Diploma - Engineering Mechanics - Equilibrium. by ednexa.com

1. Equilibrium
Definition of equilibrium
“Any system of forces which keeps the body at rest is said to be in
equilibrium”. Or “when the condition of the body is unaffected
even through it is acted upon by number of forces, it is said to be
in equilibrium.”
Analytical Condition of Equilibrium
A
For coplanar concurrent forces: we know that, resultant of
concurrent forces is given by the
R=


  FX 


2
formula,
F 

Y


2
Equilibrium means resultant force acting on the body is zero.
i.e. R = 0


 0  F 
X

2
 F 

Y


2
 Fx  0 and  Fy  0
Hence, analytical conditions of equilibrium for coplanar
concurrent forces are
i.
∑Fx = 0 i.e. Algebraic sum of component of all forces
along x-axis must be equal to zero.
ii.
∑Fy = 0 i.e. Algebraic sum of component of all forces
along y-axis must be equal to zero. In other words, these

2. analytical conditions of equilibrium can be started as follows:
“If any numbers of forces acting on the body are in
equilibrium, the algebraic sum of
their components in two
directions at right angle to each other must be equal to zero.”
B.
For coplanar non-concurrent forces: we know that, resultant
of non-concurrent

R  


forces
is given by the formula
2








 FX   FY
2




For body to be in equilibrium, the resultant force acting on it
must be equal to zero.
i.e.
R=0
2
2
∴
0=
∴
∑Fx = 0 and ∑Fy = 0


F 

X




F 

Y


According to varignon’s theorem of moments, the algebraic
sum of moment of all
forces about any point is equal to
moment of the resultant force about the same point.
i.e.
As
∑M = R  x
R=0
∴
∑M = 0
Therefore, Analytical conditions of equilibrium
for
coplanar non-concurrent forces are,
i.
∑Fx = 0
ii.
∑Fy = 0
iii.
∑M = 0

3. Concept of Free Body Diagram (F.B.D)
In static, for considering the equilibrium of the body under the any
system of forces each body is separated from its surroundings.
Such body is known as free body. If all active and reactive forces
acting on a free body are shown, the diagram is known as free
body diagram.
Lami’s Theorem (Equilibrium Of Three Coplanar Concurrent
Forces)
Statement: Its states, “If three forces acting at a point on a body
keep it at rest, then each force is proportional to the sine of the
angle between the other two forces.”

4. Mathematically,
Notations:
P = Q = R
sinα sinβ sinγ
α = Angle between Q and R
β = Angle between P and R
γ = Angle between P and Q
P, Q and R = Three concurrent forces.
Proof: To prove that,
P = Q = R
sinα sinβ sinγ
Construct a parallelogram OACB such that
OA = P, OB = Q, and OC = R.
In ∆ OAC, applying sine rule, we get,
OA = AC = OC
sin  π - α sin  π - β sin  π - γ 

5. ∴ P = Q = R = K where K is a constant.
sinα sinβ sinγ
∴ P = Ksinα, Q = Ksinβ and R = Ksinγ
∴ P α sin , Q α sinβ and R α sinγ
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