MECHANICS OF SOLID

Mechanics of Solid(2130003)
Active Learning Assignment
“Coplanar Non-concurrent Forces”
Branch :- Mechanical
Prepared By :- Harsh Kapadia

Introduction:-
Moments & couples;
Characteristics of moment and couple;
Equivalent Couples;
Force couple system;
Varignon’s theorem;
Equilibrium conditions of coplanar non-
concurrent force system.

 Moment of a force:-
 The moment of a force is equal to the product of the force
and the perpendicular distance of line of action of force
from the point about which the moment is required.
 M = P × x
 Where, M = moment
P = force
x = Perpendicular distance between the line of
action of force and the point about which
moment is required.
 Moment produce the turning effect of the body.

 Examples of moment are:-
1)To tight the nut by spanner.
2)To open or close the door.
 Sign for moment :-
Generally , clockwise moment is taken as positive and
anticlockwise moment is taken as negative.
 Unit of moment is N.m. or kN.m. .

 Couple :-
 Two equal and opposite forces whose lines of action are
different form a couple.
 Resultant or net force of couple is zero . Hence , couple
acting on a body do not create any translatory motion of
the body . Couple produces only rotational motion of the
body .
 Types of couple :-
 There are two types of couple,
1) Clockwise couple,
2) Anticlockwise couple.

Clockwise couple
It rotates the body in
clockwise direction.
It is taken as positive.
Anticlockwise couple
It rotates the body in
anticlockwise direction.
It is taken as negative.
Arm of couple:-
The perpendicular distance between the lines of action
of two forces forming couple is known as the arm of
couple.
Moment of couple :-
Moment of couple = Force × arm of couple
M = P × a
Unit of couple is N.m. or kN.m.

Examples of couple are :-
1)Forces applied to the key of a lock , while locking or
unlocking it.
2)Forces applied on the steering wheel of a car by two hands
to steer the car towards left or right .
3)To open or close wheel valve of a water supply pipe line .
Characteristics of a couple :-
A couple has the following characteristics.
1)The algebraic sum of the forces , forming the couple is
zero .
2)The algebraic sum of the moment of the forces , forming
the couple , about any point is the same and equal to the
couple itself .

Moment
Moment = Force ×
perpendicular distance
M= P × x .
To balance the force causing
moment , equal and opposite
force is required .
 for example,
•To tight the nut by spanner
•To open or close the door.
Couple
Two equal and opposite
forces whose lines of action
are different form a couple .
Couple can not be
balanced by a single force .
It can be balanced by a
couple only .
For example,
•To rotate key in the lock
•To open or close the wheel
valve of water line
•To rotate the steering wheel
of car

 Equivalent couples :-
 Couples are said to be equivalent couples , if they
produce similar effects on a body .
 Consider three different systems acting on a
rectangular block as shown below . In the three cases
above , all the couples produce moment = 20,000 N ×
mm = 20 N × m.
 Hence , these three couples are called equivalent
couples .

 Force couple system :-
 When a system consists of forces along with moments or
couple , it is known as force couple system . This concept
is helpful in shifting a force away from its line of action .
 A given force F applied to a body at any point A can
always be replaced by an equal force applied at on other
point B together with a couple which will be equivalent to
the original force .

 Force F is acting at point A . We want to shift the force
F from A to B .
 Apply two equal and opposite forces at B parallel to F ,
the magnitude of that force must be equal to the force F
acting at A .
 Let x is the perpendicular distance between line of
action of forces .
 The forces at A and the forces in opposite direction at B
forms a couple of magnitude F × x shown in figure .
 Thus , a single force F acting at A is replaced by a force
couple-system acting at B .

 Varignon’s principles of moments :-
 It states ,“If a no. of coplanar forces are acting
simultaneously on a particle , the algebraic sum of the
moments of all the forces about any point is equal to the
moment of their resultant force about the same point .”
 Proof :-
 Consider two forces P and Q acting at a point A
represented in magnitude and direction by AB amd AC as
shown in figure .
 Let O be the point , about which the moments are taken.

 Through O , draw a line CD parallel to the direction of
force P , to meet the line of action of Q at C. Now with AB
and AC as two adjacent sides , complete parallelogram
ABCD . Join the diagonal AD of the parallelogram and
OA and OB . Diagonal AD represents in magnitude and
direction the resultant of two forces P and Q .
 Now , taking moment of forces P , Q and R about O .
 Moment of force P about O
=2 × Area of triangle AOB …(i)
Moment of force Q about O
=2 × Area of triangle AOC …(ii)
Moment of force R about O
=2 × Area of triangle AOD …(iii)

 Now , from the geometry of the figure ,
Area of AOD = Area of AOC + Area of ACD.
But area of ACD = Area of ADB
= Area of AOB
(Two triangles AOB and ADB are on the same base AB and
between the same parallel lines. )
Area of AOD = Area of AOC + Area of AOB
Multiplying both sides by 2 ,
2 × Area of AOC = 2 × Area of AOC +
2 × Area of AOB.
Moment of forces R about O=Moment of force P about O
+Moment of force Q about O
(Here , we have considered only two forces , but this
principle can be extended for any forces )

 Conditions of equilibrium for coplanar non-
concurrent forces :-
 If a body is acted upon by a no. of co-planar non-
concurrent forces , it may have one of the following states:
1. The body may move in any one direction.
2. The body may rotate about itself without moving.
3. The body may move in any one direction , and at the same
time it may also rotate about itself .
4. The body may be completely at rest .
 Now we will discuss the above four states.
1. If the body moves in any direction , it means that there is a
resultant force acting on it . A little consideration will
show , that if the body is to be rest or in equilibrium , the
resultant force causing movement must be zero. I .e.,
H=0 and V=0.Ʃ Ʃ

2. If the body rotates about itself without moving it means
that there is a single resultant couple acting on it . A little
consideration will show that if the body is to be at rest or
in equilibrium , the moment of the couple causing
rotation must be zero . i.e.,
M=0.Ʃ
3. If the body moves in any direction and at the same time it
rotates about itself , it means that there is a
resultant force and also a resultant couple acting on it . A
little consideration will show that if the body is to be at
rest or in equilibrium the resultant force causing
movement and the resultant moment of the couple
causing rotation must be zero . I .e. ,
H=0 , V=0 , M=0 .Ʃ Ʃ Ʃ

4. If the body is completely at rest it means that is neither a
resultant force nor a couple acting on it . In this case the
following conditions are satisfied .
H=0 , V=0 , M=0.Ʃ Ʃ Ʃ

MECHANICS OF SOLID

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