Engineering Mechanics

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Introduction

P 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.

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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  m.a 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 M .n M .n F or F  G. 2 2 r r 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 G.M .n G.M W= Or W = m.g if g = 2 r r2  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 ForceIt 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

Radian per second

Rad/s2

-

Acceleration

square

Angular

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

s 1

Hz

Impulse

Newton-second

Kg.m/s

-

N.m

-

Velocity

Moment

of Newton-metre

force 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

-

of Cubic metre

M3

-

of Litre

10-3 m3

L

N.m

J

Volume solids Volume liquids Work

Joule

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�.

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