1. Applied Mechanics, Statics and Dynamics. Importance of Study.

Applied mechanics (also engineering mechanics) is a branch of the physical sciences and the practical
application of mechanics. Applied mechanics describes the response of bodies (solids and fluids) or
systems of bodies to external forces. Some examples of mechanical systems include the flow of
a liquid under pressure, the fracture of a solid from an applied force, or the vibration of an ear in response
to sound.
The branch of science which deals with study of different laws of mechanics as applied to the
solution of engineering problems is called APPLIED MECHANCIS.

2. Statics and Dynamics:

As you know the word static means that the things which is at rest and if we talk of it in mechanics then
mechanics deals with the force so static mechanics deals with a forces and there effect while acting upon
the body at rest. In same way if we talk of dynamics mechanics then we deals with a forces and there
effect while acting upon body in motion.

static condition -e.g suppose you are sitting on chair and you ask to leave the chair from one place to
another ,you do it easily but you ask to leave same chair from one place to another without touching your
fit to ground it will not possible to move. This is because to move your body we have to apply external
force to it .
dynamic force: moving cars

Mechanics can also be divided as Mechanics of Rigid Bodies and Mechanics of deformable bodies.

Rigid Body: bodies which do not change their size and shape under the effect of forces acting over them.
In the real world no solid body is PERFECTLY RIGID as everybody change its size and shape under the
effect of force but many times the deformation is negligible enough for the body to be considered Rigid.
Deformable Bodies : Bodies which undergoes deformation in size and shape under the effect of forces
acting over them.
2. Forces, Definition, Effects, Different Systems, Principle of Transmissibility and Superimposition
of Forces. Resolution and Composition of Forces.

FORCE-

• According to Newton's first law Force is an external agent which changes or tend to change
the state of rest or motion of a body .Force is represented in magnitude and direction so it
vector quantity.
• Scalar :only magnitude ,vector: both magnitude and direction.
• Pull force:- force acting away from the body .
• Push force:-Force acting towards the body.

Pull force Push Force

Due to application of force internal stresses will induces in body. where as P is the force and R is the
resistance which is equal and opposite to force P.

Characteristics of force:

1) A force has magnitude measured in KN or 1KN =1000KN

2) A force has direction and direction is measured in angle .Angle measured with the horizontal zero
degree in the standard co-ordinate system.
3)A force has a point of application.
4)A force has a sense of pull or push.
Unit of force:

-SI unit of force "Newton(N)"

Effect of force:
1. A force may causes a body to change its state of motion.
2.A body in motion can brought to rest.
Representation of Force:
Vector presentation:
A force can be represented graphically by a vector as shown above.Hence the force is
Represente by Force AB of 60 N @ 45° upward.
Bow's Notation:
A force can be designated by two capital letters written one on either side of the force as shown above. So
force P1 is Force CD and Force P1 Force AB

SYSTEM OF FORCES:-
When number of forces act simultaneously on the object it is known as system of forces.

A)Coplanar force -Forces act in same plane.

e.g.
Plane can either be xy,xz or yz.
B) Non coplanar force -Forces act in different plane. e.g flying kite

C) Parallel forces-two types of parallel forces-

• Like parallel forces- Line of action of forces are parallel to each other and have same direction.

Eg. Bullock cart ,People sitting on bed.

• Unlike parallel forces- Line of action of forces are parallel to each other have different
direction.

E.g. Bullock cart used to extract sugar cane juice.

D)Collinear forces-When the line action of forces act along same line.E.g.Two people standing at
opposite end of rope and pulling it .Suspended load by cable.

E)Non collinear forces-When the line of action of forces does not act along same line . E.g. see saw.

F)Concurrent force-When line of action forces pass through common point .It may be pull or push
forces.E.g. truss
G)Non concurrent force -When line of action of forces does not pass through common point or varies.

Principle of superposition -

The principle state that when two or more forces act on a particle at the same time, the resultant
force is the vector sum of the two.

eg. If force P moves a body in a the north direction by1 m and together force Q moves it by 2m in the
same direction when applied independently ,when P and Q are applied together the body would move by
5 m in the north direction.
When single force acts on body which is free to move ,the body moves in the direction of the force and
the distance travelled by the body in unit time directly proportional to the magnitude of the force .So
when many forces are acting on the body, the body moves in a direction of the resultant and the distance
travelled is proportional to the magnitude of the resultant.

Principle of transmissibility
Point of application of force on rigid body can be change along the same line of action by maintaining
the magnitude and direction without affecting the effect of force on body .
E.g. The person pushing the car with 100kn force will causes the same effect by person pulling the
car by 100 kn force.

Resultant force- When a number of coplanar forces are acting on a rigid* body, then these forces can be
replaced by a single force which has the same effect on the rigid body as that of all the forces acting together, then
this single force is known as the resultant of several forces. Hence a single force which can replace a number of
forces acting on a rigid body, without causing any change in the external effects on the body, is known as the
resultant force.or For the system of forces, when single force represents the same effect of the forces,
then it is resultant force.

a)collinear forces-
b)concurrent forces-

R1 cos ɵ1+R2 cosɵ2+R3 cosɵ3=R

c)non- concurrent forces-

P1+P2 cosɵ1+P2 cos ɵ2=R

COMPOSITION OF FORCE:

A resultant force is a single force which can replace two or more forces and produce the same effect on
the body as the forces. Many forces can be composed into one single resultant Force and this is
known as composition of forces.

Equilibrant Force(E)- Equilibrant force is a force which brings the body in equilibrium state.
It is considered to be the equal, opposite and collinear of the resultant force.

Law of parallelogram of forces

If two forces acting at a point are represented in magnitude and direction by the two adjacent sides of a
parallelogram, then their resultant is represented in magnitude and direction by the diagonal passing
through the same point.

Consider two forces Vector P and Vector Q acting at a point O inclined at an angle θ and represented
in magnitude and direction by the sides OA and OB of a parallelogram OACB as shown in Fig.
The resultant Vector R is the diagonal OC of the parallelogram. The magnitude of the resultant is
Triangular law of force-
The resultant of two forces acting at a point can also be found by using Triangle Law of Forces.
If two forces acting at a point are represented in magnitude and direction by the two adjacent sides of a
triangle taken in order, then the closing side of the triangle taken in the reversed order represents the
resultant of the forces in magnitude and direction

forces P and Q act at an angle ɵ. In order to find the resultant of P and Q. one can apply the head to tail
method, to construct the triangle.

In Figure OA and AB represent P and Q in magnitude and direction. The closing side OB of the triangle
taken in the reversed order represents the resultant R of the forces P and Q. The magnitude and the
direction of R can be found by using sine and cosine laws of triangles.
If P, Q and R are the three forces acting at a point and they are represented by the three sides of a triangle
then, P/OA = Q/AB = R/OB
Polygonal law of forces-
If a number of forces acting at a point be represented in magnitude and direction by the sides of a polygon
in order, then the resultant of all these forces may be represented in magnitude and direction by the
closing side of the polygon taken in opposite order

The figure illustrates five forces A, B, C, D and E, with arrows indicating their directions. The next figure
provides the resultant of the five forces by R, which is the closing side of the polygon, but in the opposite
direction shown by the arrowheads.

LAMI’S THEOREM.-
Lami’s theorem states that if three forces acting at a point are in equilibrium, each force is proportional to
the sine of the angle between the other two forces.

Lami’s theorem is an equation that relates the magnitudes of three coplanar, concurrent and non-collinear
forces, that keeps a body in static equilibrium.
Consider three forces A, B, C acting on a particle or rigid body making angles α, β and γ with each other.

MOMENT:
The moment of a force about a point O is the cross product of r and F, where F is the force and r is the
position vector of any point on the line of action of force (OR perpendicular distance from the point )with
respect to O.
 M=r × F

Just as force has a tendency to translate the body, moment has a tendency to rotate the body about the
point.
Our sign convention will be as follows. If the force has tendency to rotate the body in counterclockwise
sense, moment is considered +ve. If the force has tendency to rotate the body in clockwise sense ,moment
is considered -ve.

UNIT: N-M ...................... (Force in N and Distance in M)

COUPLE: When two equal opposite forces acting on body which are equal in magnitude but not through
the same point so they produce a turning effect ,these two unlike ,parallel ,equal and non collinear forces
form as couple.
The moment (or torque) of a couple is calculated by multiplying the size of one of the force (F) by the
perpendicular distance between the two forces (s).
E.g. a steering wheel in a car;

Characteristics of couple:
1. The algebraic sum of the forces, having the couple, is zero.
2. The algebraic sum of moment of the forces, constituting couple, about any point is the same, and equal
to the moment of couple itself.
3. A couple can't be balanced by a single force, but can be balanced only by a couple, however of
opposite sense.
4. Any number of coplanar couples can be reduced to single couple, whose magnitude will be equal to
algebraic sum of moments of all the couples.

PRINCIPLE OF RESOLUTION OF FORCE:

As many forces can be composed into one single resultant ,so can a single force be replaced by two forces
acting in direction s which will produced the same effect as the single force .this breaking of the force into
two forces is called the resolution.
Algebraic sum of components of all forces along any axis must be equal to component of its resultant.

A force can be resolved into

1.Two mutually perpendicular components
2.Two non -perpendicular components.
When the force is resolved into two mutually perpendicular components ,generally the two components
are horizontal and vertical. the horizontal component is denoted by FH and vertical components is
denoted by Fv.

As seen in following figure Fh(Horizontal Component )= FCosθ = F x Horizontal Distance

Hypotenuse
Fv (Vertical component ) = FSin θ =F x Vertical distance
Hypotenuse
Composition of vertical and horizontal Componet:
R= √FV²+FH² and to get the direction we use tanθ=FV/FH.

Varignon's theorem
Moment of a force about any point is equal to the sum of the moments of the components of that force
about the same point.
To prove this theorem, consider the force R acting in the plane of the body shown in Fig. at a distance x 2
from A. The forces P1 &P2 represent any three forces acting at distance x1 and x3.The moment
of R about point A is

Ma = R x X2 = (P1xX1) - (P2xX3)

Varignon's theorem need not be restricted to the case of two components, but it applies equally well to
three or more. where we take the clockwise moment sense to be positive and anticlockwise moment
negative .
Resultant of Concurrent Coplanar Forces. -concurrent coplanar forces are those forces which act in the
same plane and they intersect or meet at a common point.

(a) Analytical method:

The resultant of three or more forces acting at a point is found analytically by a method which is known
as rectangular components methods (Refer to Art. 1.8.1). According to this method all the forces acting at
a point are resolved into horizontal and vertical components and then algebraic summation** of
horizontal and vertical components is done separately. The summation of horizontal component is written
as ΣH and that of vertical as ΣV. Then resultant R is given by
The angle made by the resultant with horizontal is given by
tan θ = ( ΣV / ΣH)
Let four forces F1 F2, F3 and F4 act at a point 0 as shown in Fig. 1.44.

The inclination of the forces is indicated with respect to horizontal direction. Let
θ1 =Inclination of force F1 with OX
θ2 = Inclination of force F2 with OX’
θ3 = Inclination of force F3 with OX’
θ4 =Inclination of force F4 with OX.
The force F 1 is resolved into horizontal and vertical components and these components are shown in Fig.
1.44(a). Similarly, Figs. 1.44(b), (c) and (d) show the horizontal and vertical components of forces F2,
F3 and F4 respectively. The various horizontal components are:
F1 cos θ1 à (+)
F2 cos θ 2 à (-)
F3 cos θ 3 à (-)
F4 cos θ4 à (+)
Summation or algebraic sum of horizontal components :
ΣH= F1 cos θ1 -F2 cos θ2 -F3 cos θ3 + F4 cos θ4
Summation or algebraic sum of vertical components:
ΣV = F1 sin θ1 + F2 sin θ2 -F3 sin θ3 -F4 sin θ4
And the angle (θ) made by resultant with x-axis is given by tan θ = (ΣH / ΣH)
b) Graphical method. -
The resultant of several forces acting at a point is found graphically with the help of the polygon law of
forces, which may be stated as
“If a number of coplanar forces are acting at a point such that they can be represented in magnitude and
direction by the sides of a polygon taken in the same order, then their resultant is represented in
magnitude and direction by the closing side of the polygon taken in the opposite order.

Let the four forces F1, F2 , F 3 and F4 act at a point 0 as shown in Fig. 1.45. The resultant 1s obtained
graphically by drawing polygon of forces as explained below and shown in Fig. 1.45(a).

(i) Choose a suitable scale to represent the given forces.

(ii) Take any point a. From a, draw vector ab parallel to force OF1. Cut ab =force F 1 to the scale.
(iii) From point b, draw be parallel to OF2. Cut be= force F2:
(iv) From point C, draw cd parallel to OF3. Cut cd =force F3.
(v) From point d, draw de parallel to OF4 Cut de= force F4 .
(vi) Join point a to e. This is the closing side of the polygon. Hence are represents the resultant in
magnitude and direction.

Magnitude of resultant R = Length ae x scale.

The resultant is acting from a to e.