### Dr. Terry G. Anderson

#### tga@math.appstate.edu

Research / Teaching Interests:
Mathematical Modeling, ODEs, PDEs, Integro-DEs, Difference Equations, Numerical Methods.

Mathematical Chemistry Research Paper: Electrogenerated Chemiluminescence ## A Nonlinear Integro-Differential Equation Model

### Introduction

Thomson's Jumping Ring is a common laboratory experiment in undergraduate physics courses. A conducting metal ring (typically copper or aluminum) is placed over an iron-cored solenoid connected to a 60 Hz, 110 V AC outlet. When current flows through the solenoid, the ring will jump several feet into the air. If the ring is first cooled with liquid nitrogen, it will jump higher due to lowered electrical resistance. (Caution is advised for cooled rings due to forceful ricochets off the ceiling!) For examples, see any of the following physics demonstration sites:

A mathematical model which connects the current i(t) and the height z(t) of the ring is developed from Faraday's Law, Lenz's Law, Kirchoff's Loop Rule, and Newton's Second Law of Motion. Upon elimination of i(t) from the system, the result is an initial value problem which involves a second-order nonlinear integro-differential equation for z(t): Coefficients in this equation depend upon the initial acceleration a(0) = z''(0), resistance R, inductance L, and height h of the iron core. The dependence of z on time and a(0) is investigated numerically via the Trapezoidal Rule and a fourth-order Runge-Kutta algorithm. A computer algebra system (Maple) is used for all computations and plots. The worksheet environment allows for easy explorations in changing parameters and requires only minimal programming in a high level language.

Projects for further study are suggested: extension of the model to include dependence of the inductance L on the height z of the moving ring; use of alternative numerical methods (e.g., successive approximations, series solutions, or adaptive methods); rewriting the integro-DE as a system of three first-order ODEs; and parameter dependence / sensitivity. 