Conical Pendulum – Linearization Analysis

Abstract

A theoretical analysis is presented, showing the derivations of seven different linearization equations for the conical pendulum period T, as a function of radial and angular parameters.  Experimental data obtained over a large range of fixed conical pendulum lengths (0.435 m – 2.130 m) are plotted with the theoretical lines and demonstrate excellent agreement.  Two of the seven derived linearization equations were considered to be especially useful in terms of student understanding and relative mathematical simplicity.  These linear analysis methods consistently gave an agreement of approximately 1.5% between the theoretical and experimental values for g, the acceleration due to gravity.  An equation is derived theoretically (from two different starting equations), showing that the conical pendulum length L appropriate for a second pendulum can only occur within a defined limit: L ³ [ g / (4 p ²)].  It is therefore possible to calculate the appropriate circular radius R or apex angle (0 £ f £ p / 2) for any length L in the calculated limit, so that the conical pendulum will have a one second period.  A general equation is also derived for the period T, for periods other than one second.