The document discusses the equilibrium of rigid bodies. It provides the following key points: 1) A rigid body is in equilibrium when the external forces acting on it combine to form a system equivalent to zero force and zero couple. 2) The necessary and sufficient conditions for equilibrium are that the net force F and net moment M must both equal zero. 3) To analyze equilibrium, a free-body diagram must be drawn showing all external forces acting on the body. Examples of equilibrium under single, two, and three forces are provided.

1. Equilibrium of Rigid Bodies
Dr. S. V. Chaudhari

2. • We know that the external forces acting on a rigid body can be
reduced to a force-couple system at any point O. When the force and
the couple are both equal to zero, the external forces form a system
equivalent to zero, and the rigid body is said to be in equilibrium.
• That means the external forces combinedly nullify their effect
• In other words the forces are balancing themselves

3. • The necessary and sufficient conditions for the equilibrium of a rigid
body:
• 𝐹 = 0, and 𝑀 = 0
• Resolving each force and each moment into its rectangular
components – we obtain
• For the system of plane forces :
• 𝐹𝑥 = 0, 𝐹𝑦 = 0 𝑎𝑛𝑑 𝑀 = 0
• For system of space forces:
• 𝐹𝑥 = 0, 𝐹𝑦 = 0, 𝐹𝑧 = 0 𝑎𝑛𝑑 𝑀 𝑥 = 0, 𝑀 𝑦 = 0, 𝑀𝑧 = 0

4. • To write the equations of equilibrium for a rigid body, it is essential to
first identify all of the forces acting on that body and then to draw the
corresponding free-body diagram.
• Free body diagram: It is an isolated part of the system or the system
as a whole which shows all the forces acting on the system.
These forces include all the externally applied forces, the self
weight, reactive forces at supports and forces exerted by the other
bodies at the points of contact.
F2
F1
RA
RB
RC
A
B
C
O
F1
F2
𝜃
𝜃
Sphere
mg

5. Some particular cases of Equilibrium
(i) Equilibrium with a single force:
The equilibrium is not possible under the action of single force
unless the force itself is zero.
∴ 𝐹 = 0
(ii) Equilibrium with two forces:
If two forces F1 and F2 are acting on a body and the body is in
equilibrium, then the two forces must be equal in magnitude, opposite
in direction and collinear in action.
F1
F2

6. (iii) Equilibrium under three forces:
The three forces must be coplanar and they must be either
concurrent or parallel.
F1
F1 F2
F3
F2
F3

7. Lami’s theorem
• If a body is in equilibrium under the action of three concurrent forces,
then each force is proportional to the sine of the angle between the
remaining two forces.
• By Lami’s theorem
𝐹1
𝑠𝑖𝑛𝛼
=
𝐹2
𝑠𝑖𝑛𝛽
=
𝐹3
𝑠𝑖𝑛𝛾
F1
F2
F3
𝛾
𝛼 𝛽

8. Reactions at supports and connections
Ref: VECTOR MECHANICS FOR ENGINEERS: STATICS & DYNAMICS,
NINTH EDITION, by Beer and Johanston, Mc Graw Hills.2009.

9. Examples:
• A uniform rod of weight W is held in
position by means of a rope against a
smooth wall as shown in the figure.
Find the angle made by the rod with
wall for equilibrium. Also find the
length of the rope.
0.4 m
A
B
C
Smooth
wall

10. Solution:
0.4 m
A
B
C
W
M
𝜃
G
RB

11. • sin 𝐺𝐵𝑀 =
𝑀𝐺
𝐺𝐵
=
0.2
0.4
= 0.5
• ∴ ∠GBM = 300
• ∴ 𝜃 = 600
• Now in ∆𝐴𝐵𝐶, ∠𝐵 = 1200
• Applying cosine rule,
• 𝑐𝑜𝑠𝐵 =
𝐴𝐵2+𝐵𝐶2−𝐴𝐶2
2(𝐴𝐵)(𝐵𝐶)
• 𝐶𝑂𝑆 120 =
0.82+0.42−𝐴𝐶2
2(0.8)(0.4)
• ∴ 𝐴𝐶 = 1.0583 𝑚
A
B
C
W
M
𝜃
0.4 m
0.2m
G
RB