ENT151/4
STATICS
CHAPTER 2
FORCE VECTORS
2.1 Scalars & Vectors
Scalar - a quality characterized by positive or negative members (mass, volume, length)
Vector – a quality that has both a magnitude and a direction. Vector represented by Ā
Magnitude epresented by |A|, positive quantity.
- Magnitude, length of arrow (A)
- Direction, angle between axis and the arrow’s line.
2.2
Vectors Operations
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School of Mechatronics, KUKUM
ENT151/4
STATICS
Multiplication & division :
Addition of a vector:
parallelogram law
Substraction of a vector :
R=A−B
= A + (− B)
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School of Mechatronics, KUKUM
ENT151/4
STATICS
Resolution of vector :
Adition of force :
FR
=
(F1 + F2) + F3
EXAMPLES
Example 2-1 (pp 22)
Use cosine law to find R, sine law to find θ.
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School of Mechatronics, KUKUM
ENT151/4
STATICS
Note
EXAMPLES
Example 2-2 (pp 23)
Example 2-3 (pp 24)
Example 2-4 (pp 25)
2.3
Additional of Coplaner Forces
Introduction Coplanar Forces :
In this section, each force is resolved into its rectangular components Fx and Fy, which lie
along the x and y axes, respectively.
According to the parallelogram law, we know that:
F = Fx + Fy
and
F’ = F’x + F’y
The sense of direction of each force component is represented graphically by the
arrowhead.
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School of Mechatronics, KUKUM
ENT151/4
STATICS
F = Fx + Fy
F’ = Fx’ + Fy’
F’ = Fx’ i + Fy’ j
A notation for representing the directional sense of the rectangular components is
required for analytical work.
This notation can be done by:
(i) Scalar notation
(ii) Cartesian vector notation
Scalar notation :
Since we know that x and y axes have designated positive and negative directions, the
magnitude and directional sense of the rectangular components of a force can be
expressed in terms of algebraic scalars.
As shown in Figure (a),
The components of F can be represented by positive scalars Fx and Fy since their sense of
direction is along the positive x and y axes, respectively.
Note:
This scalar notation is to be used only for computational purposes, not for graphical
representations in figures.
Cartesian vector notation :
For two dimensions, the Cartesian unit vectors i and j are used to designate the directions
of the x and y axes, respectively.
Referring to Figure (c),
The magnitude of each component of F is always a positive quantity, which is
represented by the (positive) scalars Fx and Fy.
Therefore, F can be expressed as the Cartesian vector,
F = Fx i + Fy j
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School of Mechatronics, KUKUM
ENT151/4
STATICS
And for Figure (d),
F’ can be expressed as;
F’ = F’x i + F’y (-j)
or simply
F’ = F’x i - F’y j
Note:
The Cartesian vector notation is better because the methods of vector algebra are easier to
apply and this becomes particularly advantageous for solving problems in three
dimensions.
Coplanar force resultant :
Procedure for calculating the resultant of coplanar forces:
(i) Each force is first resolved into its x and y components
(ii) The respective components are added using scalar algebra
since they are collinear
(iii) The resultant force is formed by adding the resultants of the x
and y components using the parallelogram law.
Example:
Referring to Figure (e),
Calculate the resultant force for the three concurrent forces using Cartesian vector
notation.
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School of Mechatronics, KUKUM
ENT151/4
STATICS
Step (i);
Each force is first represented as a Cartesian vector,
F1 = F1x i + F1y j
F2 = -F2x i + F2y j
F3 = F3x i - F3y j
Step (ii);
All forces are added to get the vector resultant,
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
Generally,
The x and y components of the resultant of any number of coplanar forces can be
represented symbolically by the algebraic sum of the x and y components of all the
forces;
FRx = ∑ Fx
FRy = ∑ Fy
(Eq. 1)
Note:
It is important to use the sign convention (+ or -) established for the components when
applying these equations.
If this convention is followed, then the signs of the resultant components will specify the
sense of these components.
As shown in Figure (g),
The resultant components can be sketched along the x and y axes in their proper
directions and the resultant force can be calculated from vector addition.
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Zol Bahri Razali
School of Mechatronics, KUKUM
ENT151/4
STATICS
From the Pythagorean theorem, the magnitude of resultant force, FR is;
FR = √ [(FRx)2 + (FRy) 2 ]
The direction angle θ, which specifies the orientation of the force, is calculated from
trigonometry;
θ
= tan -1 ( FRy / FRx )
EXAMPLES
Example 2-5 (pp 35)
2.4
Example 2-6 (pp 36)
Cartesian Vector
Rectangular components of a vector :
Generally, a vector A may have one, two or three rectangular components along the x, y,
z coordinate axes, depending on how the vector is oriented relative to the axes.
Referring to Figure (a),
The vector A is resolved into components as A = A’ + Az and then,
A’ = Ax + Ay by using the parallelogram law.
Then, these equations are combined;
A = Ax + Ay + Az
(Eq. 1)
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School of Mechatronics, KUKUM
ENT151/4
STATICS
Unit vector :
The direction of vector A can be specified using a unit vector.
If A is a vector having a magnitude A = 0, then the unit vector having the same direction
as A is represented by;
uA = A / A
(Eq. 2)
A = AuA
(Eq. 3)
so that,
Note:
The unit vector is dimensionless since the units will cancel out (vector A and magnitude
A have same set of units) (refer to Eq. 2).
From Eq. 2,
Vector A is expressed in terms of both its magnitude and direction separately; (refer to
Figure (b))
A (a positive scalar) defines magnitude of A, and
uA (a dimensionless vector) defines the direction and sense of A.
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School of Mechatronics, KUKUM
ENT151/4
STATICS
Cartesian unit vector :
The set of Cartesian unit vectors, i, j, k, is used to designate the directions of the x, y, z
axes respectively for three dimensions.
The sense (or arrowhead) of these vectors will be described analytically by a plus or
minus sign, depending on whether they are pointing along the positive or negative x, y, z
axes.
Figure (c) shows the positive Cartesian unit vectors.
Cartesian vector representation :
From Eq. 1, A = Ax + Ay + Az;
As shown in Figure (d), the three components of A act in positive i, j and k directions,
Thus, A can be written in Cartesian vector form as;
A = Ax i + Ay j + Az k
(Eq. 4)
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Zol Bahri Razali
School of Mechatronics, KUKUM
ENT151/4
STATICS
Note:
The magnitude and direction of each component vector are separated.
The advantage for writing vectors in Cartesian vector form is this form will simplify the
operations of vector algebra, particularly in three dimensions.
Magnitude of a Cartesian vector :
The magnitude of vector A is equal to the positive square root of the sum of the squares
of its components.
Referring to Figure (e),
For the colored right triangle, A = √( A’2 + Az2) and
for the shaded right triangle, A’ = √(Ax2 + Ay2)
Then, these equations are combined;
A = √( Ax2 + Ay2 + Az2)
(Eq. 5)
Direction of a Cartesian vector :
The orientation of A is defined by the coordinate direction angles α (alpha), β (beta)
and γ (gamma).
It is measured between the tail of A and the positive x, y, z axes located at the tail of A
(refer to Figure (f)).
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School of Mechatronics, KUKUM
ENT151/4
STATICS
Note:
Regardless of where A is directed, each of these angles will be between 0º and 180º.
As shown in Figure (g), (h), (i), consider the projection of A onto the x, y, z axes.
To calculate α, β and γ, use the equations below;
cos α = Ax / A ;
cos β = Ay / A ;
(Eq. 6)
cos γ = Az / A ;
These numbers are known as the direction cosines of A.
Then, the coordinate direction angles α, β, γ can be calculated from the inverse cosines.
Other method to obtain the direction cosines of A is to form a unit vector in the direction
of A (as Eq. 2).
Provided A is expressed in Cartesian vector form,
A = Ax i + Ay j + Az k,
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Zol Bahri Razali
School of Mechatronics, KUKUM
ENT151/4
STATICS
Then,
uA = A / A
= (Ax / A) i + (Ay / A) j + (Az/ A) k
(Eq. 7)
where A = √( Ax2 + Ay2 + Az2)
After that, substitute Eq. 6 into Eq. 7,
We get,
uA = (cos α)i + (cos β)j + (cos γ)k
(Eq. 8)
This equation shows that the i, j, k components of uA represent the direction cosines of
A.
Since the magnitude of a vector is equal to the positive square root of the sum of the
squares of the magnitudes of its components and
uA has a magnitude of 1;
cos2 α + cos2 β + cos2 γ = 1
(Eq. 9)
Note:
This equation can be used to calculate one of the coordinate direction angles if the other
two are known.
If the magnitude and coordinate direction angles of A are given,
A may be expressed in Cartesian vector form as;
A = AuA
= A(cos α)i + A(cos β)j + A(cos γ)k
= Ax i + Ay j + Az k
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(Eq. 10)
Zol Bahri Razali
School of Mechatronics, KUKUM
ENT151/4
STATICS
2.5 Position Vectors
x , y , z coordinates ;
A (x , y , z)
Position vector, is defined as a fixed vector which locates a point in space relative to
another point.
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School of Mechatronics, KUKUM
ENT151/4
STATICS
=
=
=
=
rA + rB
rB - rA
( xB i + yB j + zB k ) - ( xA i + yA j + zA k )
(xB - xA) i + (yB - yA) j + (zB - zA) k
EXAMPLE
Example 2- 12 (pp 57)
2.6 Force Vector Directed Along a Line
F – force
r - position vector
u - unit rector = r / r
r - radius from origin
The force, F directed along the cord AB
The position vector, r directed from point A to point B on the cord.
This common direction is specified by the unit vector, u = r / r.
F = Fu = F ( r / r)
where : F is unit of force
r is unit of length
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School of Mechatronics, KUKUM
ENT151/4
STATICS
EXAMPLES
Example 2-13 (pp 59)
Example 2-14 (pp 60)
2.7 Dot Product
The dot product of vectors A and B, written A . B is defined as the product of the
magnitudes of A and B and the cosine of the angle between their tails.
A . B = A B Cos θ
0° ≤ θ ≤ 180°
Law of operation
1. A . B = B . A
2. a (A . B ) = ( aA ) . B = A . ( aB ) = (A . B )a
3. A . ( B + D ) = ( A . B ) + ( A . D )
Cartesian Vector Formulation
i . i = (1) (1) cos 0° = 1
i . j = (1) (1) cos 90° = 0
Therefore :
i . i = 1
i . j = 0
j . j = 1
i . k = 0
k . k = 1
k . j = 0
A . B = ( Ax i + Ay j + Az k ) . ( Bx i + By j + Bz k )
= Ax Bx ( i . i ) + Ax By ( i . j ) + Ax Bz ( i . k ) +
A y Bx ( j . i ) + A y By ( j . j ) + A y Bz ( j . k ) +
Az Bx ( k . i ) + Az By ( k . j ) + Az Bz ( k . k )
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School of Mechatronics, KUKUM
ENT151/4
STATICS
By dot operation;
A . B = Ax Bx + Ay By + Az Bz
Thus to determine the dot product of two Cartesian Vectors, multiply their corresponding
x , y , z components and sum their products algebraically. The result is a scalar.
Application Dot Product
1.
Angle formed between two vectors
A . B = A B Cos θ
If
A.B = 0;
Hence
2.
θ = cos-1 ( A . B / A B )
θ = cos-1 0
= 90°
A┴ B
The components of a vector parallel and perpendicular to a line.
A ╖ = A Cos θ
= A.u
where u is a unit vector
A┴ = A-A╖
since A = A ┴ + A ╖
To determine θ ,
θ = cos-1 ( A . u / A )
Then
A ┴ = A Sin θ
or A ┴ = √ ( A2 - A╖2 )
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School of Mechatronics, KUKUM
ENT151/4
STATICS
EXAMPLES
Example 2-16 (pp 71)
Example 2-17 (pp 72)
EXERCISES
Exercise 2 - 6
Exercise 2 - 14
Exercise 2 - 21
Exercise 2 - 26
Exercise 2 - 31
Exercise 2 - 55
Exercise 2 - 63
Exercise 2 - 87
Exercise 2 - 88
Exercise 2 - 114
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School of Mechatronics, KUKUM