MTE 4050 Physical Metallurgy – Mechanical Properties Fall 2023 How to Solve It: Guidance on Solving Problems Introduction Engineering is primarily about solving problems. Sometimes, the solution to a problem is a very practical one: changing the settings on an instrument, repairing/replacing a component, adopting a different process. Often, an integral part of technical problem solving is developing a mathematical description of the circumstances; this type of description can inform the decisionmaking process along lines that are very rigorous and verifiable. For this reason and others, the ability to solve mathematical problems is a crucial engineering skill. Taking an engineering course is an opportunity to practice problem-solving skills. The problems assigned for a class are intended to develop your thinking and methods when it comes to any such problems, not just those specifically related to this class or the topics involved. Solving a problem requires the generation of a solution. This is obvious, but a good solution has qualities that we can easily list. A solution to a problem should: • • • Supply necessary clarifications to the problem. Demonstrate an understanding of the concepts and methods involved. Communicate what has been done and why. Any written solution, when read by you, other classmates, or the instructor/grader will convince the reader that the author understands exactly how the answer was obtained, and what its ramifications might be with respect to the original problem, based on what is written alone.1 Answers can be • • • numerical answers with or without units, yes/no answers, or answers in the form of a selection from a menu of possible choices. The answers at the back of the textbook are not solutions. Constructing a good solution is not necessarily straightforward, and the ingredients of a good solution can differ from topic to topic. Because of this, there is no truly universal procedure for generating one. However, there are useful heuristic approaches that can help organize your thinking.2 The heuristic summarized below is my interpretation of a well-known one developed by George Pólya at Stanford.3 1 This sentiment is frequently expressed with a terse command to “show your work”. A “heuristic technique” or “heuristic” is a methodical approach to problem solving that is not guaranteed to be optimal or perfect, but only to guide you through the process. 3 How to Solve It 2nd ed. (Princeton University Press, Princeton, NJ, USA), 1957. 2 J. B. Puthoff Cal Poly Pomona MTE 4050 How to Solve it (0. Assemble materials and motivation. Get your paper, pencil, book, calculator/computer, software, tolerable workspace, etc. Take a moment and remind yourself that taking the time to find the solution will expand your capabilities in your chosen field. You have to want to know what the solution is!) 1. Understand the problem. You will not be able to solve a problem you do not understand. You might be able to obtain an answer that is numerically correct through some trial and error, but a solution to the problem should indicate that the answer follows from a verifiable procedure. For this reason, you must understand every step. When attempting to grasp the problem, try the following a) Read the problem statement. Focus less on numerical values of parameters that it provides and more on the words. What is the problem asking? What concepts are involved? Recognize that if the problem has multiple explicit parts, you will require multiple solutions, each with their own answer. b) Reread the problem statement, this time focusing on the data. This likely consists of numerical values for various parameters. Sometimes the variable (or mathematical symbol) associated with a parameter is provided, sometimes it is not. c) Make a figure. Sometimes, drawing a simple figure or diagram can clarify certain aspects of the problem: variables you don’t know, relationships between variables, etc. 2. Devise a plan. After you have identified the central question, recognized the given variables and their values, and maybe made a figure, come up with some strategy that leads you to the answer. a) Think of other problems. You may have seen this problem before or you might have solved one that is similar. Maybe find an example problem from the course materials whose solution you could emulate. Check the unknown variable! b) Think of equations that you know that involve the variables. Sometimes, one or more are stated along with the problem; these are not necessarily the only ones involved! c) Next, check assumptions given. These might inform you that there are or are not specific relationships between certain variables or that some variables have little influence (or be equal to zero). As you develop your plan, identify where these assumptions might be introduced. 3. Carry out the plan. As you do this, ask yourself: Can you verify at each step that the partial solution is correct? Do you understand why in each step that you used the equation or assumption that you did? Maybe justify each step using a short sentence or phrase explaining why this step is required at this point. J. B. Puthoff Cal Poly Pomona MTE 4050 4. Examine the solution. Cast a critical eye over your solution, thinking about how the various steps individually and the overall solution address the problem. a) Check completeness. Did you use all of the data? Did you use all of the assumptions? b) Check against expectations. Given what you know about the data and your previous exposure to similar problems, assess whether the magnitude of the final answer makes sense. Did you expect something large or something small, and did the answer conform to your expectations? Check the units of the answer. Do they make sense? If you obtained a yes/no answer, does the answer agree with your instincts? Why or why not? c) Identify unnecessary steps. Were all of the steps strictly necessary? Is there a different way to obtain the correct answer that requires fewer steps? d) Apply the solution elsewhere. If this is a problem that has multiple parts, can you identify a way that the solution of this part can be applied to the others? J. B. Puthoff Cal Poly Pomona MTE 4050
