Dangling by a Wire?   - by Karl A. Smith Subject Area: Modeling, Problem Solving and Engineering Grade Level: College/High School. Some background in algebra, physics and materials is helpful for solving this problem. College students enjoy the challenge. Instructional Objectives: The academic objectives are for students to develop skills for formulating equilibrium relationships and building models to solve problems. Additionally, they learn about materials engineering. The teamwork skill objective is for students to learn to probe to improve their depth of understanding. Time Required: Approximately 45 minutes. Lesson Summary Teacher Explanation to Whole Class: In this problem we are going to use estimation and modeling to determine the smallest diameter steel wire that could to used to suspend a typical American automobile. Small Group Task: As a triad, students are to: a. Individually estimate the diameter of the smallest steel wire that would support a typical American automobile hanging by it. b. Turn to the group and exchange estimates; discuss strategies for determining a better answer. c. Create a model of the situation and prepare one answer for the group. d. Report to the whole class on the group's answer and model. Teacher Monitoring: The teacher monitors to ensure that all students understand the assignment and are working skillfully. Simultaneous Explaining: After 10 minutes students stop work, find a partner in an adjacent group, and explain their group's model for arriving at an answer. Whole Class Discussion: A member of several different groups should be randomly selected to explain how their group solved the problem. Follow-Up: Each group writes a report on how they solved the problem--the formulation of their model, including the assumptions, sensitivity of the model, and next steps. Pre-Instructional Decisions Group Size: Three. Assignment to Group: Teacher assigned groups if enough time (and background information on students) available, otherwise random. Roles: Recorder: Person in group who has driven the heaviest automobile. Recorder gets copy of problem and records group's answer and method. Group member on Recorder's right is the Prober. The prober asks for rationale and elaboration, and questions the group's assumptions, model, etc. The Encourager makes sure each group member participates in the process. Remember these roles are in addition to each person's responsibility to help the group solve the problem. Materials: None required. Although pieces of fine steel wire and materials science textbooks are helpful. Explaining Task and Cooperation Task: "Individually estimate the diameter of the smallest steel wire that could suspend a typical American automobile." "You have 60 seconds to make an individual estimate. Record your answer and the strategy you used to arrive at your answer. Volunteer your answer so we can determine the range of individual estimates." After 60 seconds, call for volunteers to give their estimate of the diameter. Record their answers. Often the range is quite large, from "the thickness of a hair" to one inch. Ask how a better answer could be found. Probe what quantities should determine how much load a wire can hold. Record the list of quantities. It will usually include the material and the cross-sectional area of the wire and the way the wire was made. "Join with your triad and make a joint estimate. You have ten minutes to do so. Develop a model to use for refining your estimate. Record your answer and your model." If (or when) students say they're done, ask them if they would be willing to hang a car by their wire from a helicopter hovering at 5,000 feet. After 10-15 minutes ask students to stop working, find a partner in an adjacent group, and explain their group's answer and model. After 5-10 minutes, ask the students to pause and randomly call on individuals to present their group's answer and method. Record the diameter and their method on the overhead. Explore the relationship between the load and the cross-sectional area. The bigger the wire is, the more load it should be able to hold. The load in the wire divided by the cross-sectional area is called the stress in the wire. This stress is surprisingly constant across the area. By experimenting with various materials, we can find the stress that each can take without breaking, and the resulting values are referred to as breaking strengths, or failure stresses, of the materials. These quantities are not absolute, as different batches of the same material have some slight variation. But they are close enough to be usable by engineers. The breaking strengths of various steels range from 60,000 pounds per square inch (psi) of area to 200,000 pounds per square inch. Let's assume that the steel wire we're using has a breaking strength of 100,000 psi (ascertained by calling the manufacturer, or testing a piece ourselves). Let's examine a little section at the top of the wire, and derive a relationship between the quantities of interest. At equilibrium, the forces pulling up on the wire must be the same as the forces pulling down. Furthermore, let's assume that the weight of the wire is very much less than the automobile's weight, so we will ignore it. The equilibrium relationship tells us that sA = W    s = W/A    s = W/ r2 where s = stress = load/area; A = area ( r2 for a circle); W = automobile weight For a breaking stress of 100,000 psi, and a load of 2000 lbs, this relationship will tell us that for our example, the diameter is 0.16 inch, or slightly over 5/32 inch, since Would you actually go out and hang by this wire? No way for me! First or all, any flaw in the wire or error in my assumptions would drop me from the helicopter. Second, I would worry about how steady a platform the helicopter was and how the attachments might weaken the wire. I would definitely want a safety factor. How big a safety factor would be a matter of judgment, since it would depend upon how much I value life, money, and so on. However, if I could design the attachment, if I trusted the weather and the helicopter pilot, and if the manufacturer assured me that the strength of the wire was at least 100,000 psi, I might be talked into hanging on a wire with a diameter of 5/16 inch, since the cross-sectional area would be about four times as large. What steps would you take to refine your answer? For example, were we justified in ignoring the weight of the wire? Let us call the length of the wire l inches. Our relationship now becomes s= [W + ( x r2 x l x D)]/A (where D is the density of the wire. For steel, D = 0.283 lbs/in3). Let us assume that our wire is 1/4 mile ling. The diameter would have to be 0.17 inches. Not much change in diameter from before. We were probably all right in neglecting the weight of the wire Explore the effect of other assumptions and parameters. For example, what would the diameter of an aluminum wire have to be, or what is the effect of acceleration (Could you jump out of the helicopter?) Ask the students to summarize their major learnings, and to think of similar situations where their learnings apply. Criteria for Success: Sixty-second answer from each individual, and signed answer sheet from each group which contains smallest diameter steel wire and a description of their method. Positive Interdependence: The cooperative goal is for triads to decide on one answer and model which all members can explain. The intergroup goal is for triads to check each other's work and provide any suggestions and assistance other triads need to obtain a better answer. Individual Accountability: Teacher observes each triad at work. Partners check each other's work. Another triad checks each triad's work. Teacher randomly selects individual members of the groups to present. Expected Behaviors: Everyone participates in group discussion. Each member of group can explain answer and method. Intergroup Cooperation: Whenever it is helpful, check procedures, answers, and strategies with other groups. Monitoring and Intervening Monitoring: Circulate among the groups to check that everyone is participating and that roles are being followed (only one person recording, one person probing, and one person encouraging.) Monitor to catch misunderstanding in the early stages and to help establish the rules. Intervening: Remind groups that all are expected to participate, and understand and be able to explain their formulation and solution. Model and coach checking by asking questions of randomly selected individuals (Roger's favorite--individual oral exam). Evaluating and Processing Evaluating: Each student shares the triad's answer and model with a member of another triad. Processing: Remind groups that every member has two functions: complete the task and maintain good working relationships. Ask groups to discuss effectiveness by individually listing two things that went well and one thing the group needs to work on. Quickly survey the groups and list the things that went well and the things that need work. Closure: Summarize the lesson. Discuss the key features of modeling - assumptions, representation, sensitivity, resolution (how good an answer is needed), etc. Problem adapted from Adams, J. L. (1991). "Flying buttresses, entropy, and o-rings: The world of an engineer." Cambridge, MA: Harvard University Press.

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