👉 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.
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| Fig.-1 |
Second way :-
👉 The Parallelogram law of forces is used to determine the resultant of two forces acting a point in a plane and inclined to each other at an angle.
Derivation :-
Consider two forces P and Q acting on a body at O as shown in Fig. 1 . The force P is represented in magnitude and direction
whereas the force Q is represented in magnitude and direction by
. Let the angle between the two forces be θ. The resultant of these two forces is obtained by the diagonal
of the parallelogram OACB, as shown in Fig. 1.
The relationship between P, Q and R can be derived as follows:
Drop perpendicular from C and let it meet OA extend at point D. In ∆CAD, side CA is parallel and equal to OB, i.e., it represents force Q.
The resultant R , of P and Q , is given by
Put OA = P
AD = AC . cosθ = Q cosθ
CD = AC . sinθ = Q sinθ
&space;+2PQcos%5Ctheta&space;&space;%7D&space; "https://latex.codecogs.com/svg.image? = sqrt{P^{2}+Q^{2}(sin^{2}theta +cos^{2}theta) +2PQcostheta }")
…………….(1)
The inclination of the resultant R to the direction of force P is given by
…………….(2)
Note:- It is not necessary that for the law of parallelogram for forces to be valid, one of the two forces should be along the x-axis. The forces P and Q be in any direction as shown in Fig. 2.
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| Fig.-2 |
If the angle between the forces is θ, then
The direction of resultant will be
Special cases:
(1). When the two forces are equal and θ is the angle between them: (P=Q)
…………….(3)
And
…………….(4)
i.e., the resultant bisects the angle between the forces.
(2). When the two forces act at right angles; i.e., θ=90°
( cos90°=0) …………(5)
and
 "https://latex.codecogs.com/svg.image?alpha = tan^{-1} left [ frac{Qsin90^{o}}{P+Qcos90^{o}} right ]= tan^{-1}left ( frac{Q}{P} right )")
( sin90°=1)
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| Fig.-3 |
(3). When the two forces act in the same line and same sense, i.e. , θ=0°
(cos0°=1)
= P + Q …………..(6)
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| Fig.-4 |
(4). When the two forces have the same line of action but opposite senses, i.e., θ=180°
( cos180°=-1)
= P – Q …………(7)
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| Fig.-5 |
👉 The results obtained vide equations (5),(6) and (7) have been represented in figures-(3,4,5). These results lead us to conclude that when the force acting on a body are collinear, their resultant is equal to the a
lgebraic sum of the forces.
