A unit on oscillations, determinism and chaos for introductoryphysics studentsPriscilla W. Lawsa)Department of Physics and Astronomy, Dickinson College, Carlisle, Pennsylvania 17013!Received 15 September 2003; accepted 19 December 2003"This article describes a unit on oscillations, determinism and chaos developed for calculus-basedintroductory physics students as part of the laboratory-centered Workshop Physics curriculum.Students begin by observing the motion of a simple pendulum with a paper clip bob with andwithout magnets in its vicinity. This observation provides an introduction to the contrasting conceptsof Laplacian determinism and chaos. The rest of the unit involves a step-by-step study of apendulum system that becomes increasingly complex until it is driven into chaotic motion. The timeseries graphs and phase plots of various configurations of the pendulum are created using acomputer data acquisition system with a rotary motion sensor. These experimental results arecompared to iterative spreadsheet models developed by students based on the nature of the torquesthe system experiences. The suitability of the unit for introductory physics students in traditionallaboratory settings is discussed. ©2004 American Association of Physics Teachers.#DOI: 10.1119/1.1649964$I. INTRODUCTIONMany contemporary fields of physics require a knowledgeof quantum mechanics or relativity. For this reason mostcalculus-based introductory physics courses rarely give stu-dents any real insight into emerging fields of research. Thefact that the field of nonlinear dynamics is almost entirelyclassical in nature provides us with an opportunity to givestudents first-hand experience with an active field of contem-porary physics research. For this reason, we have developeda unit on Oscillations, Determinism and Chaos1as a culmi-nating experience for calculus-based introductory physicsstudents as they complete the mechanics portion of theWorkshop Physics curriculum.2,3A. The Workshop Physics ProjectThe Workshop Physics Project began in the fall of 1986with a grant from the Fund for Improvement of Postsecond-ary Education !FIPSE". As a result of continued support fromboth FIPSE and the National Science Foundation, curricularmaterials have been produced including an Activity Guide,4computer hardware and software, and apparatus to help in-structors teach introductory physics without lectures. Themajor objective of Workshop Physics courses is to help stu-dents understand the basis of knowledge in physics as asubtle interplay between observations, experiments, defini-tions, mathematical descriptions, and the construction oftheories. To this end, students use the Activity Guide to makepredictions and observations, do guided derivations, andlearn to use flexible computer tools to develop mathematicalmodels of phenomena.Instead of spending time in lectures and separate labora-tory sessions, students in calculus-based Workshop Physicscourses center their work on the Activity Guide. The fourmodules of the Guide contain 28 units covering topics inmechanics, thermodynamics, electricity and magnetism, andnuclear physics. At Dickinson College students spend 6hours a week in a laboratory environment, and are able tocomplete 27 of these units in two semesters—approximately1 unit each week. Although Workshop Physics studentsspend an equivalent amount of time solving problems anddoing equation verification experiments as those who studyunder the lecture method, they have considerably more ex-perience making observations, collecting data, and usingcomputer tools.B. The role of the Oscillations, Determinism and ChaosunitThe unit on Oscillations, Determinism and Chaos5com-pletes a series of 15 mechanics units that cover kinematics,Newton’s laws, momentum, mechanical energy, rotationalmotion, and simple harmonic motion. Most of the laboratorywork in the final unit on chaos involves recording and ana-lyzing the motion of a physical pendulum that is made in-creasingly complex until it becomes chaotic.In previous units, students gain considerable experiencewith mathematical modeling by using the dynamic graphingcapability of Excel® to fit their data to analytic functions!linear, quadratic, inverse, and sinusoidal". The chaos unitintroduces students to the use of the spreadsheet to modelmore complex systems using the Euler method for numericalintegration. Students also use !but do not develop" aspreadsheet-based second-order Runge–Kutta method to ex-plore other possible behaviors of their chaotic pendulum sys-tem and to test the sensitivity of the system to initial condi-tions.Because an overarching goal of the chaos unit is to ex-plore the viability of Laplacian determinism, the unit servesboth a philosophical and theoretical capstone to the study ofNewtonian mechanics.II. THE CHAOTIC PHYSICAL PENDULUM SYSTEMA. The experimental apparatusThe apparatus that students spend most of their time usingis a physical pendulum consisting of an aluminum diskmounted on the low friction shaft of a rotary motion sensor.This sensor is a digital encoder that transmits up to 1440logic pulses per revolution to a digital interface. When asmall mass is bolted to the edge of the disk and displacedfrom vertical equilibrium, the system becomes a physical446 446Am. J. Phys. 72 !4", April 2004 http://aapt.org/ajp © 2004 American Association of Physics Teacherspendulum #Fig. 1!a"$. Adjustable eddy damping is added bymeans of a small magnet attached to a threaded bolt #Fig.1!b"$.Students can modify the pendulum so that a string,springs, and a driver motor are coupled to it via a small drivewheel attached to the pendulum disk !Fig. 2". For certaincombinations of the springs, disk mass, edge mass, eddydamping, and driver motor frequency, the pendulum be-comes chaotic.B. Commercially available chaotic pendulaIn 1989 Priscilla Laws, Desmond Penny, and Brock Millerbegan developing the chaotic physical pendulum system atDickinson College. We used a rotary encoder developed byRobert Teese and Ronald Thornton and a data acquisitionsystem distributed by Vernier Software and Technology.6After several years of testing in Workshop Physicscourses, personnel at PASCO improved and adapted compo-nents of the Dickinson College apparatus for use with theirown driver motor and data acquisition system. These com-ponents are available for the study of large angle oscillations,magnetic damping, driven harmonic motion, and chaoticmotion.7,8The PASCO pendulum apparatus,