M082 · Constraints and symmetry breaking

Bead on a Rotating Hoop

Where does a bead on a spinning hoop settle? Below a critical spin rate it stays at the bottom. Above that rate the bottom turns unstable, and the bead settles at one of two mirror-image spots partway up the hoop; which side depends on which way it was nudged at release. The critical rate equals the frequency at which the bead rocks at the bottom of a hoop that is not spinning.

A motor spins a vertical hoop about its vertical diameter, shown in a three-dimensional view you can rotate. Drag the bead along the hoop, turn the motor speed up past the critical value, and watch one potential well split into two. Follow the motion in the frame of the hoop, read the full force from the guide, and compare the reduced energy with the laboratory energy and the motor work.

Laboratory bead speed $0\,\mathrm{m/s}$

Vertical angular momentum $0\,\mathrm{N\,m\,s}$

Motor power on bead $0\,\mathrm{W}$

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 $0\,N$

Set the release, then record the motion

Where does a bead on a spinning hoop settle?

Show physical vectorsArrows show the bead's total velocity in the laboratory, gravity, the guide's reaction force and the equivalent damping force.Show recent laboratory pathShows the path over the last two natural periods; the apparatus never shows the future trajectory.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\,\%

Drag the purple outline to set the release angle; green rings mark the stable equilibria. The bifurcation chart comes from the analytic solution, and the bead's path is integrated numerically. A perfectly symmetric release stays at the bottom, even when the bottom is unstable, until something disturbs it. The effective potential includes the term from the rotating frame, so it is more than gravitational energy. Every drag has a matching keyboard slider.

The bead is a point held on a rigid circular guide, and the motor turns at a fixed, constant speed. The three-dimensional apparatus is drawn in the laboratory frame, while the editable coordinate panel stays in the plane that turns with the hoop. A bead that has settled still moves around with the motor. Blue arrows show the full laboratory velocity, including the part carried by the hoop; purple arrows show the guide force, with both its radial component and its component along the motor direction. Lengths in the scene are rescaled to fit, and the supports and bead size are schematic. Force arrows share one adaptive scale, separate from the velocity scale. The energy view compares the motor work and the heat, each integrated on its own, with the mechanical energy. Viscous damping along the hoop is optional. The guide can push outward or pull inward along the radius, and it supplies a Coriolis force along the motor direction. The motor exchanges real energy with the bead in the laboratory frame even when the reduced energy is conserved.

Start with the essentials

Focus question: Can a stable bottom turn into two off-center resting places?
One-sentence intuition: The reduced energy is an energy defined in the rotating frame; the laboratory energy also includes the motion of the bead as the hoop carries it around.

Core mathematical model

Full velocity in the laboratory

\mathbf v=R\dot\theta\,\mathbf e_\theta+\Omega R\sin\theta\,\mathbf e_\phi

The first unit vector points toward increasing angle along the hoop; the second points in the direction the motor turns. Even a bead at rest on the hoop moves with it, and the blue arrow shows that carried velocity.

Vertical angular momentum and the two energies

L_z=mR^2\Omega\sin^2\theta,\qquad H=E_{\mathrm{lab}}-\Omega L_z

At constant motor speed, the total motor work equals the motor speed times the change in vertical angular momentum. That gives an independent check on the work-energy balance.

Equation of motion along the hoop

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

The motor speed is fixed. It does not slow down when the bead takes energy from the motor or gives it back.

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.

Critical speed and stable resting angles

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

At the critical speed the quadratic term in the potential vanishes, but a quartic well remains.

Motor power in the laboratory

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

The Lab integrates this power on its own and compares the result with the change in total laboratory energy plus the heat produced.

Two components of the guide force

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

The radial component is positive outward. The component along the motor direction supplies the torque that keeps the motor speed constant.

Common difficulties

Two energies have different roles

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

Better mental modelThe laboratory energy includes the motion of the bead around the vertical axis. The two energies change by different amounts, and the difference is the motor work.

An unstable state needs a push to leave

Typical misconceptionAn unstable state must leave the bottom spontaneously.

Better mental modelA bead placed exactly at the bottom is an exact solution, and it stays there. A small offset to one side is enough to select a branch.

A captive bead can be pushed or pulled

Typical misconceptionA negative radial reaction proves contact was lost.

Better mental modelThe guide holds the bead from both sides, so it can push or pull along the radius. A negative value only means the force points inward; it is not a test for losing contact, as it would be for a bead resting on one surface.

Run the experiment

01
Find the single well
Choose Below the critical speed and press Observe the full relaxation.
What to observe: With damping, the bead settles back toward the bottom.
02
Go past the critical speed
Raise Motor speed relative to critical above the critical value.
What to observe: The potential well splits in two, and the chart of the exact solution shows two stable branches.
03
Test the symmetry
Compare the Two stable branches and Mirror release presets, then choose Exactly on the unstable bottom.
What to observe: The mirror-image releases end on opposite branches. A bead placed exactly at the bottom stays there until something disturbs it.
04
Follow the motor work
Set Tangential damping ratio to zero, press Energy balance and scrub through the record. Compare the laboratory velocity in the top view and the overview.
What to observe: The reduced energy stays constant, while the laboratory energy changes by the work the motor does.