An improved calculation of the mass for the resonant spring pendulum

When the appropriate mass is used to oscillate a spring, the vertical oscillations couple to a pendular swing. Previous calculations of various aspects of this resonance assumed a massless spring as a simple pendulum. This paper improves the estimate of the mass necessary to induce this resonance by describing a massive spring as a physical pendulum and obtains an expression for the mass in terms of the spring constant and various lengths associated with the spring. Several approximations will be considered to simplify the complete expression. Comparisons of the predictive power of these expressions are made for various values of the spring constants. An Appendix discusses the assumption of uniform coil density of a hanging massive spring.