Track constraint of a bead on a large circular hoop under the influence of gravity and elasticity

In theoretical mechanics and engineering, the problem of constrained object motion is frequently encountered, where track geometries inherently restrict the trajectories of moving bodies. In real-world scenarios, friction introduces additional complexity, rendering the forces acting on objects and their motion along tracks more intricate and thus requiring in-depth analysis. This study examines the constrained motion of a bead on a large circular hoop situated in a vertical plane, subject to gravitational and elastic forces. Using Newton’s second law in the natural coordinate system, we derived the governing equations for the bead’s motion on the hoop and solved them numerically. Results show that the amplitude of the bead’s oscillations increases with higher initial angular velocities. Additionally, an increase in the spring stiffness coefficient causes the elastic force to gradually overcome gravity, altering the system’s stable equilibrium point. The friction coefficient significantly influences both the number of rotations and the oscillatory dynamics of the bead. When non-zero frictional forces are present, the system exhibits stable focal points and unstable saddle points, with the bead demonstrating a higher likelihood of settling at the stable focal points. This research provides valuable insights for physics education and engineering applications involving constrained mechanical systems.