INTRODUCTION TO ENGINEERING MECHANICS - SPP.pptx

Prof. Samirsinh P. Parmar
samirddu@gmail.com, spp.cl@ddu.ac.in
Asst. Prof. Dept. of Civil Engineering
Dharmasinh Desai University, Nadiad, Gujarat, India
Lecture-0
CL- ENGG. MECHANICS, DOCL- SPP, DDU, NADIAD 1

Content of the presentation
General Introduction to Engineering Mechanics
SI System of Units
Concept of Force
System of forces
Idealization in Mechanics
Fundamental Concepts
Scaler and Vector Quantities
Accuracy in Calculation
Problem Solving Approach
Reference BOOKS:

General
Introduction to
Engineering
Mechanics

Engineering Mechanics

1. Statics:
It is the branch which deals with the forces and their effects on an object or a body at
rest.
•For example, if we have an object or a body at rest and we deal with the forces and their
effects that are acting on the body than we are dealing with static branch of engineering
mechanics.
2. Dynamics:
It is the branch which deals with the forces and their effects on the bodies which are in
motion.
•For example, if we have a body that is moving and we are dealing with the forces and
their effects on the moving body than we are dealing with dynamics branch.
General Introduction to

Types of Dynamics
Dynamics is also divided into two branches and these are:
(i) . Kinetics:
Kinetics is defined as the branch of dynamics which deals with the bodies that are in
motion due to the application of forces.
(ii) . Kinematics:
It is defined as the branch of dynamics which deals with the bodies that are in motion,
without knowing the reference of forces responsible for the motion in the body.
General Introduction to

SI
System of
units

Mechanics: Units
W  mg
F  ma
Four Fundamental Quantities
→ N = kg.m/s2
→ N = kg.m/s2
1 Newton is the force
required to give a mass of 1
kg an acceleration of 1 m/s2
Quantity Dimensional SI UNIT
Symbol Unit Symbol
Mass M
Length L
Kilogram Kg
Meter M
Time T
Force F
Second s
Newton N
Basic Unit

Fundamental units of S.I system

Principal S.I. units

S.I. Prefixes

UNIT CONVERSION

Concept
of
Force

Concept of Force

The necessity of force:
To move a stationary object i.e. to move a body which is at rest.
To change the direction of the motion of an object
To change the magnitude of the velocity (speed) of the motion of an object
To change the shape of an object.

Effects of force
It may set a body into motion
It may bring a body to rest.
It may change the magnitude of motion
It may change the direction of motion
It may change the magnitude and direction of motion
It may change the shape of an object

Characteristics of Force
It has four characteristics
1. Direction
2. Magnitude
3. Point on which it acts
4. Line of action

Line of Action of force
•The line of action of a force f is a geometric representation
of how the force is applied.
• It is the line through the point at which the force is applied in the
same direction as the vector f→.

System of
Forces
When two are or more forces acts act on a body,
they are called system of forces.
1. Coplanar Force system – 2D and Non –
Coplanar system – 3D
2. Concurrent and Non – Concurrent Force system
3. Collinear and Non- Collinear Force system
4. Parallel – Like and Unlike

System of Forces

Coplanar Force System – 2D

Non- Coplanar Force System – 3D

Concurrent and Non – Concurrent Force system
Concurrent Forces Non- Concurrent Forces

Collinear and Non- Collinear Force system
Collinear Forces Non – Collinear Forces

Parallel Force system

Idealization in
Mechanics
Models or idealizations are used in
order to simplify application of the
theory of mechanics.
Here we will consider three important
idealizations
1. Rigid Body
2. Particle
3. Concentrated force

Mechanics: Idealizations
To simplify application of the theory
Particle: A body with mass but with dimensions
that can be neglected
Size of earth is insignificant
compared to the size of its
orbit. Earth can be modeled
as a particle when studying its
orbital motion

Rigid Body: A combination of large number of particles in which all particles
remain at a fixed distance (practically) from one another before and after
applying a load.
Material properties of a rigid body are not required to be considered when
analyzing the forces acting on the body.
In most cases, actual deformations occurring in structures, machines,
mechanisms, etc. are relatively small, and rigid body assumption is suitable for
analysis

Concentrated Force: Effect of a loading which is assumed to act at
a point (CG) on a body.
•Provided the area over which the load is applied is very small
compared to the overall size of the body.
Ex: Contact Force between a wheel
and ground.
40 kN 160 kN

Fundamental
Concepts

Mechanics: Fundamental Concepts
Length (Space): needed to locate position of a point in space, & describe size of the
physical system. → Distances, Geometric Properties
Time: measure of succession of even. → basic quantity in Dynamics
Mass: quantity of matter in a body →measure of inertia of a body (its resistance to
change in velocity)
Force: represents the action of one body on another → characterized by its
magnitude, direction of its action, and its point of application

Force is a Vector quantity.

Newtonian Mechanics
Length, Time, and Mass are absolute concepts independent of each other
Force is a derived concept not independent of the other fundamental concepts.
Force acting on a body is related to the mass of the body and the variation of its
velocity with time.
Force can also occur between bodies that are physically separated (Ex: gravitational,
electrical, and magnetic forces)

Remember:
• Mass is a property of matter that does not change from one
location to another.
• Weight refers to the gravitational attraction of the earth on a body
or quantity of mass. Its magnitude depends upon the elevation at
which the mass is located
• Weight of a body is the gravitational force acting on it.

Scalar
and
Vector
Quantities
BASIS FOR
COMPARISON
SCALAR QUANTITY VECTOR QUANTITY
Meaning
Any physical quantity that does
not include direction is known
as a scalar quantity.
A vector quantity is one, that has
both magnitude and direction.
Quantities One-dimensional quantities Multi-dimensional quantities
Change
It changes with the change in
their magnitude.
It changes with the change in their
direction or magnitude or both.
Operations Follow ordinary rules of algebra. Follow the rules of vector algebra.
Comparison of two
quantities
Simple Complex
Division
Scalar can divide another
scalar.
Two vectors can never divide.

What is a scalar?
Scalar quantities are measured with numbers and units.
length
(e.g. 102 °C)
time
(e.g. 16 cm)
temperature
(e.g. 7 s)

What is a vector?
Vector quantities are measured with numbers and units, but also have a specific direction.
acceleration
(e.g. 30m/s2
upwards)
displacement
(e.g. 200 miles
northwest)
force
(e.g. 2 N
downwards)

Speed or velocity?
Distance is a scalar and displacement is a vector. Similarly, speed is a scalar
and velocity is a vector.
Speed is the rate of change of distance in the direction of travel.
Speedometers in cars measure speed.
Velocity is a rate of change of displacement and has both magnitude and
direction.
average
speed
average
velocity
Averages of both can be useful:
distance
time
displacement
time
= =

Goals for Chapter 1
• To learn three fundamental quantities of physics and the units to measure them
• To understand vectors and scalars and how to add vectors graphically
• To determine vector components and how to use them in calculations
• To understand unit vectors and how to use them with components to describe
vectors
• To learn two ways of multiplying vectors
40
CL- ENGG. MECHANICS, DOCL- SPP,
DDU, NADIAD

Unit consistency and conversions
• An equation must be dimensionally consistent. Terms to be added or
equated must always have the same units. (Be sure you’re adding
“apples to apples.”)
• Always carry units through calculations.
• Convert to standard units as necessary.
41
DDU, NADIAD

Vectors and scalars
• A scalar quantity can be described by a single number.
• A vector quantity has both a magnitude and a direction in space.
• In this book, a vector quantity is represented in boldface italic type with an
arrow over it: A.
• The magnitude of A is written as A or |A|.
42
DDU, NADIAD
 

Drawing vectors—Figure 1.10
• Draw a vector as a line with an arrowhead at its tip.
• The length of the line shows the vector’s magnitude.
• The direction of the line shows the vector’s direction.
43
DDU, NADIAD

Adding two vectors graphically
• Two vectors may be added graphically using
either the parallelogram method or the head-to-
tail method.

Adding more than two vectors graphically—
• To add several vectors, use the head-to-tail method.
• The vectors can be added in any order.

The negative of a vector is defined as the vector that, when
added to the original vector, gives a resultant of zero
The negative of the vector will have the same magnitude, but
point in the opposite direction
• Represented as
• A  A 0
A
Negative of a Vector

Subtracting vectors
CL- ENGG. MECHANICS, DOCL- SPP, DDU, NADIAD
• Figure shows how to subtract vectors.
47

48
Multiplying a vector by a scalar
• If c is a scalar, the
product cA has
magnitude |c|A.
• Multiplication of a
vector by a positive
scalar and a negative
scalar.


Addition of two vectors at right angles

Components of a vector—
• Adding vectors graphically provides limited accuracy. V
ector components
provide a general method for adding vectors.
• Any vector can be represented by an x-component Ax and a y- component Ay.
• Use trigonometry to find the components of a vector: Ax = Acos θ and
Ay = Asin θ, where θ is measured from the +x-axis toward the +y-axis.

Components of a Vector
The x-component of a vector
is the projection along the x-axis
Ax  Acos
The y-component of a vector
is the projection along the y-axis
Ay  Asin
This assumes the angle θ is
measured with respect to the positive direction of x-axis
• If not, do not use these equations, use the sides of the triangle
directly
CL- Engg. Mechanics, DoCL- SPP, DDU, 51

Components of a Vector, 4
The components are the legs of the right triangle whose
hypotenuse is the length of A
• May still have to find θ with respect to the positive x-axis
A  A2
 A2
and   tan1
x y
Ay
Ax

Positive and negative components—Figure
• The components of a vector can be positive or negative numbers, as shown in the figure.

Finding components—Figure

Components of a Vector, final
The components can be positive or negative
The signs of the components will depend on the angle

Adding Two Vectors Using Their
Components
Rx = Ax + Bx
Ry = Ay + By
The magnitude and direction
of resultant vectors are:

Adding vectors using their components
• For more than two vectors we can use the components of a set of vectors
to find the components of their sum:
Rx  Ax Bx Cx  , Ry  Ay By Cy 

Adding vectors using their components—Ex.

Example 2

Unit vectors
in terms of its components as
A =Axî+ Ay j + Az k
.
• Aunit vector has a magnitude
of 1 with no units.
• The unit vector î points in the
+x-direction, j points in the +y-
direction, and k points in the
+z-direction.
• Any vector can be expressed


Adding vectors using unit-vector notation
In three dimensions if
then
and so Rx= Ax+Bx, Ry= Ay+By, and Rz =Az+Bz
R  Axî  Ay ĵ  Azk̂ Bxî  By ĵ  Bzk̂
R  Ax  Bx î  Ay  By ĵ  Az  Bz k̂
R  R î  R ĵ  R k̂
x y z
R  R2
 R2
 R2
x y z
  cos1 Rx
, etc.
R
R  A  B

Unit vector notation , adding vectors
In two dimensions, if
then
and so Rx= Ax+Bx, Ry= Ay+By,
The magnitude and direction are
R  A  B

Example 3
If A = 24i-32j and B=24i+10j, what is the
magnitude and direction of the vector C = A-B?

The scalar product
• The scalar product
(also called the “dot
product”) of two
vectors is
A B ABcos.
• Figures illustrate
the scalar product.

Dot Products of Unit Vectors
î î  ĵ ĵ  k̂k̂ 1
î  ĵ  î k̂  ĵk̂  0
Using component form with vectors:

Calculating a scalar product
•
• Find the scalar product of two vectors shown in the figure.
The magnitudes of the vectors are:A= 4.00, and B = 5.00
66

Calculating a scalar product – Example 4
•
• Find the scalar product of two vectors shown in the figure.
The magnitudes of the vectors are:A= 4.00, and B = 5.00

Finding an angle using the scalar product – Ex. 5
• Find the angle between the vectors.
• Use equation:

Finding an angle using the scalar
product – Ex. 5
• Find the angle between the vectors.
• Use equation:
69

The Vector Product Defined
Given two vectors, A and B
The vector (cross) product of Aand B is defined
as a third vector,
The magnitude of vector C is AB sin 
•  is the angle between Aand B
C  A B

More About the Vector Product
The direction of C is
perpendicular to the plane
formed by Aand B
The best way to
determine this direction is
to use the right-hand rule

Using Determinants
The components of cross product can be calculated as
Expanding the determinants gives
AB  AyBz  AzBy î  AxBz  AzBx ĵ  AxBy
If Az = 0 and Bz=0 then
=
î ĵ k̂
A A
î 
Ax
ĵ 
Ax
k̂
A. A
 y z
B. B
Ay
B B
Az
B B
x y z
y z x y
x z
x y z
A B  A
B B B
 AyBx k̂

Vector Product Example 6
Given
Find
Result
A  2î  3ĵ; B  î  2ĵ
=
=

The vector product—Summary
• The vector
product (“cross
product”) of
two vectors has
magnitude
|AB| ABsin
and the right-
hand rule gives
its direction.
74

Calculating the vector product— ex. 6
• V
ector has magnitude 6 units
and is in the direction of the +x
axis. V
ector has magnitude 4
units and lies in the xy – plane
making an angle of 300 with the x
axis. Find the cross product
Use ABsin to find the magnitude
and the right-hand rule to find the
direction.
75

Accuracy of
Calculations

NUMERICAL ACCURACY
The accuracy of a solution depends on
1. Accuracy of the given data.
2. Accuracy of the computations performed. The solution cannot be more
accurate than the less accurate of these two.
3. The use of hand calculators and computers generally makes the
accuracy of the computations much greater than the accuracy of the
data. Hence, the solution accuracy is usually limited by the data
accuracy.

Problem
Solving
Approach
Problem Solving
Strategy

Problem Solving Strategy
• Read the problem
– Identify the nature of the problem
• Draw a diagram
– Some types of problems require very specific types of diagrams

Problem Solving cont.
• Label the physical quantities
– Can label on the diagram
– Use letters that remind you of the quantity
• Many quantities have specific letters
– Choose a coordinate system and label it
• Identify principles and list data
– Identify the principle involved
– List the data (given information)
– Indicate the unknown (what you are looking for)

Problem Solving, cont.
• Choose equation(s)
– Based on the principle, choose an equation or
set of equations to apply to the problem
• Substitute into the equation(s)
– Solve for the unknown quantity
– Substitute the data into the equation
– Obtain a result
– Include units

Problem Solving, final
• Check the answer
– Do the units match?
• Are the units correct for the quantity being found?
– Does the answer seem reasonable?
• Check order of magnitude
– Are signs appropriate and meaningful?

Problem Solving Summary
• Equations are the tools of physics
– Understand what the equations mean and how to use them
• Carry through the algebra as far as possible
– Substitute numbers at the end
• Be organized

Reference
BOOKS

Pre-
Requisite
√ Trigonometry equations
√ Calculation on calculation
√ Unit Conversion
√ Balancing equation for Units

References:
https://www.studocu.com/row/document/balochistan-university-of-
information-technology-engineering-and-management-
sciences/engineering-surveying/engineering-mechanics-17/2544731
https://sites.pitt.edu/~qiw4/Academic/ENGR0135/Chapter2.pdf

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