M082 · Constraints / symmetry breaking

Bead on a Rotating Hoop

A motor rotates a vertical circular hoop about its vertical diameter. Drag the captive bead, tune motor speed through the critical value, and watch a single potential well split into two. Follow motion in the hoop frame, read the full constraint force, and compare reduced energy with laboratory energy and motor work.

Recorded time $0\,\mathrm{s}$

Principal angle $0\,\mathrm{rad}$

Unwrapped angle $0\,\mathrm{rad}$

Angular velocity $0\,\mathrm{rad\,s^{-1}}$

Relative kinetic energy $0\,J$

Effective potential energy $0\,J$

Reduced mechanical energy $0\,J$

Viscous dissipation $0\,J$

Recorded energy-balance defect $0\,J$

Step-halving state difference $0$

Actual integration step $0\,\mathrm{ms}$

Critical motor speed $0\,\mathrm{rad\,s^{-1}}$

Motor speed $0\,\mathrm{rad\,s^{-1}}$

Stable equilibrium magnitude $0\,\mathrm{rad}$

Local undamped oscillation frequency $0\,\mathrm{rad\,s^{-1}}$

Bottom equilibrium stability $\text{Stable}$

Laboratory mechanical energy $0\,J$

Motor work since release $0\,J$

Laboratory work-energy defect $0\,J$

Guide reaction outward along radius $0\,N$

Guide reaction in motor direction $0\,N$

RELEASE / ACQUIRE / VERIFY

Where will the bead settle after symmetry breaks?

Bead mass

m

0.50\,\mathrm{kg}

Hoop radius

R

0.80\,\mathrm{m}

Gravity

g

9.81\,\mathrm{m\,s^{-2}}

Motor speed relative to critical

\Omega/\Omega_c

1.35\,

Bead release angle

\theta_0

8.00\,{}^\circ

Initial angular velocity in hoop

\dot\theta_0

0.00\,\mathrm{rad\,s^{-1}}

Tangential damping ratio

\zeta

0.12\,

Maximum numerical step

h_{\max}

10.00\,\mathrm{ms}

Inspect recorded time

t/t_{\mathrm{end}}

0.00\,\%

Purple outline sets the release angle; green rings mark stable equilibria. The bifurcation chart is analytic, while the bead trajectory is integrated. An exactly symmetric release remains at the unstable bottom until perturbed. Effective potential includes the rotating-frame term; it is not gravitational energy alone. Every drag has an equivalent keyboard slider.

Captive point bead, rigid circular guide and prescribed constant motor speed. The main apparatus is viewed in the co-rotating plane; the plan-view inset tracks rotation about the vertical axis. Tangential viscous damping is optional. The guide supports a signed radial reaction and a Coriolis azimuthal reaction. The motor exchanges real laboratory energy even when reduced energy is conserved.

Start with the essentials

Focus question: Can a stable bottom turn into two off-center resting places?
One-sentence intuition: Reduced energy is a rotating-frame bookkeeping quantity; laboratory energy includes motion carried around by the hoop.

Core mathematical model

Constrained equation

\ddot\theta=\sin\theta(\Omega^2\cos\theta-g/R)-2\zeta\sqrt{g/R}\,\dot\theta

The prescribed motor rate does not slow down when the bead exchanges energy.

Effective potential

U_{\mathrm{eff}}=mgR(1-\cos\theta)-\tfrac12mR^2\Omega^2\sin^2\theta

The centrifugal contribution can turn the bottom into a local maximum.

Threshold and stable branches

\Omega_c=\sqrt{g/R},\quad \cos\theta_*=(\Omega_c/\Omega)^2\quad(\Omega>\Omega_c)

At the threshold the quadratic restoring coefficient vanishes, but a quartic well remains.

Laboratory motor work

P_{\mathrm{motor}}=2mR^2\Omega^2\sin\theta\cos\theta\,\dot\theta

The independent power integral is compared with the change in full laboratory energy plus heat.

Two guide-reaction components

N_r=-mR\dot\theta^2-mR\Omega^2\sin^2\theta-mg\cos\theta,\quad N_\phi=2mR\Omega\cos\theta\,\dot\theta

Radial reaction is positive outward. Azimuthal reaction supplies the torque needed to maintain motor speed.

Common difficulties

Two energies have different roles

Typical misconceptionA constant reduced energy means the motor does no work.

Better mental modelLaboratory energy includes the carried azimuthal motion; the motor-work ledger resolves the difference.

Instability needs a perturbation

Typical misconceptionAn unstable state must leave the bottom spontaneously.

Better mental modelExact symmetry is an exact solution. Drag the release slightly to select a branch.

Captive means bilateral

Typical misconceptionA negative radial reaction proves contact was lost.

Better mental modelThe captive guide can push or pull radially. The signed readout is not a unilateral surface-contact test.

Run the experiment

01
Find the single well
Acquire the below-threshold preset.
What to observe: The damped bead returns toward the bottom.
02
Cross the critical value
Raise the speed ratio above the threshold.
What to observe: The potential splits and the analytic chart shows two stable branches.
03
Test symmetry
Compare split and mirror releases, then use the exactly perched preset.
What to observe: The mirrored paths exchange branches; the exact bottom remains there until perturbed.
04
Read the motor ledger
Remove damping and scrub through the record.
What to observe: Reduced energy stays constant while laboratory energy changes with motor work.