Mechanics introduction

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WHAT IS MECHANICS?
 Mechanics may be defined as that branch of physical science which is
concerned with the study of resultant effect of action of forces on bodies,
both in the state of rest and in motion.
 Mechanics is subdivided into three branches, Mechanics of Rigid Bodies,
Mechanics of Deformable Bodies and Mechanics of Fluids.
 In this book we shall study Mechanics of Rigid Bodies. In Rigid Body
Mechanics bodies are assumed to be perfectly rigid i.e. there is no
deformation of bodies under the action of loads to which they are subjected.
 Though the engineering structures and machines do deform under the
action of loads, their deformation is so little that it does not affect the
conditions of equilibrium or equations of motion which are applied in their
study'.
 Study of mechanics of Rigid bodies forms a basis for the study of other two
branches i.e. Mechanics of Deformable Bodies and Mechanics of Fluids. It
is therefore a basic subject in engineering study.
HISTORICAL BACKGROUND
 The study of mechanics was developed very early in history. Early
contributions were made by Aristotle (384 -322B.c) and Archimedes (2g7 -
2t2 B.C). In his writings, Aristotle dealt with the principle of lever which
enables one to lift heavy object with comparatively lesser force. At that age
the requirements of engineering were mainly confined to construction work.
It is therefore surprising that the study of motion of bodies on inclined
plane, lifting of loads by use of lever and pulleys have been recorded in
ancient writings. On the other hand Archimedes established the
phenomenon of Buoyancy.
 It was Galileo Galilie (1564 - 1642) who introduced time factor in the study
of the effect of forces on bodies. His experiments with motion of pendulum
and falling bodies contributed to the wider and deeper study of the subject
late on.
Introduction

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 The major and most significant contribution to mechanics came from Sir
Issac Newton (1642 - L727) who propounded the theories of Fundamental
laws of motion and the Lows of universal gravitational acceleration. During
the same period Varignon (1654 - 17221a French mathematician
introduced what is now referred as Varignon's theorem. All this happened
much before the introduction of vector algebra.
 In 1687 Varignon and Newton presented the Lau of parallelogram of forces.
Further application and derivation of theorems based on these laws were
made by D’Alembert 17 L7 - 1783), Euler (1736 - 1813), Lagrange and
others.
 Plank (1858 - 1943) and Bhor (1885- 19621made contributions in the area
of Quantum Mechanics. In 1.905, Einstein formulated his theory of
relativity, referred to as Relativistic Mechanics, which challenged the
Newton's law of motion. However it was found that Einstein's theory had
certain limitations and therefore could not be applied under normal
conditions. Newton's laws therefore form the base of study of mechanics
and are therefore at times referred to as Newtonian Mechanics.
FUNDAMENTAL CONCEPTS
The study of mechanics has to start with knowing of fundamental. Concepts
involving length, time, mass and force. In Newtonian Mechanics length, time
and mass are the absolute concepts independent of each other.
Length
The concept of length means the position occupied by a point in space with
respect to a certain reference point like the origin. The three lengths in three
different directions define the position of the point. It therefore describes the
size of the system formed by number of points in space.

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Time
Whenever an event takes place, the time involved should also be known along
with the position. Though time of an event is not necessary in statics, it is
required in dynamics which deals with bodies in motion.
Mass
It is the quantity of matter contained in a body. This quantity does not change
on account of the position occupied by the body. This property is used to
compare bodies e.g. two bodies having different mass would have different
attractions to the earth or they would offer different resistance to the change
in their velocities during motion.
Force
A force is the action of one body on another body. The action could be a push'
or a 'pull'. A force is exerted when there is an actual contact between the two
bodies for e.g. a boy hitting a nail with a hammer. A force can also act even if
there is no contact between the two bodies, such as the magnetic or
gravitational force. Force is a vector quantity and is completely defined by its
magnitude, its direction and point of application fundamental.
FUNDAMENTAL PRINCIPLES
The study of Mechanics of Rigid Bodies is based on the six fundamental
principles presented below. These principles have their origin in experimental
evidences. Various theorems used in mechanics have been derived from
them.
1. Newton's First Law of Motion
Everybody continues in its state of rest or of uniform motion in a straight
line unless it is acted upon by an unbalanced force.
2. Newton's Second Law of Motion
The rate of change of momentum of a body is directly proportional to the
impressed force and takes place in the direction of the force. It is
mathematically written as
.F ma
where 'F' is the resultant force acting on a body of mass 'm' moving with
acceleration 'a'.

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3. Newton's Third Law of Motion
For every action there is an equal and opposite reaction.
4. Newton's Law of Gravitational Attraction
 After formulating the three laws of motion, Newton put forth his iaw of
gravitational attraction which states "The force of attraction between any
two bodies in the universe is directly proportional to the product of their
masses and inversely proportional to the square of the distance between
them”.
 If M and n are the masses of two bodies separated by a distance r
between them, they mutually attract each other with a force F, given by
2 2
. .
F or F G.
r r
M n M n
 
Where, G is the Universal constant of gravitational attraction.
 An important case of attraction is between a particle of mass m located at
or near the surface of earth and the earth itself of mass M. If R is the
radius of earth then the force of attraction defined as the weight of the
particle would be
W = 2
. .
r
G M n
Or W = m.g if g = 2
.
r
G M
 The value of g varies with the altitude above the surface of earth and also
with the latitude i.e. location on earth, since the earth is not exactly
spherical. For all practical computations g = 9.81 m/s2 can be used.
5. The principle of transmissibility of Force-
It states "A force being a sliding vector, continues to act along its line of
action and therefore makes no charge if it acts from a different point on its
line of action on a rigid body". This principle has been illustrated in detail
with figures in article 2.5
6. Law of parallelogram of forces.
It states "If two forces acting simultaneously on a body at a point Are
represented in magnitude and direction by the two adjacent sides of a
parallelogram then their resultant is represented in magnitude and
direction by the diagonal of the parallelogram which passes through the
point of intersection of the two sides representing the forces". This law has
been further discussed in article 2.9

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IDEALIZATIONS IN MECHANICS
In order to simplify the applications of theories in mechanics some
assumptions or idealizations are made which result in simplified solutions.
These are discussed below.
 Particle
A particle is a body whose shape and size is neglected because those being
negligible and insignificant as compared to other dimensions and lengths
involved in the analysis of the body. For example, in the motion analysis of
a vehicle on a highway between two stations kilometers apart, the shape
and size of the vehicle become insignificant and therefore the vehicle can be
idealized as a particle
 Rigid Body
A rigid body is a body whose shape and size is taken into consideration
during its analysis. Such a body is said to be made up of number of
particles which remain at fixed distances from each other. On application of
the loads or during motion the shape and size of the body does not change.
For example, the pillar supports of a building structure do deform under
the action of the loads they carry, but the deformations are very small as
compared to the lengths of the pillars and therefore the pillars can be
idealized as a rigid body.
 Point Load (Concentrated Force)
Point load or concentrated force is an idealization that a force acts at a
point on the body though in fact it must be acting over a certain area. This
idealization could be satisfactorily used when the area on which the force
acts is small, for example, the normal reaction force which the types of the
wheels receive from the ground can be idealized to act at a point though
actually it acts over a certain area. This idealization would hold true since
the area of contact is very small as compared to the size of the wheel.

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THE INTERNATIONAL SYSTEM OF UNITS
Of the four fundamental concepts i.e. length, time, mass and force, three of
them length, time and mass have fundamental units. The fourth concept i.e.
force has a derived unit which is based on the three fundamental units.
Fundamental units also known as the basic units are arbitrarily defined and
are independent of each other. In this book SI units have been extensively
used. The SI unit of length is meter (m), of mass is kilogram (kg), of time is
seconds (s) Derived units depend on the fundamental units. The SI unit of
force is Newton (N). 1N= 1kg m/s2.
The SI units used in mechanics are listed below.
Quantity SI Unit Formula Symbol
Length Metre Basic Unit M
Mass Kilogram Basic Unit Kg
Time Second Basic Unit S
Acceleration Metre per second square m/s2 -
Angle Radian - Rad
Angular
Acceleration
Radian per second
square
Rad/s2 -
Angular
Velocity
Radian per second rad / s -
Area Square metre M2 -
Couple Newton-metre N.m -
Density Kilogram per cubic metre Kg/m3 -
Energy Joule N.m J
Force Newton Kg.m/s2 N
Frequency Hertz 1
s 
Hz
Impulse Newton-second Kg.m/s -
Moment of
force
Newton-metre N.m -
Power Watt J/s W
Pressure Pascal N/m2 Pa

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Stress Pascal N/m2 Pa
Velocity Meter per second m/s -
Volume of
solids
Cubic metre M3 -
Volume of
liquids
Litre 10-3 m3 L
Work Joule N.m J
PROCEDURE OF PROBLEM ANALYSIS
Solving problems is the best way of learning the subject of mechanics.
While solving the problems follow the guidelines listed below. This should
help in successful solution of the problem.
1. After reading the problem carefully, the student should try to relate the
data of the problem to an actual physical engineering situation.
2. Even if the problem figure is given, draw your own necessary figure and
superimpose the problem data on it. The figure should be neat and clear. It
should show the various forces acting on the body. Such figures Are known
as free body diagrams and these play a very vital role in problem solution.
3. Apply the relevant fundamental principles and express the requirement in
mathematical equations. Break the working in suitable steps and record
them in an orderly manner. Check the equations dimensionally and use a
consistent set of units throughout. There may be more than one equation
which would require proper mathematical solution.
4. Try to reason the answer, or cross check the same by solving the same
problem by any other alternate method if possible, to check the correctness
of solution. Record the answers at the end of the solution.
Exercise 1
1) What is Mechanics?
2) State in brief the historical background of the development of the study of
mechanics.
3) Explain the fundamental concepts of Length, Time and Mass.
4) State and explain Newton's Law of Gravitational Attraction.
5) Explain "Idealization in Mechanics”.

Mechanics introduction

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Mechanics introduction