Pythagorean triangles analysis of the conical pendulum based on lengths, forces and times, to obtain equations for the principal physical parameters

Abstract

An original approach using three appropriate Pythagorean triangles is presented for the detailed mathematical analysis of an ideal conical pendulum. The triangles that are used in this analysis relate specifically to the physical dimensions of the conical pendulum, the magnitudes of the forces acting during the conical pendulum motion and a triangular construction involving the conical pendulum period. It is assumed that known values for the conical pendulum mass, local value for the acceleration due to gravity and the conical pendulum string length are readily available. The following physical parameters can then be calculated directly from accurate experimental measurements of the conical pendulum period only: string tension force, centripetal force (and consequently the centripetal acceleration), orbital radius, orbital speed, apex angle, magnitude of the angular momentum, potential and kinetic energies. The analysis concludes with the very straightforward derivation of a well-known simple trigonometric identity, which serves to support the internal self-consistency of the Pythagorean triangles technique that is presented here.