For this Exercise Set, I have chosen a rigid pendulum, rather than a mass suspended from a massless string, so that angular displacements of magnitude greater than π/2 π/2 can be studied, without concern for the mass remaining at the same location relative to the support. The stable Euler Cromer method is employed; however, the instructor could have the students start out with an “Exercise 0” that asks students to use the simple Euler method to model the pendulum. The Euler algorithm is unstable for oscillatory systems (total energy grows every time step), and this exercise could provide a valuable lesson in the control of artificial behavior in computational models, and the importance of using stable algorithms. Even if this “Exercise 0” were implemented, it might be best if students have already had experience modeling and studying the dynamics of a simpler oscillatory system, such as the Simple Hanging Harmonic Oscillator in the PICUP collection, before encountering the Rigid Pendulum exercise set.

Exercise 1 asks the students to build the computational model of the hanging spring-mass system. Depending on instructor preference, a working version of the computational model could be directly provided to the student; or the student could be required to modify, or add to, a nearly completed computational model; or the student required to build the model from scratch; or really anything in between these scenarios. Another possibility for producing a working model, time permitting, is to build it, or parts of it, together with the students in class. To carry out the subsequent exercises the student must have the working program, and also access to either a plotting program, or a programming environment with built-in graphics capabilities.

The student should be made aware of the analytic solution of the (undamped and unforced) rigid pendulum accessible via the small angle approximation. The second exercise in this set has the students compare this small angle solution to their computational one. Also, the concept of phase space is presented in the exercises as simply another way of plotting and observing the system dynamics. Adding the damping and driving terms in exercises 4 and 5, and guiding the students to explore the behavior for different parameter values, serves as a basic introduction to nonlinear dynamics.

This computational approach to the rigid pendulum has the educational advantage of allowing introductory students to study the dynamics of a nonlinear, chaotic system with relative computational ease.