Investigating Orbits of a Textbook Mass-Pulley System

A common classical mechanics introductory textbook problem highlighting coupled motion consists of a moving puck on a frictionless surface connected to a hanging mass via string threaded through a hole in the surface's center. A rigorous analysis via numerical integration of system equations for different puck-hanger mass ratios and initial velocities reveals complex trajectories with possible precession. In near-frictionless simulations, we observe orbits in the form of decaying circular spirals and ellipses that gradually transition into circular spirals. We further investigate these trajectories by developing an apparatus using a puck of dry ice to induce the Leidenfrost effect on the tabletop, creating a near-frictionless interaction. We vary the mass ratio from 0.4 to 2.0 by attaching different weights to the string tied to the puck. We implement a spherical pendulum to impart a speed from 0 to 3.5m/s at azimuthal angles from 0° to 90°. Decaying circular spirals are the dominant trajectories for our tested conditions, as predicted by numerical simulation. However, for a small set of initial azimuthal angles, we observe short-lived elliptical orbits that apsidally precess before falling into the same inward spirals noted previously.