From a Thornton and Marion’s textbook problem to physical model: exploring coupled oscillatorsin classical mechanics

Abstract Problem 12-3 of Chapter 12 in Thornton and Marion’s fifth edition of Classical Dynamics of Particles and Systems, calls for the solution of a pair of equations of motion for a system of one-dimensional coupled harmonic oscillators of mass M and natural frequency ω 0. The solution to this problem can be easily found by employing the theoretical approaches delineated in the chapter. However, the problem statement itself contains an implicit puzzle. The text asserts that the coupled equations of motion are derived when a mass m is added to both oscillators, which, in turn, are coupled in a certain way that is not specified. In this study, we demonstrate that if the two original harmonic oscillators with added mass m are coupled by a spring of force constant given by κ 12 = m M ω 0 2 / ( M − m ), the coupled equations of motion of problem 12-3 are obtained. A concise discussion is presented regarding the solutions of Problem 12-3 and an experimental proposal for the implementation of the aforementioned system. The pedagogical implications of the study are discussed.