Effective Torsion and Spring Constants in a Hybrid Translational-Rotational Oscillator

Abstract

A torsion oscillator is a vibrating system that experiences a restoring torque given by τ = −κθ when it experiences a rotational displacement θ from its equilibrium position. The torsion constant κ (kappa) is analogous to the spring constant k for the traditional translational oscillator (for which the restoring force F is proportional to the linear displacement x of the mass). An effective torsion oscillator can be constructed by integrating a spring's translational harmonic properties into an Atwood2 arrangement where a disk serves as the pulley for the system and the spring(s) exert restoring torques on the oscillating disk. Both effective torsion constants and effective spring constants can be expressed in terms of adjustable parameters of the system. These expressions enable one to theoretically describe the motion of the hybrid oscillator and to calculate its period. A comparison of the translational and rotational interpretations teaches of their analogous mathematical properties and challenges the intuitive skills of those considering such systems.