Bead on a rotating hoop

A point bead slides without friction on a vertical circular hoop of radius R. A motor holds the hoop at constant angular speed ω about its vertical diameter. The view rotates with the hoop, so its circle stays fixed. The signed angle θ is measured from the bottom. The initial tangential velocity is zero; there is no added kick or damping.

With g = 9.81 m/s², θ̈ = sin θ (ω² cos θ − g/R). The threshold is ωc = √(g/R). Below it, the bottom is stable. At the threshold it remains stable through a quartic potential minimum, although its linear oscillation frequency is zero. Above it, the bottom is unstable and the two stable positions are θ* = ±acos(g/(ω²R)). The top, shown at both −180° and +180° on the potential plot, is one physical point and is always unstable. Exactly balanced starts at the bottom or top remain there with zero initial velocity. Without damping, a displaced bead does not settle into a minimum.

The potential per unit mass is Ueff/m = gR(1 − cos θ) − ½ω²R²sin²θ, with zero at the bottom. The conserved effective quantity is Eeff/m = ½R²θ̇² + Ueff/m. It is not the inertial mechanical energy: the motor maintains the imposed spin. The displayed signed drift is (Eeff(t)/m − Eeff(0)/m) / max(gR, ω²R²).

Filled circles and solid branches mark stable equilibria; hollow circles and dashed branches mark unstable ones. A diamond marks the critical stable point. The red point is the current bead. The energy line and equilibrium table relate its motion to the potential minima. The bifurcation diagram shows the bottom branches; the always-unstable top is listed in the table and marked on the other two views.

Run continues the current motion; Step advances by 0.05 s. The experiment stops at 60 s. Pause, a hidden page or navigation stops playback. Input changes discard the old result; Reset keeps the inputs. Valid inputs are retained across navigation. Playback normally follows real time; long frames advance by at most 0.05 s, so it can slow down on a busy device.

Model: David Tong, Classical Dynamics, §2.5.1; frictionless and critical behavior in Dutta and Ray (2012). No friction term from the latter model is included here.