Law of Parallelogram of forces

1. 15CE31T- ENGINEERING MECHANICS AND
STRENGTH OF MATERIALS
Suresh Angadi Education Foundation’s
ANGADI INSTITUTE OF TECHNOLOGY & MANAGEMENT
POLYTECHNIC
SAVAGAON ROAD, BELAGAVI – 590 009.
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Syllabus Covered :
Composition of Forces: - Definition
Resultant force
UNIT-1 INTRODUCTION TO ENGINEERING
MECHANICS
( Marks : 15 Weightage : 10% )

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Resultant of Forces
There are usually several different forces acting on an object. The overall motion
of the object will depend on the size and direction of all the forces.
The motion of the object will depend on the resultant force. This is calculated
by adding all the forces together, taking their direction into account.

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 The process of replacing a force system by its resultant is called composition.
 The Resultant of a pair of concurrent (occurring at the same time and gathering
at the same point) forces can be determined by means of Parallelogram Law,
which states that:
Two forces on a body can be replaced by a single force called the resultant by
drawing the diagonal of the parallelogram with sides equivalent to the two
forces.
Composition of Forces
 Stevinus (1548-1620) was the first to demonstrate that forces could be combined
by representing them by arrows to some suitable scale, and then forming a
parallelogram in which the diagonal represents the sum of the two forces. In fact,
all vectors must combine in this manner.
For example if F1 and F2 are two forces, the resultant ( R ) can be found by
constructing the parallelogram.

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Law of Parallelogram of Forces
“If two forces, acting at a point be represented in magnitude and direction by
the two adjacent sides of a parallelogram, then their resultant is represented
in magnitude and direction by the diagonal of the parallelogram passing
through that point.”
Let two forces P and Q act at a point O as shown in Fig. 1 The force P is
represented in magnitude and direction by OA whereas the force Q is presented
in magnitude and direction by OB. Let the angle between the two forces be ‘a’.
The resultant of these two forces will be obtained in magnitude and direction by
the diagonal (passing through O) of the parallelogram of which OA and OB are
two adjacent sides. Hence draw the parallelogram with OA and OB as
adjacent sides as shown in Fig. 2. The resultant R is represented by OC in
magnitude and direction

5. Magnitude of Resultant (R)
From C draw CD perpendicular to OA produced.
Let α = Angle between two forces P and Q = ∠AOB
Now ∠DAC = ∠AOB (Corresponding angles)= α
In parallelogram OACB, AC is parallel and equal
to OB.
• ∴ AC = Q.
In triangle ACD,
AD = AC cos α = Q cos α
CD = AC sin α = Q sin α.
In triangle OCD,
• OC2 = OD2 + DC2.
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But OC = R,
OD = OA + AD = P + Q cos α
And
DC = Q sin α.
∴ R2 = (P + Q cos α)2 + (Q sin α)2
= P2 + Q2 cos2 α + 2PQ cos α + Q2 sin2 α
= P2 + Q2 (cos2 α + sin2 α) + 2PQ cos α
= P2 + Q2 + 2PQ cos α (∵ cos2 α + sin2 α = 1)
∴ R = P2+ Q2+ 2PQcos α

6. • Direction of Resultant
Let θ = Angle made by resultant with OA.
Then from triangle OCD,
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