We have found that incorporating computer programming into introductory physics requires problems suited for numerical treatment while still maintaining ties with the analytical themes in a typical introductory-level university physics course. In this paper, we discuss a numerical adaptation of a system commonly encountered in the introductory physics curriculum: the dynamics of an object constrained to move along a curved path. A numerical analysis of this problem that includes a computer animation can provide many insights and pedagogical avenues not possible with the usual analytical treatment. We present two approaches for computing the instantaneous kinematic variables of an object constrained to move along a path described by a mathematical function. The first is a pedagogical approach, appropriate for introductory students in the calculus-based sequence. The second is a more generalized approach, suitable for simulations of more complex scenarios.
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The Lagrangian approach is as follows. We start with kinetic and potential energies of and V = mgy, where y is the object's height. Applying the Euler-Lagrange equation to the Lagrangian , we obtain , where is the functional form of the wire curve. Here, primes (dots) indicate derivatives with respect to x (time). This result is essentially our central result given by Eq. (6), but this approach is not appropriate for introductory physics.
As an example, imagine an object at the top of a hemisphere as in Problem #16 (p. 301) of Ref. 7. The question asks at what position the object will leave the hemisphere as it slides down. This is an excellent problem to test the correspondence between the analytical and numerical solutions to the problem. The numerical solution is particularly insightful, forcing the student to consider how to examine the normal force, with the need to watch for it to become “zero” (i.e., sufficiently small).
For example, see Ref. 7, p. 273.
