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.

This paper uses elementary techniques drawn from renormalization theory to derive the Lorentz-Dirac equation for the relativistic classical electron from the Maxwell-Lorentz equations for a classical charged particle coupled to the electromagnetic field. I show that the resulting effective theory, valid for electron motions that change over distances large compared to the classical electron radius, reduces naturally to the Landau-Lifshitz equation. No familiarity with renormalization or quantum field theory is assumed.

The relationship between transmission resonances and Bloch states for a periodic array consisting of N delta function potentials is discussed. The transmission resonances are derived for matter waves incident on the periodic array, while the Bloch states are calculated using periodic boundary conditions for the array. It is shown that approximately half of the transmission resonances map into pairs of degenerate Bloch states. Wave functions are shown for both the transmission resonances and the Bloch states for arrays of five and six delta function potentials. The origin of the band structure of the Bloch states is interpreted in terms of the wave functions and eigenenergies for a particle confined to move on a ring, subjected to a periodic array of delta function potentials on the ring.