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1
EQUILIBRIUM OF CONCURRENT COPLANAR FORCE SYSTEMS
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CONTENTS 1 BASIC CONCEPTS (a) Definition
Conditions for Equilibrium (b) Space
Diagram Free Body Diagram (FBD) (c) A few
guidelines for drawing FBD 2 NUMERICAL
PROBLEMS (a) Solved Examples (b)
Exercise
2
EQUILIBRIUM OF CONCURRENT COPLANAR FORCE SYSTEMS
T1
Definition-
If a system of forces acting on a body, keeps the
body in a state of rest or in a state of uniform
motion along a straightline, then the system of
forces is said to be in equilibrium. 
ALTERNATIVELY, if the resultant of the force
system is zero, then, the force system is said to
be in equilibrium.
3
Conditions for Equilibrium of Concurrent
Coplanar Force System
T2
A coplanar concurrent force system will be in
equilibrium if it satisfies the following two
conditions
i) ? Fx 0 and ii) ? Fy 0
i.e. Algebraic sum of components of all the
forces of the system, along two mutually
perpendicular directions, is ZERO.
4
Graphical conditions for Equilibrium
T3
Triangle Law If three forces are in equilibrium,
then, they form a closed triangle when
represented in a Tip to Tail arrangement, as
shown in Fig 1.(a).
Polygonal Law If more than three forces are in
equilibrium, then, they form a closed polygon
when represented in a Tip to Tail arrangement, as
shown in Fig 1.(b).
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LAMIS THEOREM
T4
If a system of Three forces is in equilibrium,
then, each force of the system is proportional to
sine of the angle between the other two forces
(and constant of proportionality is the same for
all the forces). Thus, with reference to Fig(2),
we have,
a
F3
F2
?

Note While using Lamis theorem, all the three
forces should be either directed away or all
directed towards the point of concurrence.
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T5
SPACE DIAGRAMS FREE BODY DIAGRAMS
 Space Diagram(SPD)The sketch showing the
physical conditions of the problem, like, the
nature of supports provided size, shape and
location of various bodies forces applied on
the bodies, etc., is known as space diagram.
Eg. Fig 3(a) is a space diagram
Fig 3 (a) SPD
Weight of sphere 0.5 kN
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T6
Free Body Diagram(FBD) It is an isolated
diagram of the body being analyzed (called free
body), in which, the body is shown freed from all
its supports and contacting bodies/surfaces.
Instead of the supports and contacting
bodies/surfaces, the reactive forces exerted by
them on the free body is shown, along with all
other applied forces.
Eg. Fig 3(b) is a Free Body Diagram of Fig 3(a).
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T6
Note Free Body Diagrams should be NEAT, LEGIBLE
SUFFICIENTLY BIG. Only the details required
for the analysis of the problem are to be shown.
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A Few Guidelines for Drawing FBD
T7
1)      Tensile Force It is a force trying to
pull or extend the body. It is represented by a
vector directed away from the body. 2)     
Compressive Force It is force trying to push or
contract the body. It is represented by a vector
directed towards the body. 3)      Reactions at
smooth surfaces The reactions of smooth
surfaces, like walls, floors, Inclined planes,
etc. will be normal to the surface and pointing
towards the body. 4)      Forces in Link
rods/connecting rods These forces will be acting
along the axis of the rod, either towards or away
from the body. (They are either compressive or
tensile in nature).
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T8
5) Forces in Cables (Strings or Chords) These
can only be tensile forces. Thus, these forces
will be along the cable and directed away from
the body. 6) Tension in cables on either side
of a smooth pulley will be equal in magnitude.
(Eg. As shown in Fig)
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NUMERICAL EXAMPLES
P (1)
 (1) A sphere of 100N weight is tied to a wall
by a string as shown in fig (1). Find the tension
in the string and the reaction of the wall.
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P (2)
 Using Lamis theorem,
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P (3)
 (2) Determine the magnitude and nature of the
forces in the bars AB and AC shown in Fig (2).
Neglect size and weight of the pulley.
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(P4)
 
Angle between FAB and FAC 90º
Taking FAC as X-axis and FAB as Y axis


15
(P5)
 
FAC is ve , FAC is towards A, So it is
Compressive.
14.64kN
FAB is ve. FAB is towards A, So it is Tensile.

16
P (6)
 (3) Two cylinders A B of weight 400N and 200N
respectively, rest on smooth planes as shown in
Fig(3). Find the force P required for
equilibrium.
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P (7)
The forces in the system are as shown.
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P(8) FBD

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P(9)
Considering FBD of A and Using Lamis theorem,

20
P (10)
Considering FBD of B, We have,
 
-------Eqn(1)


-----------------Eqn(2)
Adding Eqn(1) and Eqn(2), We get,

21
P (11)
 (4) Two cylinders P and Q of diameters 100mm and
50mm, weighing 200N and 50N respectively, are
placed in a trench as shown in Fig (4). Assuming
smooth surfaces determine reactions at all
contact points.
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P (12)
Cos a 45/75 a 53.130
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P (13) FBD
a 53.130
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P (14)
Considering FBD of Q and Using Lamis theorem,
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P (15)
Considering FBD of P, We have,
 



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P (16)
 (5) Three cylinders A , B and C of diameters
500mm each are arranged as shown in Fig (5). The
weights of A and B are 500N each and weight of C
is 600N. Determine reactions at all contact
points and tension in the string holding A and B.
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P (17)
From Fig(5a), AC 250 250 500mm AO 800/2
400mm Cos a 400/500 a 36.87 0

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P (18) FBD
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P (19)
Considering FBD of C and Using Lamis theorem,

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P (20)
Considering FBD of A, we have,
 
By Symmetry, RP RQ 800 N (Or, Using FBD of
B, RQ 800 N)
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P (21)
 (6) Two Spheres, each of radius 1m and mass 1000
kg, rest on smooth surfaces as shown in Fig (6) .
Determine reactions at all contact points.
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P (22)
Weight of Spheres Mass x g 1000 x 9.81
9810 N 9.81 kN
Reaction between A B will be parallel to Plane
QR, as radius is the same.
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P (23) FBD
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P (24)
Considering FBD of B, We have,


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P (25)
Considering FBD of A, we have,
 

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P (26)
 (7) Two Spheres A B, weighing 300N 600N and
having diameters 800mm 1200mm, respectively,
rest on smooth surfaces as shown in Fig (7) .
Determine reactions at all contact points.
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P (27)
AC parallel to plane AB radial line. Sin ?
BC/AB Sin ? 200/1000 11.53 0
Force between A B will be at an angle of
11.53º to axis parallel to Plane QR.
( Radii are NOT the same in this case.)
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P (28) FBD
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P (29)
Considering FBD of B, We have,
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P (30)
Considering FBD of A, We have,
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P (31)
(8) A roller of radius 300mm, weighing 5kN is to
be pulled over a kerb of height 150mm, by
applying a horizontal force P applied at the
circumference by means of a rope wrapped around
it, as shown in Fig (8). Find (i) The magnitude
of force P when it is horizontal (ii) The
direction and magnitude of the least force P
required to pull the roller over the kerb.
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P (32)
NOTE When the roller is about to roll over the
kerb, it loses contact at B. So, there will be no
reaction at B. There will be a reaction at A
,(say, RA), only.
Therefore at the instant of rolling over the
kerb, there will be only 3 forces in equilibrium,
viz., Weight (W), P RA.
Since, there are only 3 forces in equilibrium,
they will be Concurrent.
Thus reaction at A passes through C, point of
concurrence of W P.
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P (33)
In Fig (8), From Triangle ADO,
From Triangle ACD,
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P (34)
CASE (i) Force P is Horizontal In Fig (8A),
FBD, using Lamis Theorem,
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P (35)
CASE (ii) Force P is Least. (at angle a w r t
Hz.) In Fig (8B), FBD, using Lamis Theorem,
For P to be min., Sin(120- a)1, or, a 30
Thus,
NOTE For min. value P is at right angles to RA.
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P (36)
(9) Determine, the tension in the strings AB, BC,
CD and inclination of the segment CD to the
vertical, in the system shown in Fig (9).
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P (37) FBD
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P (38)
Considering FBD of Joint B and Using Lamis
theorem,
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P (39)
Considering FBD of Joint C, We have,

Dividing Eqn(1) by (2), we get,
(NOTE For this FBD, if we use Lamis Theorem,we
have to expand Sin(50?) and solve for ?, which
can take more time.)
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P (40)
(10) A wire is fixed at two points A and D as
shown in Fig (10). Determine inclination of the
segment BC to the vertical and the tension in all
the segments.
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P (41) FBD
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P (42)
Considering FBD of Joint B and Using Lamis
theorem,
---(1)
---(2)
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P (43)
Considering FBD of Joint C and Using Lamis
theorem,
---------Eqn(3)

---------Eqn(4)
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P(44)
Equating R.H.S. of Eqns (1) and (3), we get,
(Continued in next slide)
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P(45)
(Continuation)
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EXERCISE PROBLEMS
E 1
1 A 10kN roller rests on a smooth horizontal
floor and is held by the bar AC as shown in
Fig(1). Determine the magnitude and nature of the
force in the bar AC and reaction from the floor
under the action of the forces applied on the
roller. AnsFAC0.058 kN(T),R14.98 kN
7kN
C
45
5kN
A
30
Fig(1)
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E 2
2 A 1kN roller resting on a smooth incline as
shown in Fig (2) is held by
a cable. If the tension in the cable is limited
to 0.518kN, determine the maximum inclination to
which the plane can be raised. Ans ? 300 wrt
Hz.
15
?
Fig (2)
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E 3
3 A 10 kN weight is suspended from a rope as
shown in Fig(3). Determine the magnitude and
direction of the least force P required to pull
the rope, so that, the weight is shifted
horizontally by 0.5m. Also, determine, tension in
the rope in its new position. Ans P 2.43 kN,
? 14.480 T 9.7kN.
2m
P
?
Fig(3).
10kN
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E 4
4 Three spheres A, B, C of diameters, 500mm,
500mm, 800mm and weighing 4kN, 4kN, 8kN,
respectively, are placed in a trench as shown in
Fig(4). Find the reactions at all contact points.
Ans FAC4.62kN, RA1 2.46kN, RA2 7.16kN( )
FBC?, RB1?, RB2? ( )
C
B
A
70
70
650 mm
Fig(4).
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E 5
5 Three cylinders A, B, C of diameters, 200mm,
200mm, 100mm and weighing 400N, 400N, 200N,
respectively, are placed in a trench as shown in
Fig(5). Find the reactions at all contact points.
Ans FAB257.11N, FAC162.50N,
RA1 459.62N, RA2 460.06N, RB306.42N, RC
182.72N.
C
B
A
50
40
Fig(5).
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E 6
6 Two rollers A and B of same diameter and
weight 1000N, 600N, respectively, interconnected
by a light weight rod are placed on smooth planes
as shown in Fig(6). Determine the inclination ?
of the rod and the reaction of the planes.
Ans ? 23.410,RA RB 923.7 N
B
?
A
30
30
Fig(6).
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E 7
7 Determine the value of P and the nature of the
forces in the bars for equilibrium of the system
shown in Fig(7). Ans P 3.04 kN, Forces in
bars are Compressive.
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45
45
75
P
2kN
Fig(7).
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E 8
8 A cable fixed as shown in Fig(8), supports
three loads. Determine the value of the load W
and the inclination of the segment BC. Ans
W25kN, ? 54.780
A
D
30
B
60
?
C
22.5
20
Loads are in kN
W
Fig(8)