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A TEXTBOOK OF

MULTICOLOUR ILLUSTRATIVE EDITION

R.S. KHURMI

S. CHAND & COMPANY LTD.

(AN ISO 9001 : 2000 COMPANY)

RAM NAGAR, NEW DELHI - 110 055

(SI UNITS)

I take an opportunity to present this standard treatise

entitled as A TEXTBOOK of APPLIED MECHANICS to

the Students of Degree, Diploma and A.M.I.E. (I) classes.

This object of this book is to present the subject matter in

a most concise, compact, to-the-point and lucid manner.

While writing this book, I have constantly kept in

mind the requirements of all the students regarding the

latest as well as the changing trends of their examination.

To make it more useful, at all levels, the book has been

written in an easy style. All along the approach to the

subject matter, every care has been taken to arrange matter

from simpler to harder, known to unknown with full details

and illustrations. A large number of worked examples,

mostly examination questions of Indian as well as foreign

universities and professional examining bodies, have been

given and graded in a systematic manner and logical

sequence, to assist the students to understand the text of

the subject. At the end of each chapter, a few exercises

have been added, for the students, to solve them

independently. Answers to these problems have been

provided, but it is too much to hope that these are entirely

free from errors. In short, it is expected that the book will

embrace the requirements of the students, for which it

has been designed.

Although every care has been taken to check

mistakes and misprints, yet it is difficult to claim perfection.

Any error, omission and suggestion for the improvement

of this volume, brought to my notice, will be thankfully

acknowledged and incorporated in the next edition.

Feb. 24, 1967 R.S. Khurmi

PREFACE TO THE FIRST EDITION

To

My Revered Guru and Guide

Shree B.L.Theraja

A well-known author, among Engineering

students, both at home and abroad,

to whom I am ever indebted for

inspiration and guidance

1. Introduction 1–

1.1. Science 1.2. Applied Science 1.3. Engineering Mehanics 1.4. Beginning and Development of Engineering Mechanics 1.5. Divisions of Engineering Mechanics 1.6. Statics 1.7. Dynamics 1.8. Kinetics 1.9. Kinematics 1.10. Fundamental Units 1.11. Derived Units 1.12. Systems of Units 1.13. S.I. Units (International System of Units.) 1.14. Metre 1.15. Kilogram 1.16. Second 1.17. Presentation of Units and Their Values 1.18. Rules for S.I. Units 1.19. Useful Data 1.20. Algebra 1.21. Trigonometry 1.22. Differential Calculus 1.23. Integral Calculus 1.24. Scalar Quantitie 1.25. Vector Quantities

2. Composition and Resolution of Forces 13–

2.1. Introduction 2.2. Effects of a Force 2.3. Characteristics of a Force 2.4. Principle of Physical Independence of Forces 2.5. Principle of Transmissibility of Forces 2.6. System of Forces 2.7. Resultant Force 2.8. Composition of Forces 2.9. Methods for the Resultant Force 2.10. Analytical Method for Resultant Force 2.11. Parallelogram Law of Forces 2.12. Resolution of a Force 2.13. Principle of Resolution 2.14. Method of Resolution for the Resultant Force 2.15. Laws for the Resultant Force 2.16. Triangle Law of Forces 2.17. Polygon Law of Forces 2.18. Graphical (vector) Method for the Resultant Force

3. Moments and Their Applications 28–

3.1. Introduction 3.2. Moment of a Force 3.3. Graphical Representation of Moment 3.4. Units of Moment 3.5. Types of Moments 3.6. Clockwise Moment 3.7. Anticlockwise Moment 3.8. Varignon’s Principle of Moments (or Law of Moments) 3.9. Applications of Moments 3.10. Position of the Resultant Force by Moments 3.11. Levers 3.12. Types of Levers 3.13. Simple Levers 3.14. Compound Levers

4. Parallel Forces and Couples 43–

4.1. Introduction 4.2. Classification of parallel forces. 4.3. Like parallel forces 4.4. Unlike parallel forces 4.5. Methods for magnitude and position of the resultant of parallel forces 4.6. Analytical method for the resultant of parallel forces. 4.7. Graphical method for the resultant of parallel forces 4.8. Couple 4.9. Arm of a couple 4.10. Moment of a couple 4.11. Classification of couples 4.12. Clockwise couple 4.13. Anticlockwise couple 4.14. Characteristics of a couple

5. Equilibrium of Forces 55–

5.1. Introduction 5.2. Principles of Equilibrium 5.3. Methods for the Equilibrium of coplanar forces 5.4. Analytical Method for the Equilibrium of Coplanar Forces 5.5. Lami’s Theorem 5.6. Graphical Method for the Equilibrium of Coplanar Forces 5.7. Converse of the Law of Triangle of Forces 5.8. Converse of the Law of Polygon of Forces 5.9. Conditions of Equilibrium 5.10. Types of Equilibrium.

6. Centre of Gravity 78–

6.1. Introduction 6.2. Centroid 6.3. Methods for Centre of Gravity 6.4. Centre of Gravity by Geometrical Considerations 6.5. Centre of Gravity by Moments 6.6. Axis of Reference 6.7. Centre of Gravity of Plane Figures 6.8. Centre of Gravity of Symmetrical Sections 6.9. Centre of Gravity of Unsymmetrical Sections 6.10. Centre of Gravity of Solid Bodies 6.11. Centre of Gravity of Sections with Cut out Holes

CONTENTSCONTENTS CONTENTSCONTENTSCONTENTS

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7. Moment of Inertia 100–

7.1. Introduction 7.2. Moment of Inertia of a Plane Area 7.3. Units of Moment of Inertia 7.4. Methods for Moment of Inertia 7.5. Moment of Inertia by Routh’s Rule 7.6. Moment of Inertia by Integration 7.7. Moment of Inertia of a Rectangular Section 7.8. Moment of Inertia of a Hollow Rectangular Section 7.9. Theorem of Perpendicular Axis 7.10. Moment of Inertia of a Circular Section 7.11. Moment of Inertia of a Hollow Circular Section 7.12. Theorem of Parallel Axis 7.13. Moment of Inertia of a Triangular Section 7.14. Moment of Inertia of a Semicircular Section 7.15. Moment of Inertia of a Composite Section 7.16. Moment of Inertia of a Built-up Section

8. Principles of Friction 124–

8.1. Introduction 8.2. Static Friction 8.3. Dynamic Friction 8.4. Limiting Friction 8.5. Normal Reaction 8.6. Angle of Friction 8.7. Coefficient of Friction 8.8. Laws of Friction 8.9. Laws of Static Friction 8.10. Laws of Kinetic or Dynamic Friction 8.11. Equilibrium of a Body on a Rough Horizontal Plane 8.12. Equilibrium of a Body on a Rough Inclined Plane 8.13. Equilibrium of a Body on a Rough Inclined Plane Subjected to a Force Acting Along the Inclined Plane 8.14. Equilibrium of a Body on a Rough Inclined Plane Subjected to a Force Acting Horizontally 8.15. Equilibrium of a Body on a Rough Inclined Plane Subjected to a Force Acting at Some Angle with the Inclined Plane

9. Applications of Friction 149–

9.1. Introduction. 9.2. Ladder Friction. 9.3. Wedge Friction. 9.4. Screw Friction. 9.5. Relation Between Effort and Weight Lifted by a Screw Jack. 9.6. Relation Between Effort and Weight Lowered by a Screw Jack. 9.7. Efficiency of a Screw Jack.

10. Principles of Lifting Machines 171–

10.1. Introduction 10.2. Simple Machine 10.3. Compound Machine 10.4. Lifting Machine 10.5. Mechanical Advantage. 10.6. Input of a Machine 10.7. Output of a Machine 10.8. Efficiency of a Machine 10.9. Ideal Machine 10.10. Velocity Ratio 10.11. Relation Between Efficiency, Mechanical Advantage and Velocity Ratio of a Lifting Machine 10.12. Reversibility of a Machine 10.13. Condition for the Reversibility of a Machine 10.14. Self-locking Machine. 10.15. Friction in a Machine 10.16. Law of a Machine 10.17. Maximum Mechanical Advantage of a Lifting Machine 10.18. Maximum Efficiency of a Lifting Machine.

11. Simple Lifting Machines 185–

11.1. Introduction 11.2. Types of Lifting Machines 11.3. Simple Wheel and Axle. 11.4. Differential Wheel and Axle. 11.5. Weston’s Differential Pulley Block. 11.6. Geared Pulley Block. 11.7. Worm and Worm Wheel 11.8. Worm Geared Pulley Block. 11.9. Single Purchase Crab Winch. 11.10. Double Purchase Crab Winch. 11.11. Simple Pulley. 11.12. First System of Pulleys. 11.13. Second System of Pulleys. 11.14. Third System of Pulleys. 11.15. Simple Screw Jack 11.16. Differential Screw Jack 11.17. Worm Geared Screw Jack.

12. Support Reactions 217–

12.1. Introduction. 12.2. Types of Loading. 12.3. Concentrated or Point Load 12.4. Uniformly Distributed Load 12.5. Uniformly Varying Load 12.6. Methods for the Reactions of a Beam 12.7. Analytical Method for the Reactions of a Beam 12.8. Graphical Method for the Reactions of a Beam 12.9. Construction of Space Diagram. 12.10. Construction of Vector Diagram 12.11. Types of End Supports of Beams 12.12. Simply Supported Beams 12.13. Overhanging Beams 12.14. Roller Supported Beams 12.15. Hinged Beams 12.16. Beams Subjected to a Moment. 12.17. Reactions of a Frame or a Truss 12.18. Types of End Supports of Frames 12.19. Frames with Simply Supported Ends 12.20. Frames with One End

( viii )

19. Relative Velocity 400–

19.1. Introduction. 19.2. Methods for Relative Velocity. 19.3. Relative velocity of Rain and Man. 19.4. Relative Velocity of Two Bodies Moving Along Inclined Directions. 19.5. Least Distance Between Two Bodies Moving Along Inclined Directions. 19.6. Time for Exchange of Signals of Two Bodies Moving Along Inclined Directions.

20. Projectiles 417–

20.1. Introduction. 20.2. Important Terms. 20.3. Motion of a Body Thrown Horizontally into the Air. 20.4. Motion of a Projectile. 20.5. Equation of the Path of a Projectile. 20.6. Time of Flight of a Projectile on a Horizontal Plane. 20.7. Horizontal Range of a Projectile. 20.8. Maximum Height of a Projectile on a Horizontal Plane. 20.9. Velocity and Direction of Motion of a Projectile, After a Given Interval of Time from the Instant of Projection. 20.10. Velocity and Direction of Motion of a Projectile, at a Given Height Above the Point of Projection. 20.11. Time of Flight of a Projectile on an Inclined Plane. 20.12. Range of a Projectile on an Inclined Plane.

21. Motion of Rotation 445–

21.1. Introduction. 21.2. Important Terms. 21.3. Motion of Rotation Under Constant Angular Acceleration. 21.4. Relation Between Linear Motion and Angular Motion. 21.5. Linear (or Tangential) Velocity of a Rotating Body. 21.6. Linear (or Tangential) Acceleration of a Rotating Body. 21.7. Motion of Rotation of a Body under variable Angular Acceleration.

22. Combined Motion of Rotation and Translation

22.1. Introduction. 22.2. Motion of a Rigid Link. 22.3. Instantaneous centre. 22.4. Motion of a Connecting Rod and Piston of a Reciprocating pump. 22.5. Methods for the Velocity of Piston of a Reciprocating Pump. 22.6. Graphical Method for the Velocity of Piston of a Reciprocating Pump. 22.7. Analytical Method for the Velocity of Piston of a Reciprocating Pump. 22.8. Velocity Diagram Method for the Velocity of Piston of a Reciprocating Pump. 22.9. Motion of a Rolling Wheel Without Slipping.

23. Simple Harmonic Motion 470–

23.1. Introduction. 23.2. Important Terms. 23.3. General Conditions of Simple Harmonic Motion. 23.4. Velocity and Acceleration of a Particle Moving with Simple Harmonic Motion. 23.5. Maximum Velocity and Acceleration of a Particle Moving with Simple Harmonic Motion.

24. Laws of Motion 481–

24.1. Introduction. 24.2. Important Terms. 24.3. Rigid Body. 24.4. Newton’s Laws of Motion. 24.5. Newton’s First Law of Motion. 24.6. Newton’s Second Law of Motion. 24.7. Absolute and Gravitational Units of Force. 24.8. Motion of a Lift. 24.9. D’Alembert’s Principle. 24.10. Newton’s Third Law of Motion. 24.11. Recoil of Gun. 24.12. Motion of a Boat. 24.13. Motion on an Inclined Planes.

25. Motion of Connected Bodies 503–

25.1. Introduction. 25.2. Motion of Two Bodies Connected by a String and Passing over a Smooth Pulley. 25.3. Motion of Two Bodies Connected by a String One of which is Hanging Free and the Other Lying on a Smooth Horizontal Plane. 25.4. Motion of Two Bodies Connected by a String One of which is Hanging Free and the Other Lying on a Rough Horizontal Plane. 25.5. Motion of Two Bodies Connected by a String One of which is Hanging Free and the Other Lying on a Smooth Inclined Plane. 25.6. Motion of Two Bodies connected by a String, One of which is Hanging Free and the Other is Lying on a Rough Inclined Plane. 25.7. Motion of Two Bodies Connected by a String and Lying on Smooth Inclined Planes. 25.8. Motion of Two Bodies Connected by a String Lying on Rough Inclined Planes.

( x )

26. Helical Springs and Pendulums 528–

26.1. Introduction. 26.2. Helical Springs. 26.3. Helical Springs in Series and Parallel. 26.4. Simple Pendulum. 26.5. Laws of Simple Pendulum. 26.6. Gain or Loss in the No. of Oscillations due to Change in the Length of String or Acceleration due to Gravity of a Simple Pendulum. 26.7. Gain or Loss in the No. of Oscillations due to Change in the Position of a Simple Pendulum. 26.8. Compound Pendulum. 26.9. Centre of Oscillation (or Centre of Percussion). 26.10. Conical Pendulum.

27. Collision of Elastic Bodies 553–

27.1. Introduction. 27.2. Phenomenon of Collision. 27.3. Law of Conservation of Momentum. 27.4. Newton’s law of Collision of Elastic Bodies. 27.5. Coefficient of Restitution. 27.6. Types of Collisions. 27.7. Direct Collision of Two Bodies. 27.8. Loss of Kinetic Energy During Collision. 27.9. Indirect Impact of Two Bodies. 27.10. Direct Impact of a Body with a Fixed Plane. 27.11. Indirect Impact of a Body with a Fixed Plane.

28. Motion Along a Circular Path 572–

28.1. Introduction. 28.2. Centripetal Acceleration. 28.3. Centripetal Force. 28.4. Centrifugal Force. 28.5. Centrifugal Force Acting on a Body Moving Along a Circular Path. 28.6. Superelevation. 28.7. Effect of Superelevation in Roadways. 28.8. Effect of Superelevation in Railways. 28.9. Equilibrium Speed for Superelevation. 28.10. Reactions of a Vehicle Moving along a Level Circular Path. 28.11. Equilibrium of a Vehicle Moving along a Level Circular Path. 28.12. Maximum velocity to Avoid Overturning of a Vehicle Moving along a Level Circular Path. 28.13. Maximum Velocity to Avoid Skidding Away of a Vehicle Moving along a Level Circular Path.

29. Balancing of Rotating Masses 586–

29.1. Introduction. 29.2. Methods for Balancing of Rotating Masses. 29.3. Types of Balancing of Rotating Masses. 29.4. Balancing of a Single Rotating Mass. 29.5. Balancing of a Single Rotating Mass by Another Mass in the Same Plane. 29.6. Balancing of a Single Rotating Mass by Two Masses in Different Planes. 29.7. Balancing of Several Rotating Masses. 29.8. Analytical Method for the Balancing of Several Rotating Masses in one Plane by Another Mass in the Same Plane. 29.9. Graphical Method for the Balancing of Several Rotating Masses in One Plane by Another Mass in the Same Plane. 29.10. Centrifugal governor. 29.11. Watt Governor.

30. Work, Power and Energy 599–

30.1. Introduction. 30.2. Units of Work. 30.3. Graphical Representation of Work. 30.4. Power. 30.5. Units of Power. 30.6. Types of Engine Powers. 30.7. Indicated Power. 30.8. Brake Power. 30.9. Efficiency of an Engine. 30.10. Measurement of Brake Power. 30.11. Rope Brake Dynamometer. 30.12. Proney Brake Dynamometer. 30.13. Froude and Thornycraft Transmission Dynamometer. 30.14. Motion on Inclined Plane. 30.15. Energy. 30.16. Units of Energy. 30.17. Mechanical Energy. 30.18. Potential Energy. 30.19. Kinetic Energy. 30.20. Transformation of Energy. 30.21. Law of Conservation of Energy. 30.22. Pile and Pile Hammer.

31. Kinetics of Motion of Rotation 622–

31.1. Introduction. 31.2. Torque. 31.3. Work done by a Torque. 31.4. Angular Momentum. 31.5. Newton’s Laws of Motion of Rotation. 31.6. Mass Moment of Inertia. 31.7. Mass Moment of Inertia of a Uniform Thin Rod about the Middle Axis Perpendicular to the Length. 31.8. Moment of Inertia of a Uniform Thin Rod about One of the Ends Perpendicular to the Length. 31.9. Moment of Inertia of a Thin Circular Ring. 31.10. Moment of Inertia of a Circular Lamina. 31.11. Mass Moment of Inertia of a Solid Sphere. 31.12. Units of Mass Moment of Inertia. 31.13. Radius of Gyration. 31.14. Kinetic Energy of Rotation.

( xi )

1.1. SCIENCE

In this modern age, the word ‘science’ has got

different meanings for different people. An ordinary

man takes it as ‘something’ beyond his understanding,

whereas others may take it as ‘mysteries of research’

which are understood only by a few persons

working amidst complicated apparatus in a laboratory.

A non-scientist feels that it is a ‘subject’ whose

endeavour is aimed to improve the man’s life on the

earth. A business executive has the idea that it is

‘something’ which solves our day to day manufacturing

and quality control problems, so that the nation’s

economic prosperity keeps on improving.

In fact, ‘science’ may be defined as the growth

of ideas through observation and experimentation. In

Contents

1. Science.
2. Applied Science.
3. Engineering Mehanics.
4. Beginning and Development of Engineering Mechanics.
5. Divisions of Engineering Mechanics.
6. Statics.
7. Dynamics.
8. Kinetics.
9. Kinematics.
10. Fundamental Units.
11. Derived Units.
12. Systems of Units.
13. S.I. Units (International System of Units.).
14. Metre.
15. Kilogram.
16. Second.
17. Presentation of Units and Their Values.
18. Rules for S.I. Units.
19. Useful Data.
20. Algebra.
21. Trigonometry.
22. Differential Calculus.
23. Integral Calculus.
24. Scalar Quantities.
25. Vector Quantities.

1^ Introduction

CCCCC HHHHH AAAAA PPPPP TTTTT EEEEE RRRRR

Contents

(^2) „„„„„ A Textbook of Engineering Mechanics

this sense, the subject of science does not, necessarily, has to contribute something to the welfare of

the human life, although the man has received many benefits from the scientific investigations.

1.2. APPLIED SCIENCE

Strictly speaking, the world of science is so vast that the present day scientists and

technologists have to group the various spheres of scientific activities according to some common

characteristics to facilitate their training and research programmes. All these branches of science,

still have the common principle of employing observation and experimentation. The branch of

science, which co-ordinates the research work, for practical utility and services of the mankind, is

known as Applied Science.

1.3. ENGINEERING MECHANICS

The subject of Engineering Mechanics is that branch of Applied Science, which deals with the

laws and principles of Mechanics, alongwith their applications to engineering problems. As a matter

of fact, knowledge of Engineering Mechanics is very essential for an engineer in planning, designing

and construction of his various types of structures and machines. In order to take up his job more

skilfully, an engineer must persue the study of Engineering Mechanics in a most systematic and

scientific manner.

1.4. BEGINNING AND DEVELOPMENT OF ENGINEERING MECHANICS

It will be interesting to know, as to how the early man had been curious to know about the

different processes going on the earth. In fact, he used to content himself, by holding gods responsible

for all the processes. For a long time, the man had been trying to improve his ways of working. The

first step, in this direction, was the discovery of a circular wheel, which led to the use of animal driven

carts. The study of ancient civilization of Babylonians, Egyptians, Greeks and Roman reveal the use

of water wheels and wind mills even during the pre-historic days.

It is believed that the word ‘Mechanics’ was coined by a Greek philosopher Aristotle

(384–322 BC). He used this word for the problems of lever and the concept of centre of gravity. At

that time, it included a few ideas, which were odd, unsystematic and based mostly on observations

containing incomplete information. The first mathematical concept of this subject was developed by

Archimedes (287–212 BC). The story, for the discovery of First Law of Hydrostatics, is very popular

even today in the history of the development of Engineering Mechanics. In the normal course, Hieron

king of Syracuse got a golden crown made for his use. He suspected that the crown has been made

with an adultrated gold. The king liked the design of the crown so much that he did not want it to be

melted,in order to check its purity. It

is said that the king announced a huge

reward for a person, who can check

the purity of the crown gold without

melting it. The legend goes that

Archimedes, a pure mathematician,

one day sitting in his bath room tub

realised that if a body is immersed in

water, its apparent weight is reduced.

He thought that the apparent loss of

weight of the immersed body is equal

to the weight of the liquid displaced.

It is believed that without further

Sir Issac Newton (1643–1727)

Contents

(^4) „„„„„ A Textbook of Engineering Mechanics

1.9. KINEMATICS

It is that branch of Dynamics, which deals with the bodies in motion, without any reference

to the forces which are responsible for the motion.

1.10. FUNDAMENTAL UNITS

The measurement of physical quantities is one of the most important operations in engineering.

Every quantity is measured in terms of some arbitrary, but internationally accepted units, called

fundamental units.

All the physical quantities, met with in Engineering Mechanics, are expressed in terms of three

fundamental quantities, i. e.

1. length, 2. mass and 3. time.

1.11. DERIVED UNITS

Sometimes, the units are also expressed in other units (which are derived from fundamental

units) known as derived units e.g. units of area, velocity, acceleration, pressure etc.

1.12. SYSTEMS OF UNITS

There are only four systems of units, which are commonly used and universally recognised.

These are known as :

1. C.G.S. units, 2. F.P.S. units, 3. M.K.S. units and 4. S.I. units.

In this book, we shall use only the S.I. system of units, as the future courses of studies are

conduced in this system of units only.

1.13. S.I. UNITS (INTERNATIONAL SYSTEM OF UNITS)

The eleventh General Conference* of Weights and Measures has recommended a unified

and systematically constituted system of fundamental and derived units for international use. This

system of units is now being used in many countries.

In India, the Standards of Weights and Measures Act of 1956 (vide which we switched over to

M.K.S. units) has been revised to recognise all the S.I. units in industry and commerce.

In this system of units, the †fundamental units are metre (m), kilogram (kg) and second (s)

respectively. But there is a slight variation in their derived units. The following derived units will be

used in this book :

Density (Mass density) kg / m^3

Force N (Newton)

Pressure N/mm 2 or N/m^2

Work done (in joules) J = N-m

Power in watts W = J/s

International metre, kilogram and second are discussed here.

• It is knwon as General Conference of Weights and Measures (G.C.W.M.). It is an international organisation of which most of the advanced and developing countries (including India) are members. This confer- ence has been ensured the task of prescribing definitions of various units of weights and measures, which are the very basis of science and technology today.

† The other fundamental units are electric current, ampere (A), thermodynamic temperature, kelvin (K) and luminous intensity, candela (cd). These three units will not be used in this book.

Contents

Chapter 1 : Introduction (^) „„„„„ 5

1.14. METRE

The international metre may be defined as the shortest distance (at 0°C) between two parallel

lines engraved upon the polished surface of the Platinum-Iridium bar, kept at the International

Bureau of Weights and Measures at Sevres near Paris.

1.15. KILOGRAM

The international kilogram may be defined as the

mass of the Platinum-Iridium cylinder, which is also kept

at the International Bureau of Weights and Measures at

Sevres near Paris.

1.16. SECOND

The fundamental unit of time for all the four systems is second, which is 1/(24 × 60 × 60) =

1/86 400th of the mean solar day. A solar day may be defined as the interval of time between the

instants at which the sun crosses the meridian on two consecutive days. This value varies throughout

the year. The average of all the solar days, of one year, is called the mean solar day.

1.17. PRESENTATION OF UNITS AND THEIR VALUES

The frequent changes in the present day life are facililtated by an international body

known as International Standard Organisation (ISO). The main function of this body is to

make recommendations regarding international procedures. The implementation of ISO

recommendations in a country is assisted by an organisation appointed for the purpose. In India,

Bureau of Indian Standard formerly known as Indian Standards Institution (ISI) has been

created for this purpose.

We have already discussed in the previous articles the units of length, mass and time. It is

always necessary to express all lengths in metres, all masses in kilograms and all time in seconds.

According to convenience, we also use larger multiples or smaller fractions of these units. As a

typical example, although metre is the unit of length; yet a smaller length equal to one-thousandth of

a metre proves to be more convenient unit especially in the dimensioning of drawings. Such convenient

units are formed by using a prefix in front of the basic units to indicate the multiplier.

A bar of platinum - iridium metre kept at a temperature of 0º C.

The standard platinum - kilogram is kept at the International Bureau of Weights and Measures at Serves in France.

Contents

Chapter 1 : Introduction (^) „„„„„ 7

The above mentioned figures are meant for numerical values only. Now we shall discuss about

the units. We know that the fundamental units in S.I. system for length, mass and time are metre,

kilogram and second respectively. While expressing these quantities, we find it time-consuming to

write these units such as metres, kilograms and seconds, in full, every time we use them. As a result of

this, we find it quite convenient to use the following standard abberviations, which are internationally

recognised. We shall use :

m for metre or metres

km for kilometre or kilometres

kg for kilogram or kilograms

t for tonne or tonnes

s for second or seconds

min for minute or minutes

N for newton or newtons

N-m for newton × metres ( i. e ., work done)

kN-m for kilonewton × metres

rad for radian or radians

rev for revolution or revolutions

1.19. USEFUL DATA

The following data summarises the previous memory and formulae, the knowledge of which is

very essential at this stage.

1.20. ALGEBRA

1. a^0 = 1 ; x^0 = 1

( i. e ., Anything raised to the power zero is one. )

2. xm^ × xn^ = xm + n

( i. e ., If the bases are same , in multiplication , the powers are added. )

m m n n

x

x

x

( i. e ., If the bases are same in division, the powers are subtracted. )

4. If ax^2 + bx + c = 0

then

• 2 – 4

b b ac

x

a

where a is the coefficient of x^2 , b is the coefficient of x and c is the constant term.

1.21. TRIGONOMETRY

In a right-angled triangle ABC as shown in Fig. 1.

1. sin

b

c

2. cos

a

c

sin

tan

cos

b

a

Fig. 1.1.

Contents

(^8) „„„„„ A Textbook of Engineering Mechanics

cosec

sin

c

b

sec

cos

c

a

cos 1

cot

sin tan

a

b

7. The following table shows values of trigonometrical functions for some typical angles:

angle 0° 30° 45° 60° 90°

sin 0

cos 1

tan 0

or in other words, for sin write

for cos write the values in reverse order ; for tan divide the value of sin by cos for the

respective angle.

8. In the first quadrant ( i. e ., 0° to 90°) all the trigonometrical ratios are positive.

9. In the second quadrant ( i. e ., 90° to 180°) only sin θ and cosec θ are positive.

10. In the third quadrant ( i. e ., 180° to 270°) only tan θ and cot θ are positive.

11. In the fourth quadrant ( i. e ., 270° to 360°) only cos θ and sec θ are positive.

12. In any triangle ABC ,

sin sin sin

a b c

A B C

where a , b and c are the lengths of the three sides of a triangle. A , B and C are opposite

angles of the sides a , b and c respectively.

13. sin ( A + B ) = sin A cos B + cos A sin B

14. sin ( A – B ) = sin A cos B – cos A sin B

15. cos ( A + B ) = cos A cos B – sin A sin B

16. cos ( A – B ) = cos A cos B + sin A sin B

tan tan

tan ( )

1 – tan .tan

A B

A B

A B

tan – tan

tan ( – )

1 tan .tan

A B

A B

A B

19. sin 2 A = 2 sin A cos A

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