E-Portfolio Mohd Fardzlee bin Abd Patah GB190077 MBV

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HUKUM NEWTON

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PENAMBAHAN VEKTOR DAYA YANG BERTEMU PADA SATU TITIK
Penambahan vektor tidak boleh disamakan dengan penambahan aljabar atau skalar kerana
penambahan vektor sebenarnya melibatkan pertimbangan kedua-dua kuantiti magnitud dan arah.
Hanya vektor-vektor yang mempunyai arah yang sama (selari) boleh dicampurkan magnitudnya
seperti penambahan kuantiti skalar. Umumnya, penambahan dua vektor mestilah mematuhi
Hukum Segi Empat Selari atau Hukum Segi Tiga.

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PELERAIAN VEKTOR DAYA
Peleraian Daya pada Arab yang Serenjang antara Satu Sama Lain
Peleraian vektor merupakan kebalikan pemaduan vektor. Vektor boleh dileraikan kepada
komponen-komponennya sama ada komponen segi empat atau lain-lain. Vektor daya F yang
membuat sudut θ terhadap paksi x boleh dileraikan kepada komponen pada arah x dan y masing-
masing ditandakan dengan Fx dan Fy seperti yang ditunjukkan dalam Rajah 2.3.1. Fr dan Fy boleh
dihitung dengan

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EXERCISE:
1. Berikan definisi berikut:
i. Skalar
ii. Vektor
2. Huraikan tiga Hukum Newton.

3. Selesaikan masalah berikut:

REFERENCE :
1. Hibbeler, R.C & S.C Fan 1997. Engineering Mechanics, Statics. SI Ed. New Jersey:
Prentice Hall. ISBN 0135995981
2. Yusof Ahmad. 1999. Mekanik Statik. Cetakan Ketiga. Malaysia: Penerbit UTM Skudai Johor

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Kolej Kolej Kemahiran Tinggi MARA
Masjid Tanah, Melaka.

INFORMATION SHEET

PROGRAMME : DIPLOMA IN AUTOMOTIVE ENGINEERING TECHNOLOGY
SESSION :
CODE/COURSE : OCTOBER – DECEMBER 2021 SEMESTER : 3
LECTURER : SHEET NO : 1
DKV21273 STATICS & WEEK : 1
DYNAMICS

MOHD FARDZLEE ABD PATAH

TOPIC : FORCE VECTOR
SUB-TOPIC :
1.1Scalars and Vectors Quantities
1.2Vector Operation
1.3Vector Addition of Force

LEARNING After completing the course, students should be able to:
OUTCOME : 1. Apply Parallelogram Law to add forces and resolve into
components.
2. Express force and position in Cartesian vector form also
explain how to determine the vector's magnitude and
direction.
3. Understand the dot product in order to determine the
angle between two vectors or the projection of one
vector onto another.

CONTENT:

Scalar Quantity
1. Scalars are quantities which are fully described by a magnitude alone.
2. Magnitude is the numerical value of a quantity.
3. Examples of scalar quantities are distance, speed, mass, volume, temperature, density and
energy.

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Vector Quantity

1. Vectors are quantities which are fully described by both a magnitude and a direction.
2. Examples of vector quantities are displacement, velocity, acceleration, force, momentum,

and magnetic field.

Example 1
Categorize each quantity below as being either a vector or a scalar.
Speed, velocity, acceleration, distance, displacement, energy, electrical charge, density, volume,
length, momentum, time, temperature, force, mass, power, work, impulse.

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Answer:
Scalar Quantities:

• speed
• distance
• energy
• electrical charge
• density
• volume
• length
• time
• temperature
• mass
• power
• work
Vector Quantities
• velocity
• acceleration
• displacement
• momentum
• force
• impulse

Vector Operations
In the physical world, some quantities, such as mass, length, age, and value, can be represented
by only magnitude. Other quantities, such as speed and force, also involve direction. You can use
vectors to represent those quantities that involve both magnitude and direction. One common use
of vectors involves finding the actual speed and direction of an aircraft given its air speed and
direction and the speed and direction of a tailwind. Another common use of vectors involves finding
the resulting force on an object being acted upon by several separate forces.

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Any quantity that has both size and direction is called a vector quantity. If A and B are two points
that are located in a plane, the directed line segment from point A to point B is denoted by .
Point A is the initial point, and point B is the terminal point.
A geometric vector is a quantity that can be represented by a directional line segment. From this
point on, a vector will be denoted by a boldface letter, such as v or u. The magnitude of a vector is
the length of the directed line segment. The magnitude is sometimes called the norm. Two vectors
have the same direction if they are parallel and point in the same direction. Two vectors have
opposite directions if they are parallel and point in opposite directions. A vector that has no
magnitude and points in any direction is called the zero vector. Two vectors are said to be
equivalent vectors if they have the same magnitude and same direction.
Figure 1 demonstrates vector addition using the tail‐tip rule. To add vectors v and u, translate
vector u so that the initial point of u is at the terminal point of v. The resulting vector from the initial
point of v to the terminal point of u is the vector v + u and is called the resultant. The vectors v and
u are called the components of the vector v + u. If the two vectors to be added are not parallel, then
the parallelogram rule can also be used. In this case, the initial points of the vectors are the same,
and the resultant is the diagonal of the parallelogram formed by using the two vectors as adjacent
sides of the parallelogram.

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Figure 1
Example of vector addition.
In order to multiply a vector u by a real number q, multiply the length of u by | q | and reverse the
direction of u if q < 0. This is called scalar multiplication. If a vector u is multiplied by −1, the
resulting vector is designated as − u. It has the same magnitude as u but opposite direction. Figure
2 demonstrates the use of scalars.

Figure 2
Examples of vectors.
Example 1: A plane is traveling due west with an air speed of 400 miles per hour. There is a
tailwind blowing in a southwest direction at 50 miles per hour. Draw a diagram that represents the
plane's ground speed and direction (Figure 3 ).

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Figure 3
Drawing for Example 1— vector representation.
The vector represented in the preceding example is known as a velocity vector. The bearing of a
vector v is the angle measured clockwise from due north to v. In the example, the bearing of the
plane is 270° and the bearing of the wind is 225°. Redrawing the figure as a triangle using the
tail‐tip rule, the length (ground speed of the plane) and bearing of the resultant can be calculated
(Figure 4).

Figure 4
Drawing for Example 1— angle representation.
First, use the law of cosines to find the magnitude of the resultant.

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Then, use the law of sines to find the bearing.

The bearing, β, is therefore 270° − 4.64°, or approximately 265.4°.

Vector Addition of Force
A variety of mathematical operations can be performed with and upon vectors. One such operation
is the addition of vectors. Two vectors can be added together to determine the result (or resultant).
This process of adding two or more vectors has already been discussed in an earlier unit. Recall in
our discussion of Newton's laws of motion, that the net force experienced by an object was
determined by computing the vector sum of all the individual forces acting upon that object. That is
the net force was the result (or resultant) of adding up all the force vectors. During that unit, the
rules for summing vectors (such as force vectors) were kept relatively simple. Observe the following
summations of two force vectors:

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These rules for summing vectors were applied to free-body diagrams in order to determine the net
force (i.e., the vector sum of all the individual forces). Sample applications are shown in the
diagram below.

In this unit, the task of summing vectors will be extended to more complicated cases in which the
vectors are directed in directions other than purely vertical and horizontal directions. For example, a
vector directed up and to the right will be added to a vector directed up and to the left. The vector
sum will be determined for the more complicated cases shown in the diagrams below.

There are a variety of methods for determining the magnitude and direction of the result of adding
two or more vectors. The two methods that will be discussed in this lesson and used throughout the
entire unit are:

• the Pythagorean theorem and trigonometric methods
• the head-to-tail method using a scaled vector diagram
The Pythagorean Theorem
The Pythagorean theorem is a useful method for determining the result of adding two (and only
two) vectors that make a right angle to each other. The method is not applicable for adding more
than two vectors or for adding vectors that are not at 90-degrees to each other. The Pythagorean
theorem is a mathematical equation that relates the length of the sides of a right triangle to the
length of the hypotenuse of a right triangle.

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To see how the method works, consider the following problem:
Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's
resulting displacement.
This problem asks to determine the result of adding two displacement vectors that are at right
angles to each other. The result (or resultant) of walking 11 km north and 11 km east is a vector
directed northeast as shown in the diagram to the right. Since the northward displacement and the
eastward displacement are at right angles to each other, the Pythagorean theorem can be used to
determine the resultant (i.e., the hypotenuse of the right triangle).

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EXERCISE:

1. What is :-
i. Scalar?
ii. Vector?

2. Find the sum of the two forces indicated below
F1 = 150U@ 50̊
F2 = 250U@125̊

REFERENCE :
1. Hibbeler, R.C & S.C Fan 1997. Engineering Mechanics, Statics. SI Ed. New Jersey:
Prentice Hall. ISBN 0135995981
2. Yusof Ahmad. 1999. Mekanik Statik. Cetakan Ketiga. Malaysia: Penerbit UTM Skudai Johor

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Kolej Kolej Kemahiran Tinggi MARA
Masjid Tanah, Melaka.

INFORMATION SHEET

PROGRAMME : DIPLOMA IN AUTOMOTIVE ENGINEERING TECHNOLOGY
SESSION :
CODE/COURSE : OCTOBER – DECEMBER 2021 SEMESTER : 3
LECTURER : SHEET NO : 2
DKV21273 STATICS & WEEK : 2
DYNAMICS

MOHD FARDZLEE ABD PATAH

TOPIC : FORCE VECTOR
SUB-TOPIC :
1.4Addition of a System of Coplanar
1.5Cartesian Vectors
1.6Additional and Subtraction of Cartesian Vectors

LEARNING After completing the course, students should be able to:
OUTCOME : 1. Apply Parallelogram Law to add forces and resolve into
components.
2. Express force and position in Cartesian vector form also
explain how to determine the vector's magnitude and
direction.
3. Understand the dot product in order to determine the
angle between two vectors or the projection of one
vector onto another.

CONTENT:
Addition of a System of Coplanar Forces

The magnitude of a vector is always a (+ or -) quantity?
The magnitude of a vector is always a positive quantity.

Rectangular Components
Rectangular Components
A force resolved in to two (2D) or three (3D) components.

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Rectangular components can be represented using either Scalar Notation or Cartesian Vector
Notation.
Scalar Notation
Scalar Notation
Rectangular Components of a force form a right triangle ∴ the magnitudes of the rectangular
components can be determined by SOH-CAH-TOA.
Scalar Notation Format: F = Fx + Fy
Cartesian Vector Notation
Cartesian Vector Notation
Rectangular Components of a force can be represented in terms of Cartesian unit vectors i and j
which are use to designate the x and y axis respectively.
Since the magnitude of each component of F is always a positive quantity represented by positive
scalars Fx and Fy, we can express F as a Cartesian vector.
Cartesian Vector Notation Format: F = Fx +Fy

Cartesian Vector Notation diagram

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Coplanar Force Resultants
To determine resultant of several coplanar forces:

- Resolve force into x and y components
- Addition of the respective components using scalar algebra
- Resultant force is found using the parallelogram law
Either Scalar Notation or Cartesian Vector Notation can be used to determine the resultant of
several coplanar forces by resolving each force into its x and y components.
F₁ = Fx₁ i + Fy₁ j
F₂ = Fx₂ i + Fy₂ j
F₃ = Fx₃ i + Fy₃ j
The vector resultant is therefore
FR = F₁ + F₂ + F₃
If Scalar Notation is used, we have
FRx = F₁x + F₂x + F₃x → FRx = ∑Fx
FRy = F₁y + F₂y + F₃y → FRy = ∑Fy

Coplanar Force Resultants diagram
Resolving the Magnitude and Direction of the Resultant Force Magnitude
FR = √(FR²x + FR²y)

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θ = tan⁻¹(FRy ÷ FRx)

Coordinate axes and Cartesian coordinates
By three dimensional space we mean the space we live in. To fix a point P in three dimensional
space requires a system of axes and three numbers. First select any point, call it the origin and
mark it as O. All measurements will from now on originate from this point O. Next place three
mutually perpendicular axes OX, OY , OZ through O. This axis system is drawn on a page like this:

Note that although it may not look it, the angles XOY , XOZ, Y OZ are all right-angles!
To fix any given point P in three dimensional space, we refer it to the axis system. Let us first show
the point P and the coordinate axes.

1. Drop a line from P perpendicular to the XOY plane (think of this plane as the floor), meeting
the XOY plane at a point Q (the foot of the perpendicular).

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2. Now we are in the XOY plane with a point Q and an axis system in two dimensions.
3. Drop a perpendicular from Q to OX and OY .

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4. Transfer the two-dimensional picture from 3 into the three dimensional diagram from 1.

(Some of the right angles are marked: you should mark all the other right angles in the picture.)
5. Measure the lengths OR, OS, QP, and denote them by x, y, z respectively. We call the three
numbers (x,y,z), in the order given, the Cartesian coordinates of the point P.

Notice that the order in which the numbers are written is important: (1, 2, 1) and (2, 1, 1) are the
Cartesian coordinates of different points.

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To make the diagrams uncluttered, only half of each axis has been drawn. However, if the axes are
extended infinitely in both directions, it can be seen that this axis system creates eight octants in
space, just as the two dimensional axis system creates four quadrants in the plane. Points in the
eight different octants have Cartesian coordinates corresponding to all possible combinations of
positive and negative values of x, y and z. The figures above illustrate points in the octant
corresponding to positive values of x, y and z.
Cartesian form of a vector
We begin with two dimensions. We have the following picture illustrating how to construct the
Cartesian form of a point Q in the XOY plane.

Vectors i and j are vectors of length 1 in the directions OX and OY respectively.
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