Rotating Hoop Pitchfork

Theory

A bead of mass m sliding without friction on a vertical circular hoop of radius a, which is rotating with angular speed ω about its vertical diameter, experiences both gravity and a centrifugal force. The position of the bead is specified by the angle θ, measured from the bottom of the hoop.

The effective potential energy of the bead in the rotating frame is: V_(eff)(θ) = mg a (1 - cosθ) - 1/2 m (ω a sinθ)²

Dividing by mga gives the reduced effective potential: V_(red)(θ) = 1 - cosθ - 1/2 ((ω² a)/g) sin²θ

We define the critical angular speed \omega_c = √(g/a) and the dimensionless rotation ratio r = ω / \omega_c. The reduced potential becomes: V_(red)(θ) = 1 - cosθ - (r²)/2 sin²θ

The equilibria are found where dV_red/dθ = 0: sinθ (1 - r² cosθ) = 0

This gives θ = 0 (the bottom), θ = π (the top), and cosθ = 1/r². The latter solution only exists when r > 1, indicating a pitchfork bifurcation where the stable bottom equilibrium becomes an unstable maximum, and two new stable equilibria emerge at θ = ± arccos(1/r²).

Worked Example

Let a = 1 m, r = 1.5, and we want to find the stable equilibria.

1. The ratio r = 1.5 > 1, so the bottom equilibrium (θ = 0) is unstable.
2. The stable angles are θ = ±arccos(1 / 1.5^2) = ± arccos(1 / 2.25) = ± arccos(0.444...).
3. In degrees, this corresponds to θ ≈ ± 63.61^\circ.

Let's calculate the reduced potential at the probe angle θ = 45^\circ:

V_red(45) = 1 - cos(45) - 0.5 * (1.5)^2 * sin^2(45)
          = 1 - 0.7071 - 1.125 * 0.5
          = 1 - 0.7071 - 0.5625
          = -0.26960678...

Experiment

Use the interactive module to explore the bifurcation:

1. Open the Rotating Hoop Pitchfork module.
2. Set the Rotation ratio r to 0.5 and observe that the bottom state is a "Stable minimum".
3. Increase the Rotation ratio r to 1.5 and watch as the stable branches emerge, and the bottom state becomes an "Unstable maximum".

Limitations

This model assumes the hoop rotates at a constant angular speed and neglects friction, air resistance, and the mass of the hoop itself. The omitted top equilibrium (θ = ± π) is outside the displayed bottom domain and is always unstable in this analysis.

References

For more detailed analysis of the pitchfork bifurcation in rotating hoops:

• Bifurcation Theory and the Rotating Hoop