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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