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MECHANICS: Mechanics can be defined as the branch of physics concerned with the
state of rest or motion of bodies that subjected to the action of forces
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Classification of Mechanics
The engineering mechanics are classified as shown
Engineering Mechanics

Mechanics of Rigid bodies

Statics

Dynamics

Mechanics of Deformed bodies

Mechanics of fluid

Statics

Dynamics

BRANCHES OF MECHANICS:
Mechanics can be divided into two branches
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Static
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Dynamics
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Either the body at rest or in uniform motion is called statics
b) Dynamics:
It is the branch of mechanics that deals with the study of forces on body in motion
is called dynamics
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i) Kinetics
ii) kinematics
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Rigid Body
A body is said to be rigid, if the relative positions of any two particles do not change under the action of the
forces acting on it
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1
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After the application of
forces F1, F2, F3, the body takes the position as shown in Fig
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1(b)
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If the body is treated as rigid, the relative position of A’’ AB are the same i
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B and
A’’ AB
B=

Many engineering problems can be solved by assuming bodies rigid
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Particle
A particle may be defined as an object which has only mass and no size
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However in dealing with problems involving distances considerably larger compared to the size of
the body, the body may be treated as a particle, without sacrificing accuracy
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A
— ship in mid sea is a particle in the study of its relative motion from a control tower
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In
3
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Magnitude of force is defined by Newton’second law
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Noting
that rate of change of velocity is acceleration, and the product of mass and velocity is momentum we can derive
expression for the force as given below:
From Newton’s second law of motion
Force α rate of change of momentum
α rate of change of (mass × velocity)
Since mass do not change,
Force α mass × rate of change of velocity
α mass × acceleration
Fαm×a
=k×m×a
where F is the force, m is the mass and a is the acceleration and k is the constant of proportionality
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For
example, in S
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system, unit of force is Newton, which is defined as the force that is required to move one
kilogram (kg) mass at an acceleration of 1 m/sec2
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81 m/sec2 on earth surface) and unit of
force is defined as kg-wt
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81 newtons
It may be noted that in usage kg-wt is often called as kg only
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4 What are the characteristics of forces?
Characteristics of a Force
It may be noted that a force is completely specified only when the following
characteristics are specified
—
Magnitude
—
Point of application
—
Line of action
—
Direction
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, AB is a ladder kept against a wall
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The force applied by the person on the ladder has the following
characters:
—
magnitude is 600 N
— point of application is C which is at 2 m from A along the ladder
the
— line of action is vertical
the
— direction is downward
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the
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A force may produce the following effects in a body, on which it acts :
1
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i
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if a body is at rest, the force may set it in motion
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2
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3
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4
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1
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” is also called the law of inertia, and consists of the
It
following two parts :
1
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It appears to be selfevident, as a train at rest on a level track will not move unless pulled by an engine
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2
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It cannot be exemplified because it is, practically,
impossible to get rid of the forces acting on a body
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NEWTON’SECOND LAW OF MOTION
S
It states, “he rate of change of momentum is directly proportional to the impressed force and takes place, in the
T
same direction in which the force acts
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Now consider a body moving in a straight line
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Force α rate of change of momentum
α rate of change of (mass × velocity)
Since mass do not change,
Force α mass × rate of change of velocity

α mass × acceleration
Fαm×a
=k×m×a
F=m×a
if K=1
3
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Consider the two bodies in contact with
each other
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According to this law the second body develops a reactive
force R which is equal in magnitude to force F and acts in the line same as F but in the opposite direction
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In Fig
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4
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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
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is
d’

5
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”
6
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Let F be the force acting on a rigid body at point A as shown in Fig
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4
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In using law of transmissibility it should be carefully noted that it is applicable only if the body can be treated as
rigid
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The process of finding a single Force which will have the same effect as a set of Forces acting on a body is
known as composition of Forces
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e
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The

various forces acting on a body may be grouped into:
(a) Applied Forces
(b) Non-applied Forces
(a) Applied Forces
These are the forces applied externally to a body
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Depending upon type of their contact with the body, the applied forces may be classified as:
(i) Point Force
(ii) Distributed Forces
(i) Point Force: It is the one which has got contact with the body at a point
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However, when the contact area is small
compared to the other dimensions in the problem, for simplicity of calculation the force may be
considered as a point load
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Correspondingly they are known as linear, surface and body forces
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It is usually represented with abscissa
representing the position on the body and ordinate representing the magnitude of the load
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The force ‘ at any small
dF’
length ‘ is given by
ds’
dF = w ds
Surface Force: Force acting on the surface of a body is known as surface force
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The force dF acting on any area dA is given by
dF = p dA
Where, ‘is the intensity of force per unit area
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Example of
this type of force is the weight of a block acting on the body under consideration
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(b) Non-applied Forces
There are two types of non-applied forces: (a) Self weight and (b) Reactions
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W = mg
Where, m is mass of the body and g is gravitational acceleration (9
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When analyzing equilibrium conditions of a body, self
weight is treated as acting through the centre of gravity of the body
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Reactions: These are self-adjusting forces developed by the other bodies which come in contact with the body
under consideration
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The reactions adjust themselves to bring the body to equilibrium
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8 Define various Systems of forces
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If
all the forces in a system lie in a single plane, it is called a coplanar force system
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In a system of parallel
forces all the forces are parallel to each other
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Various systems of forces, their characteristics and examples are given in Table
below
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9 what do you mean by resultant force
A resultant force is a single force, which produce same affect so that of number of forces can produce is called
resultant force
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COMPOSITION OF FORCES
The process of finding out the resultant Force of given forces (components vector) is called composition of
forces
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Graphical methods
1
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Triangle law of forces or triangular method
3
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Analytical method

1
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OR
When two forces are acting at a point such that they can by represented by the adjacent sides of a parallelogram
then their resultant will be equal to that diagonal of the parallelogram which passed through the same point
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5(a)
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If parallelogram ABCD is drawn as shown in Fig
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Drop perpendicular CE

to AB
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OR
If two forces are acting on a body such that they can be represented by the two adjacent sides of a triangle taken
in the same order, then their resultant will be equal to the third side (enclosing side) of that triangle taken in the
opposite order
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C) POLYGON METHOD
According to this method” more than two forces acting on a particle by represented by the side of polygon
if
taken in order their resultant will be represented by the closing side of the polygon in opposite direction“
OR
If more than two forces are acting on a body such that they can by represented by the sides of a polygon Taken
in same order, then their resultant will be equal to that side of the polygon, which completes the polygon
(closing side taken in opposite order)
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2
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Resolve all the forces horizontally and find the algebraic sum of all the horizontal
components (i
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, ΣH)
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Resolve all the forces vertically and find the algebraic sum of all the vertical components (i
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, ΣV)
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The resultant R of the given forces will be given by the equation :

4
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11 What do you understand by free body diagram? Explain its significance also
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B
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Procedure of drawing Free Body Diagram
To construct a free-body diagram, the following steps are necessary:
Draw Outline Shape
Imagine that the particle is cut free from its surroundings or isolated by drawing the outline shape of the particle
only
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There are two classes of forces that act on the particle
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Identify Each Force
The forces that are known should be labeled complete with their magnitudes and directions
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Q
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Moment of a force about a point is the measure of its rotational effect
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The point
about which the moment is considered is called moment centre and the perpendicular distance of the point from
the line of action of the force is called moment arm
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The magnitude of moment or
tendency of the force to rotate the body about the axis O_O
perpendicular to the plane of the body is proportional both to the magnitude of the force and to the moment arm
d, therefore magnitude of the moment is defined as the product
of force and moment arm
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13 discuss type of moment
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Clockwise Moment
When the force tends to rotate the body in the same direction in which the hands of clock move is called
clockwise moment the clockwise moment is taken as positive or other wise mentioned
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Unit of moment
S
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m
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14 Describe application of moment
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1
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Levers
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s
It states that, The algebraic sum of the moments of a system of coplanar forces about a moment centre in their
plane is equal to the moment of their resultant force about the same moment centre
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let R be the resultant of forces F1 and F2 and B the moment centre
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Then
in this case, we have to prove that:
Rd = F1 d1 + F2 d2
Join AB and consider it as y axis and draw x axis at right angles to it at A [Fig
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Denoting by θ the angle that
R makes with x axis and noting that the same angle is formed by perpendicular to R at B with AB1, we can

write:
Rd = R × AB cosθ
= AB × (R cosθ)
= AB × Rx
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Similarly, if F1x and F2x are the components of F1 and F2, in x direction, respectively, then
F1 d1 = AB F1x
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(c)
From Eqns
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(d)
From equation (a) and (d), we get
Rd = F1 d1 + F2 d2
If a system of forces consists of more than two forces, the above result can be extended as given below:
Let F1, F2, F3 and F4 be four concurrent forces and R be their resultant
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Fig ]
If R1 is the resultant of F1 and F2 and its distance from O is a1, then applying varignon’theorem:
s
R1 a1 = F1 d1 + F2 d2
If R2 is the resultant of R1 and F3 (and hence of F1, F2 and F3) and its distance from O is a2, then applying
Varignon’theorem:
s
R2a2 = R1a1 + F3d3
= F1 d1 + F2 d2 + F3 d3
Now considering R2 and F4, we can write:
Ra = R2 a2 + F4 d4
Since R is the resultant of R2 and F4 (i
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F1, F2, F3 and F4)
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(2
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Q
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A pair of two equal and unlike parallel forces (i
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forces equal in magnitude, with lines of action parallel to
each other and acting in opposite directions) is known as a couple
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e
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Mathematically:
Moment of a couple = P × a
where P = Magnitude of the force, and a = Arm of the couple
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Clockwise couple, and 2
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The algebraic sum of the forces, constituting the couple, is zero
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The algebraic sum of the moments of the forces, constituting the couple, about any point is the same, and
equal to the moment of the couple itself
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A couple cannot be balanced by a single force
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4
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of coplaner couples can be reduced to a single couple, whose magnitude will be equal to the
algebraic sum of the moments of all the couples
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18 how will you resolve a force into a couple and a force together
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In Fig
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Now it can be shown that F at A may be resolved into force F at B and a couple of magnitude M = F × d, where
d is the perpendicular distance of B from the line of action of F through A
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Hence the system of forces
in Fig
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(a)
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The system in Fig
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(c)
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Enumerate principle of equilibrium
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Such a set
of forces, whose resultant is zero, are called
equilibrium forces
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PRINCIPLES OF EQUILIBRIUM
Though there are many principles of equilibrium, yet the following three are important from the subject point of
view :
1
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As per this principle, if a body in equilibrium is acted upon by two forces, then they
must be equal, opposite and collinear
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Three force principle
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3
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As per this principle, if a body in equilibrium is acted upon by four forces, then the
resultant of any two forces must be equal, opposite and collinear with the resultant of the other two forces
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20 Explain condition of equilibrium
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A little consideration will show,
that as a result of these forces, the body may have any one of the following states:
1
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2
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3
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4
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Now we shall study the above mentioned four states one by one
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If the body moves in any direction, it means that there is a resultant force acting on it
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Or in other words, the horizontal component of all the forces (Σ H) and vertical
component of all the forces (ΣV) must be zero
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If the body rotates about itself, without moving, it means that there is a single resultant couple acting on it
with no resultant force
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Or in other words, the resultant moment of all the forces (Σ
M) must be zero
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If the body moves in any direction and at the same time it rotates about itself, if means that there is a resultant
force and also a resultant couple acting on it
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Or in other words, horizontal component of all the forces (Σ H), vertical component of all the
forces (Σ V) and resultant moment of all the forces (Σ M) must be zero
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If the body is completely at rest, it necessarily means that there is neither a resultant force nor a couple acting
on it
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21 state and prove LAMI’s Theorem
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” Mathematically,

where, P, Q, and R are three forces and α, β, γ are the angles as shown in Fig
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Let the opposite angles
to three forces be α , β and γ as shown in Fig
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We know that the resultant of two forces P and Q will be given
by the diagonal OC both in magnitude and direction of the parallelogram OACB
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From the geometry of the figure, we find
BC = P and AC = Q
Angle AOC = (180° –
β)
and Angle ACO = Angle BOC = (180° – α)

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Discuss various types of beam
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The distance
between two adjacent supports is called span
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The applied
loads make every cross-section to face bending and shearing
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The system of forces consisting of applied loads and reactions keep the beam in equilibrium
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•

Continuous beam: A beam is said to be continuous beam if it is resting
on more then two supports

•

Cantilever beam: If a beam is fixed at one end and other end of beam is
free in air such a beam is known as the cantilever beam
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•

Overhanging beam: a beam is said to be a over hanging beam if some portion of beam is hanged out
from any of the support
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Concentrated Loads: If a load is acting on a beam over a very small length, it is approximated as acting at the
mid point of that length and is represented by an arrow as shown in Fig
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It is
represented as shown in Fig
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For finding reaction, this load may be assumed as total load acting
at the centre of gravity of the loading (middle of the loaded length)
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, the
given load may be replaced by a 20 × 4 = 80 kN concentrated load acting at a distance 2 m from the left
support
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varies uniformly from C to D
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In the load diagram, the ordinate represents the load intensity and the abscissa represents the
position of load on the beam
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Thus, total load in this case is 1/2×3×20=30 kN and the
centre of gravity of this loading is at 1/3×3=1 m from D, i
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, 1 + 3 – = 3 m from A
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External Moment: A beam may be subjected to external moment atcertain
points
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9
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In this chapter the beams subjected to

concentrated loads, udl and external moments are dealt with
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Simply supported beams, 2
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Hinged beams
Q
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Shear Force at a section in a beam (or any structural member) is the force that is trying to shear
“
off the section and is obtained as algebraic sum of all the forces acting normal to the axis of beam
either to the left or to the right of the section”

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Q
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Classify it
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and is
capable of taking loads at joints
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Triangular truss is the simplest perfect
truss and it has three joints and three members
1
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Hence the following expression may be written down as the relationship between number
of joints j, and the number of member m, in a perfect truss
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imperfect truss- If a pin jointed truss which has more or less number of members to resist the loads

is called a imperfect truss
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Such trusses cannot retain their shape when loaded
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m >2j –
3
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In the theory that is going to be developed in this chapter, the following assumptions are made:
(1) The ends of the members are pin-connected (hinged);
(2) The loads act only at the joints;
(3) Self-weights of the members are negligible;
(4) Cross-section of the members is uniform
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METHOD OF SECTION
In the method of section, after determining the reactions, a section line is drawn passing through not
more than three members in which forces are not known such that the frame is cut into two separate
parts
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Equilibrium of any one of these two parts is considered
and the unknown forces in the members cut by the section line are determined
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Since there
are only three independent equation of equilibrium, there should be only three unknown forces
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Thus, the method of section is the application of non concurrent force system analysis whereas the method of
joints, described in previous article was the application of analysis of concurrent force system
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Following steps are followed in method of joint:Step 1: Determine the inclinations of all inclined members
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If such a joint is not available, determine the
reactions at the supports, and then at the supports these unknowns may reduce to only
two
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Hence, the unknown forces can be determined
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At the other end, mark the arrows in the reverse direction
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Step 6: Repeat steps 4 and 5 till forces in all the members are found
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Note that if the arrow marks on
a member are towards each other, then the member is in tension and if the arrow marks are away from each
other, the member is in compression
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28 Write steps to solve a truss by method of section
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29 Define Centroid & CG
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) have only areas, but no mass
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Centre of gravity can be defined as the point through which the resultant of force of gravity of the body
acts
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This term
is applicable to solids
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The central
points obtained for volumes, surfaces and line segments are termed as centroids
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30 Differentiate between Centroid & CG
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(2) Centre of gravity of a body is a point through which the resultant gravitational force (weight)
acts for any orientation of the body whereas centroid is a point in a line plane area volume
such that the
moment of area about any axis through that point is zero
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31 derive expression for Centroid of different sections by first principle
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4
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Height of the element at a distance x from O is y =
kx2

FIRST MOMENT OF AREA AND CENTROID
From equation, we have

From the above equation we can make the statement that distance of centre of gravity of a body
from an axis is obtained by dividing moment of the gravitational forces acting on the body, about the
axis, by the total weight of the body
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The moment of area ΣAix: is
termed as first moment of area also just to differentiate this from the term ΣAix2
i , which will be dealt
latter
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Q
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Moment of Inertia
The moment of a force (P) about a point, is the product of the force and perpendicular distance (x) between the
point and the line of action of the force (i
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P
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This moment is also called first moment of force
...
e
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x (x) = Px2, then this quantity is called moment of the moment of a force or second moment of force
or moment of inertia Sometimes, instead of force, area or mass of a
figure or body is taken into consideration
...
But all such second moments are broadly termed as moment of inertia
...
e
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,
1
...

2
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2
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Principal Moment of Inertia
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Polar Moment of Inertia
Moment of inertia about an axis perpendicular to the plane of an area is known as
polar moment of inertia
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Thus, the moment of inertia
about an axis perpendicular to the plane of the area at O in Fig
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Radius of gyration

Radius of gyration is a mathematical term defined by the relation

where k = radius of gyration,
I = moment of inertia,
and A = the cross-sectional area
Suffixes with moment of inertia I also accompany the term radius of gyration k
...
The relation between radius of gyration and moment of
inertia can be put in the form:
From the above relation a geometric meaning can be assigned to
the term ‘
radius of gyration
...
) such that there is no change in the moment
of inertia
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33 State & prove :1
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The moment of inertia of an area about an axis pependicular to its plane (polar moment of inertia) at
any point O is equal to the sum of moments of inertia about any two mutually perpendicular axis
through the same point O and lying in the plane of the area
...
, if z-z is the axis normal to the plane of
paper passing through point O, as per this theorem,
The above theorem can be easily proved
...
Let the
coordinates of dA be x and y
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Parallel axis theorem
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Referring to Fig
...

A = the area of the plane figure given and
yc = the distance between the axis AB and the parallel centroidal axis GG
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33 Derive expression for Moment of Inertia of different sections
...
Moment of inertia
of these sections about an axis can be found by the following steps:
(1) Divide the given figure into a number of simple figures
...

(3) Find the moment of inertia of each simple figure about its centroidal axis
...
This gives moment of inertia of the simple figure about the

reference axis
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Q
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Ans: Dynamics is the part of mechanics that deals with analysis of bodies in motion
...
Kinematics is the study of the relationship between displacement
velocity and acceleration without considering the forces
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Q
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Ans: Dynamic equilibrium or D’Alembert principle based on the Newton’s second law of motion
∑F= ma , which can also be written in form of ∑F- ma=0, which means that resultant of external forces(∑F) and
the force (-ma) is 0
...
The inertia of a body can be defined as the
resistant to change in the condition of rest or of uniform motion of the body
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36 Discuss Type of motion
...

b) Rotary motion: It is divided into centroidal and non centroidal motion
c) Plane motion: It is combination of translatory motion and rotary motion
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37 Write down equation of rectilinear motion

1
...
S=ut+1/2at
2
2
3
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Trigonometry
The measurement of the triangle sides and angles is called trigonometry
...
For example mass, time, volume density,
temperature, length, age and area etc
Vector quantity
Vector quantity is that quantity, which has magnitude unit of magnitude as well as direction, is called vector
quantity
...
For example velocity, acceleration, force, weight, displacement,
momentum and torque etc
are all vector quantities
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