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Engineering Problem SolvingENG1101

Physics 101 For The Design Project Mathematical Model

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Last Time

● Macros

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Individually – no name

● For each of the following topics indicate your comfort level, 5 – very comfortable to 1 – no clue● Flowcharts● Conditional Statements● Loops● Fuctions● Macros

● Any questions or areas of confusion

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Today's Agenda

● Mathematical Model

● Newton's Laws

● Forces● Gravitational

● Drag

● Spring

● Friction

● Moments

● Rotational Motion

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Design Project MathmaticalModel

● Due: 9B● Derivations

● Due: 10B● Spreadsheet with VBA Code

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Equation Derivations

● Due: 9B● Use Problem Presentation Method for EACH

equation derived.● Problem● Diagram● Theory● Assumptions● Solution ● Result

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Lext

Ltrap

α

θ

β

Diagram of Vehicle

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As a Team

● Beta (β) − how do you determine angle to drive wheel

● Dop – how do you determine distance between 2 points?

● Ls – how do you determine length of one side of a triangle (not right triangle) given the opposite angle (θ) and two sides (Larm, Dop)?

● Alpha (α) – how do you determine the angle of a triangle given 3 sides (Ls, Larm, Dop)?

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As a Team

● Beta (β) − how do you determine angle to drive wheel? Trigonometry

● Dop – how do you determine distance between 2 points? Pythagorean Theorem

● Ls – how do you determine length of one side of a triangle (not right triangle) given the opposite angle (θ) and two sides (Larm, Dop)? Law of Cosines

● Alpha (α) – how do you determine the angle of a triangle given 3 sides (Ls, Larm, Dop)? Law of Cosines

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Rotational Motion & a Mousetrap Vehicle● Distance Traveled: Relate Length of string

pulled to distance traveled

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Arc length vs Circumference● Arc length: distance along the edge of a

circle as related to the angle that distance covers● s = r * θ

s = length of arcr = radius

θ = angle in radians (1 revolution = 2π radians)

● Circumference:● distance around a circle●C= 2rπ

C = circumferencer = radius

π = 3.14...

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As a Team....

Ra

Rb

a

b

● List assumptions and known equations.

● Write an expression that describes the arc length Sa with relation to the arc length Sb, as the wheel rotates through an angle θ θ

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As a Team

● What are Newton’s 3 Laws?

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Newton’s First Law:"Every body persists in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it." - Newton

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Newton's Second LawThe acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

(Physics 1100 Text by Serway and Beichner)

● Expressed mathematically as:Σ F = ma or often as Fnet=ma

● As a vector expression:Σ Fx = max Σ Fy = may Σ Fz = maz

Common applications● Force required to move an object● Projectile trajectories● Forces on springs

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Newton's Third Law . . .If two objects interact, the force F12 exerted by object 1 on object 2 is equal in magnitude to and opposite in direction to the force F21 exerted by object 2 on object 1.

F12 = -F21

For every action there is an equal and opposite reaction.●Common applications

● Analyzing forces acting on objects● Analyzing gravitational attraction between bodies

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What forces are acting on an object?● Gravitation: attraction between two objects

● Spring: results when a spring is compressed or stretched

● Drag: exerted when an object moves through a fluid (liquid or gas)

● Friction: two solid surfaces contact each other

F CDAv2

2

F Fnormal=

F k x=

F ma=

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Forces/Tension & a Mousetrap Vehicle● Gravitational: F=ma● Spring: F=-k*x or Force=m*Angle+b● String: Tension = Spring Force/cos (α-90)

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New

ton

String

CautionSpring Angle Measurements

● When you made measurements, you were "opening" spring. (θ)

● For your vehicle design, your spring will be "closing". (β)

180−θ = β

Angle meas= θ

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As a Team

● Work on Derivation of Ftrap● Hint: Think Spring Analysis

Ftrap

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Moments1) When turning a wrench, which

point will take less effort? A or B2) When opening a door, is it

easier to push near the hinges or away from the hinges? C or D

Moment: tendency of a force to cause rotation about a point

AB

C DMoment = Force x Distance from axis● M = F x Dp

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As a Team...● If the force is 120 lbs at point A=3 ft, what

would the force at point B = 7 ft need to be to produce the same moment?

x=? lbf120 lbf

7 ft

3 ft

AB

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As a Team...● M = F x Dp

● Using the terms given, write an equation for FB if it were to produce the same moment as FA about the axis.

FBFA

DB

DA

AB

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As a Team

● Work on Derivation of Farm

Farm

Ftrap

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Moments & a Mousetrap Vehicle● Extended Mousetrap Arm

● FArm = Ftrap * Ltrap/LArm

Farm

Ftrap

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Tension in the String

● Farm is pulling the string● What is the tension in the string, Ts

Farm

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Forces● Forces are vectors

● Have magnitude and direction

● May break forces into their components ● (x direction, y direction , z direction)

● Because system is not accelerating, sum of forces in x, y and z directions are zero

F ma= a 0= F 0=Σ

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Break F2 into components

30o

F

Fx = F cos (30o)

Fy = F sin (30o)

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Break F2 into components

30o

F2

F2x = F2 cos (30o)

F2y = F2 sin (30o)

Once we have components – we no longer use the original vector.To get the net effect of force vectors acting on an object we add all the “y”s together and then all the “x”s together.

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As a Team

● What is the tension in the string, Ts?

Farm

Ts x

y

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Motive Force

● Using Moments, what force (Fwheel) produces the equivalent moment as Ts?

Ts

Fwheel Fmot

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Friction & a Mousetrap Vehicle● Air Resistance: neglected● Static Friction: Ffs = μsFn

● Kinetic Friction of Vehicle: Ffk = μkFn

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Ff = μFn = Force due to Friction

Where:● Ff = force due to friction (N)● Fn = force perpendicular to friction

surface (N)

● μ = coefficient of friction ●Static or Kinetic

Ff

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Coeff. of Coeff. of Materials Static Friction Kinetic Friction

Steel on Steel 0.74 0.57 Aluminum on Steel 0.61 0.47 Copper on Steel 0.53 Rubber on Concrete 1 0.8 Wood on Wood 0.25-0.5 0.2 Glass on Glass 0.94 0.4 Waxed wood on Wet snow 0.14 0.1 Waxed wood on Dry snow - 0.04 Metal on Metal (lubricated) 0.15 0.06 Ice on Ice 0.1 0.03 Teflon on Teflon 0.04 0.04 Synovial joints in humans 0.01 0

μs μk

Source: Serway Physics for Scientists and Engineers 4th edition (p. 126.)

Coefficients of Friction

What are the model's coefficients comparable to?

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What if you are pulling a box?Resistance the box has to being moved● Kinetic Friction slows an object down

●Ffk = μkFn

● Static Friction prevents an object from starting to move or keeps an object stationary

●Ffs = μsFn

Ff

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Team HomeworkDUE: Beginning Next class – Bring to ClassDo not turn-in to TA Mailbox

● Derive equations need for the Model● On engineering paper● Use problem presentation method● Draw and label sketches● Be prepared to share your derivation with the class

● List any questions regarding the Model. I will answer them during next class.

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INSTRUCTOR NOTE

● Warning – Long lecture. May not get through it all.

● I recommend giving out the model at the beginning of class.

● That way the students know what equations they will need to derive and will hopefully take notes.

● Homework for this class are the derivations.● The next lecture is designed to have the

students put their derivations on the board.

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INSTRUCTOR NOTE

● Point out that they already (almost) have the equation for Ftmt.

● Warn them about the differences in how theta is defined (see slide).

● Also you may want to warn them that string tension is one of the harder equations to derive.

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As a Team

● What is the tension in the string, Ts?

Farm

Farm

FarmTs

Ts

x

y

x

y