Introduction of system of coplanar forces (engineering mechanics)

Engineering Mechanics
Nilesh G.
Presented by Nilesh Gaddapawar

Classification of Mechanics
Nilesh G.

Mechanics of Deformable Bodies:
 The mechanics of deformable bodies deals
with how forces are distributed inside bodies,
and with the deformations caused by these
internal force distributions. These internal
force produce "stresses" in the body, which
could ultimately result in the failure of the
material itself.
 Principles of rigid body mechanics often
provide the beginning steps in analyzing
these internal stresses, and resulting
deformations. These will be studied in
courses called Strength of Materials or
Mechanics of Materials.
Nilesh G.

Mechanics of Fluids:
 The mechanics of fluids is the branch of mechanics that
deals with liquids or gases.
 Fluids are commonly used in engineering
applications. They can be classified as incompressible,
or compressible. While all real fluids are compressible
to some degree, most liquids can be analyzed as
incompressible in many engineering applications.
 Applications of fluid mechanics abound, from hydraulics
and general flow in pipes to air flow in ducts to
advanced applications in turbines and aerospace.
 The study of the mechanics of fluids will be studied in
courses called Fluid Mechanics, Compressible Flow,
Hydraulics, and others.
Nilesh G.

Mechanics of Rigid Bodies:
A rigid body is a body which does not
deform under the influence of forces.
In all real applications, there is always
deformation, however, many stuctures
exhibit very small deformations under
normal loading conditions, and rigid
body mechanics can be used with
sufficient accuracy in those cases.
Nilesh G.

Fundamentals
 Length: Length is the quantity used to describe the position of a point in space
relative to another point. The universally accepted standard unit for length is the
meter.
 Time: Time is the interval between two events. The generally accepted standard
unit for time is the second.
 Space: The geometric region in which the study of body is involved is called
Space.
 Displacement: It is defined as the distance moved by a body/particle in the
specified direction.
 Velocity: It is the rate of change of displacement with respect to time.
 Acceleration : It is the rate of change of velocity with respect to time.
 Momentum: The product of mass and velocity is called momentum.
 Continuum: A body consists of several matters.
 Particle: A particle may be defined as an object which has only mass and no size.
Such a body cannot exist theoretically. However in dealing with problems involving
distances considerably larger compared to the size of the body, the body may be
treated as particle. Examples of such situations are
 A bomber aeroplane is a particle for a gunner operating from the ground.
 A ship in mid sea is a particle in the study of its relative motion from a control
tower.
Nilesh G.

Vectors and Scalars
quantities
All physical quantities (e.g. speed and force)
are described by a magnitude and a unit.
VECTORS – also need to have their direction
specified
examples: displacement, velocity,
acceleration, force.
SCALARS – do not have a direction
examples: distance, speed, mass, work,
energy.
Nilesh G.

Addition of vectors
With two vectors acting at
an angle to each other:
Draw the first vector.
Draw the second vector with
its tail end on the arrow of the
first vector.
The resultant vector is the line
drawn from the tail of the first
vector to the arrow end of the
second vector.
This method also works with
three or more vectors.
4N
3N
Resultant vector
= 5N
4N
3N
Nilesh G.

Comparison of Mass and
Weight
Sr.
no.
Mass Weight
01 Mass is a property of matter.
The mass of an object is the
same everywhere.
Weight depends on the effect of
gravity. Weight varies according to
location.
02 Mass can never be zero. Weight can be zero if no gravity acts
upon an object, as in space.
03 Mass does not change
according to location.
Weight increases or decreases with
higher or lower gravity.
04 Mass is a scalar quantity. It
has magnitude.
Weight is a vector quantity. It has
magnitude and is directed toward the
center of the Earth or other gravity
well.
05 Mass may be measured using
an ordinary balance.
Weight is measured using a spring
balance.
06 Mass usually is measured in
grams and kilograms.
Weight often is measured in Newton,
a unit of force.
Nilesh G.

Comparison of Distance and
Displacement:
Sr.
no.
Distance Displacement
01 Distance is the length of the path
travelled by a body while moving
from an initial position to a final
position.
Displacement is the shortest
distance between the initial position
and the final position of the body.
02 Distance is a scalar quantity. Displacement is a vector quantity.
03 Distance measured is always
positive.
Displacement can be positive or
negative depending on the
reference point.
04 The total distance covered is
equal to the algebraic sum of all
the distances travelled in
different directions.
The net displacement is the vector
sum of the individual displacements
in different directions.
05 There is always a distance
covered whenever there is a
motion.
Displacement will be zero if the
body comes back to its initial
position.
06 Unit: metre (m) Unit: metre (m)
Nilesh G.

For example:
Q. Suppose you are observing an ant on the table, as
shown in the diagram below. The ant moves from one
corner of the table to the other corner. The blue
irregular line shows the path of the ant.
For figure (B) &
(C), Find
Distance ?
Displacement?
Answer :
•For Length of this blue line is the distance covered
by the ant.
•The straight green line, which is the minimum
distance between the two corners of the table is the
displacement of the ant. Called the displacement.
Figure
(B)
Figure
(A)
Figure
(C)
Nilesh G.

Comparison of Speed and
Velocity
Sr. no Speed velocity
01 Speed is refers to "how fast an object
is moving."
Velocity refers to "the rate at which an
object changes its position."
02 Speed is a scalar quantity. Velocity is a vector quantity.
03 Speed is the rate of motion, or the
rate of change of position.
velocity is the rate of change of
displacement.
04 Speed is thus the magnitude
component of velocity.
Velocity contains both the magnitude
and direction components.
05 Explanation :
How fast my hand is moving to
slapped on your face, this is speed
Explanation :
When you get the slap and changes
your face from right to left.. i.e the rate
at which your face changes its position,
this is Velocity....
06 speed= total distance/time taken velocity= displacement(shortest root
from initial to the final position) /time
taken
including direction.
07 Unit: km/hr like 60km/hr, Unit: 60km/hr in east direction.
Nilesh G.

Units of Measure:
 The force unit is called a newton (N), and is defined as the
force required to accelerate a mass of 1 kg at a rate of 1
meter/sec. So, we can write:
1 N = (1 kg)(1 m/s2) or 1 N = 1kg.m/s2
 The weight of an object is the gravitational force which is
exerted on that object which causes it to accelerate
downward at the acceleration due to gravity So, we can
write for the weight of a 1 kg mass:
W = mg
W = (1 kg)(9.807 m/s2)
W = 9.807 N
Nilesh G.

SI prefixes:
Multiplier Prefix Symbol
109 giga G
106 mega M
103 kilo k
10-2 centi c
10-3 milli m
10-6 micro µ
10-9 nano n
10-12 pico p
Nilesh G.

FUNDAMENTAL LAWS
OF MECHANICS
Nilesh G.

Sir Isaac Newton
Sir Isaac Newton 25 December 1642 – 20 March
1727 was an English physicist and
mathematician (described in his own day as a
"natural philosopher") who is widely recognised
as one of the most influential scientists of all
time and as a key figure in the scientific
revolution.
Nilesh G.

1. Newton's first law
“An object at rest stays at rest and an object in
motion stays in motion with the same speed and in
the same direction unless acted upon by an
unbalanced force. This law is often called the law of
inertia.”
Objects tend to "keep on doing what they're doing."
In fact, it is the natural tendency of objects to resist
changes in their state of motion. This tendency to
resist changes in their state of motion is described
as inertia.
Example :The motion of a kite when the wind
changes can also be described by the first law.
Nilesh G.

Explanation to first law :
Nilesh G.

“ The rate of change of momentum is
directly proportional to impressed
force and it takes place in the
direction of force acting on it”.
 Force ∝ rate of change of momentum.
But momentum = mass × velocity
As mass do not change,
 Force ∝ mass × rate of change of
velocity
i.e., Force ∝ mass × acceleration
F ∝ m × a
2. Newton's second law
Nilesh G.

“For every action there is an equal
and opposite reaction”.
3. Newton's third law
Nilesh G.

4.Newton’s Law of
Gravitation
“Everybody attracts the other body. The force of attraction
between any two bodies is directly proportional to their
masses and inversely proportional to the square of the
distance between them”.
 According to this law the force of attraction between the bodies of mass m1
and mass m2 at a distance d as shown in Fig.
 where G is the constant of proportionality and is
known as constant of gravitation.
Nilesh G.

5. Principle or Law of
transmissibility of forces:
“The state of rest or of Uniform motion of a rigid
body is unaltered if the point of application of
the force is transmitted to any other point along
the line of action of the force”.
Let F be the force acting on a rigid body
at point A. According to the law of
transmissibility of force, this force has
the same effect on the state of body as
the force F applied
at point B.
Nilesh G.

6. Law of superposition.
 The law of transmissibility of forces can be proved using
the law of superposition, which can be stated as “the
action of a given system of forces on a rigid body is not
changed by adding or subtracting another system of
forces in equilibrium.”
Consider the rigid body shown in Fig. (a). It is subjected to a force F at A. B is
another point on the line of action of the force. From the law of superposition it is
obvious that if two equal and opposite forces of magnitude F are applied at B
along the line of action of given force F, [Ref. Fig. (b)] the effect of given force on
the body is not altered. Force F at A and opposite force F at B form a system of
forces in equilibrium. If these two forces are subtracted from the system, the
resulting system is as shown in Fig. 1.7 (c).
Nilesh G.

7.Parallelogram Law of Forces
This law states that “ if two forces acting simultaneously on a body at
a point are presented in magnitude and direction by the two adjacent
sides of a parallelogram, 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”.
In Fig. the force F1 = 4 units and force F2 = 3 units are acting on a
body at point A. Then to get resultant of these forces parallelogram
ABCD is constructed such that AB is equal to 4 units to linear scale
and AC is equal to 3 units. Then according to this law, the diagonal
AD represents the resultant in the direction and magnitude. Thus the
resultant of the forces F1and F2 on the body is equal to units
corresponding to AD in the direction α to F1.
Nilesh G.

 DERIVED LAWS
The Triangle Law of Forces may be stated as
“If two forces acting on a body are represented
one after another by the sides of a triangle, their
resultant is represented by the closing side of
the triangle taken from first point to the last
point”.
The polygon of law of forces and it may be
stated as “If a number of concurrent forces acting
simultaneously on a body are represented in
magnitude and direction by the sides of a polygon,
taken in a order, then the resultant is represented in
magnitude and direction by the closing side of the
polygon, taken from first point to last point”.
Nilesh G.

Classifiaction of Mechanics of rigid
bodies
Mechanics of
rigid bodies
Statics Dynamics
Kinematics Kinetics
Nilesh G.

 Static : It is the branch of mechanics
which deals with the particles and
bodies at rest.
 Dynamics : It is the branch of
mechanics which deals with the
particles and bodies are in motion.
 Kinematics : It is the study of motion of
particles and bodies without reference
to masses and effects of forces.
 Kinetics: It is the study of motion of
particles and bodies with reference to
masses and effects of forces.
Nilesh G.

What is a force
 According to Newton‟s I
law, Force is defined as an
action or agent, which
changes or tends to change
the state of rest or of
uniform motion of a body in
a straight line.
 Units of force: The gravitational (MKS)
unit of force is the kilogram force and is
denoted as „kgf‟.
 The absolute (SI) unit of force is the
Newton and is denoted as “N”. Note: 1
kgf = “g”N (But g = 9.81m/s2) Therefore 1
kgf = 9.81 N or 10 N.
Nilesh G.

Characteristics of a force
 It has four important characteristics,
which can be listed as follows.
1) Magnitude: It can be denoted as 10
kgf or 100 N.
2) Point of application: It indicates the
point on the body on which the force
acts.
3) Line of action: The arrowhead placed
on the line representing the direction
represents it.
4) Direction: It is represented by a co-
ordinate or cardinal system.
Nilesh G.

Explanation to a force
 In figure, AB is a ladder kept against a wall.
At point C, a person weighing 600 N is
standing. The force applied by the person on
the ladder has the following characteristics.
 Magnitude os 600 N
 Direction is vertically downwards.
 The line of action is vertical CD or C to D
 The point of application is at C.
 The magnitude of the force is written near the
arrow.
 The line of arrow shows the line of action of
the force and the arrow head shows the
direction of the force.
 The point of intersection of the force with the
body shows the point of application of the
force on the body.
Nilesh G.

CLASSIFICATION OF
FORCE
Force system
Coplanar force
system
Concurrent/
non
concurrent
Co-linear / non
co-linear
parallel./ non
parallel
Like / unlike
parallel
Non coplanar
force system
Nilesh G.

Resolution of FORCE
It is often convenient to split a
single FORCE into two
perpendicular components.
Consider force F being split into
vertical and horizontal
components, FV and FH.
In rectangle ABCD opposite:
sin θ = BC / DB = DA / DB = FV / F
Therefore: FV = F sin θ
cos θ = DC / DB = FH / F
Therefore: FH = F cos θ
F
FV
FH
θ
C
BA
D
FV = F sin θ
FH = F cos θ
Nilesh G.

Question
Calculate the vertical and
horizontal components if F
= 4N and θ = 35o.
FV = F sin θ
= 4 x sin 35o
= 4 x 0.5736
FV = 2.29 N
FH = F cos θ
= 4 x cos 35o
= 4 x 0.8192
FH = 3.28 N
F
FV
FH
θ
Nilesh G.

Composition of forces
The reduction of a given system of forces to
the simplest system that will be its equivalent
is called composition of forces.
RESULTANT FORCE: It is possible to find a single force
which will have the same effect as that of a number of forces
acting on a body. Such a single force is called resultant force.
The process of finding out the resultant force is called composition of
forces.
Nilesh G.

Composition of Co-planar concurrent
force system.
COMPOSITION OF TWO FORCES: It is possible
to reduce a given system of forces i.e., two forces
to the simplest system as its equivalent (resultant
force) with the help of parallelogram law of forces.
Nilesh G.

COMPOSITON OF FORCES BY
RESOLUTION(Principle of resolved
parts)
Nilesh G.

Procedure to find magnitude and
direction of resultant force : -
 Algebraic sum of the components of forces in X direction
 Algebraic sum of the components of forces in Y direction
Now the system of forces is equal to two mutually
perpendicular forces , the magnitude and direction of
resultant force.
Magnitude:
Direction:
Nilesh G.

The moment of a force
Also known as the turning effect of a
force.
The moment of a force about any point is
defined as: force x perpendicular
distance from the turning point to the
line of action of the force
moment = F x d
Unit: Newton-metre (Nm)
Moments can be either CLOCKWISE or
ANTICLOCKWISE
Force F exerting an
ANTICLOCKWISE
moment through the
spanner on the nut
Nilesh G.

Question
Calculate the moments of the 25N
and 40N forces on the door in the
diagram opposite.
moment = F x d
For the 25N force:
moment = 25N x 1.2m
= 30 Nm CLOCKWISE
For the 40N force:
moment = 40N x 0.70m
= 28 Nm ANTICLOCKWISE
hinge
door
40N
25N
1.2 m
Nilesh G.

Couples and Torque
“Two equal unlike parallel
forces separated by a
definite distance form a
couple”.
In the diagram above:
couple = F d
= One of the forces x the distance between the forces
Torque is another name for the total moment of a
couple.
d
F
F
Nilesh G.

Chracteristics of a Couple
1. The algebraic sum of magnitude of
the member forces forming a couple
is always zero.
2. The couple can be balanced by a
couple only, which has same
moment and opposite sense.
3. Moment of a couple is fixed and it
does not depend on moment center
Nilesh G.

Comparison of Torque and
Moment
Sr.No. Torque Moment
01 Torque is a movement force. Moment is a static force.
02 Torque is often presented as
Nm/revolution
moment is typically presented
as Nm.
03 tends to be used when there is
an axle or pivot to be turned
around
tends to be used in
essentially non-rotational
situations, such as analysis of
forces on a beam.
04 Twisting effect produced by
force
Turning effect produced by
force
05
Nilesh G.

Varignon’s theorem (Principle of moments)
Statement : “The sum of moment of all forces about a point is
equal to the moment of their resultant about the same
point”.Proof: consider a force F acting at a point A and having components F1
and F2 in any two directions.
Let us choose any point O, lying in the plane of the forces, as a moment
center. Draw the perpendiculars d , d1 and d2 from moment center o to
the forces F, F1 and F2 respectively.
Let α,α1 and α3 be the angle made by the forces F, F1 and F2 respectively
Nilesh G.

Moment of force F about O,
F × d = F × OA× cosα =OA × F × cosα = OA × Fx -----------------(1)
Moment of force F1 about O,
F1 × d1 = F1 × OA × cosα1 =OA × F1 × cosα1 = OA × Fx1 --------(2)
Moment of force F1 about O,
F2 × d2 = F2 × OA × cosα2 =OA × F2 × cosα2 = OA × Fx2 --------(3)
Adding (2) & (3),
F1 × d1 + F2 × d2 = OA ( Fx1 + Fx2 ) ----------------------------------(4)
Let ,
x-components of the resultant force F = sum of the x-components of the forces F1 and F2
Fx = Fx1 + Fx2 -----------------------------------------------------------(5)
From (4) & (5),
F1 × d1 + F2 × d2 = OA × Fx -------------------------------------------(6)
From (1) & (6),
F1 × d1 + F2 × d2 = F × d
i.e. ∑ MF@o = MR@o
Application : It is used for finding the point of application of the
resultant in case of non concurrent and parallel force system.
Nilesh G.

Introduction of system of coplanar forces (engineering mechanics)

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