[3]

Mechanics can be defined as that branch of physical science, which deals with theresponse of bodies, which are either at condition of rest or in condition of motion,under the action of forces (or, moments). In this branch, the subject of Mathematicsplays a very important role. You might be knowing about the comment made byLeonardo da Vinci, who once said — ‘‘Mechanics is the paradise of mathematicalscience because here we come to the fruits of mathematics’’.

The subject of Mechanics can be classified as :

In this branch, we deal with thetheoretical aspects of the various principles of Mechanics. Basically, it is a branch ofPhysics and Mathematics.

In this branch, we are interested in the applications ofprinciples of Mechanics we learned from Pure Mechanics, to our real life problems. Inall Engineering disciplines, we learn this particular variety of Mechanics.

Applied Mechanics can further be classified as :

Mechanics of Rigid Bodies (popularly known as Engineering Mechanics)

Mechanics of Deformable Bodies (also known as Solid Mechanics,Mechanics of Materials or Strength of Materials*)

Mechanics of Fluids (also known as Fluid Mechanics)

Continuum Mechanics (which deals with both solid and fluid mechanics)

Engineering Mechanics is further sub divided into—

(a) which principally deals with bodies, which are at condition of restand

(b) which principally deals with the effects of forces and momentson the bodies, which are in motion.

Mechanics—Basic Definitions

1. Pure/Classical/Theoretical Mechanics

2. Applied Mechanics :

* See the book ‘‘Fundamentals of Strength of Materials, Publisher : John-Wiley, India’’ bythe authors.

Statics

Dynamics

4 An Introduction to Engineering Mechanics (Statics)

In this book, we shall be dealing with Mechanics of Rigid Bodies (both Statics andDynamics).

Often for our convenience we have to make certain simplifications and assumptionsregarding the various concepts of mechanics. These idealisations and assumptions arediscussed below.

This is an idealisation of a body. In describing the effect of forces acting on abody, if it is observed that its physical dimensions (i.e. length, breadth and height)are not of importance and really not affecting its response against the applied force,then the body can be assumed as a geometric point. For analyzing such bodies, wecan represent them by points wherein the entire masses of the respective bodies areassumed to act. We call them as mass particles or point masses.

This is an idealised concept of a material. Arigid body is defined as the collection of infinitenumber of many mass particles such that distancebetween any two such particles always remainsthe same even under the action of forces andmoments however large they may be. Figure 1.1shows schematically a rigid body.

In Fig 1.1, forces etc. are acting on arigid body and even after the application of these forces, it is noted that the distancebetween any two constituent mass particles say, and is remaining the same. Thatis, the distance is same what it was before the application of these forces.

Many great mathematicians, scholars and physicists, from age-old times, developedthe subject of Mechanics. Galileo, Archimedes, Leonardo da Vinci etc. are to bementioned but a few from this overwhelming list. Later on, Sir Isaac Newton singlehandedly developed this subject by giving a profound mathematical foundation. Thiswas possible for him as he invented the subject of Calculus along with the greatGerman mathematician Sir Gottfried Wilhelm Leibnitz. That is why, the Mechanicswe study here is sometimes called as Newtonian Mechanics. After the Newtonian era,another great physicist, Albert Einstein, expanded the realm of mechanics fromNewtonian domain to the relativistic situations. He showed Newtonian principles areno longer valid in this domain and accordingly the subject Quantum Mechanicsemerged. One can follow the historical perspectives of this fascinating subject ofMechanics in the famous book of ‘‘History of Strength of Materials’’ by S.P.Timoshenko. (See the Bibliography at the end)

Idealisations and Basic Assumptions

Concept of Particle/Point Mass

Rigid Body

Fig. 1.1 Rigid bodyF1 F2 F3

A BAB

5Introduction

Materials, such as solids, liquids and gases which we generally come across in our

engineering study, are composed of atoms and molecules separated by empty space,

which is known as the inter-atomic spacing. In a macroscopic scale, materials have

cracks and discontinuities. However, certain physical phenomena can be modeled

assuming the materials exist as a continuum, meaning the matter in the body is

continuously distributed and fills the entire region of space it occupies. It is like the

number line in mathematics, where we know from our knowledge that between any

two numbers, there exist an infinite number of numbers. A continuum is a body that

can be continually sub-divided into infinitesimal elements with properties being

those of the bulk material.

The response characteristic diagram (i.e.applied force versus deformation graph of thebody) of an idealised rigid body can be shownin Fig 1.2.

The figure shows zero deformation for anyvalue of the applied force. Mathematically wemay define a rigid body as :

It is noted that we define a term called‘‘Stiffness’’ of a body to describe itsdeformation behaviour under the action ofload. The definition of Stiffness is: ‘‘it is the amount of force required by a body to getdeformed by unit magnitude’’. It is obvious that a rigid body has infinite stiffness.However, it is to be noted that in the reality, there is no such absolutely rigid body. Wemay come across bodies, which we think it to be apparently rigid (for example, if welook at a very robust machine or a very big building). But if we measure with a verypowerful and sensitive instrument, we shall be able to record their deformations underapplied loads/moments. Hence, it is the scale of measurement based on which wedescribe a body to be rigid or flexible.

Fig. 1.2 Response characteristicof an ideal rigid body

0 P 1.1

Continuum

The concept of Continuum is a macroscopic physical model, and its validity dependson the type of problem and the scale of the physical phenomena under consideration.A material may be assumed to be a continuum when the distance between the physicalparticles is very small compared to the dimension of the problem. The validity of thecontinuum assumption needs to be verified with experimental testing andmeasurements on the real material under consideration and under similar loadingconditions. For example, in case of fluid flow problems, there exists a non-dimensionalnumber, called Knudsen number (Kn), depending on the value of which one candecide whether the flow problem can be analysed by using the principles of mechanicsbased on the continuum or not.

6 An Introduction to Engineering Mechanics (Statics)

Sometimes, the action of a force/moment on a body isfelt over a certain region on it. For example, we look at astructural member subjected to external load whichhas a built-in end as shown in the Figure 1.3 (you willlater come to know that this is called Cantilever Beam).

In the above figure, the support reactions at thebuilt-in end of the beam represent a very complexforce distribution, which acts over a length of the beam. This is an example ofdistributed force system. On the other hand if a force or a moment acts at a specificlocation on the body, we call that force or moment to be concentrated or point force/moment. However, it should be carefully noted that in reality, force or moment cannotact at a single point. So, from this point of view, point force or moment is anotherexample of idealisation.

According to this principle, a forceacting at a location on a rigid body canbe shifted undiminished anywherealong a line passing through thatlocation and collinear with the line ofaction of the force, provided thedirection of the force is kept unaltered. Figure 1.4 shows diagrammatically thisprinciple.

In the above figure, force was originally acting at a point on the rigid body.However, we can shift the force along its line of action to anywhere. Thus, the force canbe shifted to either point or point or anywhere along the line without changing itsaction on the body.

In our Engineering practice, we come across the physical quantities like force,distance, velocity, electric current, voltage, etc. All of these quantities have certaindimensions and units. In your further study, you will come across certain otherquantities, which do not possess any dimensions—we call them non-dimensional

Concept of Distributed Force/Moment andPoint Force/Point Moment

Fig. 1.3

Principle of Transmissibility of Forces

Fig. 1.4 Principle of transmissibility of forces

F A

B C

However, it is to be strongly emphasized that this principle cannot be applied to adeformable body.

Dimensions, Law of Dimensional Homogeneity and Units

7Introduction

quantities. For example, in the study of Fluid Mechanics, we have a very importantone such non-dimensional quantity called Reynolds number (Re).

Generally, in Mechanics we have three principal dimensions— Length , Mass and Time . These are called primary quantities. All other quantities can be

expressed as a mathematical formulation of these primary quantities. Those quantitiesare called derived quantities. Accordingly the units are also primary and derivedunits.

For example we may define the dimension of acceleration as : Acceleration = Rate of change of velocity

The above equation basically follows what is known as the ‘‘Principle ofDimensional Homogeneity’’. According to this, any mathematical equation shouldhave identical dimensions on either side of the equation.

In the same way, one can easily show that dimension of force is given by,

In the Engineering and Scientific studies, across the universe, people generallyfollow the SI units (abbreviated form of : Le International d'Unités,developed in 1960). According to this system, the units of the various physicalquantities are:

Apart from the SI system, two more units are still used in different countries ofthe world as a parallel set of units. They are : FPS (used in UK and USA) and MKS units(in various European countries). The salient features of those systems are as follows:

Name Symbol Quantity

metre m length

kilogram kg mass

second s time

ampere A electric current

kelvin K thermodynamic temperature

candela cd luminous intensity

mole mol amount of substance

Name FPS MKS

length foot (ft) metre (m)

force pound (lb) kilogram (kg)

time second (s) second (s)

LM T

a vt – 1 LT – 1T – 1 LT – 2 1.2

F

F M L T – 2 1.3

Systeme.

8 An Introduction to Engineering Mechanics (Statics)

The conversion factors for changing units to various systems are as follows :

In this case our equation is :

Therefore applying the principle of dimensional homogeneity we get :

The required dimension of is =

Thus, in SI, unit of coefficient of viscosity may be expressed as : .

To convert from To Action

N kg divide by 9.81

kg N multiply by 9.81

lb kg multiply by 0.4536

kg lb divide by 0.4536

The mass is a derived unit in case of MKS and FPS systems. The unit of mass is called

metric slug, in case of MKS system, and British slug .kg s2

mlb s2

ft

Further Reading

From your lesson of high-school level Physics, you already know that, the mutual

gravitational attraction between two masses and is according to

Newton’s law of universal gravitation, given by the relation : , where

represents the distance between the masses and is the universal

gravitational constant. Applying the principle of the law of dimensional

homogeneity, formulate the dimension of .

m1 m2

F Gm1m2

r2

r G

G

1

F Gm1m2

r2

F Gm1 m2

r2

G F r2

m1 m2

MLT 2L2

M2G L3T – 2

MM – 1L3T – 2

G M – 1L3T – 2

kg 1 m3 s 2

9Introduction

In this case our equation is :

Applying the principle of dimensional homogeneity we get :

The required dimenstion of is

Thus, in SI, unit of coefficient of viscosity may be expressed as :

In this case our equation is :

Applying the principle of dimensional homogeneity we get :

According to the F-L-T system, the dimension of is = .

Thus, in SI, unit gravitational constant may be expressed as

G F – 1L4T – 4

N 1 m4 s 4

In your lessons of Fluid Mechanics course, you will come to know that, thetangential force acting on a body due to the flow past is given by the relation :

, where represents the surfaces area of the body, is relative

velocity of the fluid, is the direction along which is velocity varies and is thecoefficient of viscosity of the fluid. Applying the principles of the law ofdimensional homogeneity, formuate the dimensions of .

F

F A uy

A u

y

2

F Auy

F Auy

F

Auy

MLT 2

L2LT 1L 1

MT 1

LML 1T 1

ML 1T 1

kg m 1 s 1

According to the F-L-T system, the dimension of is = .

Thus, in SI, unit gravitational of viscosity may be expressed as in F-L-Tformat.

FL 2T

N·m 2 s

In your lessons of Fluid Mechanics course, you will come to know that, theresistance of a body moving through a fluid, such as an aeroplane movingthrough air, is sometimes expressed by the equation : , where represents the cross-sectional area of the body at right angles to the direction ofmotion, is the density of the fluid, is the velocity of the object relative to theundisturbed fluid and is the drag coefficient. Applying the principle of thelaw of dimensional homogeneity, formulate the dimensions of .

FF 1

2CD V2A A

VCD

CD

3

F12

CD V2A

F CD V 2 A

10 An Introduction to Engineering Mechanics (Statics)

is non-dimensional

In this case our equation is :

Applying the principle of dimensional homogeneity we get :

We is non-dimensional]

The required dimension is :

Thus, in SI, unit of surface tension may be expressed as :

From the definition of specific weight of a substance as weight per unit volume,find its dimesions both in M-L-T and F-L-T systems.

[ (1) and (2) ]

Obtain the dimension of angular momentum in F-L-T system. [ FLT]

From the equation of Reynolds number , show by dimensional

analysis, that Re is non-dimensional. Here, = density of fluid, = velocity, =

diameter of conduit through which fluid is flowing and = coefficient of

viscosity of fluid.

Find the dimensions of kinetic viscosity of a fluid in M-L-T system from thedefinition that it is the ratio of viscosity of a fluid to its density. [ ]

CDF

V2 A

CDMLT 2

ML 3L2T 2L2CD

MLT 2

MLT 2non-dimensional

CD

A group of terms useful in the study of the surface effects of a liquid on a floating

object moving on it is the so-called Weber number, We given as : We = ,

where is the density of the fluid, is the surface tension of the liquid, is the

velocity of the object relative to the undisturbed fluid, is the characteristic

length of the object and Weber number is dimensionless. Applying the principle

of the law of dimensional homogeneity, formulate the dimensions of .

V2L

V

L

4

We V2L

V 2 LWe

V 2 L

ML 3L2T 2L MT 2

MT 2

kg s 2

According to the F-L-T system, the dimension of is = .

Thus, in SI, unit of surface tension may be expressed as in F-L-T format.

FL 1

N·m 1

ML 2T 2 FL 3

Re : Re vd

v d

L2T 1