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)