Statics

Chapter 2

Force Vectors

Eng. Iqbal Marie

iqbal@hu.edu.jo

Engineering Mechanics, Statics, 13th edition ; R. C. Hibbeler

Force Vectors

2.1 Scalar and Vectors

Scalar: is a quantity which has magnitude only.

Examples of scalars: speed, distance, energy, charge, volume, mass

and temperature. .

Vectors are quantities which are fully described by both a magnitude

and a direction. Vectors are physical quantities.

Examples of vectors are displacement, velocity, acceleration, force

and electric field

Vector notation:

A widely used convention is to denote a

vector quantity in bold type, such as A ,and

that is the convention that will be used. The

magnitude of a vector A is written as | A |.

2.2 Vector Operations:

Vector Addition:

a. Collinear Vectors

Multiplication and division of a vector

by a scalar

A x m = m|A|

A B

Coplanar vectors

Parallelogram Method

Polygon method:

A

B

C

D

E

2.3 Vector Addition of

Forces:

Resolving vectors into components:

To find components of R along lines a and b:

1. Start at tip of R.

2. Draw line parallel to line a until it intersects line b - this defines component in

b direction..

3. Draw line parallel to line b until it intersects line a - this defines component in

a direction.

The math you need:

For a right triangle:

a2+ b2 = c2

tan() = b/a

sin() = b/c

cos() = a/c

Sine Law

Cosine Law

A line intersecting parallel lines

Eg. Determine the magnitude of the resultant

force and its direction measured

counterclockwise from the positive x axis.

120 60

60 lb

80 lb

R

R = 602 + 802 – 2x 60x80 cos 60

80 / sin = R/ sin 60

Use Cosine Law

Use Sine Law

Resolve the 80N force into U and V components

250 N

70° 45

180-70-45 = 65°

70°

U

V

180-( 70+65) = 45°

2.4 Addition of a system of coplanar forces

Cartesian Vector Notation

( 2D)

Rx = Fx

Ry = Fy

Determine the components of the 150 N force

150 N

N

N

N

N

2.5 Cartesian Vectors

(600/13) x12 =

553.85

(600/13) x 5

= 230.77

800 cos 40= 612.83

800 sin 40 =

514.23

Rx = Fx = 612.83 – 553.85 = 58.98 N

Ry = Fy = 514.23 + 230.77 = 745 N

= 747.33 N

= 12.63

= 85.53º

= 67.3 N

= - 71.0 N

Cartesian vector representation:

3D Three Dimensional Vectors

Right hand rule

Cartesian vector magnitude:

Unit vector Representation of a Vector

vector uA is just a vector in the same direction as A,

but with magnitude = 1,

uA = A / A

uA is dimensionless. It serves only to indicate direction and sense.

Direction (orientation) of a Cartesian vector in 3D

α = angle between A and positive x axis •

β = angle between A and positive y axis •

= angle between A and positive z axis •

α

β

U

Eg. Determine the magnitude and directional

cosines of the vector.

A 700 820 900i j k

The magnitude of the vector is

The directional cosines are

Check the cosines

Express F as Cartesian vector

Adding and Subtracting 3D Cartesian Vectors

F1 =

N

F2 =

2.7 Position Vectors

2.8 Force Directed Along a Line

F

2.9 Dot Product

Scalar Formulation

Dot products of Cartesian vectors

Components of vector parallel and perpendicular to a line:

Proj. of F = F . U = ( 2i + 4j + 10 k ) . { 2/3 i + 2/3 j -1/3 k}

= 0.667 kN

(0,0,0)

(2,2,-1)

r(oA) = 2i + 2j - k

A

r (oA) = 4 + 4+ 1 =3

U(oA) = r/r = 2/3 i + 2/3 j -1/3 k

Express the position vector r in Cartesian vector form

and find the direction angles

If F1 = F2 = 300 N, determine the angles and so that the

resultant is directed along the positive x axis and has a

magnitude of FR = 200 N.

300 cos + 300 cos = 200

300 sin - 300 sin = 0

=

3 cos + 3 cos = 2

cos = 2/6

= 70.5 =