Edward F. Redish
and
Jack M. Wilson
This equation, unfortunately, requires advanced special functions for its analytic solution.[12] For small amplitudes, the equation becomes
directly equivalent to the simple harmonic oscillator equation. Essentially all introductory texts give both these equations. The first is ignored except for the construction of the correct form of the energy. Dynamics problems are done with the second equation.
In the M.U.P.P.E.T. class, we are able to consider the large amplitude equation in more detail. Because we are solving Eq. (1) numerically, we have the situation shown in Fig. 3.
Many students assume that, because the analytic expression can be expressed in closed form, it is the "better" solution. We can bring them to a dramatic contradiction of this viewpoint by asking them to consider the analytic and numerical solutions for the cases:
In the first case, the correct (numerical) solution goes "over the top", spinning round and round the pivot. The analytic solution goes over the top some distance, stops in midair, turns around, goes back over the top. The second case is a small angle oscillation, but the "analytic" solution doesn't recognize this. Instead of falling and oscillating through a small angle, it rises over the top, oscillating back and forth nearly a full circle each time.
The explanation of these strange results is fairly simple. The small angle approximation doesn't hold for large angles, so the analytic form shouldn't be applied. But these simple examples illustrate a principle of great importance.
When approximate solutions are extrapolated beyond their realm of validity they can give results that are qualitatively wrong.
Again, this result is obvious to the professional physicist, but there is essentially no example of this important result anywhere in the traditional introductory curriculum. If our students are to learn the art of approximation, they must have simple touchstone examples that clearly illustrate the possible pitfalls.
One additional point is important and illustrative about this case. Note that the mathematics of solving the approximate equation is simpler. But from the students' point of view, the ideal pendulum requires one extra logical step to set up its equation of motion and is therefore conceptually more difficult than the realistic one. Indeed, many students have a poor understanding of the small angle approximation. With M.U.P.P.E.T., we can discuss the exact case first.
A combination of dimensional reasoning and symmetry principles can be used to construct the Newton drag law:
where etais a dimensionless parameter. This has interesting consequences[13] and further discussion can be given during the section on kinetic theory.
We then use a M.U.P.P.E.T. program to study the behavior of an object under the influence of this force. The total amount of programming required from the student is to put in the equation for the force law into the program Proj1D. The output of this program is shown in Fig. 2.
The students can be asked to carry out an interesting mix of qualitative and quantitative analyses. In studying the qualitative behavior we can ask the student the following questions:
Sample problem 1: A 10 gm sheet of paper is crumpled up into a compact ball with a radius of 4 cm. When dropped, it takes 1.0 sec to fall a distance of 2 meters. Use this to determine the air resistance coefficient b in the force law
. If a wooden and a steel ball are dropped from the same height, how long would they each take to fall? What accuracy would you need in your measurements to see the difference in the rates of fall between the wooden and steel balls?
Sample problem 2: A ball of mass 0.14 kg is thrown straight up with a speed of 20 m/s. It comes down 1 second earlier than expected, if air resistance is ignored. Find the air resistance coefficient b for this object if the force has the form
Find the coefficient g if the force has the form
Design a simple experiment (with numbers!) using this ball to determine which force gives a better description of the real world.
In principle, classical systems are totally predictable once starting conditions are specified.
However, it is equally important that the student understand the contemporary lesson of chaos theory which emphasizes that:
In practice, it is usually impossible to predict the long-term motions of any classical system with a finite calculation since they are highly sensitive to the starting conditions.
In the M.U.P.P.E.T. course, the presence of the computer allows us to include a segment on chaos theory as a natural extension of Newtonian dynamics at the end of the first semester. The students find this topic of great interest, and many of them choose some non-linear problem as a research topic in the second semester. A screen from the M.U.P.P.E.T. sample program Iterate is shown in Fig. 5. This program illustrates the phenomena of bifurcation, period doubling, and repeatable randomness. The code for this program contains only 60 lines, many of which can be clipped from the basic sample programs.
When one of us (EFR) taught the M.U.P.P.E.T. course, he required project work of every student in every semester. An earlier attempt to require projects of sophomore physics majors in 1970-72 had not been successful. Only about 15% of the students were able to do projects that had any characteristics of normal scientific research. Most students were severely hampered by not having sufficiently strong analytic and mathematical skills to carry through an open-ended investigation. However, in the M.U.P.P.E.T. environment with freshman majors in 1986-1989, when each student had access to a computer and the M.U.P.P.E.T. tools, the results were strikingly different. About two thirds of the students were able to do valuable and interesting projects.
The students were told to seek a topic they were interested in and would like to know more about. It had to have some relation to the content of the course, although we tended to be flexible if a student showed strong interest in some other topic. They were told that this was not to be a project where they read, organized, and replayed other people's materials. They were supposed to design their own project, carry out an investigation, and write a report.
In an ideal project, we believe that the student should carry out the following activities:
Some of the subjects investigated by students in conjunction with our calculus-based physics course include:
At universities with very high admission standards, it is well known that freshman can do independent research (viz. winners of the Westinghouse competition). What was not widely appreciated before M.U.P.P.E.T. was that freshmen with a wide range of abilities and backgrounds can begin to design their own research projects and carry them out successfully if the physics course is tied to the computer and the students empowered in its use.
Perhaps the most surprising result was the distribution of good projects. Students who would have been identified as mediocre students by their exam grades occasionally did outstanding projects. A careful analysis of these students showed that they had "stylistic" rather than content problems. They did not perform well under exam pressure, but preferred to work slowly and carefully. Some of them had extraordinary intensity and persistence when they were interested and involved in a project. Others who did well on exams could find no topic at all to interest them and turned in very poor research projects. These observations raise the question whether it is a good idea to use a student's performance on traditional timed hour exams as a "first cut filter" to weed out those who should not be physicists. This automated hoe may be chopping some valuable flowers!
For example, in one of the laboratories, students observed and took data on the motion of a pendulum using a stroboscope and a Polaroid camera. They then used their graphing and data analysis package to plot the observed angle, and to calculate and plot the angular velocity, angular acceleration, kinetic energy, potential energy, and total energy vs. time. They then used M.U.P.P.E.T. to build a mathematical model of the system on a template we provided. With their program, they were able to compare the prediction of a mathematical model with their observations and make an estimate of the damping. With this model, students were able to extend their analysis of their investigations to consider large angle corrections, driving forces, and resonance.
Instead of listening to lectures about modeling and error analysis (as had been the previous practice -- even in the lab section of the class!) the students performed the activities themselves using our computer tools. The students' projects were significantly improved in quality and showed a better understanding of the phenomena than when they worked in the traditional mode.
Indeed, some materials were left out that are included in the traditional course. Rigid body motion was suppressed, as was fluid dynamics and much discussion of sound. In our three semester sequence we did not do any modern physics or relativity, in part since the sequence is followed immediately by a full two semester sequence on modern physics.
The additional materials, however, did not take very much time to include. The programming handouts were read in parallel with the standard reading, a few pages per week. Less than 5% of the lecture time was used to discuss programming. Two to three lecture hours per semester were actually spent in the microcomputer laboratory with the students getting started on some of the more computer oriented homework assignments.
From this small basis of computer instruction, we were able to include one to two homework problems per week that were somehow associated with the computer. (These were occasionally estimation, analytic, or qualitative problems.) These were done at the cost of reducing the number of standard "plug-and-chug" problems the students were assigned.
The University of Maryland is a reasonably typical university environment. The College Park campus of the University of Maryland is a large state university with a large student body having a wide range of interests, backgrounds, and levels of ability. Materials developed and tested at Maryland should be usable at many campuses across the nation.
M.U.P.P.E.T. at Maryland has not been restricted to its developers. The M.U.P.P.E.T. course for physics majors has been and is being taught at Maryland by faculty not involved in the development of M.U.P.P.E.T.
New courses have been developed with M.U.P.P.E.T. materials in Australia.
Prof. Ian Johnston at the University of Sydney became acquainted with the M.U.P.P.E.T. idea at the Raleigh Conference on Computers in Physics Instruction in the summer of 1988.[18] Since then, he and his colleagues have developed materials for second year physics majors on numerical methods and quantum mechanics. In 1989-90, he used the M.U.P.P.E.T. environment to integrate computational physics into the undergraduate courses at Sydney University.[19]
Johnston tested M.U.P.P.E.T. in three successive semesters. In the first test, 18 volunteers, chosen from a class of 200 second-year students, were given six four-hour microlab sessions in addition to the normal work in a course in quantum mechanics. In the second test, this was repeated with 92 students out of a class of 202. The third test involved 24 students in the third year class. These students were asked to work through four computer modeling problems in diverse areas of physics including solid state, kinetic theory, plasma physics, and Fourier transforms. Johnston and McPhedran conclude:
(1) Students do not need to be able to program before they can handle these materials. Students who had no previous programming experience (about 25% and 16% of the students in the first two trials) had to work harder at first, but had little trouble once they got started.
(2) The students' understanding of a number of traditional subjects was significantly improved by adding computer modeling problems as shown by grades in a comparison of the students in the test and traditional groups on traditional tests. This was because the computer programs allowed students to explore many more cases than they could by hand. For example, because the students in these trials had seen the shapes of many different wave functions, they could easily answer questions in ordinary texts dealing with the geometrical property of eigenfunctions and performed significantly better than students in the traditional group on such questions in exams.
(3) In the last microlab, students were asked to investigate broadly posed problems. They were able to successfully design and complete projects in a one week period, thanks in a large part to the ease of building models with the M.U.P.P.E.T. software package. This confirms our experience with projects at Maryland.
We conclude from these experiences that the M.U.P.P.E.T. environment is robust and survives being transferred to other users.
Our conclusion is that M.U.P.P.E.T. works well for majors in small classes. We have not yet tested whether these methods can be extended to large classes with other scientists such as chemists and engineers. It may be possible if the infrastructure exists to provide students with sufficiently accessible networked computer resources. Success in this environment could also be aided substantially by good coordination with other departments.
The Consortium for Undergraduate Physics Software (CUPS) is a project based at George Mason University and funded by the NSF to add computers to upperclass physics courses. A group of 27 physics faculty with software design experience are developing six manuals to accompany upperclass physics courses. Each manual contains nine simulations, each of which will add an element of new physics, not easily includable without the computer.
The Comprehensive Unified Learning Environment[20] (CUPLE) is a project to bring together in a single unified computer environment some of the successful attempts to reach more introductory physics students and to train them more effectively and professionally. CUPLE is bringing together sophisticated tools for handling graphing, student programming, laboratories, and video with modularized text materials and a database of information. The M.U.P.P.E.T. environment is being upgraded for this project to an object-oriented approach now called Window on Physics (or WinPhys for short). WinPhys is built on Turbo Pascal for WindowsTM and takes full advantage of the Graphical User Interface (GUI) Microsoft WindowsTM 3.
[2] F. Verbrugge, "Conference on Introductory Physics Courses", Amer. J. of Phys., 25, 127-128 (1957) ; "Improving the Quality and Effectiveness of Introductory Physics Courses", ibid. 417-424; F. Bitter et al., "Report of Conference on the Improvement of College Physics Courses", ibid., 28(1960) p. 568-578.
[3] R. L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-Wesley, 1989).
[4] J. S. Brown, A. Collins, and P. Duguid, "Situated cognition and the culture of learning", Educational Researcher, p. 32-42 (Jan-Feb 1989) .
[5] The most precise comparison of theory and experiment occur in quantum electrodynamics, where the g-2 value of the muon and the binding energy of the helium atom can be calculated to more than 10 significant figures. At greater than this level of accuracy, one runs into the problem of virtual production of strongly interacting particles where the theory does not yet exist to permit further improvements.
[6] We are greatful to John Risley for a discussion of this idea.
[7] E. F. Redish, J. M. Wilson, and I. P. Johnston, The M.U.P.P.E.T. Utilities (Physics Academic Softwarem 1994).
[8] We have focussed in this work on the course for physics majors. Some preliminary testing of the use of student programming in large classes with engineering students was begun in the fall of '91. This effort is to focus more on conceptual problems and building up a strong view of how one does physics than on developing professional skills.
[9] These last two libraries were developed by I. P. Johnston at the University of Sydney.
[10] E. F. Redish and Edwin Taylor, "Impulse Mechanics", AAPT Announcer 17(4), 82 (Dec. 1987)
[11] This tends to hold largely for textbook problems. The classic laboratory in which the student measures the value of g to a high accuracy with a pendulum and calculates many corrections provides one of a number of excellent counter-examples to this statement.
[12] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Fourth Edition (Cambridge U. Press, 1952).
[13] E. F. Redish, "The impact of the computer on the physics curriculum", in Computers in Physics Instruction, E. F. Redish and J. S. Risley, eds. (Addison-Wesley, 1990), p. 15-22.
[14] In the sections taught by EFR, students were instructed: "Your paper has to teach the teacher something he doesn't know in order to earn an A." This had the effect of encouraging them to seek advising elsewhere in the department.
[15] The total number of upperclass majors has remained constant at about 50.
[16] J. M. Wilson, "Combining computer modeling with traditional laboratory experiences in the introductory mechanics laboratory for physics majors", AAPT Announcer, 17(2), 80 (May, 1987).
[17] Ian Johnston, Sydney University, private communication.
[18] The Conference on Computers in Physics Instruction, Proceedings, E. F. Redish and J. S. Risley, eds. (Addison-Wesley, 1990)
[19] I. D. Johnston and R. C. McPhedran, "Computational Physics in the Undergraduate Curriculum", submitted to the The Australian Physicist.
[20] J. M. Wilson and E. F. Redish, "The Comprehensive Unified Physics Learning Environment: Part I. Background and system operation", Computers in Physics, 6(2) (Mar/April 1992), 202-209; "..: Part II. The basis for integrated studies", ibid. 6(3) (May/June 1992), 282-286 .
©1994, American Association of Physics Teachers.
U. of Md. Computers in Physics Education HomePage
This page prepared 24. March 1995 by
Edward F. Redish
