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Statics
Chapter 2
Force Vectors
Eng. Iqbal Marie
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 =