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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.
GROUP PROBLEM SOLVING
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GROUP PROBLEM SOLVING
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 :
GROUP PROBLEM SOLVING
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
GROUP PROBLEM SOLVING
GROUP PROBLEM SOLVING