CE 201 - Statics

Lecture 3

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.

F1

F2

F3

R1=F1+F2

R2=R1+F3

How about finding the components of each force along certain axe?

F1

F2

F3

x

y

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 perpendicular

F

Fx

Fy

y

x

F

Fx

Fy

y

x

Directional Sense of Rectangular Components

There are two ways to do that:

1. Scalar Notation

The 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.

F

Fx

FyF

Fx

Fy

F

Fx

FyF

FxFy

Fx and Fy are +ve

Fx is +ve and Fy is +ve

Fx is –ve and Fy is +ve

Fx and Fy are -ve

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.

x

y

F

Fx

Fy

i

j

x

y

F

Fx

Fyi

-j

F = Fx i + Fy j F = Fx i – Fy j

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.

Resultant of Coplanar Forces

F1

F2

F3

F1

F2

F3

F2x

F2y

F3x

F3y

F1y

F1x

- By Scalar Notation

FRx = F1x – F2x – F3x

FRy = F1y + F2y – F3y

- By Cartesian Vector Notation

FR = 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) j

F1

F2

F3

F2x

F2y

F3x

F3y

F1y

F1x

In general,

FRx = Fx

FRy = 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 + F2

Ry

and the direction of FR can be found from:

= tan-1 ( FRy / FRx )

Examples

• Example 2.5• Example 2.6• Example 2.7• Problem 2-31• Problem 2-34• Problem 2-38