The Time of Frictionless Motion of a Swinging Compound Pendulum

Abstract

The equations of motion of a free swinging compound (physical) pendulum were integrated to obtain a general solution for the elapsed time in the form of a trigonometric integral. The latter was reduced to Jacobian elliptic functions of the first kind, which were then solved by conventional techniques for complete and incomplete integrals. Applying the method developed to a generalized pendulum described by its degree of compounding, the period of its oscillation whilst in frictionless motion for any angle of launch was determined. The degree of compounding of the pendulum had a significant effect on its period of oscillation. This was shown graphically in the form of the variation of a dimensionless time ratio with change in angle of launch for various degrees of compounding. Five specific cases of the time intervals of motion of a compound pendulum were analyzed and solutions obtained. The general equation for the elapsed time of free swinging motion of a simple (mathematical) pendulun, launched from any position and its period of oscillation were also determined.