A Lab Light at the End of the Funnel: An Experiment for a Classical Mechanics Course

It is fairly typical for advanced-undergraduate texts to present Bertrand's theorem, which says that the only attractive central forces that are some power of the distance between the planet and the star that yield reentrant orbits are Newton's Law of Universal Gravitation and Hooke's Law. As part of this derivation, the dependence of the apsidal angle (the angle between aphelion and perihilion) on the exponent of the power law is obtained. Building on the work of Su, et al. [Am. J. Phys, 84, 943 (2016)], we present an advanced undergraduate lab activity where different possible laws of gravity can be explored by rolling ball bearings in 3-D-printed funnels. By perfoming video capture of the motion of the ball bearings, students were able to measure the apsidal angle and deduce whether a particular funnel was simulating an inverse-square law or a hypothetical gravitational force that varied inversely with distance. Resolutions to some of the technical challenges of this experiment will be presented.