This page written by
Dan Styer,
Oberlin College Physics Department;
http://www.oberlin.edu/physics/dstyer/WhyHowProblems.html.
last updated 6 August 2001.
Why do I assign problems in physics courses? How can working these problems help you?
Philosophy. Listening to lectures, reading books, running computer simulations, performing experiments, participating in discussions . . . all these are fine tools for learning physics. But you will not really become familiar with the subject until you get it under your skin by working problems. The problems in a physics course do not simply test your comprehension of the material that you learned in the text. Instead they are an important component of the learning experience, designed to extend and solidify your grasp of the concepts and content of physics. Solving problems is a more active, and hence more effective, learning technique than reading text or listening to lectures.
In answering your homework (and exam) problems, you must show your work. That is, you must present your evidence and your reasoning as well as your final conclusion. (This rule holds for all intellectual discourse. For example, suppose you are asked in an English Literature course to consider Melville's influence on American literature. You march off to the library and -- after considerable reading and analysis -- conclude that "Moby Dick changed the entire landscape of the American novel." If you typed up this single sentence and submitted it as your paper, your professor would be unimpressed.) Exactly how much detail should you give in presenting your reasoning? A good rule of thumb is to present enough detail that you could reconstruct your thought processes two or three months later, when you're using your solutions to study for the final exam.
Difficulty. No one should expect to score 100% on homework. Some of the problems are deliberately very challenging. Everyone can use improvement, and the problems are a relatively painless way for me to challenge you and show you how to improve. I assign problems that will expose you to many fascinating phenomena and useful devices. The exams I set for you are much easier because they serve an entirely different purpose. Homework is a way for me to show you some of the many vistas of physics that you don't know, whereas exams are a way for you to show me the many aspects of physics that you do know. Before each exam I will distribute a practice exam so that you will have some idea of what to expect in terms of length and difficulty.
Warm up exercises. A problem arises from my assigning such interesting homework problems. They tend to be harder than plodding, mechanical problems, and in particular, they tend to need many steps in their solution. Hence I will, for many assigned problems, suggest "warm up exercises" that are more mechanical, simpler, and involve fewer steps than the assigned problems. Working the warm-ups is not required, and if you do work them you should not hand them in. But if you find that a particular assigned problem is too difficult, then try the associated warm-ups. Working them should give you practice that will be directly relevant in helping you solve the associated assigned problem.
Model solutions. I distribute model solutions to the homeworks on the day that I collect them from you. My model solutions are only that: models. You might have solved the problem in a completely different way that is actually superior to the way I chose. I urge you to scan the model solutions on the day I give them out . . . the problems will be fresh in your mind and you'll learn from my solutions more readily. If you find that you solved the problem (correctly!) in a way different from mine, then please do let me know about it. One of the great joys I find in teaching is to learn from my students, and it happens more frequently than you might think.
Sample problem. For concreteness, many of the tips below refer to the following problem (Halliday, Resnick, and Walker, Fundamentals of Physics, sixth edition 22-9):
Two free point charges +q and +4q are a distance L apart. A third charge is placed so that the entire system is in equilibrium. (a) Find the location, magnitude, and sign of the third charge. (b) Show that the equilibrium of the system is unstable.
Explain. When you write up solutions to problems, be sure to explain your reasoning. Don't just give me the final numerical answer or the end formula . . . I already know what it is! Instead I'm interested in seeing how you overcome the roadblocks that get in your way as you progress through the problem. An appendix in the text by Halliday, Resnick, and Walker lists "Answers to odd-numbered questions, exercises, and problems". Be aware that these are merely skeleton answers, and I am interested in a full solutions, like the model solutions that I hand out to you or that you can find in the "sample problems" of the text. The benefits that accrue from active problem solving come only if you supply the reasoning yourself. The "answers" at the end of the book will help you learn physics if you work through the problem yourself and then use the skeleton answer to check your reasoning. If you instead look up the answer before attempting the problem, the "answers" section will actually be an impediment to your learning.
The explanation does not need to be terribly long or detailed, but it must exist if you are to earn full credit. For example, in the sample problem the third charge must be located on the axis: Otherwise the total force on the first two charges would have a vertical component and hence could not equal zero. You can get away with a statement as simple as "The equilibrium position is on the axis", but you can't just omit it.
Map out the logic. This point is similar to the one above. If you are using Coulomb's law to find the force between two particles of charge +q and +4q separated by a distance L, then you might say "F = k q1q2/r2 = k 4q2/L2", or "From Coulomb's law, F = k 4q2/L2", but don't just say "k 4q2/L2" without any indication of what you are calculating or how you are calculating it. (Note, however, that it is perfectly all right to have a quantity like the above pointing to a force arrow in a diagram. In that case the diagram, rather than words or equations, provides the context.)
Diagrams and definitions. In almost all cases, the first step in solving a problem is to draw a diagram showing the geometry of the situation. The diagram will organize your work and point out ways to proceed.
The diagram is often a good place to define variables that you will need as well. Using the sample problem as an example again, remember that you need to find the location of the third charge. Saying x = L/3 is not an answer unless you have first defined x to be the distance from the charge +q rather than the distance from the charge +4q. The easiest way to do this is through a diagram.
Do full problem for full credit. The sample problem has two parts. Part (a) is clearly the main part of the problem, but it's not the only part of the problem! For full credit, you must do part (b) as well. If a problem asks you to derive an equation and discuss the result in the limit that the charge vanishes, then you have to supply the discussion to get full credit. Sentences count just as much as equations and numbers!
Do partial problem for partial credit. If you can't solve a problem completely, then hand in a start. If you have a plan for solving the problem but can't execute it, then hand in the plan. If you can't find the magnitude of the electric field but can find its direction, then tell me the direction. If a problem has two variables x and y, and you can solve it only in the case that x = 2y, then hand in the solution for the special case that you have solved. You will even get some points for saying nothing more than "This problem can be solved using Coulomb's Law, but I can't figure out the details." In the world of physics research, it often happens that the questions change as you work on the answers, and you can use the same philosophy in this course. (If you follow this advice and solve a problem related to but different from the one that I assigned, then please point this out explicitly in your solution.)
Citation. If you use a specialized result from your text, then cite it. No need to cite momentum conservation, but there is a need to cite an equation giving the final velocity of the target particle in a one-dimensional elastic collision with the target initially stationary . . . now there's a specialized result!
If you can't solve the problem then at least do something: sketch the situation and define a few relevant variables. State a relevant principle. If you come up with a silly result (e.g. a negative kinetic energy), then tell me that it's silly and you've earned a point (or two). If you run out of time, then write a sentence about how you would solve the problem if you did have time. Writing down your thoughts can clarify them and lead you to your goal. If nothing else, they might earn you points.
Mechanics. The problem sets are graded by a student working under my close supervision. I have the final say on your homework grade, so if you feel that the grader has been unfair or arbitrary or wrong, then see me and I might change your grade. (In practice, however, I rarely do so because I give the grader very detailed instructions.) Sometimes I will ask the grader to look at only about half the problems that I assign to you. This way, he or she can go through the sets quickly and get your solutions back to you soon enough that you can learn effectively from them.
Your solutions do not need to be obsessively neat, but they do need to be legible . . . particularly your name! One former grader for this course said "If I can't read it, I can't give you credit." I know of no one (certainly not myself) who can solve the problems in this course on the first shot: you'll need to work the problems first in rough draft and then copy out a version to submit for grading. The copying out is not just for neatness. It helps you consolidate your thoughts and brings the logic of your solution into focus. Please staple your problems together, for otherwise the pages are likely to become separated from each other and you will get credit for only the first page of your answers.
The problems in a physics course are not dry appendages designed to keep you indoors on sunny days. They are exciting, dynamic, and central to the course structure. Enjoy them!