The ellipsing pendulum

Abstract

The motion of a pendulum executing small oscillations about its position of stable equilibrium is examined. This motion is canvassed in many introductory mechanics texts, but its true complexity is rarely addressed. In the simplest of approximations, the bob executes simple harmonic motion and traces out an elliptical path. However, more precise analysis demonstrates that this ellipse actually precesses in the direction along which the motion proceeds. A derivation of the formula giving the rate of this precession is presented here. It would seem to be new and moreover is simpler than previous derivations. It uses dimensional analysis and completes the calculation by deducing a general formula from two special cases. It thus provides an illustration of the power of dimensional analysis, and also offers the student a simpler entrée into a classical (if often oversimplified) aspect of applied mathematics.

Keywords:

• spherical pendulum
• simple pendulum
• conical pendulum
• dimensional analysis