CVEN1300CVEN1300ENGINEERING MECHANICSENGINEERING MECHANICS

INTRODUCTIONINTRODUCTION

STATICS, UNITS, CALCULATIONS & PROBLEM SOLVING

Todays Objectives:Students will be able to:

a) Identify what is mechanics (statics and dynamics) .b) Round the final answer appropriately.b) Round the final answer appropriately.d) Apply problem solving strategies.

Homework Tasks

Review Vectors Review Vectors Dot Product Cross Product Cross Product

TEAM EXERCISE AND HOMEWORKTEAM EXERCISE AND HOMEWORK:Solve Problems in the Textbook at theSolve Problems in the Textbook at theback of Chapters 1 and 2.

H ? How many? Try at least 3-4 Problems from each section!

Homework TasksHomework Tasks Based on what you know from Physics how would you design the cables A and B

required to support this weight of 100 kN? (i e what is the required cross-required to support this weight of 100 kN? (i.e. what is the required crosssectional area for the cables? The Cables are made from steel and the maximum stress these can take is 300 MPa)

FAj + FBj = 100y

FAj + FBj = 100

FAj = FBj = 50 [kN]

45o45oCable A Cable B

A B

FA = FB = 50/(cos 45o)FBFA

FBjFAj

xFA = FB = 70.7 [kN]FBiFAi

100 kN

Stress = Force/AreaArea = Force/StressArea = 70 7 x 103/300100 kN Area = 70.7 x 10 /300

= 235.7 [mm2]

IntroductionIntroductionIntroductionIntroduction

Designing & constructing devices/structures:

U d d h h i d l i h d i Understand the physics underlying the designs. Use mathematical models to predict their

behaviour. Learn how to analyse & predict the behaviors of y p

physical systems by studying mechanics.

Engineering & MechanicsEngineering & MechanicsEngineering & MechanicsEngineering & Mechanics Knowledge of previous designs experiments Knowledge of previous designs, experiments,

ingenuity & creativity to develop new designsdesigns.

Develop mathematical equations based onDevelop mathematical equations based on the physical characteristics of the device/structures designs:device/structures designs: Predict the behavior

Modify the design Modify the design Test the design prior to actual construction

Engineering & MechanicsEngineering & Mechanics Elementary Mechanics the study of

Engineering & MechanicsEngineering & Mechanics Elementary Mechanics the study of

forces & their effects

Statics the study of objects in equilibriumDynamics the study of objects in motionDynamics the study of objects in motion

Retrace historical development of ideas Retrace historical development of ideas.

Engineering & MechanicsEngineering & MechanicsEngineering & MechanicsEngineering & Mechanics

Applications in many fields of engineering:

Statics: equilibrium equations Statics: equilibrium equations Designing structures (mechanical & civil)

Dynamics: motion equationsy q Analyze responses of buildings to earthquakes

(civil) Determine trajectories of satellites (aerospace)

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

A) ti fi ldA) magnetic fieldB) heat C) forcesC) forcesD) neutronsE) lasers

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

A) Newtonian MechanicsA) Newtonian MechanicsB) Relativistic MechanicsC) Euclidean MechanicsC) Euclidean MechanicsD) Greek Mechanics

Newtons Three Laws

First law---Equilibriumq

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 thethe 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 q pparticle and its acceleration.

Newtons Three Laws

Third law---Free body diagram (FBD)y g ( )

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 lawDynamic analysis: Second law + Third law

WHAT IS MECHANICS??

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

Either the body or the forces could be large or small

name is body ) when FORCES are applied to it.

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

BRANCHES OF MECHANICS

Mechanics

Rigid Bodies Deformable Bodies Fluids

Statics Dynamics

g(Things that do not change shape) (Things that do change shape)

Incompressible CompressibleStatics Dynamics Incompressible Compressible

SYSTEMS OF UNITS

F f d t l h i l titi Four fundamental physical quantities.

Length mass time force Length, mass, time, force.

One equation relates them, F = m x aq ,

We use this equation to develop systems of units

U it bit i t th Units are arbitrary names we give to the physical quantities.

UNIT SYSTEMS

D fi 3 f th it d ll th Define 3 of the units and call them the base units.

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

W ill k i h i i We will work with one unit system in statics: SI.

NUMERICAL CALCULATIONS

Must have dimensional homogeneity. Dimensions have to be the same on both sidesDimensions have to be the same on both sides of the equal sign,

(e.g. distance = speed time.)(e.g. distance speed time.) Use an appropriate number of significant figures

(3 for answer, at least 4 for intermediate calculations). Why?

Be consistent when rounding off.

) y

g- greater than 5, round up (3528 3530)- smaller than 5, round down (0.03521 0.0352)smaller than 5, round down (0.03521 0.0352)- equal to 5, see your textbook.

CONCEPT QUIZ

1 Evaluate the situation in which mass (kg) force1. Evaluate the situation, in which mass (kg), force (N), and length(m) are the base units and recommend a solution.recommend a solution.

A) A new system of units will have to be formulatedA) A new system of units will have to be formulated

B) Only the unit of time have to be changed from ) y gsecond to something else

C) N h i dC) No changes are required.

D) The above situation is not feasibleD) The above situation is not feasible

CONCEPT QUIZ

2. Give the most appropriate reason for using three significant figures in reporting results of typical g g p g ypengineering calculations.

A) Historically slide rules could not handleA) Historically slide rules could not handle more than three significant figures.

B) Three significant figures gives better than one-percent accuracy.

C) Telephone systems designed by engineers have area codes consisting of three figures.

D) Most of the original data used in engineering calculations do not have accuracy better than onecalculations do not have accuracy better than one percent

PROBLEM SOLVING STRATEGY: IPE, A 3 Step Approach

1 INTERPRET: Read carefully and determine what is1. INTERPRET: Read carefully and determine what isgiven and what is to be found/ delivered. Ask, if notl If k ti d i di t thclear. If necessary, make assumptions and indicate them.

2. PLAN: Think about major steps (or a road map) thatyou will take to solve a given problem. Think ofalternative/creative solutions and choose the best one.

3 EXECUTE: Carry out your steps Use appropriate3. EXECUTE: Carry out your steps. Use appropriate diagrams and equations. Estimate your answers. Avoid simple calculation mistakes Reflect on / reviseAvoid simple calculation mistakes. Reflect on / reviseyour work.

CONCEPT QUIZ

1. For a statics problem your calculations show the final answer as 12345.6 N. What will you write as yyour final answer?

A) 12345.6 NB) 12 3456 kNB) 12.3456 kNC) 12 kND) 12.3 kNE) 123 kN

2. In three step IPE approach to problem solving,

E) 123 kN

2. In three step IPE approach to problem solving, what does P stand for

A) Position B) PlanC) ProblemD) PracticalD) PracticalE) Possible

VECTORSVECTORS

2D2DVECTORSVECTORS

2D VECTOR ADDITIONTodays Objective:

St d t ill b bl tStudents 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) VelocityA) Force B) Position C) Mass D) Velocity

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

A) Newtons Second

B) the arithmetic)

C) Pascals

D) h ll lD) the parallelogram

APPLICATION OF VECTOR ADDITIONAPPLICATION OF VECTOR ADDITION

There are four concurrent cable forces acting on the bracket.

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

SCALARS AND VECTORS

Scalars VectorsScalars Vectors

Examples: mass, volume force, velocity

Characteristics: It has a magnitude It has a magnitude

( iti ti ) d di ti(positive or negative) and directionAddition rule: Simple arithmetic Parallelogram law

Special Notation: None Bold font, a line, an

arrow or a carrot

VECTOR OPERATIONSVECTOR OPERATIONS

Scalar Multiplication

and Division

VECTOR ADDITION USING EITHER THETHE

PARALLELOGRAM LAW OR TRIANGLE

Parallelogram Law:

Triangle method (always tip to tail):

H d b ? H ddHow do you subtract a vector? How can you add more than two concurrent vectors graphically ?

RESOLUTION OF A VECTOR

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

CARTESIAN VECTOR NOTATION

We resolve vectors into components using the x and y

h f h

axes system

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

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 jF 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 anyother. Together,they can be directed at any inclination.

ADDITION OF SEVERAL VECTORS

Step 1 is to resolve eachStep 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 vectorbecome the resultant vector.

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

Example of this process,Example of this process,

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

EXAMPLE Given: Three concurrent forces acting on aforces acting on a bracket.

Fi d Th i d dFind: The magnitude and angle of the resultant forceforce.

Plan:

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

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

c) Find magnitude and angle from the resultant components.

EXAMPLE

EXAMPLE

yFR

x

CONCEPT QUIZ

1. Can you resolve a 2-D vector along two directions, which are not at 90 to each d ect o s, c a e ot at 90 to eacother?A) Yes, but not uniquely.B) NoB) No.C) Yes, uniquely.

2. Can you resolve a 2-D vector along three directions (say at 0, 60, and 120)?( y )A) Yes, but not uniquely.B) No.C) Y i lC) Yes, uniquely.

GROUP PROBLEM SOLVING

Given: Three concurrent forces acting onforces acting on a bracket

Fi d Th it dFind: The magnitude and angle of the resultant forceresultant force.

Plan:

a) Resolve the forces in their x y componentsa) 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

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) jx

y

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

C) 80 sin (30) i - 80 cos (30) j 30F 80 ND) 80 cos (30) i + 80 sin (30) j F = 80 N

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 NA) 30 N B) 40 N C) 50 N

D) 60 N E) 70 N

3D3DVECTORSVECTORS

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

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

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

APPLICATIONS

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

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

APPLICATIONS

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

A UNIT VECTOR

For a vector A with a magnitude of A, an unit vector is defined as

Ch i i f i

,UA = A / A .

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

i i l t (A)original vector (A).

The unit vectors in the CartesianThe 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.

3-D CARTESIAN VECTOR TERMINOLOGY

Consider a box with sides AX, XAY, 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: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)

TERMS

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 = 1This result can be derived from the definition of a coordinate direction angles and the unit vector Recall the formula for finding the unitangles 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 .

ADDITION/SUBTRACTION OF VECTORS

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

F l ifFor 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 (A B ) i (A B ) j (A B ) kA B = (AX - BX) i + (AY - BY) j + (AZ - BZ) k .

IMPORTANT NOTESIMPORTANT NOTES

Sometimes 3-D vector information is given as:Sometimes 3-D vector information is given as:

a) Magnitude and the coordinate direction angles, or

b) Magnitude and projection angles.

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

vector into the Cartesian form, i.e.,

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

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 Forceangle with the X-Y plane. Force G is pointing up and has a magnitude of 80 N with =

G

g111 and = 69.3.

i d h l f hFind: The resultant force in the Cartesian vector form.

Plan:Plan:

1)Using geometry and trigonometry write F and G1)Using geometry and trigonometry, write F and Gin the Cartesian vector form.

2) Then add the two forces.

Solution :

CONCEPT QUESTIONS

1. If you know just UA, you can determine the ________ of A uniquelyof A uniquely.A) magnitude B) angles (, and ) C) components (A A & A ) D) All of the aboveC) components (AX, AY, & AZ) D) All of the above.

2 For an arbitrary force vector the following2. For an arbitrary force vector, the following parameters are randomly generated. Magnitude is 0.9 N, = 30, = 70, = 100. What is wrong, , , gwith this 3-D vector ?A) Magnitude is too small.) g

B) Angles are too large.

C) All three angles are arbitrarily pickedC) All three angles are arbitrarily picked.

D) All three angles are between 0 to 180.

GROUP PROBLEM SOLVING

Given: The screw eye is subjected to two forces.j

Find: The magnitude and the coordinate direction anglescoordinate 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 .2) Add F1 and F2 to get FR .

3) Determine the magnitude and , , .

GROUP PROBLEM SOLVING

F1z1z

F

GROUP PROBLEM SOLVING

ATTENTION QUIZ

1. What is not true about an unit vector, UA ? A) It is dimensionless.A) It is dimensionless.B) Its magnitude is one.C) It always points in the direction of positive X- axisC) It always points in the direction of positive X- axis.D) It always points in the direction of vector A.

2. If F = {10 i + 10 j + 10 k} N andG = {20 i + 20 j + 20 k } N, then F + G = { ____ } N

A) 10 i + 10 j + 10 kB) 30 i + 20 j + 30 kC) -10 i - 10 j - 10 k) jD) 30 i + 30 j + 30 k