Thinking Like an Engineer An Active Learning Approach

Thinking Like an Engineer An Active Learning Approach, 2 e SOLVEM Instructor Slides Thinking Like an Engineer 2 e SOLVEM Stephan, Bowman, Park, Sill, Ohland Copyright © 2013 Pearson Prentice-Hall, Inc. 1

SOLVEM Methodology • L = List • V = Variables and constants • E = Equations • M = Manipulate Thinking Like an Engineer 2 e • O = Observations and / or Objectives SOLVEM • S = Sketch 2

Example SOLVEM_1 from a circular piece of material 50 centimeters in diameter? Thinking Like an Engineer 2 e equilateral triangle that can be cut SOLVEM What is the area of the largest 3

Example, continued SOLVEM_1 60° L Thinking Like an Engineer 2 e 120° H SOLVEM What is the area of the largest equilateral triangle that can be cut from a circular piece of material 50 cm in diameter? 4

Example, continued SOLVEM_1 • Objective: 60° • Observations: L • Each angle of the triangle is 60° • Each side of the triangle is a chord of the circle • The angle subtended by one of the chords is 120° (360°/3) SOLVEM 120° H Thinking Like an Engineer 2 e • Determine the area of an equilateral triangle inscribed in a circle 5

SOLVEM_1 Diameter of circle 50 cm • R Radius of circle • φ Internal angle of equilateral triangle 60° • θ Angle subtended by chord 120° • L Length of chord (base of triangle) • H Height of triangle • A Area of triangle SOLVEM • D Thinking Like an Engineer 2 e Example, continued 6

Example, continued SOLVEM_1 • Radius of circle: A = 0. 5 L H • Length of chord: L = 2 R sin(θ/2) • tan(φ) = length of opposite side / adjacent side H/(L/2) for the right triangle defined by half of the equilateral triangle Thinking Like an Engineer 2 e • Area of triangle: SOLVEM R = 0. 5 D 7

Example, continued SOLVEM_1 Thinking Like an Engineer 2 e • Height of triangle: H = L tan(φ)/2 H = (1. 73 R) tan(60°)/2 H = 1. 5 R • Area of triangle: A = 0. 5 (1. 73 R) (1. 5 R) A = 1. 30 R 2 SOLVEM • Length of chord: L = 2 R sin(60°) = 1. 73 R 8

Example, continued SOLVEM_1 A = 1. 30 R 2 A = 1. 30 (25 cm)2 A = 811 cm 2 Thinking Like an Engineer 2 e R = 0. 5 D R = 0. 5 (50 cm) = 25 cm SOLVEM • NOW, plug in numbers! 9

Example SOLVEM_2 You purchase a truck having wheels and tires such that each these tires. Inspired by a monster-truck event, you decide to jack up your truck such that the wheel-tire radius is 20 inches. Identify the effect this has on your speedometer reading. Specifically, if your speedo-meter indicates that you are traveling at 60 miles per hour, what is your actual speed? Thinking Like an Engineer 2 e The speedometer, which reads accurately, is calibrated for SOLVEM wheel-tire combination has a radius of 14 inches. 10

Thinking Like an Engineer 2 e SOLVEM Example, continued SOLVEM_2 11

Example, continued SOLVEM_2 • The number of wheel revolutions per time determines the speedometer reading. • Since more distance is covered per revolution for a bigger tire, the truck will be moving faster than the speedometer says it is in this case. • Since we are comparing one tire to another, I should be able to solve this problem as a ratio, without having to worry about unit conversions. Thinking Like an Engineer 2 e • Assuming the tires do not slip, for one revolution, the tire circumference will be the distance traveled by the truck. SOLVEM • Observations: 12

• R wheel radius 14 inches, 20 inches • Vtire the truck velocity for a particular tire • t time • revtire revolution distance for a particular tire • tire radius SOLVEM_2 Thinking Like an Engineer 2 e Example, continued 13

Example, continued SOLVEM_2 • rev 14 = circumference of wheel 14 • rev 20 = circumference of wheel 20 Thinking Like an Engineer 2 e • V 14 = (rev 14 / t) • V 20 = (rev 20 / t) SOLVEM • Circumference of wheel = 2 π R 14

Example, continued SOLVEM_2 t = K (rev 20 / V 20) • Step 2: Since unit time is the same in each case, equate the two expressions: (rev 14 / V 14) = (rev 20 / V 20) V 20 = V 14 (rev 20 / rev 14) Thinking Like an Engineer 2 e t = K (rev 14 / V 14) SOLVEM • Step 1: Solve Equations 2 & 3 for time (t), which is unimportant since we are finding a rate. 15

• Step 3: Substitute for circumference (rev 20 /rev 14) = (2 π R 20) / (2 π R 14) = (R 20 / R 14) • Step 4: Now we can use ratio of radii instead of ratio of velocities V 20 = V 14 (rev 20 / rev 14) = V 14 (R 20 / R 14) SOLVEM_2 Thinking Like an Engineer 2 e Example, continued 16

Example, continued SOLVEM_2 V 20 = 86 mph Thinking Like an Engineer 2 e V 20 = V 14 (R 20 / R 14) = (60 mph) (20 in / 14 in) SOLVEM • NOW plug in number! 17