Engineering Problem Solving ENG1101. Physics 101 For The Design Project Mathematical Model

Transcription

1 Engineering Problem Solving ENG1101 Physics 101 For The Design Project Mathematical Model 1 1

2 Last Time Macros 2 2

3 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 3 3

4 Today's Agenda Mathematical Model Newton's Laws Forces Gravitational Drag Spring Friction Moments Rotational Motion 4 4

5 Due: 9B Derivations Design Project Mathmatical Model Due: 10B Spreadsheet with VBA Code 5 5

6 Equation Derivations Due: 9B Use Problem Presentation Method for EACH equation derived. Problem Diagram Theory Assumptions Solution Result 6 6

7 Diagram of Vehicle α Lext Ltrap β θ 7 7

8 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)? 8 8

9 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 9 9

10 Rotational Motion & a Mousetrap Vehicle Distance Traveled: Relate Length of string pulled to distance traveled

11 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 arc r = radius θ = angle in radians (1 revolution = 2π radians) Circumference: distance around a circle C= 2rπ C = circumference r = radius π =

12 As a Team... List assumptions and known equations. Write an expression that describes the arc length S a with relation to the arc length S b, as the wheel rotates through an angle θ R a b R b θ a

13 As a Team What are Newton s 3 Laws?

14 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

15 Newton's Second Law The 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 F net =ma As a vector expression: Σ F x = ma x Σ F y = ma y Σ F z = ma z Common applications Force required to move an object Projectile trajectories Forces on springs

16 Newton's Third Law... If two objects interact, the force F 12 exerted by object 1 on object 2 is equal in magnitude to and opposite in direction to the force F 21 exerted by object 2 on object 1. F 12 = -F 21 For every action there is an equal and opposite reaction. Common applications Analyzing forces acting on objects Analyzing gravitational attraction between bodies

17 What forces are acting on an object? Gravitation: attraction between two objects F Spring: results when a spring is compressed or stretched F = = ma kx Drag: exerted when an object moves through a fluid (liquid or gas) F C D A v2 Friction: two solid surfaces contact each other F = F normal

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

19 Caution Spring Angle Measurements When you made measurements, you were "opening" spring. (θ) For your vehicle design, your spring will be "closing". (β) Newton Angle meas= θ 180 θ = β String

20 As a Team Work on Derivation of Ftrap Hint: Think Spring Analysis Ftrap

21 Moments 1) When turning a wrench, which point will take less effort? A or B 2) 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 Moment = Force x Distance from axis M = F x D p C D A B

22 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? 7 ft x=? lb f B 120 lb f A 3 ft

23 As a Team... M = F x D p Using the terms given, write an equation for F B if it were to produce the same moment as FA about the axis. D B F B F A D A B A

24 As a Team Work on Derivation of F arm F arm F trap

25 Moments & a Mousetrap Vehicle Extended Mousetrap Arm F Arm = F trap * L trap /L Arm F arm F trap

26 Tension in the String F arm is pulling the string What is the tension in the string, Ts F arm

27 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 ΣF =

28 Break F 2 into components F y = F sin (30 o ) 30 o F F x = F cos (30 o )

29 Break F 2 into components F 2y = F 2 sin (30 o ) 30 o F 2 F 2x = F 2 cos (30 o ) 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

30 As a Team What is the tension in the string, Ts? F arm y T s x

31 Motive Force Using Moments, what force (F wheel ) produces the equivalent moment as T s? Ts F wheel F mot

32 Friction & a Mousetrap Vehicle Air Resistance: neglected Static Friction: F fs = μ s F n Kinetic Friction of Vehicle: F fk = μ k F n

33 F f = μf n = Force due to Friction Where: F f = force due to friction (N) F n = force perpendicular to friction surface (N) μ = coefficient of friction Static or Kinetic F f

34 Coefficients of Friction Coeff. of Coeff. of Materials Static Friction Kinetic Friction μs μk Steel on Steel Aluminum on Steel Copper on Steel 0.53 Rubber on Concrete Wood on Wood Glass on Glass Waxed wood on Wet snow Waxed wood on Dry snow Metal on Metal (lubricated) Ice on Ice Teflon on Teflon Synovial joints in humans Source: Serway Physics for Scientists and Engineers 4th edition (p. 126.) What are the model's coefficients comparable to?

35 What if you are pulling a box? Resistance the box has to being moved Kinetic Friction slows an object down F fk = μ k F n Static Friction prevents an object from starting to move or keeps an object stationary F f F fs = μ s F n

36 Team Homework DUE: Beginning Next class Bring to Class Do 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

37 INSTRUCTOR NOTE 37 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. 3 7

38 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

39 As a Team What is the tension in the string, Ts? F arm y T s x F arm F arm y x T s