2 ADDITION OF A SYSTEM OF COPLANNAR FORCES If we have more than two forces, the resultant can be determined by successive applications of the parallelogram law.F1F2F3R1=F1+F2R2=R1+F3
3 How about finding the components of each force along certain axe? y
4 The forces can then be added algebraically and the resultant can be determined. Which method is EASIER?The main objective of this SECTION is to resolve each force into its rectangular components, Fx and Fy along x and y axes, respectively, where x and y must be perpendicularFFxFyyxFFxFyyx
5 Directional Sense of Rectangular Components There are two ways to do that:1. Scalar NotationThe components can be represented by algebraic scalar (+ve and –ve).If the component is in the positive direction of x or y, then it is positive.If the component is in the negative direction of x or y, then it is negative.
6 Fx and Fy are +ve Fx is –ve and Fy is +ve Fx and Fy are -ve
7 2. Cartesian Vector Notation The component forces can also be represented in terms of Cartesian Unit Vector.In two dimensions, the Cartesian unit vector i and j are used to represent x and y, respectively.xyFFxFyi-jxyFFxFyijF = Fx i + Fy jF = Fx i – Fy j
8 As can be seen, the sense of the Cartesian unit vectors are represented by plus or minus signs depending on if they are pointing along the +ve or –ve x and y axes.
10 - By Cartesian Vector Notation FR = F1 + F2 + F3 - By Scalar NotationFRx = F1x – F2x – F3xFRy = F1y + F2y – F3y- By Cartesian Vector NotationFR = F1 + F2 + F3= F1x i + F1y j – F2x i + F2y j – F3x i – F3y j= (F1x – F2x – F3x) i + (F1y + F2y – F3y) j= (FRx) i + (FRy) j= (Fx) i + (Fy) jF1F2F3F2xF2yF3xF3yF1yF1x
11 Components along positive x and y axes are positive. In general,FRx = FxFRy = FyComponents along positive x and y axes are positive.Components along negative x and y axes are negative.After finding FRx and FRy, the magnitude of yhe resultant force (FR) can be determined using Pythagorean Theorem, where:FR = F2Rx + F2Ryand the direction of FR can be found from: = tan-1 ( FRy / FRx )
12 Examples Example 2.5 Example 2.6 Example 2.7 Problem 2-31 Problem 2-34