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The Law of Parallelogram


This law states that, 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 lite diagonal of the parallelogram passing through that point


Given OA = P force and OB = Q force as per Fig. 1.5 (a).

We construct parallelogram OBCA as per Fig. 1.5 (b). Let us drop perpendicular CD on extension of OA.


Thus,               AD= Q cos α,CD=Q sin α

∆OCD is a right angled triangle,

:.                                  OC2 =OD2 +CD2


Or                                R2 = (P + Q cos α)2 + (Q sin α)2

= P2 + Q2 cos2 a + 2PQ cos a+ Q2 sin2 α

= p2 + Q2 (cos2 a+ sin2 a)+ 2PQ cos α.


R2 = P2 + Q2 + 2PQ cos α                   [since cos2 a+ sin2 a= 1]


R= √p2 +Q2 +2PQ cos α


Equation (1.1) gives the magnitude of the resultant force R.

Direction of R :

Let θ be the angle of R with P

tan θ = CD/OD =Q sin a/ P+Q cos α

θ  = tan -1 (Q sin α /P+ Q cos α)