Physics - Ch5 Vectors

8/10/2019 Physics - Ch5 Vectors

1/25

NORAIHAN BINTI SALLEH HUDIN

DEPT OF SCIENCE AND MATHEMATICS

CENTER FOR DIPLOMA STUDIES, UTHM

DAS 12603 TECHNICAL SCIENCE I

CHAPTER 5: VECTOR


2/25

INTRODUCTION

This chapter will discuss the basic concept ofvector operation, which includes:

Vector definition

Addition and subtraction of vector The concept of vector component


3/25

LEARNING OBJECTIVE

The objectives of this chapter are to impartstudents with:

The basic knowledge in vectors

The concept of vectors in engineering course

LEARNING OUTCOMES

Differentiate between the vector and the scalar

quantities.

Apply a vector analysis in engineering problem

Enhance knowledge on a vector concept and its

operation


4/25

2.1 DEFINITION OF VECTOR

QUANTITIES

Scalar quantities:

quantities that are measured require a description with only

a number (magnitude) and a unit.

Examples: mass, volume, temperature, time speed anddistance.

Vector quantities:

Require description with a number (magnitude) and adirection

Examples: displacement, velocity, acceleration, force and

weight.

The scalar part of the vector is called magnitude of thevector.


5/25

2.2 REPRESENTATION OF A

VECTOR: A BASIC CONCEPT

Some common denotation of vectors include:

Boldcase in typed writing: a.

In handwriting: or a.a

Point A: is called the initial point, tail, or base.

Point B: is called the head, tip, or endpoint.The length of the arrow represents the magnitude.

The direction in which the arrow points represents the vectors

direction.

VectorABis represented by awith magnitude a and directionAB.

A

Ba


6/25

Two vectors are equal if the length (magnitude) and the

direction of both are the same.

Combination of two or more vectors will produce new

vector with both magnitude and direction depending on

the initial state of the vectors.

The new magnitude/direction of the new vector isknown as resultant magnitude/direction.

If one of these vectors changes the magnitude/direction,

it will affect the magnitude/direction of the resultant

vector.

Vector A

Vector B


7/25

Same magnitude, different direction:

Same direction, different magnitude:

Zero magnitude, no direction

i.e. equilibrium state / steady

condition

Vector A

Vector B

Vector A Vector A

Vector B

Vector B

Vector A + Vector B (to the right)


8/25

Different on magnitude and direction:

or

Vector A

Vector B

Vector B Vector A

Vector A Vector B (to the right)

Vector A

Vector BVector B

Vector A

Vector A + Vector B


9/25

a. Naming and defining the vector magnitude and

direction

We can name a vector by its length and direction 3 km, southeast 25 km/h, at 50east of north

Or we can name a vector by x- and y-coordinates ofits endpoint, if its tail at the origin (3, 4)

(5, -2)


10/25

b. Vector unit assignment and notation

For 2D or 3D coordination system (Cartesian

coordinate), the notation of vector is assigned as i,j,

k.

Sometimes also written as

The notation i,j, k are defined as x, y and z direction

respectively in the coordinate system.

In a general form of unit vector, a unit of magnitudein any direction is defined.

Vector A has a magnitude of Axin the x-direction

and a magnitude of Ayin the y-direction.

Therefore vector Acan be written as:

A = Axi + Ayj

Similarly, in 3D system, vector unit can be written as:

A = Axi + Ayj + Azk

, , .i j k

A

Ax

Ay


11/25

Adding/subtracting a vector can be done by

adding/subtracting magnitude with the same direction

respectively.

Example:

Given A= (3i+ 2j)m, B= (-5i4j)m and C = (-2i+ 5j).

Determine the vector resultant of R = A + B + C.

Solution:

x-component:

y-component :

Therefore, the resultant force

3 ( 5 ) ( 2 )

4

x x x xR A B C

i i i

i

2 ( 4 ) 5

3

y y y yR A B Cj j j

j

4 3

x yR R R

i j m


12/25

2.3 VECTOR RESULTANT

Methods of finding a resultant vector:

Resultant Vector

Graphical method Analytical methodTrigonometry method

Parallelogram Tail-to-tip


13/25

2.3.1 GRAPHICAL METHOD

a) Adding vectors by using parallelogrammethod:

AB

Construct a parallelogram Diagonal line is the resultant ve

A A

BB

RR =A + B


14/25

b) Adding a vector by using tail-to-tip method:

A

B

R

R = A + B

A B

Move the tail one vector tothe tip of the other vector

A

B

Resultant vector goes from the tail ofthe first vector to the tip of the secon


15/25

Tail-to-tip method for the addition of more than

two vectors:

CA B

Move the tail of Bto the tip of A, then move the tail of Cto the ti

A

BC

RR = A + B + C


16/25

Example:

Two forces 3.0N and 4.0N act on a point.

a) What are the maximum and the minimum oftheir resultant force?

b) If a force of 3.0N acts on y-axis, while 4.0N actson the x-axis with the angle between them is90, what is their resultant?

Solution:

a) Maximum resultant occurs when both forces act

in the same direction:R =3.0N + 4.0N = 7.0N

Minimum resultant occurs when both forces actin the opposite direction:

R=|3.0N + (-4.0N)| = 1.0N


17/25

b) By using

paralellogram:

The vectors should be drawn according to scale.

By doing so, measuring the length of the

resultant vector will give its magnitude.

In this case, the magnitude of the resultant

vector is 5.0 N.

4.0 N

3.0 N R


18/25

2.3.2 ANALYTICAL METHOD

This method is more precise than graphical method.

Easier to perform operation on more than two vectors.

In this method, vectors are resolved into two

components: x-component and y-component.

In some cases, we can find resultant force by using

cosine rule.

Cosine rule:c2= a2+ b22ab cos


19/25

Example:

An aircraft tracking station determines the distance from

a common point O to each aircraft and the angle

between the aircrafts. If angle O between the twoaircrafts is equal to 49oand the distances from point O

to the two aircrafts are 50 km and 72 km, find distance d

between the two aircrafts.

Solution:Aircraft-1

Aircraft-2

O

49

72km

50km

d

km

d

4.54

49cos)50)(72(25072 222


20/25

2.3.3 TRIGONOMETRY METHODa) Vector resolution using Pythagoras theorem

(Finding a resultant vector when component vectors aregiven)

The original vector is split into x- and y-component.

The original vector, A, can be written as the sum of itsx- and y-component vectors: A = Ax+ Ay

The length of the original vector, A, can be accuratelycalculated by using Pythagoras theorem:

Ay

AxA

(x-component)

(y-component)

2 2 2

x yA A A

2 2

x yA A A


21/25

Example:

A plane flies at 241.5km/h due east. The wind blows

due north at 64.4 km/h. What is the net speed of the

plane?

Solution:

Use Pythagoras theorem tofind R:

241.5 km/h

(plane)

64.4 km/h

(wind)

R

2 2

2 2(241.5) (64.4)

62469.61

249.94 /

x yR A A

km h


22/25

b) Vector resolution using Trigonometric Equation

(Resolving a single vector to component vectors)

Consider the following triangle:

The magnitude of Acan be found usingPythagoras theorem:

The direction of Ais calculated using the

following equation:

A Ay

Ax

A

y

x

cos cos

sin sin

xA x A

yA y A

AA A

A

A A AA

2 2

x yA A A

tan y

Ax

A

A


23/25

Example:

A boy pulls his little brother in a wagon. The handle of the wagon

makes a 30 angle upward relative to the ground. If the boy pulls

with a force of 100N along the handle, how much force isactually being used horizontally and vertically?

Example:

A coplanar forces acting at a point O as shown in the figure below.

a) Resolve the forces along the x- and y-axis.

b) Determine the resultant force, FR, and its direction, R.

A = 8N

C = 6N

B = 5N

y

x45

30

30


24/25

SUMMARY

Vectors are fundamental in the physical sciencesand engineering. It can be used to represent any

quantity that has both magnitude and direction

Vectors are usually denoted in boldcase, as A, or

or a.

In vector unit system, the notation of i,jand kare

defined as x, yand zsystem.

The addition and subtraction operation of a vectorcan be performed by either a graphical or

analytical methods.

A graphical method consists of two methods:

parallelogram and tip-to-tail method.

a


25/25

EXERCISE:1. Two tugboats are towing a ship. Each exerts a force of 6 tons,

and the angle between the two ropes is 60. What is theresultant force on the ship?

2. A boat moving at 5km/h is crossing a river. The current of theriver is flowing at 3km/h. In what direction should the boat headto reach a point on the other bank of the river directly oppositeits starting point?

3. Going from one city to another, a driver drives his car 30kmnorth, 50km west and 20km southeast. How far has he beentravelling from his starting point?

4. A boy rides his bicycle 5m east before turning left and rides for

another 10m. How far has he travelled from where he starts?5. A woman in a car on a level road sees an airplane travelling in

the same direction climbing at an angle of 30 above thehorizontal. By driving at 110 km/h, she is able to stay directlybelow the airplane. Find the airplanes velocity.

6. A car weighing 12kN is on a hill that makes an angle of 20with the horizontal Find the components of the cars weight

Physics - Ch5 Vectors

Documents