Resultant of forces

1. Part 1

2. Engineering Mechanics
Mechanics of Solids Mechanics of Fluids
Rigid Bodies Deformable
Bodies
Statics Dynamics
Kinematics Kinetics
Strength of
Materials
Theory of
Elasticity
Theory of Plasticity
Ideal Viscous
Fluids Fluids
Compressible
Fluids
Branches of Mechanics

3.  Definition of Mechanics :
In its broadest sense the term ‘Mechanics’ may be defined as the
‘Science which describes and predicts the conditions of rest or
motion of bodies under the action of forces’.
Mechanics is the physical science concerned with the behavior of
bodies that are acted upon by forces.

4. Statics is the study which deals with the condition of bodies in
equilibrium subjected to external forces.
In other words, when the force system acting on a body is
balanced, the system has no external effect on the body, the body
is in equilibrium.

5.  Dynamics is also a branch of mechanics in which the forces
and their effects on the bodies in motion are studied. Dynamics
is sub-divided into two parts: (1) Kinematics and (2) Kinetics
 Kinematics deals with the geometry of motion of bodies without
and application of external forces.
 Kinetics deals with the motion of bodies with the application of
external forces.

6.  Concept of Rigid Body :
It is defined as a definite amount of matter the parts of
which are fixed in position relative to one another under the
application of load.
Actually solid bodies are never rigid; they deform under the
action of applied forces. In those cases where this deformation is
negligible compared to the size of the body, the body may be
considered to be rigid.

7.  Force
It is that agent which causes or tends to cause, changes or tends to
change the state of rest or of motion of a mass.
A force is fully defined only when the following four characteristics
are known:
(i) Magnitude
(ii) Direction
(iii) Point of application
(iv) Sense.

8.  Example: Characteristics of the force 100 kN are :
(i) Magnitude = 100 kN
(ii) Direction = at an inclination of 300 to the x-axis
(iii) Point of application = at point Ashown
(iv) Sense = towards pointA
300
100 kN
A

9.  It is stated as follows : ‘The external effect of a force on a
rigid body is the same for all points of application along its line
of action’.
P P
A B
For example, consider the above figure. The motion of the block will be the
same if a force of magnitude P is applied as a push at Aor as a pull at B.
P P
O
The same is true when the force is applied at a point O.

10. Force system
Coplanar Forces Non-Coplanar
Forces
Concurrent Non-concurrent
Concurrent Non-concurrent
Like parallel Unlike parallel
Like parallel Unlike parallel

11.  A coplanar force system consists of forces that lie in the same
plane.
 These forces can be concurrent or non-concurrent.
 Coplanar force systems are commonly encountered in two-
dimensional problems and are often analyzed using methods
such as vector addition or graphical techniques like the
method of polygons.

12.  In a concurrent force system, all the forces have their lines of
action intersecting at a single point.
 The resultant force of a concurrent force system can be
determined by finding the vector sum of all the individual
forces.
 Examples of concurrent force systems include the forces acting
on a pin-jointed structure or the forces exerted by a group of
ropes attached to a common point.

13.  In a non-concurrent force system, the lines of action of the
forces do not intersect at a single point.
 This type of force system requires additional considerations
such as moments and couples to determine the overall effect
on the object.
 An example of a non-concurrent force system is the forces
acting on a beam or a truss structure.

14.  Resultant, R : It is defined as that single force which can
replace a set of forces, in a force system, and cause the same
external effect.
R

=
R  F1  F2  F3
external effect on particle,A is same
F3
F1
F2
A A

15.  Resultant of two forces acting at a point
Parallelogram law of forces : ‘If two forces acting at a point are
represented in magnitude and direction by the two adjacent sides
of a parallelogram, then the resultant of these two forces is
represented in magnitude and direction by the diagonal of the
parallelogram passing through the same point.’
B
C
A
O
P2
P1
R



16. In the above figure, P1 and P2, represented by the sides OA and
OB have R as their resultant represented by the diagonal OC of
the parallelogram OACB.
It can be shown that the magnitude of the resultant is given by:
R = P1
2 + P2
2 + 2P1P2Cos α
Inclination of the resultant w.r.t. the force P1 is given by:
 = tan-1 [( P2 Sin ) / ( P1 + P2 Cos  )]
B C
A
O
P2
P1
R



17.  Resultant of two forces acting at a point at right angle
B C
A
O
P2
P1
R


If α = 900 , (two forces acting at a point
are at right angle)

C
A
B
P2
O
P1
R

2
2
2
1
R  P  P
tan 
P2
P1

18.  Triangle law of forces
If two forces acting at a point can be represented both in
magnitude and direction, by the two sides of a triangle taken in
tip to tail order, the third side of the triangle represents
both in magnitude and direction the resultant force F, the sense
of the same is defined by its tail at the tail of the first force and its
tip at the tip of the second force’.

19. 
R

F1
F2
θ
A
F1
F2
θ
F1
F2
R
=
(180 -  - ) = θ

F1 F2

R
sin  sin sin(180  )
where α and β are the angles made by the resultant force
with the force F1 and F2 respectively.

20.  Component of a force
Component of a force, in simple terms, is the effect of a force in a
certain direction. A force can be split into infinite number of
components along infinite directions.
Usually, a force is split into two mutually perpendicular
components, one along the x-direction and the other along y-
direction (generally horizontal and vertical, respectively).
Such components that are mutually perpendicular are called
‘Rectangular Components’.
The process of obtaining the components of a force is called
‘Resolution of a force’.

21.  Rectangular component of a force
F
x
F
x
F
Fx
y = Fy
F
x
Fx
Consider a force F making an angle θx with x-axis.
Then the resolved part of the force F along x-axis is given by
Fx = F cos θx
The resolved part of the force F along y-axis is given by
Fy = F sin θx

22.  Oblique component of a force

F

F1
F2


F
F1
F2
M
Theresolved partof the force F along OM and ON can
obtained by using the equation of a triangle.
F1 / Sin  = F2 / Sin  = F / Sin(180 -  - )
Let F1 and F2 be the oblique components of a
force F. The components F1 and F2 can be
found using the ‘triangle law of forces’.

23. x
y
+ve
y
+ve
x
The adjacent diagram gives the sign convention for
force components, i.e., force components that are directed along
positive x-direction are taken +ve for summation along the x-
direction. Also force components that are directed along +ve y-
direction are taken +ve for summation along the y-direction.
 Sign Convention for force components: