The looping pendulum: optimization with analytical and computational analysis

Abstract

A looping pendulum is a setup that consists of two loads connected by a light string, which in turn is put over a fixed rod, with the lighter load being held by a person, and the heavier close to the rod. When the lighter load is released, experience shows there is an instant when the heavier load stops falling, and the lighter winds around the rod, bringing the system to rest. This setup is used in science demonstrations to prevent a fragile object, such as a ceramic mug, from falling to the ground, using washers as the lighter load. In this paper we propose a mathematical model for this demonstration, which we use to analyze the main parameters that influence the effectiveness of the setup. We find its equations of motion analytically, and use numerical methods to relate relevant parameters (such as falling distance and trajectory) to the initial configuration, determine the conditions that allow the phenomenon to happen, and display results as plots and parameter spaces. Lastly, through careful experimentation we found strong evidence for the validity of our theoretical predictions. We found the looping pendulum to be an extremely useful problem for introductory college physics courses interested in strengthening analytical and computational physics skills, as its construction is easy, inexpensive, and its analysis encompasses various mechanics concepts, solving differential equations numerically, and data collection through video software.