Week 1---Introduction+Vectors

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CVEN1300ENGINEERING MECHANICS

INTRODUCTION

Introduction

Designing & constructing devices/structures:

Understand the physics underlying the designs. Use mathematical models to predict their

behaviour. Learn how to analyse & predict the behaviors of

physical systems by studying mechanics.

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Engineering & Mechanics Knowledge of previous designs, experiments,

ingenuity & creativity to develop new designs.

Develop mathematical equations based on the physical characteristics of the device/structures designs: Predict the behavior Modify the design Test the design prior to actual construction

Elementary Mechanics the study of forces & their effects

Statics the study of objects in equilibrium

Dynamics the study of objects in motion

Engineering & Mechanics

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Applications in many fields of engineering:

Statics: equilibrium equations Designing structures (mechanical & civil)

Dynamics: motion equations Analyze responses of buildings to earthquakes

(civil) Determine trajectories of satellites (aerospace)

Engineering & Mechanics

1. The subject of mechanics deals with what happens to a body when ______ is / are applied to it.

A) magnetic fieldB) heat C) forcesD) neutronsE) lasers

2. ________________ still remains the basis of most of todays engineering sciences.

A) Newtonian MechanicsB) Relativistic MechanicsC) Quantum Mechanics

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Newtons Three Laws

First law---Equilibrium

When the sum of the forces acting on a particle is zero, its velocity is constant. In particular, if the particle is initially stationary, it will remain stationary.

Second law---Equation of motion

When the sum of the forces acting on a particle is NOT zero, the sum of the forces is equal to the rate of change of the linear momentum of the particle. If the mass is constant, the sum of the forces is equal to the product of the mass of the particle and its acceleration.

Newtons Three Laws

Third law---Free body diagram (FBD)

The forces exerted by two particles on each other are equal in magnitude and opposite in direction.

Static analysis: First law + Third law

Dynamic analysis: Second law + Third law

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WHAT IS MECHANICS??

Either the body or the forces could be large or small.

Study of what happens to a thing (the technical name is body) when FORCES are applied to it.

BRANCHES OF MECHANICS

Statics Dynamics

Rigid Bodies(Things that do not change shape)

Deformable Bodies(Things that do change shape)

Incompressible Compressible

Fluids

Mechanics

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SYSTEMS OF UNITS

Four fundamental physical quantities.

Length, mass, time, force.

One equation relates them, F = m x a

We use this equation to develop systems of units

We will work with one unit system in statics: SI.

UNIT SYSTEMS

Define 3 of the units and call them the base units.

Derive the 4th unit (called the derived unit) using F = m x a.

We will work with one unit system in statics: SI.

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VECTORS

2DVECTORS

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2D VECTOR ADDITIONTodays Objective:

Students will be able to :

a) Resolve a 2-D vector into components

b) Add 2-D vectors using Cartesian vector notations.

READING QUIZ

1. Which one of the following is a scalar quantity?A) Force B) Position C) Mass D) Velocity

2. For vector addition you have to use ______ law.

A) Newtons Second

B) the arithmetic

C) the parallelogram

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APPLICATION OF VECTOR ADDITION

There are four concurrent cable forces acting on the bracket.

How do you determine the resultant force acting on the bracket ?

SCALARS AND VECTORS

Scalars Vectors

Examples: mass, volume force, velocity

Characteristics: It has a magnitude It has a magnitude

(positive or negative) and directionAddition rule: Simple arithmetic Parallelogram law

Special Notation: None Bold font, a line, an

arrow or a carrot

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VECTOR ADDITION USING EITHER THE

PARALLELOGRAM LAW OR TRIANGLE

Parallelogram Law:

Triangle method (always tip to tail):

How do you subtract a vector? How can you add more than two concurrent vectors graphically ?

Resolution of a vector is breaking up a vector into components. It is kind of like using the parallelogram law in reverse.

RESOLUTION OF A VECTOR

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CARTESIAN VECTOR NOTATION

Each component of the vector is shown as a magnitude and a direction.

We resolve vectors into components using the x and y axes system

The directions are based on the x and y axes. We use the unit vectors i and j to designate the x and y axes.

For example,

F = Fx i + Fy j or F' = F'x i - F'y j

The x and y axes are always perpendicular to each other. Together,they can be directed at any inclination.

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ADDITION OF SEVERAL VECTORS

Step 3 is to find the magnitude and angle of the resultant vector.

Step 1 is to resolve each force into its components

Step 2 is to add all the x components together and add all the y components together. These two totals become the resultant components.

Example of this process,

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You can also represent a 2-D vector with a magnitude and angle.

EXAMPLE Given: Three concurrent forces acting on a bracket.

Find: The magnitude and angle of the resultant force.

Plan:

a) Resolve the forces in their x-y components.

b) Add the respective components to get the resultant vector.

c) Find magnitude and angle from the resultant components.

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EXAMPLE

EXAMPLE

x

y

FR

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GROUP PROBLEM SOLVING

Given: Three concurrent forces acting on a bracket

Find: The magnitude and angle of the resultant force.

Plan:

a) Resolve the forces in their x-y components.

b) Add the respective components to get the resultant vector.

c) Find magnitude and angle from the resultant components.


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y

x

FR

QUIZ

1. Resolve F along x and y axes and write it in vector form. F = { ___________ } N

A) 80 cos (30) i - 80 sin (30) j

B) 80 sin (30) i + 80 cos (30) j

C) 80 sin (30) i - 80 cos (30) j

D) 80 cos (30) i + 80 sin (30) j

2. Determine the magnitude of the resultant (F1 + F2) force in N when

F1 = { 10 i + 20 j } N and F2 = { 20 i + 20 j } N .

A) 30 N B) 40 N C) 50 N

D) 60 N E) 70 N

30

xy

F = 80 N

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3DVECTORS

Todays Objectives:Students will be able to :a) Represent a 3-D vector in a Cartesian coordinate

system.b) Find the magnitude and coordinate angles of a 3-D

vectorc) Add vectors (forces) in 3-D space

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READING QUIZ

1. Vector algebra, as we are going to use it, is based on a ___________ coordinate system.

A) Euclidean B) left-handed

C) Greek D) right-handed E) Egyptian

2. The symbols , , and designate the __________ of a 3-D Cartesian vector.

A) unit vectors B) coordinate direction angles

C) Greek societies D) x, y and z components

APPLICATIONS

Many problems in real-life involve 3-Dimensional Space.

How will you represent each of the cable forces in Cartesian vector form?

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APPLICATIONS

Given the forces in the cables, how will you determine the resultant force acting at D, the top of the tower?

A UNIT VECTOR

Characteristics of a unit vector:a) Its magnitude is 1.b) It is dimensionless.c) It points in the same direction as the

original vector (A).

The unit vectors in the Cartesian axis system are i, j, and k. They are unit vectors along the positive x, y, and z axes respectively.

For a vector A with a magnitude of A, a unit vector is defined as UA = A / A .

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3-D CARTESIAN VECTOR TERMINOLOGY

Consider a box with sides AX, AY, and AZ meters long.

The vector A can be defined as A = (AX i + AY j + AZ k) m

The projection of the vector A in the x-y plane is A.The magnitude of this projection, A, is found by using the same approach as a 2-D vector:

A = (AX2 + AY2)1/2 .The magnitude of the position vector A can now be obtained as

A = ((A)2 + AZ2) = (AX2 + AY2 + AZ2)

The direction or orientation of vector A is defined by the angles , , and .

These angles are measured between the vector and the positive X, Y and Z axes, respectively. Their range of values are from 0 to 180

Using trigonometry, direction cosines are found using the formulas

These angles are not independent. They must satisfy the following equation.

cos + cos + cos = 1

This result can be derived from the definition of a coordinate direction angles and the unit vector. Recall, the formula for finding the unit vector of any position vector:

or written another way, u A = cos i + cos j + cos k

TERMS

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ADDITION/SUBTRACTION OF VECTORS

Once individual vectors are written in Cartesian form, it is easy to add or subtract them. The process is essentially the same as when 2-D vectors are added.

For example, if

A = AX i + AY j + AZ k and

B = BX i + BY j + BZ k , then

A + B = (AX + BX) i + (AY + BY) j + (AZ + BZ) k

or

A B = (AX - BX) i + (AY - BY) j + (AZ - BZ) k

IMPORTANT NOTES

Sometimes 3-D vector information is given as:

a) Magnitude and the coordinate direction angles,

i.e. , , and or

b) Magnitude and projection angles.

You should be able to use both these types of information to change the representation of the

vector into the Cartesian form, i.e.,

F = {10 i 20 j + 30 k} N

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EXAMPLE Given: Two forces F and G are applied to a hook. Force F is shown in the figure and it makes 60angle with the X-Y plane. Force G is pointing up and has a magnitude of 80 N with = 111 and = 69.3.

Find: The resultant force in the Cartesian vector form.

Plan:

1)Using geometry and trigonometry, write F and Gin the Cartesian vector form.

2) Then add the two forces.

GG=80 KN

Solution :

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Solution :


Given: The screw eye is subjected to two forces.

Find: The magnitude and the coordinate direction angles of the resultant force.

Plan:

1)Using the geometry and trigonometry,

write F1 and F2 in the Cartesian vector form.

2) Add F1 and F2 to get FR .

3) Determine the magnitude and , , .

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F1z



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Week 1---Introduction+Vectors