Outline of coplanar concurrent forces with its equilibrium conditions, resultant forces and Lami's theorem.

1. Mechanical – 3B1_GROUP NO. 3
COPLANAR CONCURRENT FORCES

2. CONTENTS : -
 Introduction
 Equilibrium Conditions
 Resultant of Coplanar Concurrent Forces System
1. Analytical Method
2. Graphical Method
 Lami’s Theorm
Type equation here.

3. INTRODUCTION : -
 DEFINITION : -
Coplanar Concurrent Forces :-
When the acting forces line of action lies in a same
plane and meeting at a point to each other, said to
be Coplanar Concurrent Forces.

4. EQUILIBRIUM CONDITIONS : -
 If a system of forces acting on a body, keeps the body in
a state of rest or in a state of uniform motion, then the
system of forces is said to be in equilibrium.
 Alternatively, we can say that the resultant force is zero,
the system of forces will be in equilibrium.

5. 1) The Coplanar Concurrent Forece is said to be in equilibrium
condition if it satisfies following conditions :-
2) Algebraic sum of components of all the forces of the system,
along two mutually perpendicular directions, is ZERO.
H = 0 and V = 0

6. RESULTANT OF COPLANAR CONCURRENT
FORCES SYSTEM : -
 A resultant force is the
single force and associated torque
obtained by combining a system
of forces and torques acting on a
rigid body.
 The defining feature of a resultant
force is that it has the same effect
on the rigid body as the other
original system of forces.

7. The Methods to find out Resultant Force : -
Methods Of
Resultant
force
Analytical
Methods
Parallelogram
law of Force
Resolution of
Forces
Triangle law
of Forces
Graphical
Methods
Polygon law
of Forces
Triangle law
of Forces

8. PARALLELOGRAM LAW OF FORCES : -
If two forces, acting at a point, are
represented in magnitude and
direction by the two sides of a
parallelogram drawn from one of its
angular points, their resultant is
represented both in magnitude and
direction by the diagonal of the
parallelogram passing through that
angular point.

9.  Assume two forces P and Q on partical at point O,
 As shown in figure,
𝑅 = 𝑃2 + 𝑄2 + 2𝑃𝑄 cos 𝜃
tan 𝛼 =
𝑄𝑠𝑖𝑛𝜃
𝑃+𝑄𝑐𝑜𝑠𝜃
Where,
R = resultant force
𝜃 = Angle between P & Q
𝛼 = Angle between P & R

10. RESOLUTION OF FORCES : -
 The process of splitting up the given force in two or more
components, in particular direction, without changing
effect on the body is called resolution of a force.
 Generally a given force is split up in two manually
perpendicular force components,
1. Horizontal components ( x-components )
2. Vertical components ( y-components ).

11. Here,
For force R,
Horizontal component = Rcos𝜃
Vertical component = Rsin𝜃

12. TRIANGLE LAW OF FORCES : -
 If two forces acting on a body, it can represented by two
adjacent sides of triangle with magnitude and directions
and resultant can be given by other side of triangle; which
is the statement of law of triangle.
 Assume that,
we have a pair of forces acting on a point than Triangle
law of forces can be applied as,

13.  For image,
 Resultant can be given by,
R = 𝑃2 + 𝑄2 − 2𝑃𝑄𝑐𝑜𝑠𝛽.
 The direction of the resultant can
be given by,
 = 𝑠𝑖𝑛−1
(
𝑄
𝑅
sin 𝛽 ).

14.  Graphically,
Triangle Law of Forces can be represented by taking two
forces at two adjacent sides of triangle, which are acting
on a body or particle and the resultant force can be
represented by the other side of the triangle.
The angle can be also obtained from figure by using
specific instrument.

15. POLYGON LAW OF FORCES : -
 If a number of forces acting at
a point be represented in
magnitude and direction by the
sides of polygon taken in
order, then the resultant of all
these forces may be
represented in magnitude and
direction by the closing side of
polygon taken in opposite
order.

16.  Graphically ,
It can be represented by
making a polygon as shown
as shown in figure.

17. GRAPHICAL METHOD : -
 Resultant of Coplanar Concurrent Forces can be
graphically represented by using Space diagram and
Vector diagram.
 Space Diagram : - If a force acting in a system are
represented by direction in the form of diagram by
denoting every force with specific character , the diagram
is known as space diagram.
 Vector Diagram : - A space diagram of coplanar
concurrent forces with magnitude and direction of all
forces, the diagram is known as vector diagram.

18. Space Diagram Vector Diagram

19. LAMI’S THEORM : -
 If three coplanar forces acting at a
point be in equilibrium , then each
force is proportional to the sine of
angle between other two sides.
 Applying Lami’s theorm on given
figure, we have;
𝐴
sin 𝛼
=
𝐵
sin 𝛽
=
𝐶
sin 𝛾