Dr. Terry G. Anderson

Mathematical Sciences Department
Appalachian State University
Boone, NC 28608
tga@math.appstate.edu

Thomson's Jumping Ring: A Nonlinear Integro-Differential Equation Model

Thomson's Jumping Ring is a common laboratory experiment in undergraduate physics courses (see, e.g., UC- Berkeley's and NCSU's WWW pages of physics demos and the Amateur Scientist article in references). 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 the cooled copper ring due to forceful ricochets off the ceiling!)

A mathematical model which connects the current i(t) and the height z(t) of the ring is developed from Faraday's, Lenz's, and Kirchoff's Voltage Laws, and from 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 the 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.

Thomas L. Marchioro II

uces_info@krellinst.org

17 July, 1997