Introduction 1

INTRODUCTION

TO

MECHANICS OF SOLIDS

Introduction 2

SCALAR AND VECTOR QUANTITIES:

Scalar Quantities  A scalar quantity is defined as a quantity that has  magnitude only.

e.g.  time, speed, temperature, volume, density, mass, energy etc. (the units tell nothing about direction)

Vector Quantities A vector quantity is defined as a quantity that has both magnitude and direction.

e.g. displacement, velocity, acceleration, lift, drag, thrust, etc.

A vector is represented by a directed line.

e.g. O A, Vector (OA)

The length OA represents magnitude of the vector OA and direction of vector is from O (i.e. Starting point) to A (i.e. End point)

Introduction 3

Composition of vectors

The operation of adding vectors graphically consists of drawing one vector with appropriate length and direction, and from the head of this vector, another vector is drawn with appropriate length and direction. The straight line drawn from the origin of the first vector to the head of the last vector represents the sum of the vectors and is called the resultant.

 

bc Here, ac = ab + bc

ac Vector ac is composed of vectors

ab ab and bc or ac is the resultant

of ab and bc

Introduction 4

Resolution of vectors

Resolution of vectors is exactly the opposite process of Composition. It is the process of finding a number of component vectors equivalent to the given single vector.

To find the resultant of multiple forces component of each force in two mutually perpendicular directions (x and y) are calculated the components in each direction are then added to get components of the resultant vector in x- and y-direction. The resultant is obtained from these mutually perpendicular components.

Let P1, P2, P3,and P4 are the forces of a system. Components of these forces in x- and y- direction are Px and Py

RX = P1x + P2x + P3x + P4x ; RY = P1y + P2y + P3y + P4y

Resultant R = (RX2 +RY

2)1/2

Its inclination with x-axis a = tan-1 (RY /RX)

Introduction 5

Law of Parallelogram of forces :

If two forces acting simultaneously on a particle are represented by two adjacent sides of a parallelogram the diagonal passing through their point of interaction gives magnitude and direction of their resultant force.

Mathematically the resultant force, R = (F1

2 + F22 + 2 F1F2 cos )1/2 F2 R

tan α = F2 sin θ αF1 + F2 cos θ F1

Where F1 and F2 are the forces whose resultant is to be calculatedθ= angle between the forces F1 and F2

α= angle of the resultant force with any one force say F1

Introduction 6

Law of Triangle of forces :

If two forces acting simultaneously on a particle are represented by two sides of a triangle taken in order, third side of the triangle taken in opposite order represents their resultant in magnitude and direction.

Law of polygon of forces :

If number of forces acting simultaneously on a particle are represented in order magnitude and direction by sides of a polygon closing side of the polygon taken in opposite order gives the magnitude and direction of their resultant force.

F2F1

R

F1

F2

F3

F4

F5R

Introduction 7

The four system of units commonly used and universally recognized are as follows :

F.P.S. units : In this system the fundamental units of length mass and time are foot, pound and second

C.G.S. units : In this system the fundamental units of length mass and time are centimeter, gram and second

M.K.S. units : In this system the fundamental units of length mass and time are meter, kilogram and second

S.I. units : we will discuss this system if units in detail.

The measurement of physical quantities is one of the most important operation in engineering. Every quantity is measured in terms of some arbitrary but internationally accepted units called fundamental units.

The three fundamental quantities are : Length, Mass and Time

Introduction 8

4.SI units

In 1960 the International System of Units was proposed as a replacement for the Metric System. The seven base units for the SI system are given below.

SI Basic Units

Physical Quantity Name of Unit Symbol

Length meter m

Mass kilogram kg

Time second s

Temperature kelvin K

Electric current ampere D

Amount of substance mole mol

Luminous intensity candela cd

Introduction 9

Prefix - Symbol Meaning

Femto - f 1/1,000,000,000,000,000 (10-15)

Pico - p 1/1,000,000,000,000 (10-12)

Nano - n 1/1,000,000,000 (10-9)

Micro - µ 1/1,000,000 (10-6)

Milli - m 1/1,000 (10-3)

Centi - c 1/100 (10-2)

Deci - d l/10(10-1)

Deca – da 10 (101)

Hecto – h 100 (102)

Kilo - k 1,000 (103)

Mega - M 1,000,000 (106)

Giga - G 1,000,000,000 (109)

Tera - T 1,000,000,000,000 (1012)

Introduction 10

Space :

Space is a region in all direction and contains everything within.

Time :

It is the measure of duration between successive events.

Particle :

It is an idealized body which may have negligible mass and whose size and shape can be ignored without introducing any appreciable error in description and prediction of its state of rest or motion.

Rigid body :

A body is considered rigid if the relative distance between particles of the body remains unchanged before and after application of force.

Force :

It is defined as an agent that produces or tend to produce, destroys or tend to destroy motion.

Introduction 11

Assignment

1) Define : Space, Time, Force, Particle, Rigid body

2) Write the basic SI quantities & give their units with symbol.

3) List the units and symbols of any 25 engineering quantities.

e.g. Force (F) Newton (N) = kg·m / s2

4) Reduce the following SI units to the units indicated:

a. 50 mm to fm b. 30 kg to gm

c. 5 X103 N to GN d. 19 X 10–10 m to nm

e. 500 N to kN f. 2000 N/mm2 to Mpa g. 1 dm2 to m2

5) Find magnitude and direction of forces acting at a point

20 N actig 300 towards North East, 25 N towards North, 30 N towards North West and 35 N at 400 towards South West. (Ans:45/6 N at 132.30)

6) Two forces of 100 N and 150 N are simultaneously acting on a point, calculateee their resultant force if angle between the forces is 450. (Ans:232 N)

Introduction 12

Introduction 13