 13.1: Experimental set up to studydepression of metre scale (used ascantilever) with load suspended atfree end of the cantilever

y

Rigid butadjustableG clamp

Load ( )Mg

Beam

1

2

3

4

5

6

7. After taking 6-7 observations with increasing load, graduallyremove the slotted weights one by one and record the readingwhile unloading.

8. Plot a graph between the depression and the load.

OBSERVATIONSLength of the cantilever L = ... cm

Width of the metre scale cantilever b = ... cm

Thickness of the metre scale d = ... cm

Reading of the free end of the cantilever with no load, l0=

Table A 13.1: Effect of load on depression of cantilever

169

UNIT NAMEACTIVITY

RESULTThe depression 'y' is directly proportional to the load M.

PRECAUTIONS

1. The beam should be rigidly clamped at one end.

2. Loading and unloading of the slotted weights should be donecarefully without disturbing the position of the hanger onthe beam.

3. The vertical scale should be adjusted close to the pointer in sucha way that the pointer moves along it freely.

SOURCES OF ERROR

1. The scale should not be loaded beyond its elastic limit.

(This can be easily checked by comparing the zero load readingafter removing the maximum suspended load with that taken atthe beginning of the experiment).

2. There should be no vibratory motion of the beam when reading isrecorded.

3. While noting down the observation, the eye should be normal tothe tip of the pin and the graduated scale.

4. Observations should be repeated while removing masses.

B. Bending of metre scale loaded in the middle

APPARATUS AND MATERIAL REQUIREDMetre scale, two wedges to rest the ends of the metre scale, thread,slotted weights 200 g each, hanger for slotted weights, a graduatedscale with a stand to hold the scale vertical, a plane mirror, a pointerand plasticine.

DESCRIPTION OF THE DEVICEFig. A 13.2 shows the arrangement. A horizontal metre scale is heldon two wedges, a hanger is provided at the middle of the metre scalefor applying load. A pointer is fixed at the mid point to measure thedipression. A graduated (least count 1 mm) scale with a plane mirrorstrip attached to it is held in vertical position in a stand behind thehorizontal metre scale.

13

170


THEORY

Let a beam be loaded at the centre and supported near its ends asshown in Fig A 13.2. A bar of length 'L', breadth 'b' and thickness 'd'when loaded at the centre by a load 'W ' sags by an amount given by

3

34W l

yb d Y

=

where 'Y' is the Young’s modulus of the material of the rod/ beam, W,the load (= mg), where 'm' is the mass of the hanger with weights.

The depression 'y' is directly proportional to the load.

PROCEDURE

1. Place the metre scale on two wedges with (5–10 cm) lengthprojecting out on either side. Metre scale supported at both endsis like a beam.

2. Tie a loop of thread in the middle of the load such that a hangerto support slotted weights each of 200 g can be suspended onit. Ensure that the thread is tied tightly with the rod and doesnot slip.

3. Place a graduated scale (with least count 0.1 cm) vertically in astand at the centre of the metre scale used as beam. To facilitatereadings the vertical scale should be kept on the far side of themetre scale. Fix a pin to the hanger such that its pointed end isclose to the edge of the vertical scale which has graduation markson it.

4. Suspend the hanger of mass 200 g and record the position of thepointer fixed to the hanger. The mirror strip on the vertical scaleshould be used to remove any parallax.

Fig. A 13.2: Experimental set up to study depression,i.e., sag of a beam with load in the middle

1

2

3

4

5

6

Depression

Wedge

Scale

Beam

WHanger with weightsin middle of beam

Plane mirror for taking readingwithout any parallex

Wedge

171

UNIT NAMEACTIVITY

5. Keep on adding 200 g slotted masses to the hanger and recordthe readings of the pointer each time.

6. Take about six observations.

7. Now, remove masses of 200g one by one recording the position ofthe pointer each time while unloading.

8. Calculate the depression for the load M gram and hence depressionper unit load.

9. Plot a graph between the values of depression y againstcorresponding values of load and interpret the result.

OBSERVATIONS

Width of the beam, b =

Thickness of the beam, d =

Length of the beam between the wedges, L =

Table A 13.2 Depression of the beam for different loads

RESULTThe depression of the metre scale at its middle is ... mm/g. Thedepression 'y ' is directly proportional to the load M.

SOURCES OF ERROR1. The rod should not be loaded beyond elastic limit.

13

S.No.

LoadM (g)

Mean y/M(cm/g)

Loadincreasingr′1 (cm)

Loaddecreasingr′2 (cm)

Meanreading

1 2

2

r rr

+′ ′=

(cm)

Reading of the centre ofcantilever

1 0 r0 0

2 200 r1 r1 – r0

3 400 r2 r2 – r0

4

5

6

Depressionfor load M(g), y (cm)

Depressionper unitload y/M(cm/g)

172


2. There should be no vibratory motion of the rod when reading isrecorded.

3. While taking readings, the eye should be normal to tip of thepointer and the metre scale.

4. The beam should be of uniform thickness and density throughoutits length.

5. The masses used must have standard value as engraved on them.

PRECAUTIONS1. The beam should be symmetrical on the knife edges.

2. Loading and unloading of the slotted weights should be donecarefully without disturbing the centre point.

3. Mirror strip used to eliminate parallax error should not disturbthe experimental setup.

173

UNIT NAMEPROJECT

PROJECTPROJECTPROJECTPROJECTPROJECT

AIM

To investigate whether the energy of a simple pendulum is conserved.


A tall laboratory clamp stand with clamps, a split cork, a brick (orany heavy metallic weight) to be used as bob, strong cotton thread/string (about 1.5 m to 2.0 m), stop-watch, ticker timer, paper tape,balance, wooden block, cellotape, metre scale and graph paper.

PRINCIPLE

Energy can neither be created nor destroyed, though it can betransformed from one form to another, and the sum of all formsof energies in theuniverse remainsconstant (Law ofconservation ofenergy). In anyisolated mechanicalsystem with practicallyn e g l i g i b l e / n odissipation of energy toovercome viscousdrag/air resistance /friction, (as in case ofa pendulum), thesum of the kinetic andpotential energiesremains constant.For small angularamplitude (θ ≤ 15°), the pendulum executes simple harmonic motion(SHM) with insignificant damping, i.e., loss of energy. Hence, an oscillatingsimple pendulum provides a convenient arrangement to investigate/validate the law of conservation of energy for a mechanical system.

11111

Fig. P 1.1: An oscillating pendulum

(b)

O

xF Eh

D

L h–

A

x

PROJECTS

174


The oscillations of a simple pendulum of effective length L with meanposition at point A and extreme positions at points B and C, are shownin Fig. P 1.1. In the extreme positions, i.e., at B and C the oscillatingbob is raised to a certain height h ( = AD) above the mean positionwhere it possesses maximum potential energy but minimum kineticenergy. In the mean position, at A, it possesses maximum kineticenergy and minimum potential energy. At any intermediate positioni.e., at E and F the bob will possess energy in the form of both kineticand potential energies. The effective length L ( = l + r ) of the pendulumis taken from the point of suspension O to the centre of gravity of thebob (Fig P 1.1; also refer Experiment E 6). For small angular amplitudes(θ ) (about 8° to 10°) the arc length EA = (FA) is about the same aslinear distance ED = (FD) = x, the points E and F are symmetricallyabove point D.

From the geometry of the Fig. P 1.1, it follows

DF. DE = OD. DA

x × x = (L – h ) h

For small values of x and h (and x << L and h << x)

=2x

hL

Then the potential energy of the bob (brick) of mass m at point E (or F)

= mgh = 2mgx

L

The kinetic energy E possessed by the bob moving with velocity v at

point E (or F) is = 2mv

12

Then total energy of the bob is given by

=mg

E mv + xL

2 212

Using this relation, now investigate whether the total energy E of theoscillating simple pendulum remains constant.

DEVICE FOR MEASURING SHORT TIME INTERVALS IN THELABORATORY: TICKER TIMER

Ticker-timer is a device used for the measurement of short time-intervals in the laboratory. It can measure short time intervals ofabout 0.02s to much higher degree of accuracy as compared to thatof a stop-watch (with least count of 0.1s). Ticker-timers are availablein different designs.

(P 1.1)

(P 1.2)

(P 1.3)

(P 1.4)

175

UNIT NAMEPROJECT

A simple type of ticker-timer,as s hown in Fig.P1.2,consists of a steel/metallicstrip T which can be madeto vibrate at a knownfrequency with the help of anelectromagnet. The pointedhammer of the vibratingsteel strip, T strikes a smallcarbon paper disc C underwhich a paper tape, ispulled by the oscillatingobject. The dot marks aremarked on the paper tapeby the pointed hammerwhen the strip vibrates.

The dot marks are obtained on the paper tape at regular (or equal)intervals of time. Each dot mark refers to a complete vibration of thevibrating steel strip. The time interval between the two consecutive dotmarks can be taken as a unit of time for a tick. The time period of thevibrating strip is obtained from its given (known) frequency of vibration.When it is run on 6V step-down ac supply, its frequency is the same asthat of ac mains (50 Hz, in India).

In this way, the measured time interval for one tick (between the twoconsecutive dot marks) can be converted into the basic unit, second,for time measurement. Thus, the ticker-timer can be used to measureaccurately time interval of the order of 0.02 s in the laboratory.

PROCEDURE

1. Find the mass of the pendulum bob.

2. Determine r and l by metre scale.The length of the pendulumL = l + r.

3. Take the ticker-timer and place it atabout the same level as the centre ofthe bob as shown in Fig. P 1.3. Fixthe ticker-timer on a wooden blockwith tape, to ensure that its positionis not disturbed when tape is pulledthrough it.

4. Attach the tip of the paper tape ofthe ticker-timer to the bob with thehelp of cellotape such that it ishorizontal and lies in the plane inwhich centre of gravity of the bob liesin its rest position.

Fig. P1.2: Ticker-timer

Fig. P 1.3: Experimental setup for studyingconservation of energy

1

176


5. Pull the bob towards the timer such that its angulardisplacement (θ < 10o ) is about one tenth of its length from thevertical position. Take care that the ticker tape is sufficientlylight and is so adjusted that it easily moves by the pull of bobas soon as it begins to move.

6. Start the ticker-timer carefully and let the bob oscillate. Whilethe bob moves towards the other side, it pulls the paper tapethrough the ticker-timer. Ticker timer, thus, records the positionsof the bob at successive time intervals.

7. Switch off the ticker-timer when the brick reaches the otherextreme end. Take out the paper tape and examine it.Extreme dot marks on the record of the tape represent theextreme positions B and C of the pendulum. The centrepoint A of this half oscillation is the centre of the twoextreme dot marks, and may be marked by the half metrescale, as in Fig. P1.4.

AB C

Extreme Centre

r

Extreme

Fig. P 1.4: Position of the oscillating bob marked on paper tape

8. Measure the displacements of the bob corresponding toeach dot (about 10 to 12) on either sides from the centremarked A as x

1, x

2, x

3, ... Find the time t

1, t

2, t

3, ... when

each selected dot was made by counting the number of dotsfrom the central point A, representing the mean position ofthe pendulum. If central point A is not coinciding with adot marked by the ticker -timer, appropriate fraction oftime-period of ticker-timer has to be added for findingcorrect t1, t2, t3, ...

9. Record observations in the tabular form in SI units and propersignificant figures.

10. Calculate the corresponding velocity for each selected positionof the dot as v

i (= Δx

i / Δt

i ). For this take one earlier and one

later dot. The distance between these two dots is Δxi and Δti istime to cover this distance. Then find magnitude of

kinetic energy ΔΔ

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

i

i

xmmv =

t

2

212 2

and potential energy

177

UNIT NAMEPROJECT

mghi [= mg (xi2/L) ] of the pendulum bob. Find the sum of kinetic

and potential energies in each case. Express the result in SIunits and proper significant figures.

11. Plot a graph between the displacement (xi ) of the pendulum bob(distance of dots from the central dot) against the time.

12. Find the velocity (v) from the slope of the graph at five or sixpoints on the left and also on the right of the mean position.Calculate the corresponding kinetic energy (mv2/2 ) for eachposition of the points on the graph.

13. Plot another graph between kinetic energy and the position (x) ofthe bob. Find out the position of the point for which kinetic energyis minimum.

14. Calculate also the potential energy, PE ixmg

L

⎛ ⎞=⎜ ⎟⎝ ⎠

2

, at the

corresponding points at which you have calculated the kineticenergy. Plot the graph of potential energy (PE) against thedisplacement position (x) on the same graph on which you haveplotted kinetic energy versus position graph.

15. Find the total mechanical energy E as the sum of kinetic energyand potential energy of the pendulum at each of the displacementpositions x. Express the result in SI units with proper significantfigures. Plot also a graph between the total mechanical energy Eagainst displacement position (x) of the pendulum on the samegraph on which you have plotted the graphs in Steps 13 and 14,i.e., for K.E. and P.E.

OBSERVATIONSMeasuring the mass of bob and effective length of simplependulum

(a) Effective length of the simple pendulum

Least count of the metre scale = ... mm = ... cm

Length of the top of the brick from the point of suspension, l = ... cm = ... m

Diameter of the bob, 2r = ... cm

Effective length of the simple pendulum L = ( l + r ) = ... cm = ... m

(b) Mass of the ... g

Time period (T) of ticker-timer = ... s

Fraction of T to be added for finding corrected Ti on left = ...

Fraction of T to be added for finding corrected Ti on right = ...

1

178


Table P1.1: Measuring the displacement and time usingticker-timer and the recorded tape

(c) Plotting a graph between displacement and time

Take time t along x-axis and displacement x along y-axis, using theobserved values from Table P1.1. Choose suitable scales on these axesto represent t and x. Plot a graph between t and x as shown in Fig. P1.5.What is the shape of x-t graph?

CALCULATION

(i) Find out from the graph (Fig. P1.5), thevelocity of bob at five or six differentpoints on the either side of the meanposition O of the graph.

Compute the values of kinetic energy,using Eq. (P1.3), corresponding toeach value of velocity obtained fromthe graph. Record these values inTable P1.2.

(ii) Plot a graph by taking displacement(distance) x along x-axis and kineticenergy (K.E.) along y-axis using thevalues from Table P1.2 as shown inFig. P1.6.

(iii) Compute the values of potential energyusing Eq. (P1.2), for each value ofdisplacement in Step (ii) above.

S. No. S. No. of doton tape (i)

Displacement(Distance of

dot fromcentre, xi)

(cm)

Number ofvibrations ofticker-timer

betweencentral and

ith point

T1 (s) Velocity v

(m s–1 )

1

2

3

2nd right

4th right

6th right

------

2nd left

4th left

6th left

------

Fig. P1.5: Graph between displacement andtime of the oscillating bob

(s)t

sine curve

d(c

m)

Variation of vsd t

179

UNIT NAMEPROJECT

Table P 1.2: Finding potential, kinetic and total energy of theoscillating bob

(iv) Plot a graph by takingdisplacement (distance) xalong x-axis and potentialenergy (P.E.) along y-axis onthe same graph (Fig. P1.6).

(v) Compute the total energyE

T as the sum of the kinetic

energy and potentialenergy at each of thedisplacement positions, x.Plot a graph by taking thedisplacement along x-axisand total energy E

T along

y-axis on the same graphFig. P1.6).

RESULTThe total energy, as the sum of kinetic and potential energies, of thebob of the simple pendulum is conserved (remains the same) at all thepoints along its path.

DISCUSSION

1. Refer to points 3 to 5 under discussion given in Experiment E 6on page 65.

2. Eq. P1.1 that expresses the relation between x, h and L for a simplependulum, holds true under the conditions h << x << L for smallangular amplitudes (θ < 10°) of the pendulum.

3. Linear displacement x of the bob, about (1/8)th to (1/10)th ofthe effective length of the pendulum corresponds to angulardisplacement (θ ) of about 8° to 10° for small angular amplitudes,

x

En

ergy

Total energy

P.E

K.E

Displacement, x

ET

y

O

Fig. P 1.6: Graph between displacement andenergy of the oscillating bob

1

2

3

4

S.No.

Total Energy =Potential Energy+ Kinetic Energy(J)

Velocity, v(ms–1)

Potential Energy,

mg x

L

2

(J)

Kinetic Energy,

mv212 (J)

1

180


the displacement (distance) of a dot mark on the paper tape fromthe central point/position truly represent correspondingdisplacement of the pendulum bob from its central (mean)position.

4. The shape of the graphs shown in Fig. P1.5 and Fig. P1.6correspond to ideal conditions in which no energy is lost due tofriction and air drag. The graph drawn on the basis of observeddata may differ due to error in data collection and friction.

SELF ASSESSMENT

1. Identify the shape of displacement time graph, you have drawnfor the oscillating simple pendulum. Interpret the graph.

2. Identify the shape of kinetic energy-displacement and potentialenergy-displacement graphs, you have drawn for the simplependulum.

Study the change in potential energy and kinetic energy at eachof the displacement positions. Interpret these graphs and see howthese compare.

3. What is the shape of the graph between the total (mechanical)energy and displacement you have drawn for the simplependulum? Interpret the graph to show what it reveals?

181

UNIT NAMEPROJECT

PROJECTPROJECTPROJECTPROJECTPROJECT 22222

AIM

To determine the radius of gyration about the centre of mass of ametre scale used as a bar pendulum.


A metre scale with holes at regular intervals, knife edge shaped axle,a rigid support, two glass plates (to be used for suspension plane),spring balance, spirit level, telescope fixed on a stand, stop-watchand graph paper.

PRINCIPLE

A rigid body oscillating in a vertical plane about a horizontal axispassing through it is known as compound pendulum. The point inthe body through which the axis of rotation passes is known as centreof suspension.

The time period of a compound pendulum is given by

IT

mgl= π2

where m is the mass of the rigid body, l is the distance of the point ofsuspension from the centre of gravity, I is the moment of inertia ofthe body about the axis of oscillation and g is the acceleration dueto gravity.

If K is the radius of gyration of the body about an axis through thecentre of gravity, then the moment of inertia about the centre ofsuspension is

I = m (K2 + l2)

2K

m l ll

⎛ ⎞= +⎜ ⎟⎝ ⎠

(P 2.2)

(P 2.1)

182


Hence

K Kml l l

l lT

mgl g

⎛ ⎞ ⎛ ⎞+ +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠= π = π

2 2

2 2

or L

Tg

= π2

where L = (l + K2/l)

Eq. (P 2.4) can be written as

l . L = (l2 + K2) ⇒l2 – l L + K2 = 0

Eq. (P 2.5) is quadratic in l and therefore has two roots, say l1 and

l2 then

l1 + l

2 = L and l

1 l

2 = K 2

or K l l= 1 2

PROCEDURE1. Take a metre scale. Draw a line in the

middle along its length. Drill holes ofabout 1.6 mm diameter on this lineseparated by a distance of 2 cm,starting from one end to the other.

2. Determine the centre of gravityof the scale by balancing it overa wedge.

3. Pass the knife edge shaped axle inthe hole near one of the ends of themetre scale and let it rest on thesuspension base having glass platesat its top.

4. Ensure that the glass plates fixedon the suspension plane behorizontal and in the same level sothat when we suspend the metrescale by placing the knife edge wemay be sure that the scale hangsvertically (Fig. P2.1).

5. Make a reference line, drawn on thepaper strip, near the lower end of thependulum and focus it with atelescope. Adjust the telescope untilits vertical crosswire focuses on thereference line.

(P 2.3)

(P 2.4)

(P 2.5)

Fig. P 2.1: A metre scale oscillating about a pointclose to C.G.

183

UNIT NAMEPROJECT

6. Displace the lower end of the scale horizontally through a smalldistance from its equilibrium position and then release it. Thependulum (metre scale) will begin to oscillate. Take care that theangular amplitude of oscillation is within 5° or 6° and pendulumoscillates in a vertical plane without any jerk.

7. Count zero when the reference mark on oscillating pendulumpasses across the vertical crosswire of telescope and start the stop-watch at that instant (The counting of oscillations could be donevisually, in case a telescope is not available).

8. Continue counting 2, 3, 4,... successively when the reference lineprogressively passes the vertical crosswire from the same side andnote the time for 20 oscillations. Repeat the observations at leastthree times.

9. Measure from the lower end, the distance of the pointof suspension.

10.Repeat Steps 7 and 9 after shifting the knife edge to the successiveholes leaving two holes on either side of the centre of gravity of thependulum. Take length of pendulum on one side of C. G. as positivewhile on the other side as negative. Record your observations intabular form.

OBSERVATIONSTable P. 2.1: Measurement of time period of compound pendulum

HoleNo.

One side of C.G.

Distancefrom C.G.,

l1

(cm)

Other side of C.G.

Time for 20oscillations

t1 t2 t3

Time period

T =t + t + t1 2 3

3

(s)

HoleNo.

Distancefrom C.G.,

l2

(cm)

Time for 20oscillations

t ′1 t′2 t′3

Time period

T =t + t + t′ ′ ′

′ 1 2 3

3

(s)

CALCULATION1. Plot a graph between l and T by taking the l along x-axis

and T along y-axis. The graph will consist of two symmetricalcurves Fig. P 2.2. The point on the x-axis about which thegraph is symmetrical is the centre of gravity of the metrescale pendulum.

2

184


2. Draw a line parallel to x-axis cutting the graphat points, P, Q, R and S

(a) From the graph, CP= ... cm, CS= ... cm

1

CP +CSl = = ...cm

2

(b) From the graph, CQ = ... cm, CR = ... cm

2

CQ +CRl = = ...cm

2

(c) The radius of gyration K = 1 2l l

x

y

P Q C R S

O

Fig. P 2.2: Graph between distance fromC.G. and time period

RESULTThe radius of gyration about the axis passing through the centre ofmass of the metre scale is found as K = ... cm.

PRECAUTIONS1. Pendulum should be hung vertically and knife edge be kept

horizontal so that the pendulum oscillates in a vertical plane.

2. Note the time by the stop-watch leaving 5 or 6 initial oscillationsso that effect of any irregularities in the oscillations get subsided.

3. Increase the number of observations for a given length ofpendulum if time for 20 oscillations is to be measured withoutusing a telescope.

4. Keep the fans off or else air droughts will shift the position of thescale and its oscillations will not remain in the same plane.

SOURCES OF ERROR1. The metre scale may not have uniform mass distribution.

2. The wedge may not be sharp.

3. The holes drilled may not be colinear or have equally smooth innersurface.

DISCUSSION

1. If a metallic bar is used in place of wooden scale we would havebetter results as its inertia will hold it in position in a better way.

185

UNIT NAMEPROJECT

Moreover a metallic bar of homogeneous material and uniformcross-section can be easily made.

2. To draw smooth symmetrical graphs, we may make use of curvedsurface on the inside of set squares or by suitably bending plastictongue cleaners or broomsticks.

SELF ASSESSMENT

1. How would you establish that the compound pendulum executesSHM?

2. By knowing the radius of gyration of the metre scale about itscentre of mass, determine the moment of inertia of the same scaleabout an axis passing through the centre of mass.

3. Why do we get two L – T plots symmetrical about y-axis?


1. Increase the angular amplitude slowly and see how your resultchanges.

2. Note the angular amplitude at which the variation in your results isappreciable. How will you explain the changes?

2

186


Fig.P 3.1: Dots on tape

PROJECTPROJECTPROJECTPROJECTPROJECT33333

AIMTo investigate changes in the velocity of a body under the action of aconstant force and to determine its acceleration.


Ticker-timer, a horizontal table, a bumper (a heavy rectangularblock of wood), a trolley, three G-clamps, long paper tape, a pulley,strong thread, a few bricks, hanger, slotted weights, plug-key anda spring balance.

PRINCIPLE

The acceleration of a moving body is constant when force acting on itis kept constant. The principle and working of a ticker-timer hasalready been discussed in Project P1. Suppose the experimentalarrangment allows you to mark the position of a moving object as adot on the tape of the ticker-timer. The time interval between twosuccessive dots is the same but the dots may not be necessarily equallyspaced. Equally spaced dots would represent uniform motion whileunequally spaced dots would represent non-uniform motion.

For calculation of speed of a given object from the tape, take one of thetapes used in the experiment. Let S1, S2, S3, ..... be the distancesbetween two successive dots, say of ten dots, on the tape measuredfrom point A by a metre scale as shown in the Fig. P 3.1.

The frequency of the vibrator of ticker-timer

= Frequency of the A.C. supply

= 50 Hz.

187

UNIT NAMEPROJECT

The time interval between two successive dots = 150

s

The time taken for covering 10 dots i.e., for displacements S1, S2, S3, ...

= 150 × 10 = 0.2 s

The average speed v1 over the distance 11

S (cm)S

0.2s= = ... cm s–1

The average speed v2 over the distance =

SS 2

2

(cm)0.2s

= ... cm s–1

So, the increase in speed in the time interval of 0.2 s

= 2 1S S0.2s 0.2s

– = ... cm s–1

The average acceleration = ( )S – S

×2 1

0.2 0.2 = ... cm s–2

PROCEDURE

1. Setup the ticker-timer at one end of a long horizontal table andfix the bumper at its other end with the help of G-clamps as shownin Fig. P 3.2.

2. Place the trolley between the timer and the bumper. Attach oneend of a strong thread of suitable length to the trolley and pass itover a frictionless pulley fixed on the bumber. Attach a hanger atthe free end of the thread.

3. Adjust the length of the thread in such a manner that when thetrolley is brought near the timer, the hanger stands at its highestposition near the pulley.

4. Bring the trolley near the ticker-timer and release it, observeits motion.

5. Place one or two small bricks on the trolley if it moves too fast.Adjust the weights on the hanger so that the trolley moves with amoderate speed.

6. Hold the trolley in position near the timer. Check that thetape is passing under the carbon paper disc. Switch theticker-timer on and release the trolley. Ensure that the trolleygains speed till the pan touches the ground, thereafter it isstopped by the bumper.

7. Encircle the mark on the tape which was under the point ofthe vibrator of the timer at the instant when the pan touchesthe ground because there after the force ceases to act on the

3

188


trolley, label this mark as P. Encircled markP is the limiting position upto which thetrolley was accelerated by constant forcebefore it touched the ground.

Fig. P 3.3: Graph between speed and timeunder a constant force

S.No.

Time interval (inunits of tickinterval) (s)

Distance, s(cm)

Average velocityvav = s/t (cm s–1 )

Time (in midof interval), t

(tickinterval) (s)

1

2

3

0 – 10

10 – 20

20 – 30

S1

S2

S3

...

...

...

5

15

25

8. Remove the part of tape where dots are marked,from the timer.

9. Choose a dot, close to the starting point, mark itas A and take it as the reference point formeasurement of displacements.

10. Divide the entire motion of the trolley in about10 equal intervals of time. To do this, count thetotal number of dots marked on the tape duringthe motion of trolley. From A, mark the positionsas B, C, D etc. at the end of 10 ticks on the paper.

11. Measure the distance AB, BC, CD etc. and record them as shownin Table P 3.1. Compute average speed between different timeintervals (Table P 3.1). This can be taken as instantaneousvelocity at the mid point of the time interval tabulate. Thecomputed values of the average speeds against the mid point ofthe time intervals.

12. The instantaneous speed at the mid point of time intervals wouldbe nearly the same as the average speed during the interval ineach case.

13. Plot a graph showing the values of speed against time whichdepicts the motion of the trolley under a constant force. Find theslope of speed-time graph to calculate the instantaneousaccleration (Fig. P 3.3).

OBSERVATIONS

(a) Mass of the pan ... g.

(b) Mass of the pan + Mass of the weights in the pan = ... g.

(c) Mass of the trolley + mass placed in the trolley = ... g.

Table P 3.1: Instantaneous speed of the body

189

UNIT NAMEPROJECT

Table P 3.2: Acceleration of the body


1. Study the variation of acceleration for different masses placed onthe trolley for constant force.

2. Study the variation of acceleration for different forces, by changingthe mass placed on the pan.

S.No.

Point chosen Time, t (No. ofticks)

Interval Slope, MLNM

=

accelerationcm/ tick–2

ML (cm/tick)

NM(tick)

RESULT1. The speed of the trolley increases with time as constant force acts

on it.

2. The acceleration of the trolley is found to be ... roughly constantwithin the limitations of the experiment.

PRECAUTIONS

1. Make sure that the ticker-timer and bumper are rigidly fixed.

2. The ticks in the beginning when the trolley just begins to moveand at the time when the force ceases to act, be encircled properlyfor the purpose of measurement of distances and calculation ofvelocity and acceleration.

SELF ASSESSMENT

Is the acceleration calculated equal to ‘g’ ? If not, why? With increasein mass in the pan, does the acceleration approach to accelerationdue to gravity?

3

190


Fig. P 4.1:

PROJECTPROJECTPROJECTPROJECTPROJECT 44444AIM

To compare the effectiveness of different materials as insulators of heat.

APPARATUS AND MATERIAL REQUIREDA cylindrical metallic container, a cylindrical plastic container (withheight same as that of metal container but having a much largerradius), a thermometer, an insulating lid for plastic container with ahole for inserting a thermometer, different insulating materials inpowder or liquid forms.


Insulators of heat are those substances, which do not allow the flowof heat through them easily.

PRINCIPLEThe underlying principle of comparing the effectiveness of differentmaterials as insulators of heat is to compare their thermal

Thermometer

Water

Plastic container

Insulating lid

Metal container

A

D

B

C

Insulating material

191

UNIT NAMEPROJECT

conductivities. A material having a lower thermal conductivity will bemore effective as an insulator.

PROCEDURE

1. Place the metal container A inside the plastic container B leavingequal gap all around it. Fill the gap, between the two containerswith the insulating material you want to study (Fig P4.1).

2. Pour in container A hot water (having temperature nearly 60 °C).

3. Cover both the containers with a non-conducting lid.

4. Fix a thermometer, in a hole provided in the lid, in such a waythat the thermometer bulb is well within the water.

5. Record time for every 5 °C fall in temperature.

6. Repeat the above procedure for different insulating materials.

7. Plot temperature v/s time graph for different materials on thesame graph paper.

OBSERVATIONS

Least count of the thermometer = ... °C

Table P 4.1: Fall in temperature with time for differentmaterials as insulators

4

1.

Temparature

Time

Temparature

Time

Temparature

Time

Temparature

Time

2.

3.

4.

S. No.Name of

theMaterial

Variation of temparature with time

192


PLOTTING OF GRAPH AND INTERPRETATION

Plot a graph between time t and temperature θ for different materialson the same graph paper, taking time on x-axis and temperatureon y-axis.

Steeper the graph, faster the rate of cooling of water thereby implyinglower efficiency of the material used as thermal insulator.

RESULT

From the cooling curves of water drawn for different insulatingmaterials surrounding it can be inferred that the effectiveness ofdifferent materials as insulators of heat in decreasing order is

(a)

(b)

(c)

(d)

PRECAUTIONS1. Make sure that the gaps C and D are kept the same for all the

materials.

2. This method can be used only for the insulating materials availablein the powdered/liquid form as the effect of trapped air can beminimised for them.

3. Packing of insulating material in the gaps C and D should be equallyuniform in all the cases.

4. Insulating lid should fit tightly to minimise heat loss.


1. Repeat the same procedure with the cold water (instead of hot water).

2. Repeat the same procedure with other insulating materials otherthan the ones you have used in this Activity.

193

UNIT NAMEPROJECT


AIM

To compare the effectiveness of different materials as absorbersof sound.

APPARATUS AND MATERIALS REQUIRED

An audio frequency oscillator, cathode ray oscilloscope (CRO),two transformers, a microphone, a speaker (8 Ω), absorbingmaterials such as glass sheet, cardboard, plywood and fibreboard having roughly the same thickness, 4 cardboard sheets ofdifferent thicknesses, screw gauge, vernier calipers anda metre scale.

PRINCIPLEWhen sound waves travel through a material, part of its mechanicalenergy is absorbed by the material. The degree of absorption of soundenergy by a material depends upon

(i) the nature of material and

(ii) the thickness of the material through which sound waves are madeto pass.

PROCEDURE

1. Take sheets of different absorbing materials such as glass sheet,cardboard, plywood and fibre board sheets.

2. Measure the thickness of each material with the help of screwgauge/vernier calipers/metre scale.

3. Make the circuit arrangement of various components as shown inFig. P 5.1. High impedence coils L1 and L4 of the two transformersare to be connected to an audio frequency oscillator and a CROrespectively. Speaker and the microphone are to be connected tothe low resistance coils L

2 and L

3 of the two transformers in order

to achieve impedence-matching of the coils.

194


4. Adjust the CRO such that a suitable wave form appears onthe screen.

5. Feed an audio signal of known frequency from the audio oscillatorto the speaker and note the amplitude of the corresponding audiosignal on the CRO, without any sheet between the speaker andmicrophone.

6. Without changing the distance between speaker and microphone,insert one by one sheets of different materials, i.e., glass, cardboard,plywood, fibreboard (having same thickness) in between thespeaker and the microphone and each time note the amplitude ofthe corresponding audio signal on the CRO graduated screen.

7. Record the observations in tabular form to analyse the relationbetween the degree of absorption of sound energy and the natureof the absorbing material.

8. Insert four sheets of different thicknesses of the same material (saycardboard) one by one in between the speaker and the microphone.

9. Repeat Steps 5 and 6 of the experiment.

10.Record the observations in tabular form to analyse the degree ofabsorption of sound with the thickness of the absorbing material.

OBSERVATIONS

1. Least count of screw gauge/vernier valipers = ... mm

2. Thickness of cardboard = ... mm

Thickness of glass sheet = ... mm

Thickness of fireboard = ... mm

Thickness of plywood = ... mm

3. Frequency of the audio signal used = ... Hz

Fig. P. 5.1: Circuit arrangement for comparing effectiveness of differentmaterials as absorbers of sound

195

UNIT NAMEPROJECT

Table P 5.1: Degree of absorption of sound in differentabsorbing materials of same thickness.

Table P 5.2: Variation in degree of absorption of sound fordifferent thicknesses of the same absorbing material

CALCULATION1. Find the ratio of amplitude of the waveform before and after

insertion of the absorbing material from the experiment datarecorded in Table P 5.1.

2. Find the ratio of amplitude of the waveform before and afterinsertion of the absorbing material of different thicknesses andinfer its dependence on absorption of sound.

RESULT

1. Degree of absorption of sound waves is maximum in .... (material)and minimum in ... (material).

1. Glass

Card board

Fibre board

Plywood

2.

3.

4.

No. ofobservations

Name ofabsorbingmaterial

Amplitude of wave on CRO (mm)

Before insertionof absorbingmaterial A

0

After insertionof absorbingmaterial A

1

1

0

AA

1.

2.

3.

4.

No. ofobservations

Thicknessof

absorbingmaterial

Amplitude of wave on CRO (mm)

Before insertionof absorbingmaterial A

0

After insertionof absorbingmaterial A

1

1

0

AA

5

196



1. Plot a graph between the density (along x-axis) and the ratio of theamplitudes of the waveform (along y-axis) after and before insertionof the absorbing material (Table P 2.1). Study the nature of thegraph and interpret it.

2. Plot a graph between the thickness (along x-axis) of theabsorbing material and the ratio of the amplitude of the waveform (along y-axis) after and before insertion of the absorbingmaterial (Table P 5.2). Study the nature of the graph andinterpret it.

2. Degree of absorption of sound waves increases/decreases withincrease in the thickness of absorbing material (cardboard).

PRECAUTIONS

1. The amplitude of the input audio signal is kept constant whileperforming the experiment, with different absorbing materials ofsame thickness.

2. The thickness of absorbing material should not be so highthat the corresponding output signal on the screen of CRO is nolonger measurable.

3. The respective positions of the speaker, microphone andabsorbing material sheets for all sets of experiment should bekept unchanged.

197

UNIT NAMEPROJECT


AIM

To compare the Young’s modules of elasticity of different specimen ofrubber and compare them by drawing their elastic hysteresis curve.


Two samples of rubber bands of about 10 cm length, a rigid support,number of slotted weights (10 g), a hanger (10 g), a scale and a fine pointer.


1. Elastic hysteresis: When the stress-strain curve is not retraced onreversing the strain, the phenomenon is known as elastic hysteresis.

2. Residual strain: On removing the deforming force, if the length ofthe specimen does not reduce to its original length, this results inresidual strain.

PRINCIPLE

1. The graph of stress versus strain (or elongation) for rubber is nota straight line. Hence, the Young’s modules of elasticity for rubbercannot be defined uniquely. For a given stress, it is defined as theslope of the stress-strain curve at particular stress-strain point.

2. The area enclosed by the hysteresis curve is a measure of energyloss during the loading and unloading cycle.

PROCEDURE1. Suspend a rubber band from a rigid support and attach a hanger

of mass (10 g) along with a fixed pointer at the lower end.

2. Fix a scale S vertically such that the pointer moves freely on thescale and note the reading on the scale.

3. Place 10g slotted weight in the hanger and wait till the rubberband becomes stationary. Read the position of the pointer.

198


4. Repeat Step 3 by increasing load in Steps of 10 g till the totalweight is 80-100 g.

5. Start removing the weight in Steps of 10 g and note thecorresponding reading of the pointer (Give time for the rubber tostabilise before taking the reading).

6. Repeat Steps 1 to 5 for different samples of rubber bands.

OBSERVATIONS

(i) Least count of the scale = ... cm

(ii) Original length of unstreched rubber band, L = ... cm

Table P 6.1: Extension of rubber band on loading

CALCULATIONS

1. Plot a graph between the load and extension by takingextension along x-axis and load along y-axis for loadingand unloading.

2. The area of hysteresis loop for specimen A = ...

The area of hysteresis loop for specimen B = ...

(This can be done by counting the squares enclosed in thehysteresis loop).

RESULT

Hysteresis of specimen A ... is (greater or less than the) hysteresis ofspecimen B.

1

2

3

Specimen A

Specimen B

S.No.

Loadsuspended =

applied force =F (N)

Extension

Loading

Reading of pointer r (cm)

Unloading Loading Unloading

1

2

3

199

UNIT NAMEPROJECT

PRECAUTIONS

1. The weights must be added or removed gently.

2. One should wait for some time after adding or removing the weightsbefore reading is taken.

EVALUATION

1. What does the area of hysteresis curve depict?

2. Interpret the hysteresis curves obtained for the specimen A and B.

3. When do the curves obtained while loading and unloadingcoincide?

4. When do the curves obtained while loading and unloadingnot coincide?

5. For which purpose is the rubber with large hysteresis loop used?

6. For which purpose is the rubber with small hysteresis loop used?

7. Is the stress-strain graph for rubber a straight line asexpected by Hooke’s law? What would happen if the elasticlimit is exceeded?

8. How would you known that elastic limit has been crossed?

6

200



AIM

To study the collision of two balls in two-dimensions.


Apparatus for collision in two dimensions, metre scale, tracing paper,carbon paper, G-clamp, a screw, cellotape, protractor, two identicalsteel spheres or marble spheres and a plumbline.


A scale or ruler with a groove (or analuminium channel) which is bentto act as a ramp so that a steel ballcan be rolled from the top. At thelower end of the ruler a set screw isfixed that has a depression on itstop. This is the resting place for thetarget steel ball. The ruler rests ona metal base which can be clampedat the edge of a laboratory table.From the set screw, a plumbline issuspended as shown in Fig. P 7.1.Fig. P 7.1: Setup to study the collision of two

balls in two-dimensions

TableWoodensupport

Aluminiumchannel

Plumb line

Depression

Carbon paper

B

A

PRINCIPLEWhen two steel spheres of mass m and m′ moving with velocities uand u′′′′′ respectively collide, their velocities change after collision. Iftheir velocities after collision are v and v′′′′′ respectively, then accordingto the law of conservation of momentum

mu + mu′′′′′ = mv + mv′′′′′

In this Activity, we study collision of two balls in two-dimensions usingthe apparatus described above and verify the law of conservation ofmomentum in two-dimensions. We allow one steel ball to roll down

201

UNIT NAMEPROJECT

the ramp and collide with a target ball (at rest) placed at the lower endof the ramp. For simplicity, we take two identical balls.

After collision the two balls moving in different directions fall downand strike the ground. The horizontal velocity of each sphere isproportional to the horizontal distance travelled by each sphere (Whythis should be so?). The horizontal distance is the distance from pointon the floor just below the initial position of the stationary ball to thepoint where it lands. This same horizontal distance can also be usedto represent the magnitude of the momentum of each ball as theyhave the same mass.

PROCEDURE

1. Arrange the apparatus as shown in Fig. P 7.1. Adjust the set screwso that the depression in it is directly in front of the groove andabout one radius of the steel ball away from the groove end. Roll asteel ball down the ramp and adjust the set screw by movingupward/downward so that the ball just clears it as it falls freely.Place the target ball on the depression in the screw. Suspend theplumb line with it.

2. Next adjust the position of the set screw so that the bullet ball willcollide with the target ball at an angle. Mark the incident and targetballs as 1 and 2. Ensure that the two balls are exactly at the sameheight from the floor at the time of collision.

3. Spread on the floor a large sheet of tracing paper on a similarsized carbon paper. The steel balls would be falling on thiscombination to make their imprints. In case large sheets of carbonpaper or plain paper are not available tape together their pieces(A-4 size) to make a large sheet.

4. Put the carbon paper on the floor, with its inked side facingup. Place the tracing paper directly over it. Place the sheetssuch that the centre of one end of the paper lies just belowthe plumb line.

5. Without placing a target ball on the set screw, roll the ball marked1. Mark the point on the tracing paper where the ball lands (P

0).

Repeat it several times and mark the cluster P0 1, P0 2, P0 3, etc. ...Find the centre of the cluster and mark it P

0.

6. Using identical steel ball (2) to act as a target ball, try a fewcollisions. Ensuring that the incident ball (1) is always releasedfrom the same height. Circle and label the clusters of points wherethe incident ball and the target ball hit the paper.

(You can find the centre of cluster points, by drawing a quadrilateraland intersecting diagonals to find the location of mean point.)

7

202


7. Mark point ‘O’ on the paper where the plumb line touches the paper.Draw vectors from the point O to the mean point P0, P1 and P2.

8. (a) Add the two vectors uuur

1OP and uuur

2OP representing themomentum of the incident ball and target ball to determinethe total momentum P after the collision (Fig. P7.2).

(b) Relate the total momentum P after the collision with the initial

momentum of the incident ball represented by vector uuur

0OPand the target ball.

RESULT

The total momentum of the two ball after collision is ... g cms–1 whichis almost equal to the initial momentum of the incident ball.

PRECAUTIONS1. Adjust the set screw and ensure that the two balls are exactly at

the same height from the floor at the time of collision.

2. In each trial, the incident ball should be rolled down from thesame height.

SOURCES OF ERROR

Friction between the ball and surface may introduce an error.

SELF ASSESSMENT

1. For each trial, measure the angle between the two final momentumvectors. Can you make any generalisation?

2. Suppose the target ball is replaced by a glass marble of same sizeand we carry out the experiment using the same incident ball. In

Fig. P 7.2: To find location of mean print

203

UNIT NAMEPROJECT


This experiment can also be used to verify the law of conservation ofmomentum quantitatively, the momentum of a ball can be calculatedknowing its mass and velocity. Measure the mass of each ball with abalance. The horizontal velocity is equal to the horizontal distance travelleddivided by the time taken. Note that this time is equal to the time takenby the ball to hit the floor. This time can be determined by measuring thedistance (d) from the top of the set screw to the floor and using theequation d = (gt2)/2. Further, note that t will be the same for all calculations.

Calculate the original momentum of the incident ball and final momentaof the incident and target balls for the case with balls of (1) equal massand (2) unequal mass. Find the resultant of the two final momenta ineach case and compare it with the initial momentum.

ALTERNATE METHOD FOR MAKING CHANNEL

Take plastic pipe having internal diameter slightly more than the diameterof the balls.

Cut the pipe lengthwise into two equal parts (two halves). Bend slightlyone part of the cut pipe by gently warming it and fix it on a table top asshown in the figure below.

Make a small depression near end B of the pipe with the help of a heatedthick nail/rod for resting the target ball.

this case, the horizontal distances, would represent velocityvectors? Do they still represent momentum vectors? How will youdraw momentum vectors in this case and verify the law ofconservation of momentum?

3. What happens to the momentum components corresponding toOP1 and OP2 in Fig. P7.2 in the direction perpendicular to OP0?

7

204



Fig. P 8.1: Fortin’s barometer

AIM

To study Fortin’s Barometer and use it to measure theatmospheric pressure.


Fortin’s Barometer and a thermometer.


Fortins’s Barometer

It consists of a uniform glass tube about 80 cmlong, open at one end. It is filled with mercuryand turned upside down carefully in a troughof mercury C. The lower part of the trough ismade of leather and the level of mercury in thetrough can be adjusted by means of screw A[Fig. P 8.1 (a)]. The upper side of the trough isclosed by a leather patch L in such a way that thecontact is maintained between the outside air andthe mercury in the trough. There is a small ivorypin P fixed with its pointed tip touching themercury in the trough. The function of the pin Pis to adjust the zero of the scale at the same levelas the mercury in the trough. The glass tube isenclosed in a brass tube for protection. There aretwo vertical slits diametrically opposite each otherso that the level of mercury in the tube can beseen [Fig. P 8.1 (b)]. A scale graduated incentimetre is engraved on the brass tube on bothsides along the edges of the front slit. The scalegraduation does not start from zero but from 68cm to 85 cm, as the atmospheric pressureremainswithin these limits. A brass vernier scale slidesalong the front slit and can be adjusted usingscrew B.

205

UNIT NAMEPROJECT

PRINCIPLEWhen a completely filled mercury tube is turned upside down in thetrough C, some mercury flows out of the tube in the trough leaving avacuum on the top.

The level of mercury stabilises when the atmospheric pressure exertedon the surface of mercury in the trough equalises that due to themercury column in the tube. The height of the mercury column inthe tube is proportional to atmospheric pressure under normalconditions, Column of mercury in the glass tube stands at a height ofabout 76 cm at sea level.

From theoretical point of view, a barometer could be made of anyliquid. Mercury is chosen for many reasons mainly it is so dense(13600 kg/m3) that column supported by air pressure is of amanagable height.

A water barometer would be more than 10 m in height.

PROCEDURE1. Use the plumb line to hang the barometer vertically on a wall.

2. Examine the screws A, B, pin P and vernier V.

3. Determine the least count of the vernier scale.

4. Adjust the level of mercury surface in the trough of the barometer

Fig. P 8.2: Correct adjustment of mercury surface in the reservoir

8

206


with the help of screw A and by looking at the ivory pin andits image on the mercury surface in the trough (Fig. P 8.2).

Fig. P 8.3: Eye should be at the level ofmeniscus of mercury in the tube

Vernier V

R

Correctadjustmentof Vernier Scale

OBSERVATIONS(i) Vernier constant or least count = ...

No. of divisions on the vernier = ...

No. of divisions on the main scale = ...

Least count of main scale (1 MSD) = ... cm

Least count of vernier scale

= = ...1 MSD

cmNo. of divisions on vernier scale

(ii) Room Temperature = ... °C

Table P 8.1: Measuring height of mercury columnin a barometer

RESULTAtmospheric pressure in the laboratory on dd/mm/yr (date) at ... am/pm at room temperature ...°C was measured as ...cm of Hg.Atmospheric pressure = ...N/m2

5. Adjust the vernier using screw B such that the zero of the verniertouches the convex meniscus of mercury in the tube. The eyeshould be kept at the level of the meniscus (Fig. P 8.3).

6. Note the reading on the main scale and the vernier.

7. Record the room temperature using a thermometer.

8. Repeat the procedure two more times and determine theaverage atmospheric pressure.

1

2

3

S.No.

Main scale readingbelow zero mark of

vernier scale, S (cm)

Vernier scalereading n

Height of mercurycolumn

h = (S + n × least count)

207

UNIT NAMEPROJECT

PRECAUTIONS1. The barometer is a fragile instrument and should be handled

carefully.

2. The wall mount should be firm in a room of a laboratory and notin any passage.

3. Adequate light must fall on the ivory pin and the vernier scale.

4. Least count should be calculated with care.

5. Screw A should be moved slowly and gently.

SOURCES OF ERROR1. There may be air bubbles in the barometer tube.

2. Ivory pin may not be fixed properly.

3. Room temperature may change, affecting the observations.

DISCUSSION1. The barometer should be placed in such a way on the wall that screw

A can easily be adjusted by viewing the ivory pin P. A suitable platformcan be used to stand and see the vernier reading at eye level.

2. Why does the barometer require adjustment everytime one has touse it?

SELF ASSESSMENT1. What effect would there be of the following:

(a) Ivory pin not adjusted as advised?

(b) Barometer is not vertical but tilted?

(c) The pin P and scale S not viewed at eye level?

2. If water is used instead of mercury, what problems would youencounter?


1. Take barometer readings and temperature readings at different timesduring school hours. Study the pattern for the change in atmosphericpressure over a week.

2. Plot a graph between atmospheric pressure and humidity (as givenin the newspaper) for a month. Can we relate humidity to atmosphericpressure?

8

208



PointerA

Hanger H

Helicalspring

Load (m)

Rigid support

0

10

20

30

40

50

S

P

Fig. P 9.1: Measurement of extension ofa helical spring due to a load

(P 9.1)

AIM

To study of the spring constant of a helical spring from itsload-extension graph.


Helical spring with a pointer attached at its lower end and a hook/ring for suspending a hanger; a rigid support/clamp stand; five or sixslotted masses (known) for hanger; a metre scale.

PRINCIPLEWhen an external force is applied to a body,the shape of the body changes or deformationoccurs in the body. Restoring forces (havingmagnitude equal to the applied force)are developed within the body which tendto oppose this change. On removingthe applied force, the body regains itsoriginal shape.

For small changes in length (or shape/dimensions) of a body (wire), within the elasticlimit, the magnitude of the elongation orextension is directly proportional to the appliedforce (Hooke’s law).

Following Hooke’s law, the spring constant (orforce constant) of a spring is given by

Spring constant, =Restoringforce,

Extension,F

Kx

Thus, the spring constant is the restoring force per unit extension inthe spring. Its value is determined by the elastic properties of the spring.A given load is attached to the free end of the spring which is suspendedfrom a rigid point support (a nail fixed to a wall). A load (slotted weight)is placed in the hanger and the spring gets extended/elongated dueto the applied force. By measuring the extensions, produced by theforces applied by different loads (slotted mass) in the spring and

209

UNIT NAMEPROJECT

plotting the load (force) extension graph, the spring constant of thespring can be determined.

PROCEDURE

1. Suspend the helical spring, SA, having a pointer, P, at itslower free end, A, freely from a rigid point support, as shownin Fig. P 9.1.

2. Set the metre scale close to the spring vertically. Take care thatthe pointer moves freely over the scale without touching it andthe tip of the pointer is in front of the graduations on the scale.

3. Find out the least count of the metre scale. It is usually 1 mm or0.1 cm.

4. Record the initial position of the pointer on the metre scale,without any slotted mass suspended from the hook.

5. Suspend the hanger, H (of known mass, say 20 g) from the lowerfree end, A, of the helical spring and record the position of thepointer, P on the metre scale.

6. Put a slotted mass on the hanger gently. Wait for some time forthe load to stop oscillating so as to attain equilibrium (rest)position, or even hold it to stop. Record the position of the pointeron the metre scale. Record observations in a table with properunits and significant figures.

7. Put another slotted mass on the hanger and repeat Step 6.

8. Keep on putting slotted masses on the hanger and repeat Step 6.Record the position of the pointer on the metre scale every time.

9. Compute the load/force F ( = mg ) applied by the slotted mass,M and the corresponding extension (or stretching), x in thehelical spring. Here g is the acceleration due to gravity at theplace of the experiment.

10. Plot a graph between the applied force F along x-axis and thecorresponding extension x (or stretching) on the y-axis. What isthe shape of the curve of the graph you have drawn?

11. If you find that the force-extension graph is a straight line, findthe slope (F/x) of the straight line. Find out the spring constantK of helical spring from the slope of the straight line graph.

OBSERVATIONS

Least count of the metre scale= ... mm= ... cm

Mass of the hanger = ... g

9

210


Table P 9.1: Computing spring constant of the helical spring

Mean Spring constant K = ... N/m

Plotting load - extension graph for a helical spring

Take force, F along the x-axis and extension, x along the y-axis.Choose suitable scales to represent F and x. Plot a graph between Fand x (as shown in Fig. P 9.2). Identify the shape of the load-extensiongraph OA.

CALCULATIONS

Choose two points, O and A, wide apart on the straight line OAobtained from load extension graph, as shown in Fig. P 9.2. From

the point A, draw a perpendicular AB on x-axis. Then, fromthe graph,

Slope of the straight line graph = tan θ = ABOB

= x/F

Spring constant, K = Fx

= 1(slope of the graph)

Spring constant, −= = =

−–1B O

A B

OB...Nm

ABF F

Kx x

where xA and xB are the corresponding extensions at pointsA and B (or O) respectively where F

B and F

O are the loads

(forces) at points B and O.

y

Ox

A

Ext

ensi

on (m

)

F (N)

B

S.No.

Masssuspendedfrom thespring, M

Force,F = mg

Position ofthe pointer

Extension,x

Springconstant, K

(= F/x)

(10–3 kg) (N) (cm) (10–2 m) (N m–1)

1 0

2 203 .

4 .5 .6 .

. .

. .

. .

. .

Fig. P 9.2: Load-extension graphfor a helical spring

211

UNIT NAMEPROJECT

RESULTThe spring constant of the given helical spring = ... Nm–1

PRECAUTIONS

1. The spring should be suspended from a rigid support and itshould hang freely so that it remains vertical.

2. Slotted weights should be chosen according to elastic limit ofthe spring.

3. After adding or removing the slotted weight on the hanger, waitfor sometime before noting the position of the pointer on thescale because the spring takes time to attain equilibrium position.

SOURCES OF ERROR1. If support is not perfectly rigid, some error may creep in due to

the yielding of the support.

2. The slotted weights may not be standard weights.

DISCUSSION

1. A rigid support is required for suspending the helical spring withload (or slotted mass) from it. The slotted masses may not haveexact values engraved on them. Some error in the extension islikely to creep in due to the yielding (sometimes) of the supportand inaccuracy in the values of the masses of loads.

2. The accuracy of the result depends mainly on the measurementof extension produced by the force (load) within the elastic limit.Take special care that the slotted mass is put gently on the hangeras the wire of the helical spring takes sometime to attain its new-equilibrium position.

3. If the elastic limit is crossed slightly, what changes will you expectin the spring and your result?

SELF ASSESSMENT

1. Two springs A (of thicker wire) and B (of thinner wire) of the samematerial, loaded with the same mass on their hangers, aresuspended from a rigid support. Which spring would have morevalue of spring constant?

2. Soft massive spring of mass Ms and spring constant K has

extension under its own weight. What mass correction factor for

9

212


the extension in the spring would you suggest when a mass, M isattached at its lower end?

[Hint: Extension Xm of the spring of mass M

s with the mass M

attached at its lower end would be = + sm ( )( )

2MF g

X MK K

]

3. What other factors affect spring constant, e.g. length.


1. Take spring of the same material but of different diameters of thewires. See how the spring contant varies.

2. Take springs of the same diameters of the wires but of differentmaterials. See how the spring constant varies. What inference doyou draw from your result?

213

UNIT NAMEPROJECT

PROJECTPROJECTPROJECTPROJECTPROJECT1010101010AIM

To study the effect of nature of surface on emission and absorptionof radiation.

APPARATUS AND MATERIAL REQUIREDTwo identical calorimeters with wooden lids having holes forthermometers, two thermometers, clamp stands for holdingthermometer, arrangement to coat one calorimeter black and the othershining white, stop-clock, ice.

PRINCIPLEBlack surfaces are good emitters and good absorbers of heat radiation.Bright surfaces are poor emitters and poor absorbers of heat radiation.

PROCEDUREA. For emission of radiation

1. Note the range and least count of both the thermometers.

2. Record the room temparature.

3. Paint one of the calorimeters with black paint or lamp black asshown in Fig. P 10.1(a) and the other calorimeter white withaluminium paint or by wrapping shining silver foil around thecalorimeter as shown in Fig. P 10.1(b).

4. Fill hot water in each calorimeter and insert a thermometer in each.Let them stand 30 cm apart.

5. Start the stop-clock and keep it in the middle.

6. Record the temperatures of both the calorimeters at intervals of1/2 a minute for first 10 minutes and next 10 minutes at intervalsof one minute.

B. For absorption of radiation

1. Use the two calorimeters used for Activity (A) above.

2. Fill them with cold water taken from the refrigerator or made byadding ice to tap water.

3. Insert thermometers in the calorimeters and place them in front ofan electric heater so that they receive the same amount of heat.

214


Alternatively, place them in the sun, if there is bright sunlightcoming from a window.

4. With the help of a stop-clock, take temperature vs. time data as inActivity (A).

OBSERVATIONS AND CALCULATIONSRange of thermometer A = ... °C

Least count of thermometer A = ... °C

Range of thermometer B = ... °C

Least count of thermometer B = ... °C

Table P 10.1(a) : For emission of radiation

Fig. P 10.1(a): Experimental setup for studyingemission of heat radiation fromblack surface

Fig. P 10.1(b): Experimental setup for studyingemission of heat radiation fromshining surface

Table P 10.1(b) : For absorption of radiation

A

Calorimeter A

Coated black paint

Hot water

Stand

Coated with silverpaint or wrappedwith silver foil

Calorimeter B

Hot water

Stand

B

White painted calorimeterS. No.

Time (t)(minutes)

Temperature ofwater (°C)

Time (t)(minutes)


123

Black coated calorimeter

S. No.

Time (t)(minutes)


Time (t)(minutes)


123

White painted calorimeterBlack coated calorimeter

215

UNIT NAMEPROJECT

GRAPHPlot a graph between time (on x-axis) and temperature (on y-axis) forboth the calorimeters and for both, emission and absorption, as shownin Fig. P 10.2 (a) and (b).

Fig. P 10.2(a): Temperature vs. time graph for emission of heat radiation

Fig. P 10.2(b): Temperature vs. time graph for absorption of heat radiation

y

0x

Time (min)

Tem

per

atu

re (°C

)

2 4 6 8 201816141210

90 °C

Calorimeter B

Calorimeter A

(silvered)

(Blackened)25 °C x

2 4 6 8 201816141210

y

40 °C

Tem

per

atu

re (°C

)

Calorimeter BCalorim

eter A

(silvered)

(Blackened)

Time (min)

25 °C

CONCLUSION1. Compare the rates of cooling in Activity (A) in both cases for the

same temperature range. It is found that the (blackened/silvered)calorimeter is a better emitter of heat.

2. Compare the rise in temperatures of the two calorimeters in Activity(B). It is found that the ... calorimeter is a better absorber ofradiation.

SOURCES OF ERROR1. Perfectly black and perfectly shining surfaces may not be available.

2. Variations in surrounding temperature during the period ofActivity may take place.

10

216



(P 11.1)

AIMTo study conservation of energy with a 0.2 pendulum.

APPARATUS AND MATERIAL REQUIREDA heavy spherical, bob with a hook, thread, metre scale, a peg (a pencilor a 15 cm scale), a rigid support and a stand with a clamp.

PRINCIPLEA simple pendulum of length l, mass m oscillates due to therestoring force expressed as F = – mg sin θ for small displacement(less than 15°)

sinθ = θ =x

l

The force constant k can be written as k = mg

l

and maximum kinetic energy =KE kx 212

DESCRIPTION

When the oscillation of a simple pendulum is restricted into twoparts using a peg P at any point on its string, it becomes a two-length pendulum. During one half of the journey, the bob of massm, has length l1 and dispalcement x1 at position A and for other halfit has a length l2 and displacement x2. At position B, the bob of massm has the same kinetic energy. Therefore, energy conservationdemands that

=k x k x2 21 1 2 2

1 12 2

or =l xl x

21 1

22 2

This relationship (Eq. P 11.1) can be verified for different positionsof peg P.

217

UNIT NAMEPROJECT

PROCEDURE1. Setup a simple pendulum using a heavy bob. Release the bob

gently from position A and measure the maximum displacementx1, using a metre scale (Fig. P 11.1).

2. Fix a peg P (a pencil or a scale will do) horizontally to a clampstand and bring it in contact with the string of the oscillatingpendulum. The peg should obstract the motion of thependulum when its sting is vertical, that is, along its meanposition (Fig. P 11.2).

3. The effective length of the pendulum would get reduced for a partof its oscillation after it is held by the peg (Fig. P 11.2).

4. Measure the maximum displacement x2 using metre scale, whenthe bob reaches at position C.

5. Repeat the Steps 2 to 4 for different positions of peg P.

6. Record these observations in a table and calculate l

l1

2

and x

x

2122

for

each case.

7. Establish the equality, =l x

l x

21 1

22 2

.

OBSERVATIONS AND CALCULATIONSLength of a simple pendulum, l = ... cm

ABC

Peg ,P

x1x2

l2

l1

θ

Fig. P 11.1: A simple pendulum Fig. P 11.2: A two-length pendulum

A

B

x1

l

mg

θ

11

218


S. No.

1

2

3

4

Displacement of bob Length of the pendulum

In positionA

x1 (cm)

In positionB

x2(cm)

In positionA

l1(cm)

In positionB

l2(cm)

l

l1

2

x

x

2122

RESULT

Relationship =l x

l x

21 1

22 2

, based on the conservation of energy is verified.

219

UNIT NAMEDEMONSTRATION

To demonstrate uniform motion in a straight line

It is rather difficult to demonstrate uniform motion of a freelymoving body due to the inherent force of friction. However, it ispossible to demonstrate uniform motion if a body of the forcesacting on it are balanced.

(a) Demonstration of uniform motion of a body in glycerine or casteroil in a glass or a plastic tube

Take a glass or plastic tube one metre long and about 10 mm enddiameter. Close one end of it with a cork. Fill the tube with glycerine(white) or castar oil upto the brim. Insert a steel ball or lead shot ofthree mm diameter in it and close it with a cork such that no airbubble is left in the tube. Take a wooden base 7.5 – 10.0 cm broadhaving metallic brackets near its ends. Paint the board with whitepaint or fix a sheet of white paper on it. Mount the tube on the woodenbase with the help of metallic brackets (to rest the tube like the base ofa fluorescent tube). Put marks on the base with black/blue paint orink at regular intervals of 10 cm each [Fig. D 1.1(a)]. To demonstrate

DEMONSTRATIONDEMONSTRATIONDEMONSTRATIONDEMONSTRATIONDEMONSTRATION 11111

Fig. D 1.1: Demonstration of uniform rectilinear motion of a ball undertwo balancing forces: (a) A demonstration apparatus 1 mlong (b) A low cost apparatus 50 cm long

DEMONSTRATIONS

220

LABORA TORY MANUALLABORATORY MANUAL

uniform motion keep the tube vertical and ask a student to note thetime taken by the ball to travel successive segments of 10 cm. Repeatthe experiment by inverting the tube a couple of times. It may beemphasised that if a 10 cm segment is further sub-divided intosegments of 1 or 2 cm length, then the ball should travel successivesmaller segments also is equal intervals of time*.

This demonstration can also be done with a half metre long glass tubeand a half metre scale. It may be clamped vertical in a laboratory stand[Fig. D 1.1(b)]. In this case students can also be asked to note the timetaken by the ball to travel successive segments of one cm.

The tube may be inclined slightly, say, at about 5° to the vertical. Theadvantages of this are:

(i) The ball moves closer to the scale which reduces the parallaxerror in observing its position on the scale.

(ii) The ball moving in contact with the wall of the tube is underidentical conditions throughout its motion. If you wish it tomove in the centre of the tube, i.e., along the axis of the tube,then the vertical adjustment of the tube has to done withgreater precision.

In order to perform this demonstration with the half metre tubemore effectively, students may be encouraged to devise their ownmechanism to simultaneously record the distance moved by the balland the time taken to do so. For example, let one student watch thefalling ball at close distance and give signals by tapping the table as

the ball passes successive equidistant marks at apre-decided distance from each other.

A second student may start the stop-watch at thesound of any tap. Thereafter, he observes and speaksout the time shown by the watch at each successivetap, without stopping the watch. A third student maykeep noting the data of distance covered by the balland time elapsed since the measurement was started.Ask students to plot the distance versus time graphof the motion of the ball on the basis of this data anddiscuss the nature of this graph [Fig. D 1.1(c)].

In this coordinated activity of three students, it islikely that the first one may happen to miss givingsignal at a mark when the ball passes it. He shouldonly indicate this by saying “missed” and a fewpoints less on the graph made with about 15 to 20

points are of no significance. Similarly, any tapping which hesubsequently feels, was not made at the right instant, he may indicate

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

20

30

40

50

60

70

80

10

90

100

0

(s)t

(cm

)d

Fig. D 1.1(c): Distance–time graph for motionof metal ball in glycerine

* In this experiment, the ball accelerates for some time initially and approaches theterminal velocity u

0 according to relation u = u

0 = (1-e-t/T). For a typical terminal

velocity u0 = 3 cms–1, the time constant T = 0.003s. Thus, the duration of acceleratedmotion is so small that one may not at all bother for it.

221


by saying “wrong”. Two students can also record this data, if thereis sufficient time between successive readings, the second one takingover the task of the third. With some practice and by keeping thewatch in the left hand close to the ball, even one student can recordthe data and take it up as an individual activity.

By mixing water with glycerine in a suitable ratio one can makeadjust the speed of motion of the ball such that it is neither tooslow as to cause boredom to the class nor so fast that the data isdifficult to record.

(b) By using a burette

The above demonstration may also be performed by using a longburette. It has its own scale too. However, it may be difficult forstudents sitting at the back in the classroom to see the scale. Also,the upper end is open, which implies that several balls of the samesize should be available. In fact, in the demonstration (a) above, theupper end of the tube may be kept open, if several balls of the samesize are available, since the most tricky part of it is to close the upperend leaving no air bubbles inside the tube.

The demonstration with the burette can also be made more effectivein the same manner as discussed above.

Note:

1. In the class discussion following the demonstration of a steelball falling down with uniformed speed, an important questionwill be “what are the two balancing forces under which it moveswith uniform velocity?” One is the net weight of the ball actingdownwards due to which its speed increases in the beginning.As its speed increases, the resistance of liquid, acting upwards,to the motion increases till it balances the weight. Thenonwards, the ball acquires terminal velocity and the speedremains nearly constant.

2. There are a number of situations in everyday life where an objectfalls down with uniform velocity in exactly the same manner asthe ball in a liquid.

(a) When a paratrooper descends from an aeroplane with thehelp of a parachute, resistance of air on the parachute oftenbalances her/his weight. In such an event she/he movesvertically down with uniform speed, except for somehorizontal drift due to the wind (Fig. D 1.2).

(b) Many children play with a toy parachute which is first thrownup. Then it moves down in exactly the same manner as theparatrooper with a parachute.

(c) A shuttle cock, which is used in the game of badminton,may be shot vertically upwards, when it comes down,

1

222


players often see that it is moving down with uniformdownward speed (if there is no wind) after a small initialperiod of increasing speed.

3. This demonstration may also be done by the apparatus used forfinding the viscosity of liquid by Stoke’s law. However, fordemonstrating uniform motion in a straight line, thedemonstration is easier and better by: (a) using a scale to readthe position of the ball, and (b) keeping the tube slightly inclinedtowards the horizontal.

Fig. D 1.2: Discent of a parachute is nearly uniform

223


To demonstrate the nature of motion of a ball on aninclined track

Make an inclined plane of about 50 cm length with 2 – 3 cm height atthe raised end. Alternately, one can use a double inclined trackapparatus and make the inclined plane by joining its two arms at thebase strip so that these form a single plane. Give it a low inclinationby raising one end of the base strip by about two cm with the help ofa wooden block, or a book, etc. (Fig. D 2.1). Now let a metronomeproduce sound signals at intervals of ½ seconds. Keep the ball at thehigher end of one of the inclined planes. Release it at any signal (whichmay be called 0th signal) and let students observe its position at 1st,2nd, 3rd and 4th signals after the release. For this purpose, divide theclass into four groups. Explain to them in advance, with the help of adiagram on the blackboard, that group I will observe the 1st positionof the ball, group II the 2nd position of the ball, and so on.

22222DEMONSTRATIONDEMONSTRATIONDEMONSTRATIONDEMONSTRATIONDEMONSTRATION

A′B′

Fig. D 2.1: Motion of a ball on a double inclined plane

After the demonstration, there are as many observations for eachposition of the ball as the number of students in each group. Let onestudent in each group collect the observation in his/her group,calculate the mean value and record it on the blackboard. Then itcan be shown that distances covered by the ball in successive intervalsof ½ second go on increasing by equal amounts when the ball rolldown the incline.

Note:

1. In the absence of a metronome, let a person tap on the table at asteady pace which synchronises with extreme positions of thependulum of a clock, or a simple pendulum of 25 cm length ona laboratory stand.

2. If a strobe-light is available, use it illuminate the ball movingdown the track. Then students can visually see successivelylonger distances moved by the ball in equal intervals of time.

224


To demonstrate that a centripetal force is necessary for movinga body with a uniform speed along a circle, and that magnitudeof this force increases with angular speed

(a) Using a glass tube and slotted weights

Take a glass tube about 15 cm long and 10 mm outside diameter.Make its ends smooth by heating them over a flame. Now pass astrong silk or nylon thread about 1.5 m long through the tube. Atone end of the thread tie a packet of sand or a rubber stopper andat the other a weight (W) (about three to 10 times the weight of thesand/cork). First, demonstrate that on lifting the glass tube, theweight stays on the table while the packet of sand or the stoppergets lifts up (Fig. D 3.1).

Now by holding the glass tube firmly in one hand and the weight (W)in the other, rotate the packet of sand in a horizontal circle. When thespeed of motion is sufficiently fast, the weight (W) can hang freelywithout the support of your hand. Adjust the speed of rotation suchthat the position of the weight (W) does not change. In this situation,weight (W) provides the centripetal force necessary to keep the packetor stopper moving along a circular path (Fig. D 3.2). If the speed ofmotion is increased further, the weight (W) even moves up and viceversa. Why?


Fig. D 3.1:The weight tied at the end passingdown the glass tube is much heavierthan the packet of sand

225


As a safety precaution, in this demonstration,the packet to be rotated in a horizontal circleshould be a packet of sand, or a packet of afew fine lead shots, or a rubber stopper, etc.,lest it breaks off and strikes someone. Again,the glass tube should be wrapped with twolayers of tape, lest it breaks and hurt the handof the person demonstrating the experiment.

(b) Using a roller and a turn table

If a turn table (as you might have seen in agramophone) or a potter’s wheel is available,it can also be used to demonstrate centripetalforce. A small roller is placed on the turntable and its frame is attached to the controlpeg by a rubber band (Fig. D 3.3). The rolleris free to roll radially towards or away fromthe centre. The disc is set in motion first atthe lowest speed of 16 revolution per minute.The stretching of the rubber band indicatesthat a force acts outwards along the radius.At higher speeds, 33 r.p.m., or 45 r.p.m., or 78 r.p.m., the stretchingof the rubber band could seen to be larger and larger, showing thatgreater and greater centripetal force comes into play. Note that asthe angular speed increases, the radius of circular motion of theroller also increases due to elongation of the rubber band.

Glass tubeheld in hand

Packet movingin a horizontalcircle

Nylon thread

Weight

Fig. D 3.2: On revolving the packet ofsand at a suitable speed,the weight lifts off the table;its weight is just enough toprovide the necessarycentripetal force

RollerRubber band

Central peg

Turn table

Fig. D 3.3: Elongation of rubber band indicatesthat it is exerting a centripetal forceon the roller

3

226


To demonstrate the principle of centrifuge

Bend a glass tube (about 10 to 15 mm diameter) slightly at its middleto make an angle of, say, 160°. Fill it with coloured water leaving anair bubble in it and then close its both ends with rubber stoppers.Now mount it on the turn-table with both its arms inclined tohorizontal say, at, 10° while keeping the turn-table horizontal. Thelowest portion of the tube in the middle is attached to the central pegof the turn-table (Fig. D 4.1). The air bubble then stays at the top ofone or both the arms of the glass tube.

Now rotate the turn-table and increase its speed in steps, 16 r.p.m.,then 33 r.p.m., then 45 r.p.m. and then 78 r.p.m. As the speed ofrotation increases, draw attention that the air bubble is movingtowards the centre, the lowest part of the tube.

The rotating turn-table is an accelerated frame of reference. At everypoint on it, the acceleration is directed towards the centre. Thus, anobject at rest in this frame of reference experiences an outward force.Every molecule of water in the tube experiences this force, just likethe force of gravitation. Under the action of this force, denser mattermoves outwards and the less dense inwards.

Fig. D 4.1: A bent glass tube filled with a liquid but having an air bubbleattached to the central peg of turn table at its middle


227


To demonstrate interconversion of potential and kinetic energy

Interconversion of kinetic and potential energies may be easilydemonstrated by Maxwell’s Wheel (Fig. D 5.1). It consists of a wheelrigidly fixed on a long axle passing through its centre. It is suspendedby two threads of equal length, tied to the axle on two sides of thewheel. In the lowest position of the wheel, separation between thelower ends of the two threads is slightly more than that betweenthem at the supporting at thetop.

To set it in action the wheel isrotated and moved up so thatboth threads wind up on theaxle. As the wheel moves up, itgains some potential energy. Onreleasing, it moves down and itsP.E. is converted to K.E. ofrotation of the wheel. At itslowest position when all thelength of the two threads hasunwound, all the energy of thewheel is kinetic due to which thethreads start winding up in theopposite direction.

Thus, the wheel starts movingupwards, converting its K.E.into P.E.


Wheel

Axle

Thread

Thread

Fig. D 5.1: The Maxwell’s wheel

Note: In order to ensure that loss of energy in successive up anddown motions of the wheel be small, the threads should bequite flexible, inextensible and identical to each other.

228


To demonstrate conservation of momentum

The law of conservation of momentum can be demonstrated usingtwo bifilar pendulums of the same length using bobs of differentmaterials (Fig. D 6.1). The time period T for both pendulums is thesame. Initially the two bobs A and B touch each other in their restposition. Also the suspension fibres of A and B are parallel to eachother in their rest positions.

The bob A is displaced with thehelp of a wooden strip and allowedto touch the reference peg C andthus given a displacement, a,which is noted with the scale. Thestrip is then quickly removed, sothat bob A moves smoothlytowards the rest position andcollides with the bob B. Themaximum displacement a′ and b′of the bobs A and B respectivelyafter collision are notedsimultaneously. On the right handside of B, a rider is put on the scale,which is pushed by the ball B, asit undergoes the displacement b′.Then reading the displacement of A directly and of B from thedisplaced position of the rider becomes easier.

The masses mA and mB of the bobs are measured. The velocities of thebobs, just before and just after the collision are proportional to theirdisplacements, since the time period, T, for both the pendulums isequal and the velocity of a simple pendulum in its central position isequal to (amplitude × 2π/T). Therefore, the equality of total momentumof the two bobs before and after their collision implies

mA a = mA a′ + mB b′

Having measured a, a′ and b ′, the above equality can be checked up(a′ and b′ are the displacements’ after the impact).


AC B

Fig. D 6.1: The bifilar pendulums

229


To demonstrate the effect of angle of launch on range of aprojectile

The variation in the range of a projectile with the angle of launch canbe demonstrated using a ballistic pistol or toy-gun and mounting itsuitably so that the angle of launch can be varied. While mountingthe gun care must be taken to see that the axis of the gun passesthrough the centre of the circle graduated in degrees (Fig. D 7.1). If atoy-gun is used, whose maximum range is more than the length ofthe classroom, then this demonstration may be done in an openarea such as the school play ground.


Circularprotector Wooden

circular discrigidly fixedvertically

Holes forfixing theclamp

Clamp

Plumb-line

90°

180°

270°

Fig. D 7.1: A set up to study the range of aprojectile fired with a toy pistol

As the gun is fired at different angles ranging between 0° and 90°,the corresponding ranges are measured with care. A graph for theangle of projection versus the range may be drawn.

Alternately one can also study the range of water jet projected atdifferent angles provided it is assured that water will be released atsame pressure.

230


To demonstrate that the moment of inertia of a rod changes withthe change of position of a pair of equal weights attached to the rod

Take a glass rod and hang it horizontally from its centre of gravity withthe help of a light, thin wire. Take two lumps of equal mass of plasticine,roll both of them separately to get discs of same size and uniformthickness. Now attach them near the two ends of the rod (like rings) sothat the rod is again horizontal [Fig. D 8.1(a)]. Make sure that theplasticine cylinders easily move along the rod. Give a small angulardisplacement to the rod and note the time for 10 oscillations. Find thetime period for one oscillation. Now, move the rings of plasticine byequal distances towards the centre of the rod so that it remainshorizontal [Fig. D 8.1(b)]. Give a small displacement to the rod andagain note the time period for 10 oscillations. Find the time period forone oscillation. Are the two time periods the same or different? If you


Nut closerto centre

(b)

Nut nearthe ends

Glass rod

Copper wire

C.G.

(a)

Fig. D 8.1: Setup to demonstrate that total mass remaining constant, themoment of inertia depends upon distribution of mass. Herenuts have replaced the plasticine balls: (a) the movable massare far apart, (b) the masses are closer to the C.G. of the rod

231


find that the time periods in both the situations are different, it showsthat the moment of inertia changes with the distribution of the massof a body even if the total mass remains the same.

An important caution for a convincing demonstration is that thepoint where a thin metal wire is attached to the glass rod (the pointabout which the glass rod makes rotatory oscillations) should remainfixed. The metal wire should be so tied that the rod hangs horizontallyfrom it. It ensures that the axis of rotation passes through its C.G.The wire can be fixed tightly by using a strong adhesive. Therefore,the position of plasticine discs have to be adjusted so that the glassrod hangs horizontally.

8

To demonstrate the shape of capillary rise in a wedge-shaped gapbetween two glass sheets

You would require two plane glassslides, a thick rubber band, a matchstick, a petri-dish, some potassiumpermanganate granules and a felt-tipglass marking pen.

Clean the two slides and the petri-dishthoroughly with soap and water andrinse with distilled water. Ensure thatno soap film remains on them. Fill thedish about half with distilled watercoloured by potassium permangnate.Tie one end of the pair of slides togetherwith a rubber band and put a matchstick between their free ends (Fig. D 9.1).Dip this arrangement in the colouredwater in the petri-dish. Water rises moreat the tied end as compared to that atthe match stick end because theseparation between the glass slidesincreases linearly from the tied end tothe match stick end.

Note

1. The same effect could bedemonstrated by using a numberof capillary tubes of differentdiameters arranged side by side inincreasing order of diameter, asshown in Fig. D 9.2.

2. Students may take up thisexperiment as an activity orproject work.


Rubber band

Water levelbetween the slides

Match stick

Glass slides

Coloured water

Petri-dish

Fig. D 9.1: Capillary rise of water is higher at the endtied by rubber band in the wedge-shapedgap between the glass slides

Colouredwater

Petri-dish

Capillarytubes

Waterlevels

Fig. D 9.2: Rise of water in capillary tubes of differentdiameters

To demonstrate affect of atmospheric pressure by making partialvacuum by condensing steam

To perform this demonstration you will need a round-bottom flask,a glass tubing, a cork, cork borer, a long piece of pressure rubbertube just fitting the glass tubing, a pinch cock, burner, tripot stand,laboratory stand with a clamp and large water container.

Take some water in a round bottom flask. Close its mouth tightlywith a rubber cork, in which a short glass tube is fitted. Attach apressure rubber tube, about 1.5 m long, in the open end of the glasstube. Heat the water, as shown in Fig. D 10.1(a). The steam producedin the flask expels the air from the flask, the glass tube and therubber tube. Stop heating after some time and tightly close the mouthof the rubber tube with a pinch cock immediately.

Invert the flask and clamp it as high as possible in a tall stand placedon the table [Fig. D 10.1(b)]. Dip the free end of the rubber tube incoloured water kept in the large container on the ground and releasethe pinch cock. As the flask cools, water from the container rushesthrough the glass tube into the flask. The students will naturally


Rubber tube

Short glass tubeRubber cork

Water

Fig. D 10.1 (a): On heating the water in flask,steam drives air out from it

234


become curious to know the reason why water rises through the height.It may be explained in terms of difference in pressure of air on thesurface of the water in the container and inside the flask.

Note

To make this experiment more spectacular, a student may climbon the table and raise the stand by another 2 m. Then the pressurerubber tube may also have to be longer.

Fig. D 10.1 (b): Atmospheric pressure pushes coloured water upinto the flask as steam in the flask condenses

235


To study variation of volume of a gas with its pressure at constanttemperature with a doctor’s syringe

This demonstration can be given with the help of a large (50 mL ormore) doctor’s syringe (disposable type), laboratory stand, grease orthick lubricating oil, 200 gram to 1 kg weights which fit over oneanother, cycle value-tube, rubber band, a wooden block and alaboratory stand.

Make the piston in the syringe air tight by applying a drop of thicklubricating oil or grease into the syringe. Draw out the piston in thesyringe so that the volume of air enclosed by it is equal to its fullcapacity. Next close the outlet tube of the syringe by fixing a piece ofcycle value-tube on it and folding the valve-tube. Hold the syringevertically with a laboratory stand with its base resting on a woodenblock (Fig. D 11.1).

Press the piston downward with the hand to compress the air inside.Release the piston and observe, whether the air inside regains itsinitial volume by pushing the piston up. Since, the friction betweenthe piston and the inner surface of the syringe is quite large, both


Piston

Graduatedouter bodyof syringe

Cycle valvetube

Wooden block

Rubber orcloth pad

Compressed air

5

10

0

25

35

40

20

15

Fig. D 11.1: The load is kept on the piston of the syringe to applythe force of its weight along the axis of the piston

236


being of plastic, the air inside is unable to push the piston upto itsoriginal position. When the piston comes to rest, the thrust ofatmospheric pressure plus limiting friction is acting on it downwards.Note the volume of enclosed air in this position of the piston.

Next, pull the piston up a little and release. Again it does not reachquite upto its original position. This time the thrust of atmosphericpressure minus limiting friction is acting on it downwards. Note thisvolume of air also and check that the mean of the two volumes someasured is equal to the original volume of air at atmospheric pressure.

Now balance a 1 kg weight on the handle of the piston. Note the twovolumes of enclosed air, (i) piston slowly moving up and coming torest, and (ii) piston slowly moving down and coming to rest and findtheir mean. In this manner note volume, V, of air for at least twodifferent loads, say 1 kg and 1.8 kg, balanced turn by turn on thepiston. Check up, in the end that volume for no load is same as thatat the beginning to ensure that no air leaks out during the experiment.Draw a graph between 1/V and load W for the three observations,W = 0 kg, 1 kg and 1.8 kg if a graph black-board is available.Alternately, it may be given as an assignment to students.

237



Fig. D 12.1

Fig. D 12.2

To demonstrate Bernoulli’s theorem with simple illustrations

(a) Suspend two simple pendulums from a horizontal rod clampedto a laboratory stand (Fig. D 12.1). Use paper balls or table tennisballs as bobs. Their bobs should be close to each other and atthe same height but not touching eachother. Ask the students what wouldhappen if you strongly blow into thespace between the bobs. A person/student not thinking in terms ofBernoulli’s theorem would conclude thatair pushed into this space will push thebobs away from each other. Now blow airbetween the two bobs suspended closeto each other and ask them to observewhat happens. The speed of air passingbetween them gets increased due to lessspace available and so the pressure there,gets decreased. Thus, the pressure of airon their outer faces of the bobs pushesthem closer. That is why one observesthe bobs to actually move closer.

(b) Place a sheet of paper supported bytwo books in the form of a bridge.Let the books be slightly converging(Fig. D 12.2) i.e., their separation is largeron the side facing you. Now, you blowunder the `bridge’, the paper `bridge’ ispushed down.

(c) Hold the shorter edge of a sheet of paperhorizontally, so that its length curvesdown by its weight [Fig. D 12.3(a)]. If youpress down lightly on the horizontal partof the curve with your finger the papercurves down more. Now, instead oftouching with the hand hold the horizontal edge of the sheet ofpaper close to your mouth. Blow over the paper along thehorizontal. Does the hanging sheet of paper get pushed down or

238


lifts up [Fig. D 12.3(b)]? The curved shape of paper makes thetubes of flow of the wind narrower as the wind moves ahead asshown in [Fig. D 12.3(c)]. Thereby its speed increases and pressureon the upper side of the paper decreases.

Sheet ofPaper Blown Air

Fig. D 12.3 (b)

Fig. D 12.4

(d) Fill coloured water in an insecticide/pesticide spraypump. Spray the water on a white sheet of paper.Coloured drops deposit on the paper. It is evidentthat water from the tank rises up in the tubeattached to it and is then forced ahead in the formof tiny droplets. But what makes it rise up in thetube? As the pump forces air out of a fine hole, thespeed of air in the region immediately above theopen end of the tube in the tank becomes high(Fig. D 12.4). Thus, the pressure of air in the region

is lower than the surrounding still air (which is equalto atmospheric pressure). Right in this region, just belowthe hole in the pump is the upper end of the fine tubethrough which water rises up, due to atmosphericpressure acting on the surface of water outside the tube.

Fig. D 12.3 (a)

Fig. D 12.3 (c)

239


(e) Fig. D 12.5 shows the construction of a Bunsen burner. The fuelgas issues out of the jet J in the centre of the vertical tube. Dueto the high speed of gas, its pressure gets lowered. So, through awide opening in the side of the vertical tube air rushes in, mixesup with fuel gas and the gas burns with a hot and blue flame. Ifthe air does not get mixed with fuel gas at this stage and comesinto contact with it only at the flame level, the flame will bebright yellow-orange like that of a candle, due to incompletecombustion of the gas which gives off comparatively less heatthan when it burns with a blue flame.

Fig. D 12.5

12

240


To demonstrate the expansion of a metal wire on heating

Stretch a length of any metal wire firmly between two laboratorystands, which are fixed rigidly on the table by G-clamps (Fig. D 13.1).Suspend a small weight at the centre of the wire and stretch the wireas tightly as possible, without significantly bending the iron stands.However, the wire cannot be made straight and some sagging isinevitable due to the weight suspended at the centre. Place a pointeron the hind side of the upper edge of the weight to serve as reference.

Heat the wire along its entire length by a spirit lamp or a candle. Thewire is seen to sag more and the weight moves down. Remove theflame to let the wire cool. As the wire gradually cools, the weightascends to its original position.


Smallweight

Pointer

Wire

Fig. D 13.1: A taut wire sags on heating due to its thermal expansion

Note:

The wire can also be heated electrically, if so desired. Use a step-downtransformer which gives various voltages in steps from 2 volt to 12volt. The advantage is that heating of the wire for a certain voltageapplied across it will be uniform along its whole length and theobserved sagging by this heating will be repeatable.

241



(a)

(b)

Fig. D 14.1: Qualitative comparison ofheat capacities of differentmetals

To demonstrate that heat capacities of equal masses ofaluminium, iron, copper and lead are different

This demonstration can be performed withfour cylinders of aluminium, iron, copperand lead having equal mass and cross-sectional area, a rectangular blocks ofparaffin wax, beaker/metallic vessel,thread, water and a heating device.

Since the four solid cylinders are havingequal mass and equal cross-sectionalarea, their lengths are inverselyproportional to their densities. Takewater in a beaker or a metallic vessel andboil it. Suspend the four cylinders, tiedwith threads, fully inside boiling water(cautiously, if a beaker is being used).After a few minutes all have attainedthe temperature of boiling water[Fig. D 14.1(a)].

Take out the cyl inders in quicksuccession and place them side byside on a thick block of paraffin wax[Fig. D 14.1(b)]. The cylinders sink todifferent depths in the paraffin wax. Theyall cool from temperature of boiling waterto melting point of wax during the processof sinking. Although all the cylinders havethe same mass, but the amount of heatthey give out are different.

An alternative (and more convenient to do)method is to have a wooden board withsemi cylindrical grooves resting against ablock. Equal length of each groove isinitially filled with wax. Hot cylinders areplaced on this wax in the grooves, insteadof on the wax block.

242


Note

A substantial portion of heat given out by each cylinder isradiated into the atmosphere. Moreover, they radiate atdifferent rates because of the difference in their surface areas.Therefore, by this experiment we only get a qualitativecomparison of the heat capacities of these solids.

243



Fig. D 15.1: Set up to study oscillationsof a heavy pendulum

To demonstrate free oscillations of different vibrating systems

A number of demonstrations involving vibrating systems arepresented through (a) to (j). Demonstrate as many vibrating systemsas possible and discuss the following in each case:

(i) What are the energy changes that occur during vibrations?

(ii) How can the frequency of vibration be altered?

(iii) Can the damping of the system be reduced? If so, how?

(iv) How does the force acting on the oscillating body vary withits displacement from themean position?

(a) Simple pendulum: Make arather long and heavy simplependulum following the stepsdescribed in Experiment 6. Onemay tie a brick or a 1kg weightat one end of a strong threadabout 1.5 m in length. Suspendthe pendulum from a standhaving a heavy base so that itdoes not topple over. The basecan be made heavy by puttinga heavy load on it, say a fewbricks. Alternatively, the standmay be clamped on the tablewith a G-clamp. The vertical rodof the stand may be furthersupported by tying it to threeG-clamps fixed on the table(Fig. D 15.1). A sturdy stand willhelp in keeping the pendulumoscillating for quite a long timewith very small damping.

(b) Vibrating hacksaw blade: Clamp a hacksaw blade (or a thin metalstrip) with its flat surface horizontal at the edge of the table by a

244


G-clamp (Fig. D 15.2). Load the free end byabout 20 g of plasticine or by putting a 20 gweight on the flat free end and fastening it tothe blade with a thread. Let the free end of theblade vibrate up and down. Repeat thedemonstration with a smaller load and thenwith no load on the blade. Compare theoscillations with different loads.

(c) Oscillating liquid column: Fix a U-tube oflarge diameter (about 2cm) on a stand withits arms vertical. Fill liquid of low viscositye.g., water or kerosene or methylated spiritin it. Let the liquid column oscillate up anddown in the tube (Fig. D 15.3). For thispurpose blow repeatedly into one arm ofthe U-tube with your mouth as soon asthe liquid column in the arm you areblowing attains maximum height so as togenerate a small air pressure in it eachtime so as to oscillate the liquid columnby resonance. Another method is to slightlytilt the stand to one side repeatedly, withthe U-tube fixed on it so as to oscillate theliquid column by resonance.

A low cost U-tube can be improvised withtwo straight tubes of about 3.5 cm to 4 cmdiameter and each of length about 50 cm. Fixthe tubes vertically on a wooden board about20 cm to 30 cm apart. Join their lower endswith a piece of a rubber tube, or a piece of hosepipe made of plastic. A plastic hose pipe is better

because it bends to the U-shape easily. Fill this U-tube withcoloured water upto about 10 cm below the two open ends.Oscillate the liquid in the tube by either of the two methodsdescribed above.

(d) Helical spring : Attach a suitable mass, say 1kg, at one end of ahelical spring (Fig. D 15.4). Suspend the spring vertically. Pullthe weight down through a small distance and let it go. Observeand study the vertical oscillations of the mass suspended by thelower end of the spring.

(e) Oscillations of a floating test-tube : Take a test tube and fill at itsbottom about 10 g of lead shots or iron filings or sand. Float thetube in water and adjust the load (lead shots or iron filings orsand) in the tube till it floats vertically. Keeping the tube verticalpush it a little downwards and release it so that it begins tooscillates up and down on the surface of the water (Fig. D 15.5).

Fig. D 15.2: A hacksaw blade clamped atone end vibrates up and down

Fig. D 15.3: Set up to demonstrate oscillationsof liquid column in a U-tube

Fig. D 15.4: A load attachedto the lower end of a helicalspring oscillates up and down

245


(f) Oscillations of a ball along a curve : Take about30 cm length of aluminium curtain channeland bend it into an arc of a circle. Put it on atable and provide it proper support by tworectangular pieces of thick card board orplywood to keep it standing in a vertical plane.Let a ball–bearing or a glass marble oscillatein its groove (Fig. D 15.6). Alternatively placea concave mirror (10 cm or 15 cm aperture)or a bowl or a karahi on a table with itsconcave side facing up. Let a ball bearing or aglass marble oscillate in it along an arc passingthrough its lowest point as shown by point Pin Fig. D 15.6.

(g) Oscillations of a ball on the double inclinedtrack: Adjust a double inclinedtrack on a table with its armsequally inclined to the horizontal(Fig. D 15.7). Release a steel ball–bearing (2.5 cm diameter) fromthe upper end of one of the armsand let it oscillate to and frobetween the two arms of thedouble inclined plane.

(h) Oscillations of a trolley heldbetween two springs on a table:Take a trolley and attach twoidentical helical springs at each ofits ends such that the springs are along a straight line. Place thetrolley on a table and fix the free ends of the springs to two rigidsupports on opposite ends of the table so that the springs areunder tension along the same straight line [Fig. D 15.8(a)].Displace the trolley slightly toone side keeping both springsunder tension. Release thetrolley and observe its to andfro motion along the length ofthe springs. Find the timeperiod of oscillations and alsomake a note of damping.

(i) Oscillations of a trolleyattached to a spring: Removeone of the springs from the setup arranged for demonstration(h) shown in Fig D 15.8 (a).Displace the trolley to one side and release it. Compare the timeperiod of oscillations affect of damping with the earlier case.

Fig. D 15.5: A test tube floatingin vertical positiondue to a load in it,oscillates up anddown, when it ispushed a little andthen released

Fig. D 15.6: Arrangement to demonstrate theto-and-fro motion of a steel ball ina channel in the form of an arc ofa circle

Enlarge view ofcentral curved part

Fig. D 15.7: Arrangement to demonstrate the to-and-fro motion of a ball along adouble inclined track

Lead shots

15

246


(j) Oscillations of a trolley suspended from a point and held betweentwo springs: Set up the trolley with two springs on a table asdescribed in demonstration (h) above. Attach an inflexible stringto the trolley as shown in Fig. D 15.8(b). Fix the other end of thestring to a stand kept on a stool placed on the table or to a hookon the ceiling such that the trolley remains suspended just abovethe table. Set the trolley in oscillation by displacing it slightly toone side. Study how the time period of oscillations and dampingget affected as compared to the case when the trolley was placedon the table, as in demonstration (h).

(a)

Fig. D 15.8: (a) Set up for demonstrating the to-and-fro motionof a trolley held between two identical springs(b) Arrangement to demonstrate the to-and-fro motionof a trolley suspended from a high support while it isheld between two springs on either side

(b)

Wheels nottouchingthe table

247


To demonstrate resonance with a set of coupled pendulums

Take two iron stands and keep them on the table at about 40 cmfrom each other. Tie a half metre scale (or still better a straight stripof wood about 1.5 cm wide) between them so that it is horizontalwith its face vertical and free to rotate about its upper edge(Fig. D 16.1). Near one edge of the scale suspend a pendulum with aheavy bob (say, approximately 200 g). Also suspend four or fivependulums of different lenghts with bobs of relatively lower masses.However, one of them should be exactly of the same length as theone with the heavy bob, as described.


Let all the pendulums come to a rest after setting up the arrangementdescribed above. Gently pull the bob of the heavy pendulum andrelease it so that it starts oscillating. Make sure that other pendulumsare not disturbed in the process. Observe the motion of otherpendulums. Which of the pendulums oscillates with the samefrequency as that of the heavy pendulum? How does the amplitudeof vibrations of different pendulums differ?

Fig. D 16.1: A set up to demonstrate resonance

248


To demonstrate damping of a pendulum due to resistanceof the medium

(a) Damping of two pendulums of equal mass due to air: Set up twosimple pendulums of equal length. The bob of one should be smallin size say made of solid brass. The bob of the other should be ofthe same mass but larger in size — either of a lighter material likethermocole or a hollow sphere. Give them the same initialdisplacement and release simultaneously. Observe that in thependulum with the larger bob the amplitude decreases morerapidly. Due to its larger area, air offers more resistance to itsmotion. Though both pendulums had the same energy to startwith, the larger bob looses more energy in each oscillation.

(b) Alternative demonstration by comparing damping due to air andwater: Set up a simple pendulum about half metre long with ametal bob of 25 mm or more diameter. In its vertical position thebob should be about 4 cm to 5 cm above the table. First, let thependulum oscillate in air and observe its damping. Now place atrough below the bob containing water just enough to immersethe bob in water. Let the pendulum oscillate with the bob immersedin water and note the effect of changing the medium on damping.


249


To demonstrate longitudinal and transverse waves

A few characteristic properties oftransverse and longitudinal waves canbe demonstrated with the help of aslinky, which is a soft spring made of athin flat strip of steel (about 150 to 200turns) having a diameter of about 6 cmand width 8 cm to 10 cm. Nowadaysslinky shaped spirings made of plasticsare also available. Let two students holdeach end of the slinky and stretch it toits full length (at least 5 metres) on asmooth floor. Give a sharp transversejerk at one end and let the studentobserve the pulse as it moves along thespring [Fig. D 18.1(a)].

Find the speed of the pulse bymeasuring the time taken by it to movefrom one end to the other along thestretched length of the spring. For moreaccuracy, instead of measuring timetaken by the pulse to move from oneend to the other, measure the time taken by it to make three to fourjourneys along the entire length of the spring. This would be possiblebecause each pulse moves back and forth along the spring a fewtimes before it dies.

Repeat the experiment by decreasingthe tension in the spring (by stretchingit to a smaller length) and find the speedof the pulse. Does the speed depend ontension?

The slinky can also be used todemonstrate propagation of longitudinal waves. To do so, give alongitudinal jerk at one end of the slinky, keeping the slinklystretched on the floor to about half the length (2.5 m) than whiledemonstrating movement of a transverse pulse [Fig. D 18.1(b)]. Ask


Fig. D 18.1(a): Motion of a pulse through aslinky

(a)

(b)

Fig. D 18.1(b): A compression moving alongthe length of a slinky

250


the students to observe the motion of the pulse in the form ofcompression of the spring.

The damping may be too high if the floor is not very smooth. In thatcase the experiment may be performed by suspending the slinky froma steel wire stretched between two pegs firmly fixed on opposite wallsof the room. In order to minimise the effect of sagging of the spring inthe middle, support the spring by tying it to the wire with pieces ofthread spaced at about 25 cm from each other. All pieces of threadmust be equal in length.

The transverse waves may also be demonstrated with the help of aflexible clothes line or a rubber tubing or a rope instead of a slinky.Tie one end of the rubber tubing or the clothes line to the knob of adoor and give it a jerk at the other end while keeping it stretched. Ifthe rubber tube is heavy (fill water in it) and is held loosely, the pulsewould move slowly to make better observation.

Instead of a single pulse, a series of pulses one after the other creatingan impression of a continuous wave propagation may also bedemonstrated. This can be done by using a slinky or a flexible clothesline. Stretch the slinky on the ground and ask one of the students tohold one end firmly. Instead of giving just one jerk at the other end,move the hand to and fro continuously to make waves of wavelengthabout 0.5m which can be seen to move continuously along the spring.

251


To demonstrate reflection and transmission of waves at theboundary of two media

Stretch the slinky on a smooth flooror suspend it from a stretched steelwire as described in Demonstration18.1. Keeping one end fixed, send apulse from the other end. Note thesize and direction of displacement ofpulse before and after it gets reflectedat the fixed end. Note that thereflected pulse is upside down withlittle change in its size in comparisonto the incident pulse [Fig. D 19.1(a)].

Next join the coil spring (slinky) withanother long helical spring of heaviermass end to end [Fig. D 19.1(b)].Stretch them by holding the free endof each spring and produce a pulseat the free end of the lighter spring(slinky). Observe what happens whenthe pulse arrives at the joint of twosprings. In what way (i.e., withrespect to size and direction ofdisplacement) does the reflectedpulse undergo a change? Does thepulse transmitted to the heavier spring also undergo any change?

Repeat the demonstration by sending the pulse from the end of theheavier spring. Note how the reflected and transmitted pulse undergoa change at the boundary of the two springs as compared to theincident pulse [Fig. D 19.1(c)].

How do these changes differ from those in case of incident pulsegoing from lighter to heavier spring?


(a)

Fig. D 19.1 (a): A pulse reflected at a fixed endundergoes phase change of π

252


Fig. D 19.1 (b): Reflection and transmissionof a pulse moving froma rarer medium to adenser medium

Fig. D 19.1 (c): Reflection and transmission ofa pulse moving from a densermedium to a rarer medium

Now join the slinky (coil spring) to a fine thread instead of a heavierspring. Stretch the spring and the thread and produce a pulse at thefree end of the spring. Note what happens to the pulse at the boundaryof the spring and the thread.

253


To demonstrate the phenomenon of beats due tosuperposition of waves produced by two tuning forks ofslightly different frequencies

Take two tuning forks of identical frequency. Attach a small piece ofplasticine or wax to the prongs of one of the tuning forks. This willslightly lower the frequency of the tuning fork. Now holding themone in each hand strike both the tuning forks simultaneously on tworubber pads. Place them close to each other.

Carefully listen to the combined sound produced by the two tuningforks. Gradual increase and decrease in the intensity of sound willbe heard. It is due to beats produced by the superposition of wavesof slightly different frequencies. You can also count the number ofbeats produced per second if their frequency does not exceed two orthree beats per second. The person who is listening to the beats,gives a silent signal at each minimum intensity or maximum intensity,e.g., by shaking his head in the manner we say ‘yes’. Then a secondperson with a stop-watch, either finds the time taken by 10 beats orcounts the number of beats in 5 seconds. The person with the stop-watch will also listen to the beats, though less loudly and may measurethe frequency without the aid of a signal by the first person.

If two tall tuning forks of the same frequency mounted on resonatingwooden boxes are available, all the students in a classroom may beable to listen to the beats. Place them on a desk in the centre of theclassroom. Let there be pin-drop silence in the classroom. Then strikethe tuning forks with a rubber hammer in quick succession, withroughly equal force. Make their frequencies slightly different byloading one with plasticine or wax or by tightly attaching a smallload with adhesive tape. Both tuning forks must be of rather goodquality and must give audible sound for about 8 to 10 seconds inspite of dissipation of energy in the resonating box.

DEMONSTRATIONDEMONSTRATIONDEMONSTRATIONDEMONSTRATIONDEMONSTRATION2020202020

254



To demonstrate standing waves with a spring

Stretch the wire spring (heavier one and not the slinky) to a length of6 m to 7 m, by tying its one end to a door handle. It may sag in themiddle but that will not affect the demonstration. Give a transversehorizontal jerk at the free end, a pulse will travel along the spring,and get reflected back and forth. If instead of stretching the spring inair it is stretched along the ground, then due to large damping, theresults will not be so clear and convincing.

Now generate a continuous transverse wave in the spring by givingseries of jerks to the spring at fixed time intervals. Change thefrequency of the waves by changing the time period of oscillating yourhand till stationary waves are set up. You will find that stationarywaves are produced only when an integral number of loops, i.e., 1,2,3etc. are accommodated in the entire length of the spring. In otherwords, stationary waves are produced corresponding to only somedefinite time periods.

Ask one of the students to measure the time period of standing waveswhen one loop, two loops, three loops, and so on are formed in agiven length of stretched spring. For the same extension of the spring,and thus for the same tension in the spring, how are the time periodsof stationary waves of one loop, two loops, and three loops related toeach other?

While producing stationary waves, suddenly stop moving your handto and fro and thus stop supplying energy to the spring. This is bestdone by taking the help of a stool on which your hand rests whileproducing the waves as well as when you stop your hand. Observethat the spring continues to vibrate for some time with the same timeperiod and the same number of loops. Thus, it can be demonstratedthat the stretched spring is capable of making free oscillations inseveral modes—with one loop, two loops, three loops, etc. The varioustime periods with which you can produce stationary waves in it, arealso the natural time periods of the spring.

Thus, when you are producing and observing stationary waves in thestretched spring, you can consider it as a resonance phenomenon.However, in this case, the object being subjected to forced oscillations(i.e., the stretched spring), is capable of oscillating freely with one of

255


the several time periods, unlike the simple pendulums with whichyou experimented earlier to study the phenomenon of resonance.

One can also demonstrate stationary waves with a spring when itsboth ends are free to move. Tie a thread, 3 – 4 m in length, at oneend of the spring. Tie other end of the thread to a hook on the wall ora door handle. Stretch the spring by holding it at its free end andsend a continuous transverse wave in the spring by moving the endin your hand. Do you observe that the stationary waves now producedare somewhat different than those produced when one end of thespring was fixed. Note the difference in the pattern of stationarywaves in the two situations and discuss the reason for the difference.Also ask to note the number of loops produced when a stationarywave is set in the spring.

Change the time period of the wave by adjusting to and fro motion ofyour hand to produce ½ loop, 1½ loop, 2½ loop and so on for sameextension of the spring.

How are these time periods related to the various time periods ofvibration when the end not in your hand was kept fixed and extensionof the spring was the same?

Note

Mathematically, it can be shown that superposition of two wavesof the same frequency (and thus moving with same velocity)travelling in opposite directions in an infinite medium, producestationary waves. In this mathematical treatment, there is noneed of specific frequencies at which the stationary waves areproduced. However, it is not possible to translate thatmathematical result into a simple experimental demonstration.In an experiment we have to take a finite medium, like thestretched spring of finite length. A finite medium with boundarieshas its natural frequencies and thus experiment is done at thosefrequencies. In the above demonstrations one wave is producedby hand and the other (travelling in the opposite direction) isthe reflected wave and their superposition produces stationarywaves, exemplifying the above referred mathematical result.

21

256


Name Symbol Value

Speed of light in vacuum c 2.9979 × 108 m s–1

Charge of electron e 1.602 × 10–19 CGravitational constant G 6.673 × 10–11 N m2 kg –2

Planck constant h 6.626 × 10–34 J sBoltzmann constant k 1.381 × 10–23 J K–1

Avogadro number NA

6.022 × 1023 mol–1

Universal gas constant R 8.314 J mol–1 K–1

Mass of electron me

9.110 × 10–31 kgMass of neutron m

n1.675 × 10–27 kg

Mass of proton mp

1.673 × 10–27 kgElectron-charge to mass ratio e/m

e1.759 × 1011 C/kg

Faraday constant F 9.648 × 104 C/molRydberg constant R 1.097 × 107 m–1

Bohr radius a0 5.292 × 10–11 mStefan-Boltzmann constant σ 5.670 × 10–8 W m–2 K–4

Wien’s constant b 2.898 × 10–3 m KPermittivity of free space ε0 8.854 × 10–12 C2 N–1 m–2

1/4πε0 8.987 × 109 N m–2 C–2

Permeability of free space μ0 4π × 10–7 T m A–1

≅ 1.257 × 10–6 Wb A–1m–1

APPENDIX A-1SOME IMPORTANT CONSTANTS

Other Useful Constants

Name Symbol Value

Mechanical equivalent of heat J 4.186 J cal–1

Standard atmospheric pressure 1 atm 1.013 × 105 PaAbsolute zero 0 K – 273.15 °CElectron volt 1 eV 1.602 × 10–19 JUnified Atomic mass unit 1 u 1.661 × 10–27 kgElectron rest energy mc2 0.511 MeVEnergy equivalent of 1 u 1 uc2 931.5 MeVVolume of ideal gas (0 °C and 1 atm) V 22.4 L mol–1

Acceleration due to gravity g 9.78049 m s–2

(sea level, at equator)

APPENDICESAPPENDICESAPPENDICESAPPENDICESAPPENDICES

257

UNIT NAMEAPPENDICES

APPENDIX A-2Densities of substances (20 °C)

Substance Density Substance Density(103 kgm–3) (103 kgm–3)

Alcohol (methyl) 0.81 Olive oil 0.9Alcohol (ethyl) 0.79 Quartz (crystal) 2.6Asbestos 2.4 Sea water 1.03Brass (60.40) 8.4 Stainless steel 7.8Brass (70.30) 8.5 Turpentine 0.85Cast iron 7.0 Wrought iron 7.8Caster Oil 0.95 Zinc 7.1Charcoal 0.4 Water 0 °C 0.99987Coal 1.6 – 1.4 4 °C 1.00000Copper 8.9 20 °C 0.99823Constantan 8.9 100 °C 0.9584Cork 0.24 Water, heavyDiamond 3.5 (D

2O) at max. densityGerman Silver 8.4 temperature, 11 °C 1.106Glass 2.5Glycerine 1.3 Petrol 0.70Gold (pure) 19.3 Kerosene 0.80Gold (22 carat) 17.5 Common salt sol. 1.189Gold (9 carat) 11.3 (20% by wt.)Graphite 2.3 Air (STP) 0.00129Ice 0.92 Carbon dioxide (STP) 0.00198Manganin 8.5 Hydrogen (STP) 0.00009Mild Steel 7.9 Oxygen (STP) 0.00143Milk 1.03 Nitrogen (STP) 0.00125Mercury 13.56

APPENDIX A-3Variation of atmospheric temperature and pressure with altitude

(At sea level, pressure = Standard atmosphere and temperature = 15 °C assumed)

Altitude Pressure Temperature (°C)(metres) (millibars)

(1) (2) (3)

0 1013.25 15.0250 983.58 13.4500 854.61 11.8750 926.34 10.1

1000 898.75 8.51500 825.56 5.22000 794.25 2.02500 746.82 1.23000 701.08 – 4.53500 657.64 – 7.84000 616.40 –11.04500 577.28 –14.25000 540.20 –17.56000 471.81 –29.07000 410.61 –30.58000 356.00 –37.09000 307.42 –43.5

10000 246.36 –50.0

258


APPENDIX A-4

Acceleration due to gravity at differentplaces in India along with their Latitude,

Longitude and Elevation

Place g(m/s2) Latitude Longitude Elevation (N) (E) (m)

Agra 9.7905 27°12’ 78°02’ 158Aligarh 9.7908 27°54’ 78°05’ 187Allahabad 9.7894 25°27’ 81°51’ 94Varanasi 9.7893 25°20’ 83°00’ 81Mumbai 9.7863 18°54’ 72°49’ 10Kolkata 9.7880 22°35’ 88°20’ 6Delhi 9.7914 28°40’ 77°14’ 216Equator 9.7805 00°00’ n.a. 0Jaipur 9.7900 26°55’ 75°47’ 433Udaipur 9.7881 24°35’ 73°44’ 563Srinagar 9.7909 34°05’ 74°50’ 159North Pole 9.8322 90°00’ n.a. 0Chennai 9.7828 13°04’ 80°15’ 6Thruanatapuram 9.7812 8°28’ 76°58’ 27Tirupati 9.7822 13°38’ 79°24’ 169Madurai 9.7810 9°55’ 78°7’ 133Bangaluru 9.7803 12°57 77°37’ 915Guwahati 9.7899 26°12’ 91°45’ 52Bhubaneswar 9.7866 20°28’ 85°54’ 23

APPENDIX A-5Surface tension of liquids

Substance In contact with Temp (°C) Surface Tension(10–3 Nm–1)

Water Air 10 74.22Air 20 72.55Air 30 71.18Air 40 69.56Air 50 67.91

Acetic acid Vapour 10 28.8Vapour 20 27.8Vapour 50 24.8

Ethyl Alcohol Air 0 24.05Vapour 10 23.61Vapour 20 22.75Vapour 30 21.89

Glycerol Air 20 63.04Vapour 90 58.6

Methyl Alcohol Air 0 24.49Air 20 22.61Vapour 50 20.14

Mercury Vapour 20 470Vapour 100 456

Oleic acid Air 20 32.5Kerosene Air 20 24Turpentine Air 20 27

259

UNIT NAMEAPPENDICES

APPENDIX A-6Coefficient of viscosity of liquids

Substance Temp (°C) Coefficient of viscosity (cP)

(1) Water 0 1.78720 1.00250 0.5468

100 0.2818(2) Acetic Acid 15 1.31

30 1.0460 0.70

100 0.43(3) Ethyl Alcohol 0 1.773

20 1.20050 0.83470 0.504

(4) Mercury 0 1.68520 1.55450 1.407

100 1.240200 1.052

(5) Methyl Alcohol 0 0.8220 0.59730 0.51050 0.403

(6) Glycerine 20 149525 94230 622

(7) Carbon disulphide 0 0.43620 0.437540 0.329

(8) Castor oil 10 242030 45150 125

(9) Carbon tetrachloride 0 1.34820 0.97240 0.744

APPENDIX A-7Elastic properties of solids

Substance Young’s Modulus of BulkModulus rigidity Modulus

(1010 Nm–2) (1010 Nm–2) (1010 Nm–2)

Aluminium 7.03 2.61 7.55Brass (70/30) 10.06 3.73 11.18Copper 12.98 9.83 13.78Gold 7.8 2.7 21.7Iron (soft) 21.14 8.16 16.98Silver 8.27 3.03 10.36Steel (mild) 21.19 8.22 17.92Rubber 0.05 0.00015 -Wood (oak) 1.3 - -Wook (teak) 1.7 - -Glass 5.1-7.1 3.1 3.75Quartz 5.4 3.4 -

260


APPENDIX A-8Velocity of sound

Substance Temperature Velocity of Substance Temperature Velocity of(0 °C) longitudinal (0 °C) longitudinal

wave (ms–1) wave (ms–1)

Alcohol 20 1177 Hydrogen 0 1284*Aluminium 20 5240 *Iron 20 5170Air 0 331.45 Mercury 20 1451*Brass 20 3130-3450 Nitrogen 0 334*Copper 20 3790 *Steel 20 5150(annealed) (tool)Carbon dioxide 0 259 Water 20 1484*Glass, crown 20 4710-5300 Water vapour 100 405*Glass, flint 20 3490-4550 Oxygen 0 316

*In case of solids, velocities of longitudinal waves in thin rods are quoted.For the gases for which v0, the velocity of sound at 0 °C is quoted here, vt the velocity at t 0C with fair degree

of accuracy, is t 0

tv v

+⎛ ⎞= ⎜ ⎟⎝ ⎠

12273.15

273.15

APPENDIX A-9Average speed of some selected objects

S.No. Object Speed

1. Slug or snail 1.6 mm/s2. Tortoise 10 to 15 cm/s3. Man walking 80 to 160 cm/s4. Man riding a bicycle 2.5 to 5 m/s5. 100 m race (International men’s) ~10 m/s6. Railway train (fastest in India in 38.9 m/s

1988 - Shatabdi Express)7. Cheetah (the fastest land animal) 29 m/s8. Surjit (the fastest bird) 100 m/s9. Jumbo jet aeroplane 267 m/s

10. Sound in air 331 m/s11. Near earth satellite 7.7 km/s12. Earth moving round Sun 29.9 km/s13. Light 300,000 km/s

APPENDIX A-10Coefficient of friction between some common surfaces

Surfaces in contact Condition Coefficient of dynamic friction

Glass on glass Clean and dry 0.18Wood on glass Clean and dry 0.2 to 0.3Wood on wood Clean and dry 0.25 to 0.5Wood on steel Clean and dry 0.20 to 0.25Steel on steel Clean and dry 0.17 - 0.23Steel on steel Greased 0.05Stone on concrete Dry 0.45Car tyre on concrete Moderate speed 0.40

261

UNIT NAMEAPPENDICES

APPENDIX A-12Coefficient of expansion

Solids Coefficient of Liquids Coefficient oflinear expansion volume expansion

(10–6 K–1) (10–4 K–1)

Aluminium 24 Alcohol (ethyl) 11.2

Brass 18 to 19 Alcohol (methyl) 12.2

Copper 16.7 Benzene 12.4

Constantan 18 Ether (ethyl) 16.3

Glass (Pyrex) 3 Glycerine 5.3

Glass (soft soda) 8.5Iron (cast) 10.0 Mercury 1.8Iron (wrought) 12.0 Water (15 °C) 1.5Ice 51.0 Water (99 °C) 7

Steel 11.0 Kerosene oil 10.0

Lead 23.0

Zinc 28.0

APPENDIX A-11Standard wire gauge

Size (S.W.G) Diameter (mm) Size (S.W.G.) Diameter (mm)

1 7.62 21 0.813

2 7.01 22 0.711

3 6.40 23 0.6104 5.89 24 0.559

5 5.38 25 0.508

6 4.88 26 0.457

7 4.47 27 0.417

8 4.06 28 0.3769 3.66 29 0.345

10 3.25 30 0.315

11 2.95 31 0.295

12 2.64 32 0.274

13 2.34 33 0.25414 2.03 34 0.234

15 1.83 35 0.213

16 1.63 36 0.193

17 1.42 37 0.173

18 1.22 38 0.15219 1.02 39 0.132

20 0.914 40 0.122

262


APPENDIX A-13Specific heat of substances

Substance Specific heat, Substance Specific heat,Jkg–1K–1 Jkg–1K–1

Solids Liquids

Aluminium (0 °C) 877 Ethyl Alcohol 2436Copper (0 °C) 380 Methyl alcohol 2562Copper (50 °C) 390 Benzene 1680Ice 2100 Ether 2352Iron (cast) 500 Glycerine 2478Iron (wrought) 483 Mercury 140Steel 470 Water (15 °C) 4185.5Lead 130 Brine (0 °C) 2970Brass 380 Sea water (17 °C) 3930Constantan 412Zinc (0 °C) 384Glass (crown) 670Glass (flint) 500Sand 1000 to 800

APPENDIX A-14Latent heat of fusion and vaporisation

Substance 104 Jkg–1

(i) Latent heat of fusion

Aluminium 40.2Calcium 23.0Copper 20.5Gold 6.3Iron 26.8Lead 2.5Mercury 1.17Magnesium 37.7Platinum 11.3Silver 10.5Sodium 11.3Tin 5.8Zinc 10.0Water 33.4(ii) Latent heat of vaporisationAcetic acid 40.5Benzene 39.4Carbon disulphide 35.1Ether 35.1Water (100 °C) 226.0Water (30 °C) 243.0

263

UNIT NAMEAPPENDICES

APPENDIX A-16

International practical temperature scale 1968

K °C

1. Triple point of water 13.81 – 259.34

2. Boiling point of hydrogen at pressure

of 25 cm of mercury 17.042 – 256.108

3. Boiling point of hydrogen 20.28 – 252.87

4. Boiling point of neon 27.102 – 246.048

5. Triple point of oxygen 54.361 – 218.789

6. Triple point of argon 83.798 – 189.352

7. Condensation point of oxygen 90.188 – 182.962

8. Triple point of water 273.16 0.01

9. Boiling point of water 373.15 100

10. Freezing point of tia 505.1181 231.9681

11. Freezing point of zinc 692.73 419.58

12. Freezing point of silver 1235.08 961.93

13. Freezing point of gold 1337.58 1064.43

APPENDIX A-15Boiling point of distilled water

Pressure Boiling point(105 Pa) (°C)

0.784 93.0

0.88 96.2

0.98 99.1

1.013 100.0

1.209 105.0

1.76 116.3

1.96 119.6

Notes :

1. The regulations on the IPTS-68 were adopted by the International Committee on Weightsand Measures in 1968.

2. Boiling points (condensation points) and freezing points, unless otherwise stated are atstandard atmospheric pressure (i.e., 760 mm of mercury at 0°C or 101325 Pa).


1. Advanced Practical Physics, Leslie Beckett (General Editor)John Murray,

2. Physics is Fun (Book I to IV), Jim JardineHeinemann Educational Books Ltd.

3. The Flying Circus of Physics, Jearl Walker,New York: John Wiley & Sons Inc.

4. Practical Physics for Pre- U.S. Kushwaha & S.S.Datta, Chandigarh:Engg./Pre.Medical/B.Sc. I Panjab University.students.

5. Practical Physics for Pre- U.S. Kushwaha & S.S. Datta, Chandigarh:University Students Panjab University.

6. A Lab. Manual of Physics, D.P. Khandelwal, Vani Educational7. Physics-Guide to Experiments The Nuffield Foundation,

Vol I to V Longmans, Green and Co. Ltd.8. Physics Laboratory Guide Physics Science Study Committee, Indian

Edition published by NCERT, New Delhi.9. Physics, an Experimental Science White and White, Van Nostrand

10. Source Book for Science Teaching UNESCO (Revised Edition)11. Physics Resource Materials for Regional College of Education,

Secondary School Teachers, Mysore.Vol. I and II,

12. Physics Experiments and Projects W.Bolton, Pergamon Press.13. Enjoy Physics Vinay B. Kamble,

Vikram Sarabhai Community Science Centre.14. Projects in Physics N.D.N. Belham,

London: B.T. Batsford Ltd.15. Physics Through Experiments B. Saraf and D.P. Khandelwal,

Vol. II - Mechanical Systems Vikas Publishing House.16. Great Experiments in Physics Edited by Morris H. Shamos,

New York : Holt, Rinehart and Winston.17. Laboratory Experiments in Physcis Charles E. Dull, H. Clark Metcal and John

E. Williams. New York; Holt, Rinehart and Winston.18. The Many Faces of Teaching and Edited by P.L. Linse, UNESCO - Utrecht.

Learning Mechanics in Secondaryand Tertiary Education

19. New Trends in Physics Teaching, Edited by John L. Lewis, UNESCO. Vol. III

20. New Trends in Physics Teaching, Edited by E.J. Wenham, UNESCO,Vol. IV

21. Teaching School Physics: John L. Lewis, Paris: Penguine - UNESCO.A UNESCO Source Book

22. Physics Laboratory Manual for NCERT, New Delhi.Senior Secondary Classes,Vol. I, Class XI and Vol. II, Class XII

23. A Textbook of Practical Physics, H.S. Allen and H.Moore, Macmillan & Co,,Book No. 5697, 2007.

24. Practical Physics, W.S. Franklin, C.M., EBSCO, HOST.

BIBLIOGRAPHYBIBLIOGRAPHYBIBLIOGRAPHYBIBLIOGRAPHYBIBLIOGRAPHY

265

UNIT NAMEBIBLIOGRAPHY

25. Advanced Level Practical Work Mike Crundell & Chris Mee, Hodder &for Physics Stoughton (seek books.com.air)

26. Practical Physics, G.L. Squires, Cambridge UniversityPress (2001).

27. Thinking Physics :Practical Lessons inLewis C. Epstein.Critical Thinking (Paperback)

SOME USEFUL WEBSITES

1. www.meet - physics.net/David Harrison.

2. www.upscale.utoronto.ca/general internet/Harrison/Flash

3. www.ac.wwu.edu

4. www.scienceclarified.com

5. www.met.tamu.edu

6. hyperphysics.phy-astr.gsci.edu

7. http://kidsearth.nasa.gov/archive/airpressure

8. www.Amazon.co.uk/books

9. www.tesco.com/books/

10. www.choosebooks.com

11. www.doorone.co.uk/physics+books

12. www.GLsquires-2001-books.google.com

13. www.antiqbook.co.uk/boox/seab/5697.html

14. www.practicalphysics.org/go/guidance_43.html

LOGARITHMS

N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

10 0000 0043 0086 0128 0170 5 9 13 17 21 26 30 34 38

0212 0253 0294 0334 0374 4 8 12 16 2O 24 28 32 36

11 0414 0453 0492 0531 0569 4 8 12 16 20 23 27 31 350607 0645 0682 0719 0755 4 7 11 15 18 22 26 29 33

12 0792 0828 0864 0899 0934 3 7 11 14 18 21 25 28 32

0969 1004 1038 1072 1106 3 7 10 14 17 20 24 27 31

13 1139 1173 1206 1239 1271 3 6 10 13 16 19 23 26 291303 1335 1367 1399 1430 3 7 10 13 16 19 22 25 29

14 1461 1492 1523 1553 1584 3 6 9 12 15 19 22 25 281614 1644 1673 1703 1732 3 6 9 12 14 17 20 23 26

15 1761 1790 1818 1847 1875 3 6 9 11 14 17 20 23 261903 1931 1959 1987 2014 3 6 8 11 14 17 19 22 25

16 2041 2068 2095 2122 2148 3 6 8 11 14 16 19 22 242175 2201 2227 2253 2279 3 5 8 10 13 16 18 21 23

17 2304 2330 2355 2380 2405 3 5 8 10 13 15 18 20 232430 2455 2480 2504 2529 3 5 8 10 12 15 17 20 22

18 2553 2577 2601 2625 2648 2 5 7 9 12 14 17 19 212672 2695 2718 2742 2765 2 4 7 9 11 14 16 18 21

19 2788 2810 2833 2856 2878 2 4 7 9 11 13 16 18 202900 2923 2945 2967 2989 2 4 6 8 11 13 15 17 19

20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 2 4 6 8 11 13 15 17 19

21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 2 4 6 8 10 12 14 16 18

22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 2 4 6 8 10 12 14 15 17

23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 2 4 6 7 9 11 13 15 17

24 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 2 4 5 7 9 11 12 14 16

25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 2 3 5 7 9 10 12 14 15

26 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 2 3 5 7 8 10 11 13 15

27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 2 3 5 6 8 9 11 13 14

28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 2 3 5 6 8 9 11 12 14

29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 1 3 4 6 7 9 10 12 13

30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 1 3 4 6 7 9 10 11 13

31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 1 3 4 6 7 8 10 11 12

32 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 1 3 4 5 7 8 9 11 12

33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 1 3 4 5 6 8 9 10 12

34 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 1 3 4 5 6 8 9 10 11

35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 1 2 4 5 6 7 9 10 11

36 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 1 2 4 5 6 7 8 10 11

37 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 1 2 3 5 6 7 8 9 10

38 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 1 2 3 5 6 7 8 9 10

39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 1 2 3 4 5 7 8 9 10

40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 1 2 3 4 5 6 8 9 10

41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 1 2 3 4 5 6 7 8 9

42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 1 2 3 4 5 6 7 8 9

43 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 1 2 3 4 5 6 7 8 9

44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 1 2 3 4 5 6 7 8 9

45 6532 6542 6551 6561 6471 6580 6590 6599 6609 6618 1 2 3 4 5 6 7 8 9

46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 1 2 3 4 5 6 7 7 8

47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 1 2 3 4 5 5 6 7 8

48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 1 2 3 4 4 5 6 7 8

49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 1 2 3 4 4 5 6 7 8

TABLE I


266

DATA SECTIONDATA SECTIONDATA SECTIONDATA SECTIONDATA SECTION

LOGARITHMS

TABLE 1 (Continued)

N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 1 2 3 3 4 5 6 7 851 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 1 2 3 3 4 5 6 7 8

52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 1 2 2 3 4 5 6 7 7

53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 1 2 2 3 4 5 6 6 7

54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 1 2 2 3 4 5 6 6 7

55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 1 2 2 3 4 5 5 6 7

56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 1 2 2 3 4 5 5 6 7

57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 1 2 2 3 4 5 5 6 7

58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 1 1 2 3 4 4 5 6 7

59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 1 1 2 3 4 4 5 6 7

60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 1 1 2 3 4 4 5 6 661 7853 7860 7768 7875 7882 7889 7896 7903 7910 7917 1 1 2 3 4 4 5 6 6

62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 1 1 2 3 3 4 5 6 6

63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 1 1 2 3 3 4 5 5 6

64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 1 1 2 3 3 4 5 5 6

65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 1 1 2 3 3 4 5 5 6

66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 1 1 2 3 3 4 5 5 6

67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 1 1 2 3 3 4 5 5 6

68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 1 1 2 3 3 4 4 5 6

69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 1 1 2 2 3 4 4 5 6

70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 1 1 2 2 3 4 4 5 671 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 1 1 2 2 3 4 4 5 5

72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 1 1 2 2 3 4 4 5 5

73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 1 1 2 2 3 4 4 5 5

74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 1 1 2 2 3 4 4 5 5

75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 1 1 2 2 3 3 4 5 5

76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 1 1 2 2 3 3 4 5 5

77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 1 1 2 2 3 3 4 4 5

78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 1 1 2 2 3 3 4 4 5

79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 1 1 2 2 3 3 4 4 5

80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 1 1 2 2 3 3 4 4 581 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 1 1 2 2 3 3 4 4 5

82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 1 1 2 2 3 3 4 4 5

83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 1 1 2 2 3 3 4 4 5

84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 1 1 2 2 3 3 4 4 5

85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 1 1 2 2 3 3 4 4 5

86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 1 1 2 2 3 3 4 4 5

87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 0 1 1 2 2 3 3 4 4

88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 0 1 1 2 2 3 3 4 4

89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 0 1 1 2 2 3 3 4 4

90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 0 1 1 2 2 3 3 4 491 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 0 1 1 2 2 3 3 4 4

92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 0 1 1 2 2 3 3 4 4

93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 0 1 1 2 2 3 3 4 4

94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 0 1 1 2 2 3 3 4 4

95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 0 1 1 2 2 3 3 4 4

96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 0 1 1 2 2 3 3 4 4

97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 0 1 1 2 2 3 3 4 4

98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 0 1 1 2 2 3 3 4 4

99 9956 9961 9965 9969 9974 9978 9983 9987 9997 9996 0 1 1 2 2 3 3 3 4

UNIT NAMEDATA SECTION

267

N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

00 1000 1002 1005 1007 1009 1012 1014 1016 1019 1021 0 0 1 1 1 1 2 2 2

.01 1023 1026 1028 1030 1033 1035 1038 1040 1042 1045 0 0 1 1 1 1 2 2 2

.02 1047 1050 1052 1054 1057 1059 1062 1064 1067 1069 0 0 1 1 1 1 2 2 2

.03 1072 1074 1076 1079 1081 1084 1086 1089 1091 1094 0 0 1 1 1 1 2 2 2

.04 1096 1099 1102 1104 1107 1109 1112 1114 1117 1119 0 1 1 1 1 2 2 2 2

.05 1122 1125 1127 1130 1132 1135 1138 1140 1143 1146 0 1 1 1 1 2 2 2 2

.06 1148 1151 1153 1156 1159 1161 1164 1167 1169 1172 0 1 1 1 1 2 2 2 2

.07 1175 1178 1180 1183 1186 1189 1191 1194 1197 1199 0 1 1 1 1 2 2 2 2

.08 1202 1205 1208 1211 1213 1216 1219 1222 1225 1227 0 1 1 1 1 2 2 2 3

.09 1230 1233 1236 1239 1242 1245 1247 1250 1253 1256 0 1 1 1 1 2 2 2 3

.10 1259 1262 1265 1268 1271 1274 1276 1279 1282 1285 0 1 1 1 1 2 2 2 3

.11 1288 1291 1294 1297 1300 1303 1306 1309 1312 1315 0 1 1 1 2 2 2 2 3

.12 1318 1321 1324 1327 1330 1334 1337 1340 1343 1346 0 1 1 1 2 2 2 2 3

.13 1349 1352 1355 1358 1361 1365 1368 1371 1374 1377 0 1 1 1 2 2 2 3 3

.14 1380 1384 1387 1390 1393 1396 1400 1403 1406 1409 0 1 1 1 2 2 2 3 3

.15 1413 1416 1419 1422 1426 1429 1432 1435 1439 1442 0 1 1 1 2 2 2 3 3

.16 1445 1449 1452 1455 1459 1462 1466 1469 1472 1476 0 1 1 1 2 2 2 3 3

.17 1479 1483 1486 1489 1493 1496 1500 1503 1507 1510 0 1 1 1 2 2 2 3 3

.18 1514 1517 1521 1524 1528 1531 1535 1538 1542 1545 0 1 1 1 2 2 2 3 3

.19 1549 1552 1556 1560 1563 1567 1570 1574 1578 1581 0 1 1 1 2 2 3 3 3

.20 1585 1589 1592 1596 1600 1603 1607 1611 1614 1618 0 1 1 1 2 2 3 3 3

.21 1622 1626 1629 1633 1637 1641 1644 1648 1652 1656 0 1 1 2 2 2 3 3 3

.22 1660 1663 1667 1671 1675 1679 1683 1687 1690 1694 0 1 1 2 2 2 3 3 3

.23 1698 1702 1706 1710 1714 1718 1722 1726 1730 1734 0 1 1 2 2 2 3 3 4

.24 1738 1742 1746 1750 1754 1758 1762 1766 1770 1774 0 1 1 2 2 2 3 3 4

.25 1778 1782 1786 1791 1795 1799 1803 1807 1811 1816 0 1 1 2 2 2 3 3 4

.26 1820 1824 1828 1832 1837 1841 1845 1849 1854 1858 0 1 1 2 2 3 3 3 4

.27 1862 1866 1871 1875 1879 1884 1888 1892 1897 1901 0 1 1 2 2 3 3 3 4

.28 1905 1910 1914 1919 1923 1928 1932 1936 1941 1945 0 1 1 2 2 3 3 4 4

.29 1950 1954 1959 1963 1968 1972 1977 1982 1986 1991 0 1 1 2 2 3 3 4 4

.30 1995 2000 2004 2009 2014 2018 2023 2028 2032 2037 0 1 1 2 2 3 3 4 4

.31 2042 2046 2051 2056 2061 2065 2070 2075 2080 2084 0 1 1 2 2 3 3 4 4

.32 2089 2094 2099 2104 2109 2113 2118 2123 2128 2133 0 1 1 2 2 3 3 4 4

.33 2138 2143 2148 2153 2158 2163 2168 2173 2178 2183 0 1 1 2 2 3 3 4 4

.34 2188 2193 2198 2203 2208 2213 2218 2223 2228 2234 1 1 2 2 3 3 4 4 5

.35 2239 2244 2249 2254 2259 2265 2270 2275 2280 2286 1 1 2 2 3 3 4 4 5

.36 2291 2296 2301 2307 2312 2317 2323 2328 2333 2339 1 1 2 2 3 3 4 4 5

.37 2344 2350 2355 2360 2366 2371 2377 2382 2388 2393 1 1 2 2 3 3 4 4 5

.38 2399 2404 2410 2415 2421 2427 2432 2438 2443 2449 1 1 2 2 3 3 4 4 5

.39 2455 2460 2466 2472 2477 2483 2489 2495 2500 2506 1 1 2 2 3 3 4 5 5

.40 2512 2518 2523 2529 2535 2541 2547 2553 2559 2564 1 1 2 2 3 4 4 5 5

.41 2570 2576 2582 2588 2594 2600 2606 2612 2618 2624 1 1 2 2 3 4 4 5 5

.42 2630 2636 2642 2649 2655 2661 2667 2673 2679 2685 1 1 2 2 3 4 4 5 6

.43 2692 2698 2704 2710 2716 2723 2729 2735 2742 2748 1 1 2 3 3 4 4 5 6

.44 2754 2761 2767 2773 2780 2786 2793 2799 2805 2812 1 1 2 3 3 4 4 5 6

.45 2818 2825 2831 2838 2844 2851 2858 2864 2871 2877 1 1 2 3 3 4 5 5 6

.46 2884 2891 2897 2904 2911 2917 2924 2931 2938 2944 1 1 2 3 3 4 5 5 6

.47 2951 2958 2965 2972 2979 2985 2992 2999 3006 3013 1 1 2 3 3 4 5 5 6

.48 3020 3027 3034 3041 3048 3055 3062 3069 3076 3083 1 1 2 3 3 4 5 6 6

.49 3090 3097 3105 3112 3119 3126 3133 3141 3148 3155 1 1 2 3 3 4 5 6 6

ANTILOGARITHMS

TABLE II

268


ANTILOGARITHMS

TABLE II (Continued)

N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

.50 3162 3170 3177 3184 3192 3199 3206 3214 3221 3228 1 1 2 3 4 4 5 6 7

.51 3236 3243 3251 3258 3266 3273 3281 3289 3296 3304 1 2 2 3 4 5 5 6 7

.52 3311 3319 3327 3334 3342 3350 3357 3365 3373 3381 1 2 2 3 4 5 5 6 7

.53 3388 3396 3404 3412 3420 3428 3436 3443 3451 3459 1 2 2 3 4 5 6 6 7

.54 3467 3475 3483 3491 3499 3508 3516 3524 3532 3540 1 2 2 3 4 5 6 6 7

.55 3548 3556 3565 3573 3581 3589 3597 3606 3614 3622 1 2 2 3 4 5 6 7 7

.56 3631 3639 3648 3656 3664 3673 3681 3690 3698 3707 1 2 3 3 4 5 6 7 8

.57 3715 3724 3733 3741 3750 3758 3767 3776 3784 3793 1 2 3 3 4 5 6 7 8

.58 3802 3811 3819 3828 3837 3846 3855 3864 3873 3882 1 2 3 4 4 5 6 7 8

.59 3890 3899 3908 3917 3926 3936 3945 3954 3963 3972 1 2 3 4 5 5 6 7 8

.60 3981 3990 3999 4009 4018 4027 4036 4046 4055 4064 1 2 3 4 5 6 6 7 8

.61 4074 4083 4093 4102 4111 4121 4130 4140 4150 4159 1 2 3 4 5 6 7 8 9

.62 4169 4178 4188 4198 4207 4217 4227 4236 4246 42S6 1 2 3 4 5 6 7 8 9

.63 4266 4276 4285 4295 4305 4315 4325 4335 4345 4355 1 2 3 4 5 6 7 8 9

.64 4365 4375 4385 4395 4406 4416 4426 4436 4446 4457 1 2 3 4 5 6 7 8 9

.65 4467 4477 4487 4498 4508 4519 4529 4539 4550 4560 1 2 3 4 5 6 7 8 9

.66 4571 4581 4592 4603 4613 4624 4634 4645 4656 4667 1 2 3 4 5 6 7 9 10

.67 4677 4688 4699 4710 4721 4732 4742 4753 4764 4775 1 2 3 4 5 7 8 9 10

.68 4786 4797 4808 4819 4831 4842 4853 4864 4875 4887 1 2 3 4 6 7 8 9 10

.69 4898 4909 4920 4932 4943 4955 4966 4977 4989 5000 1 2 3 5 6 7 8 9 10

.70 5012 5023 5035 5047 5058 5070 5082 5093 5105 5117 1 2 4 5 6 7 8 9 11

.71 5129 5140 5152 5164 5176 5188 5200 5212 5224 5236 1 2 4 5 6 7 8 10 11

.72 5248 5260 5272 5284 5297 5309 5321 5333 5346 5358 1 2 4 5 6 7 9 10 11

.73 5370 5383 5395 5408 5420 5433 5445 5458 5470 5483 1 3 4 5 6 8 9 10 11

.74 5495 5508 5521 5534 5546 5559 5572 5585 5598 5610 1 3 4 5 6 8 9 10 12

.75 5623 5636 5649 5662 5675 5689 5702 5715 5728 5741 1 3 4 5 7 8 9 10 12

.76 5754 5768 5781 5794 5808 5821 5834 5848 5861 5875 1 3 4 5 7 8 9 11 12

.77 5888 5902 5916 5929 5943 5957 5970 5984 5998 6012 1 3 4 5 7 8 10 11 12

.78 6026 6039 6053 6067 6081 6095 6109 6124 6138 6152 1 3 4 6 7 8 10 11 13

.79 6166 6180 6194 6209 6223 6237 6252 6266 6281 6295 1 3 4 6 7 9 10 11 13

.80 6310 6324 6339 6353 6368 6383 6397 6412 6427 6442 1 3 4 6 7 9 10 12 13

.81 6457 6471 6486 6501 6516 6531 6546 6561 6577 6592 2 3 5 6 8 9 11 12 14

.82 6607 6622 6637 6653 6668 6683 6699 6714 6730 6745 2 3 5 6 8 9 11 12 14

.83 6761 6776 6792 6808 6823 6839 6855 6871 6887 6902 2 3 5 6 8 9 11 1314

.84 6918 6934 6950 6966 6982 6998 7015 7031 7047 7063 2 3 5 6 8 10 11 13 15

.85 7079 7096 7112 7129 7145 7161 7178 7194 7211 7228 2 3 5 7 8 10 12 13 15

.86 7244 7261 7278 7295 7311 7328 7345 7362 7379 7396 2 3 5 7 8 10 12 13 15

.87 7413 7430 7447 7464 7482 7499 7516 7534 7551 7568 2 3 5 7 9 10 12 14 16

.88 7586 7603 7621 7638 7656 7674 7691 7709 7727 7745 2 4 5 7 9 11 12 14 16

.89 7762 7780 7798 7816 7834 7852 7870 7889 7907 7925 2 4 5 7 9 11 13 14 16

.90 7943 7962 7980 7998 8017 8035 8054 8072 8091 8110 2 4 6 7 9 11 13 15 17

.91 8128 8147 8166 8185 8204 8222 8241 8260 8279 8299 2 4 6 8 9 11 13 15 17

.92 8318 8337 8356 8375 8395 8414 8433 8453 8472 8492 2 4 6 8 10 12 14 15 17.93 8511 8531 8551 8570 8590 8610 8630 8650 8670 8690 2 4 6 8 10 12 14 16 18

.94 8710 8730 8750 8770 8790 8810 8831 8851 8872 8892 2 4 6 8 10 12 14 16 18

.95 8913 8933 8954 8974 8995 9016 9036 9057 9078 9099 2 4 6 8 10 12 15 17 19.96 9120 9141 9162 9183 9204 9226 9247 9268 9290 9311 2 4 6 8 11 13 15 17 19

.97 9333 9354 9376 9397 9419 9441 9462 9484 9506 9528 2 4 7 9 11 13 15 17 20.98 9550 9572 9594 9616 9638 9661 9683 9705 9727 9750 2 4 7 9 11 13 16 18 20

.99 9772 9795 9817 9840 9863 9886 9908 9931 9954 9977 2 5 7 9 11 14 16 18 20

269


270


NATURAL SINES

TABLE I

0' 6' 12' 18' 24' 30' 36' 42' 48' 54' Mean

0°.0 0°.1 0°.2 0°.3 0°.4 0°.5 0°.6 0°.7 0°.8 0°.9 Differences

1' 2' 3' 4' 5'

0 .0000 0017 0035 0052 0070 0087 0105 0122 0140 0157 3 6 9 12 15

1 .0175 0192 0209 0227 0244 0262 0279 0297 0314 0332 3 6 9 12 15

2 .0349 0366 0384 0401 0419 0436 0454 0471 0488 0506 3 6 9 12 15

3 .0523 0541 0558 0576 0593 0610 0628 0645 0663 0680 3 6 9 12 15

4 .0698 0715 0732 0750 0767 0785 0802 0819 0837 0854 3 6 9 12 15

5 .0872 0889 0906 0924 0941 0958 0976 0993 1011 1028 3 6 9 12 14

6 .1045 1063 1080 1097 1115 1132 1149 1167 1184 1201 3 6 9 12 14

7 .1219 1236 1253 1271 1288 1305 1323 1340 1357 1374 3 6 9 12 14

8 .1392 1409 1426 1444 1461 1478 1495 1513 1530 1547 3 6 9 12 14

9 .1564 1582 1599 1616 1633 1650 1668 1685 1702 1719 3 6 9 12 14

10 .1736 1754 1771 1788 1805 1822 1840 1857 1874 1891 3 6 9 12 14

11 .1908 1925 1942 1959 1977 1994 2011 2028 2045 2062 3 6 9 11 14

12 .2079 2096 2113 2130 2147 2164 2181 2198 2215 2232 3 6 9 11 14

13 .2250 2267 2284 2300 2317 2334 2351 2368 2385 2402 3 6 8 11 14

14 .2419 2436 2453 2470 2487 2504 2521 2538 2554 2571 3 6 8 11 14

15 .2588 2605 2622 2639 2656 2672 2689 2706 2723 2740 3 6 8 11 14

16 .2756 2773 2790 2807 2823 2840 2857 2874 2890 2907 3 6 8 11 14

17 .2924 2940 2957 2974 2990 3007 3024 3040 3057 3074 3 6 8 11 14

18 .3090 3107 3123 3140 3156 3173 3190 3206 3223 3239 3 6 8 11 14

19 .3256 3272 3289 3305 3322 3338 3355 3371 3387 3404 3 5 8 11 14

20 .3420 3437 3453 3469 3486 3502 3518 3535 3551 3567 3 5 8 11 14

21 .3584 3600 3616 3633 3649 3665 3681 3697 3714 3730 3 5 8 11 14

22 .3746 3762 3778 3795 3811 3827 3843 3859 3875 3891 3 5 8 11 14

23 .3907 3923 3939 3955 3971 3987 4003 4019 4035 4051 3 5 8 11 14

24 .4067 4083 4099 4115 4131 4147 4163 4179 4195 4210 3 5 8 11 13

25 .4226 4242 4258 4274 4289 4305 4321 4337 4352 4368 3 5 8 11 13

26 .4384 4399 4415 4431 4446 4462 4478 4493 4509 4524 3 5 8 10 13

27 .4540 4555 4571 4586 4602 4617 4633 4648 4664 4679 3 5 8 10 13

28 .4695 4710 4726 4741 4756 4772 4787 4802 4818 4833 3 5 8 10 13

29 .4848 4863 4879 4894 4909 4924 4939 4955 4970 4985 3 5 8 10 13

30 .5000 5015 5030 5045 5060 5075 5090 5105 5120 5135 3 5 8 10 13

31 .5150 5165 5180 5195 5210 5225 5240 5255 5270 5284 2 5 7 10 12

32 .5299 5314 5329 5344 5358 5373 5388 5402 5417 5432 2 5 7 10 12

33 .5446 5461 5476 5490 5505 5519 5534 5548 5563 5577 2 5 7 10 12

34 .5592 5606 5621 5635 5650 5664 5678 5693 5707 5721 2 5 7 10 12

35 .5736 5750 5764 5779 5793 5807 5821 5835 5850 5864 2 5 7 10 12

36 .5878 5892 5906 5920 5934 5948 5962 5976 5990 6004 2 5 7 9 12

37 .6018 6032 6046 6060 6074 6088 6101 6115 6129 6143 2 5 7 9 12

38 .6157 6170 6184 6198 6211 6225 6239 6252 6266 6280 2 5 7 9 11

39 .6293 6307 6320 6334 6347 6361 6374 6388 6401 6414 2 4 7 9 11

40 .6428 6441 6455 6468 6481 6494 6508 6521 6534 6547 2 4 7 9 11

41 .6561 6574 6587 6600 6613 6626 6639 6652 6665 6678 2 4 7 9 11

42 .6691 6704 6717 6730 6743 6756 6769 6782 6794 6807 2 4 6 9 11

43 .6820 6833 6845 6858 6871 6884 6896 6909 6921 6934 2 4 6 8 11

44 .6947 6959 6972 6984 6997 7009 7022 7034 7046 7059 2 4 6 8 10

271


NATURAL SINES

TABLE I (Continued)

0' 6' 12' 18' 24' 30' 36' 42' 48' 54' Mean

0°.0 0°.1 0°.2 0°.3 0°.4 0°.5 0°.6 0°.7 0°.8 0°.9 Differences

1' 2' 3' 4' 5'

45 .7071 7083 7096 7108 7120 7133 7145 7157 7169 7181 2 4 6 8 10

46 .7193 7206 7218 7230 7242 7254 7266 7278 7290 7302 2 4 6 8 10

47 .7314 7325 7337 7349 7361 7373 7385 7396 7408 7420 2 4 6 8 10

48 .7431 7443 7455 7466 7478 7490 7501 7513 7524 7536 2 4 6 8 10

49 .7547 7558 7570 7581 7593 7604 7615 7627 7638 7649 2 4 6 8 9

50 .7660 7672 7683 7694 7705 7716 7727 7738 7749 7760 2 4 6 7 9

51 .7771 7782 7793 7804 7815 7826 7837 7848 7859 7869 2 4 5 7 9

52 .7880 7891 7902 7912 7923 7934 7944 7955 7965 7976 2 4 5 7 9

53 .7986 7997 8007 8018 8028 8039 8049 8059 8070 8080 2 3 5 7 9

54 .8090 8100 8111 8121 8131 8141 8151 8161 8171 8181 2 3 5 7 8

55 .8192 8202 8211 8221 8231 8241 8251 8261 8271 8281 2 3 5 7 8

56 .8290 8300 8310 8320 8329 8339 8348 8358 8368 8377 2 3 5 6 8

57 .8387 8396 8406 8415 8425 8434 8443 8453 8462 8471 2 3 5 6 8

58 .8480 8490 8499 8508 8517 8526 8536 8545 8554 8563 2 3 5 6 8

59 .8572 8581 8590 8599 8607 8616 8625 8634 8643 8652 1 3 4 6 7

60 .8660 8669 8678 8686 8695 8704 8712 8721 8729 8738 1 3 4 6 7

61 .8746 8755 8763 8771 8780 8788 8796 8805 8813 8821 1 3 4 6 7

62 .8829 8838 8846 8854 8862 8870 8878 8886 8894 8902 1 3 4 5 7

63 .8910 8918 8926 8934 8942 8949 8957 8965 8973 8980 1 3 4 5 6

64 .8988 8996 9003 9011 9018 9026 9033 9041 9048 9056 1 3 4 5 6

65 .9063 9070 9078 9085 9092 9100 9107 9114 9121 9128 1 2 4 5 6

66 .9135 9143 9150 9157 9164 9171 9178 9184 9191 9198 1 2 3 5 6

67 .9205 9212 9219 9225 9232 9239 9245 9252 9259 9265 1 2 3 4 6

68 .9272 9278 9285 9291 9298 9304 9311 9317 9323 9330 1 2 3 4 5

69 .9336 9342 9348 9354 9361 9367 9373 9379 9385 9391 1 2 3 4 5

70 .9397 9403 9409 9415 9421 9426 9432 9438 9444 9449 1 2 3 4 5

71 .9455 9461 9466 9472 9478 9483 9489 9494 9500 9505 1 2 3 4 5

72 .9511 9516 9521 9527 9532 9537 9542 9548 9553 9558 1 2 3 3 4

73 .9563 9568 9573 9578 9583 9588 9593 9598 9603 9608 1 2 3 3 4

74 .9613 9617 9622 9627 9632 9636 9641 9646 9650 9655 1 2 2 3 4

75 .9659 9664 9668 9673 9677 9681 9686 9690 9694 9699 1 1 2 3 4

76 .9703 9707 9711 9715 9720 9724 9728 9732 9736 9740 1 1 2 3 3

77 .9744 9748 9751 9755 9759 9763 9767 9770 9774 9778 1 1 2 3 3

78 .9781 9785 9789 9792 9796 9799 9803 9806 9810 9813 1 1 2 2 3

79 .9816 9820 9823 9826 9829 9833 9836 9839 9842 9845 1 1 2 2 3

80 .9848 9851 9854 9857 9860 9863 9866 9869 9871 9874 0 1 1 2 2

81 .9877 9880 9882 9885 9888 9890 9893 9895 9898 9900 0 1 1 2 2

82 .9903 9905 9907 9910 9912 9914 9917 9919 9921 9923 0 1 1 2 2

83 .9925 9928 9930 9932 9934 9936 9938 9940 9942 9943 0 1 1 1 2

84 :9945 9947 9949 9951 9952 9954 9956 9957 9959 9960 0 1 1 1 2

85 .9962 9963 9965 9966 9968 9969 9971 9972 9973 9974 0 0 1 1 1

86 .9976 9977 9978 9979 9980 9981 9982 9983 9984 9985 0 0 1 1 1

87 .9986 9987 9988 9989 9990 9990 9991 9992 9993 9993 0 0 0 1 1

88 .9994 9995 9995 9996 9996 9997 9997 9997 9998 9998 0 0 0 0 0

89 .9998 9999 9999 9999 9999 1.000 1.000 1.000 1.000 1.000 0 0 0 0 0

90 1.000

272


NATURAL COSINES

TABLE II

0' 6' 12' 18' 24' 30' 36' 42' 48' 54' Mean

0°.0 0°.1 0°.2 0°.3 0°.4 0°.5 0°.6 0°.7 0°.8 0°.9 Differences

1' 2' 3' 4' 5'

0 1.000 1.000 1.000 1.000 1.000 1.000 .9999 9999 9999 9999 0 0 0 0 0

1 .9998 9998 9998 9997 9997 9997 9996 9996 9995 9995 0 0 0 0 0

2 .9994 9993 9993 9992 9991 9990 9990 9989 9988 9987 0 0 0 1 1

3 .9986 9985 9984 9983 9982 9981 9980 9979 9978 9977 0 0 1 1 1

4 .9976 9974 9973 9972 9971 9969 9968 9966 9965 9963 0 0 1 1 1

5 .9962 9960 9959 9957 9956 9954 9952 9951 9949 9947 0 1 1 1 2

6 .9945 9943 9942 9940 9938 9936 9934 9932 9930 9928 0 1 1 1 2

7 .9925 9923 9921 9919 9917 9914 9912 9910 9907 9905 0 1 1 2 2

8 .9903 9900 9898 9895 9893 9890 9888 9885 9882 9880 0 1 1 2 2

9 .9877 9874 9871 9869 9866 9863 9860 9857 9854 9851 0 1 1 2 2

10 .9848 9845 9842 9839 9836 9833 9829 9826 9823 9820 1 1 2 2 3

11 .9816 9813 9810 9806 9803 9799 9796 9792 9789 9785 1 1 2 2 3

12 .9781 9778 9774 9770 9767 9763 9759 9755 9751 9748 1 1 2 3 3

13 .9744 9740 9736 9732 9728 9724 9720 9715 9711 9707 1 1 2 3 3

14 .9703 9699 9694 9690 9686 9681 9677 9673 9668 9664 1 1 2 3 4

15 .9659 9655 9650 9646 9641 9636 9632 9627 9622 9617 1 2 2 3 4

16 .9613 9608 9603 9598 9593 9588 9583 9578 9573 9568 1 2 2 3 4

17 .9563 9558 9553 9548 9542 9537 9532 9527 9521 9516 1 2 3 3 4

18 .9511 9505 9500 9494 9489 9483 9478 9472 9466 9461 1 2 3 4 5

19 .9455 9449 9444 9438 9432 9426 9421 9415 9409 9403 1 2 3 4 5

20 .9397 9391 9385 9379 9573 9367 9361 9354 9348 9342 1 2 3 4 5

21 .9336 9330 9323 9317 9311 9304 9298 9291 9285 9278 1 2 3 4 5

22 .9272 9265 9259 9252 9245 9239 9232 9225 9219 9212 1 2 3 4 6

23 .9205 9198 9191 9184 9178 9171 9164 9157 9150 9143 1 2 3 5 6

24 .9135 9128 9121 9114 9107 9100 9092 9085 9078 9070 1 2 4 5 6

25 .9063 9056 9048 9041 9033 9026 9018 9011 9003 8996 1 3 4 5 6

26 .8988 8980 8973 8965 8957 8949 8942 8934 8926 8918 1 3 4 5 6

27 .8910 8902 8894 8886 8878 8870 8862 8854 8838 1 3 4 5 7

28 .8829 8821 8813 8805 8796 8788 8780 8771 8763 8755 1 3 4 6 7

29 .8746 8738 8729 8721 8712 8704 8695 8686 8678 8669 1 3 4 6 7

30 .8660 8652 8643 8634 8625 8616 8607 8599 8590 8581 1 3 4 6 7

31 .8572 8563 8554 8545 8536 8526 8517 8508 8499 8490 2 3 5 6 8

32 .8480 8471 8462 8453 8443 8434 8425 8415 8406 8396 2 3 5 6 8

33 .8387 8377 8368 8358 8348 8339 8329 8320 8310 8300 2 3 5 6 8

34 .8290 8281 8271 8261 8251 8241 8231 8221 8211 8202 2 3 5 7 8

3S .8192 8181 8171 8161 8151 8141 8131 8121 8111 8100 2 3 5 7 8

36 .8090 8080 8070 8059 8049 8039 8028 8018 8007 7997 2 3 5 7 8

37 .7986 7976 7965 7955 7944 7934 7923 7912 7902 7891 2 4 5 7 9

38 .7880 7869 7859 7848 7837 7826 7815 7804 7793 7782 2 4 5 7 9

39 .7771 7760 7749 7738 7727 7716 7705 7694 7683 7672 2 4 6 7 9

40 .7660 7649 7638 7627 7615 7604 7593 7581 7570 7559 2 4 6 8 9

41 .7547 7536 7524 7513 7501 7490 7478 7466 7455 7443 2 4 6 8 10

42 .7431 7420 7408 7396 7385 7373 7361 7349 7337 7325 2 4 6 8 10

43 .7314 7302 7290 7278 7266 7254 7242 7230 7218 7206 2 4 6 8 10

44 .7193 7181 7169 7157 7145 7133 7120 7108 7096 7083 2 4 6 8 10

273


NATURAL COSINES

TABLE II (Continued)

0' 6' 12' 18' 24' 30' 36' 42' 48' 54' Mean

0°.0 0°.1 0°.2 0°.3 0°.4 0°.5 0°.6 0°.7 0°.8 0°.9 Differences

1' 2' 3' 4' 5'

45 .7071 7059 7046 7034 7022 7009 6997 6984 6972 6959 2 4 6 8 10

46 .6947 6934 6921 6909 6896 6884 6871 6858 6845 6833 2 4 6 8 11

47 .6820 6807 6794 6782 6769 6756 6743 6730 6717 6704 2 4 6 9 11

48 .6691 6678 6665 6652 6639 6626 6613 6600 6587 6574 2 4 7 9 11

49 .6561 6547 6534 6521 6508 6494 6481 6468 6455 6441 2 4 7 9 11

50 .6428 6414 6401 6388 6374 6361 6347 6334 6320 6307 2 4 7 9 11

51 .6293 6280 6266 6252 6239 6225 6211 6198 6184 6170 2 5 7 9 11

52 .6157 6143 6129 6115 6]01 6088 6074 6060 6046 6032 2 5 7 9 11

53 .6018 6004 5990 5976 5962 5948 5934 5920 5906 5892 2 5 7 9 12

54 .5878 5864 5850 5835 5821 5807 5793 5779 5764 5750 2 5 7 9 12

55 .5736 5721 5707 5693 5678 5664 5650 5635 5621 5606 2 5 7 10 12

56 .5592 5577 5563 5548 5534 55]9 5505 5490 5476 5461 2 5 7 10 12

57 .5446 5432 5417 5402 5388 5373 5358 5344 5329 5314 2 5 7 10 12

58 .5299 5284 5270 5255 5240 5225 5210 5195 5180 5165 2 5 7 10 12

59 .5150 5135 5120 5105 5090 5075 5060 5045 5030 5015 3 5 8 10 13

60 .5000 4985 4970 4955 4939 4924 4909 4894 4879 4863 3 5 8 10 13

61 .4848 4833 4818 4802 4787 4772 4756 4741 4726 4710 3 5 8 10 13

62 .4695 4679 4664 4648 4633 4617 4602 4586 4571 4555 3 5 8 10 13

63 .4540 4524 4509 4493 4478 4462 4446 4431 4415 4399 3 5 8 10 13

64 .4384 4368 4352 4337 4321 4305 4289 4274 4258 4242 3 5 8 11 13

65 .4226 4210 4195 4179 4163 4147 4131 4115 4099 4083 3 5 8 11 13

66 .4067 4051 4035 4019 4003 3987 3971 3955 3939 3923 3 5 8 11 14

67 .3907 3891 3875 3859 3843 3827 3811 3795 3778 3762 3 5 8 11 14

68 .3746 3730 3714 3697 3681 3665 3649 3633 3616 3600 3 5 8 11 14

69 .3584 3567 3551 3535 3518 3502 3486 3469 3453 3437 3 5 8 11 14

70 .3420 3404 3387 3371 3355 3338 3322 3305 3289 3272 3 5 8 11 14

71 .3256 3239 3223 3206 3190 3173 3156 3140 3123 3107 3 6 8 11 14

72 .3090 3074 3057 3040 3024 3007 2990 2974 2957 2940 3 6 8 11 14

73 .2924 2907 2890 2874 2857 2840 2823 2807 2790 2773 3 6 8 11 14

74 .2756 2740 2723 2706 2689 2672 2656 2639 2622 2605 3 6 8 11 14

75 .2588 2571 2554 2538 2521 2504 2487 2470 2453 2436 3 6 8 11 14

76 .2419 2402 2385 2368 2351 2334 2317 2300 2284 2267 3 6 8 11 14

77 .2250 2233 2215 2198 2181 2164 2147 2130 2113 2096 3 6 9 11 14

78 .2079 2062 2045 2028 2011 1994 1977 1959 1942 1925 3 6 9 11 14

79 .1908 1891 1874 1857 1840 1822 1805 1788 1771 1754 3 6 9 11 14

80 .1736 1719 1702 1685 1668 1650 1633 1616 1599 1582 3 6 9 12 14

81 .1564 1547 1530 1513 1495 1478 1461 1444 1426 1409 3 6 9 12 14

82 .1392 1374 1357 1340 1323 1305 1288 1271 1253 1236 3 6 9 12 14

83 .1219 1201 1184 1167 1149 1132 1115 1097 1080 1063 3 6 9 12 14

84 .1045 1028 1011 0993 0976 0958 0941 0924 0906 0889 3 6 9 12 14

85 .0872 0854 0837 0819 0802 0785 0767 0750 0732 0715 3 6 9 12 15

86 .0698 0680 0663 0645 0628 0610 0593 0576 0558 0541 3 6 9 12 15

87 .0523 0506 0488 0471 0454 0436 0419 0401 0384 0366 3 6 9 12 15

88 .0349 0332 0314 0297 0279 0262 0244 0227 0209 0192 3 6 9 12 15

89 .0175 0157 0140 0122 0105 0087 0070 0052 0035 0017 3 6 9 12 15

90 .0000

274


NATURAL TANGENTS

TABLE III

0' 6' 12' 18' 24' 30' 36' 42' 48' 54' Mean

0°.0 0°.1 0°.2 0°.3 0°.4 0°.5 0°.6 0°.7 0°.8 0°.9 Differences

1' 2' 3' 4' 5'

0 .0000 0017 0035 0052 0070 0087 0105 0122 0140 0157 3 6 9 12 15

1 .0175 0192 0209 0227 0244 0262 0279 0297 0314 0332 3 6 9 12 15

2 .0349 0367 0384 0402 0419 0437 0454 0472 0489 0507 3 6 9 12 15

3 .0524 0542 0559 0577 0594 0612 0629 0647 0664 0682 3 6 9 12 15

4 .0699 0717 0734 0752 0769 0787 0805 0822 0840 0857 3 6 9 12 15

5 .0875 0892 0910 0928 0945 0963 0981 0998 1016 1033 3 6 9 12 15

6 .1051 1069 1086 1104 1122 1139 1157 1175 1192 1210 3 6 9 12 15

7 .1228 1246 1263 1281 1299 1317 1334 1352 1370 1388 3 6 9 12 15

8 .1405 1423 1441 1459 1477 1495 1512 1530 1548 1566 3 6 9 12 15

9 .1584 1602 1620 1638 1655 1673 1691 1709 1727 1745 3 6 9 12 15

10 .1763 1781 1799 1817 1835 1853 1871 1890 1908 1926 3 6 9 12 15

11 .1944 1962 1980 1998 2016 2035 2053 2071 2089 2107 3 6 9 12 15

12 .2126 2144 2162 2180 2199 2217 2235 2254 2272 2290 3 6 9 12 15

13 .2309 2327 2345 2364 2382 2401 2419 2438 2456 2475 3 6 9 12 15

14 .2493 2512 2530 2549 2568 2586 2605 2623 2642 2661 3 6 9 12 16

15 .2679 2698 2717 2736 2754 2773 2792 2811 2830 2849 3 6 9 13 16

16 .2867 2886 2905 2924 2943 2962 2981 ‘3000 3019 3038 3 6 9 13 16

17 .3057 3076 3096 3115 3134 3153 3172 3191 3211 3230 3 6 10 13 16

18 .3249 3269 3288 3307 3327 3346 3365 3385 3404 3424 3 6 10 13 16

19 .3443 3463 3482 3502 3522 3541 3561 3581 3600 3620 3 7 10 13 16

20 .3640 3659 3679 3699 3719 3739 3759 3779 3799 3819 3 7 10 13 17

21 .3839 3859 3879 3899 3919 3939 3959 3979 4000 4020 3 7 10 13 17

22 .4040 4061 4081 4101 4122 4142 4163 4183 4204 4224 3 7 10 14 17

23 .4245 4265 4286 4307 4327 4348 4369 4390 4411 4431 3 7 10 14 17

24 .4452 4473 4494 4515 4536 4557 4578 4599 4621 4642 4 7 11 14 18

25 .4663 4684 4706 4727 4748 4770 4791 4813 4834 4856 4 7 11 14 18

26 .4877 4899 4921 4942 4964 4986 5008 5029 5051 5073 4 7 11 15 18

27 .5095 5117 5139 5161 5184 5206 5228 5250 5272 5295 4 7 11 15 18

28 .5317 5340 5362 5384 5407 5430 5452 5475 5498 5520 4 8 11 15 19

29 .5543 5566 5589 5612 5635 5658 5681 5704 5727 5750 4 8 12 15 19

30 .5774 5797 5820 5844 5867 5890 5914 5938 5961 5985 4 8 12 16 20

31 .6009 6032 6056 6080 6104 6128 6152 6176 6200 6224 4 8 12 16 20

32 .6249 6273 6297 6322 6346 6371 6395 6420 6445 6469 4 8 12 16 20

33 .6494 6519 6544 6569 6594 6619 6644 6669 6694 6720 4 8 13 17 21

34 .6745 6771 6796 6822 6847 6873 699 6924 6950 6976 4 9 13 17 21

35 .7002 7028 7054 7080 7107 7133 7159 7186 7212 7239 4 9 13 18 22

36 .7265 7292 7319 7346 7373 7400 7427 7454 7481 7508 5 9 14 18 23

37 .7536 7563 7590 7618 7646 7673 7701 7729 7757 7785 5 9 14 18 23

38 .7813 7841 7869 7898 7926 7954 7983 8012 8040 8069 5 9 14 19 24

39 .8008 8127 8156 8185 8214 8243 8273 8302 8332 8361 5 10 15 20 24

40 .8391 8421 8451 8481 8511 8541 8571 8601 8632 8662 5 10 15 20 25

41 .8693 8724 8754 8785 8816 8847 8878 8910 8941 8972 5 10 16 21 26

42 .9004 9036 9067 9099 9131 9163 9195 9228 9260 9293 5 11 16 21 27

43 .9325 9358 9391 9424 9457 9490 9523 9556 9590 9623 6 11 17 22 28

44 .9657 9691 ‘ 9725 9759 9793 9827 9861 9896 9930 9965 6 11 17 23 29

275


NATURAL TANGENTS

TABLE III (Continued)

0' 6' 12' 18' 24' 30' 36' 42' 48' 54' Mean

0°.0 0°.1 0°.2 0°.3 0°.4 0°.5 0°.6 0°.7 0°.8 0°.9 Differences

1' 2' 3' 4' 5'

45 1.0000 0035 0070 0105 0141 0176 0212 0247 0283 0319 6 12 18 24 30

46 1.0355 0392 0428 0464 0501 0538 0575 0612 0649 0686 6 12 18 25 31

47 1-0724 0761 0799 0837 0875 0913 0951 0990 1028 1067 6 13 19 25 32

48 1-1106 1145 1184 1224 1263 1303 1343 1383 1423 1463 7 13 20 27 33

49 1.1504 1544 1585 1626 1667 1708 1750 1792 1833 1875 7 14 21 28 34

50 1-1918 1960 2002 2045 2088 2131 2174 2218 2261 2305 7 14 22 29 35

51 1.2349 2393 2437 2482 2527 2572 2617 2662 2708 2753 8 15 23 30 38

52 1.2799 2846 2892 2938 2985 3032 3079 3127 3175 3222 8 16 24 31 39

53 1.3270 3319 3367 3416 3465 3514 3564 3613 3663 3713 8 16 25 33 41

54 1.3764 3814 3865 3916 3968 4019 4071 4124 4176 4229 9 17 26 34 43

55 1-4281 4335 4388 4442 4496 4550 4605 4659 4715 4770 9 18 27 36 45

56 1-4826 4882 4938 4994 5051 5108 5166 5224 5282 5340 10 19 29 38 48

57 1.5399 5458 5517 5577 5637 5697 5757 5818 5880 5941 10 20 30 40 50

58 1.6003 6066 6128 6191 6255 6319 6383 6447 6512 6577 11 21 32 43 53

59 1.6643 6709 6775 6842 6909 6977 7045 7113 7182 7251 11 23 34 45 56

60 1-7321 7391 7461 7532 7603 7.675 7747 7820 7893 7966 12 24 36 48 60

61 1.8040 8115 8190 8265 8341 8418 8495 8572 8650 8728 13 26 38 51 64

62 1.8807 8887 8967 9047 9128 9210 9292 9375 9458 9542 14 27 41 55 68

63 1.9626 9711 9797 9883 9970 2.0057 2.0145 2.0233 2.0323 2.0413 15 29 44 58 73

64 2.0503 0594 0686 0778 0872 0965 1060 1155 1251 1348 16 31 47 63 78

65 2.1445 1543 1642 1742 1842 1943 2045 2148 2251 2355 17 34 51 68 85

66 2.2460 2566 2673 2781 2889 2998 3109 3220 3332 3445 18 37 55 73 92

67 2.3559 3673 3789 3906 4023 4142 4262 4383 4504 4627 20 40 60 79 99

68 2.4751 4876 5002 5129 5257 5386 5517 5649 5782 5916 22 43 65 87 108

69 2.6051 6187 6325 6464 6605 6746 6889 7034 7179 7326 24 47 71 95 119

70 2.7475 7625 7776 7929 8083 8239 8397 8556 8716 8878 26 52 78 104 131

71 2.9042 9208 9375 9544 9714 9887 3.0061 3.0237 3.0415 3.0595 29 58 87 116 145

72 3.0777 0961 1146 1334 1524 1716 1910 2106 2305 2500 32 64 96 129 161

73 3.2709 2914 3122 3332 3544 3759 3977 4197 4420 4646 36 72 108 144 180

74 3.4874 5105 5339 5576 5816 6059 6305 6554 6806 7062 41 811 22 163 204

75 3.7321 7583 7848 8118 8391 8667 8947 9232 9520 9812 46 93 139 186 232

76 4.0108 0408 0713 1022 1335 i653 1976 2303 2635 2972 53 107 160 213 267

77 4.3315 3662 4015 4374 4737 5107 5483 5864 6252 6646

78 4.7046 7453 7867 8288 8716 9152 9594 5.0045 5.0504 5.0970 Mean differences cease

79 5.1446 1929 2422 2924 3435 3955 4486 5026 5578 6140 to be sufficiently accurate.

80 5.6713 7297 7894 8502 9124 9758 6.0405 6.1066 6.1742 6.2432

81 6.3138 3859 4596 5350 6122 6912 7720 8548 9395 7.0264

82 7.1154 2066 3002 3%2 4947 5958 6996 8062 9158 8.0285

83 8.1443 2636 3863 5126 6427 7769 9152 9.0579 9.2052 9.3572

84 9.5144 9.677 9.845 10.02 10.20 10.39 10.58 10.78 10.99 11-20

85 1143 11.66 11.91 12.16 12.43 12.71 13.00 13.30 13.62 13.95

86 14.30 14.67 15.06 15.46 15.89 16.35 16.83 17.34 17.89 18.46

87 19.08 19.74 20.45 21.20 22.02 22.90 23.86 24.90 26.03 27.27

88 28.64 30.14 31.82 33.69 35.80 38.19 40.92 44.07 47.74 52.08

89 57.29 63.66 71.62 81.85 95.49 114.6 143.2 191.0 286.5 573.0

90 not defined

276


NOTES

277


NOTES

278


NOTES

Class XI Physics Lab Manual

Documents

Transcript of Class XI Physics Lab Manual