M057 · Rigid body / sleeping top

Spinning Top: Precession & Nutation

Inspect a uniform conical top with a captured apex in three dimensions. Separate axial spin, precession and nutation, compare two steady branches, and perturb a sleeping top on either side of its stability threshold. Read the pivot reaction and an independent numerical error ledger.

Reviewed time $\text{Pending}$

Axis tilt $\text{Pending}$

Instantaneous precession $\text{Pending}$

Nutation rate $\text{Pending}$

Physical axial spin $\text{Pending}$

Slow-precession estimate $\text{Pending}$

Exact slow branch $\text{Pending}$

Prepared initial state $\text{Pending}$

Rotational kinetic energy $\text{Pending}$

Gravitational potential $\text{Pending}$

Total energy $\text{Pending}$

Vertical angular momentum $\text{Pending}$

Axial angular momentum $\text{Pending}$

Gravitational torque $\text{Pending}$

Pivot reaction magnitude $\text{Pending}$

Vertical pivot reaction $\text{Pending}$

Max scaled energy defect $\text{Pending}$

Max scaled momentum defect $\text{Pending}$

Axis difference: two steps $\text{Pending}$

Upright spin threshold $\text{Pending}$

Spin / upright threshold $\text{Pending}$

PREPARE / MEASURE / VERIFY

Prepare the angular momentum

Show physical vectorsIndependently normalized vector lengths; magnitudes remain in the readouts.Body mass

m

0.60\,\mathrm{kg}

Base radius

R

0.14\,\mathrm{m}

Cone height

h

0.20\,\mathrm{m}

Physical axial spin

\omega_3

100.00\,\mathrm{rad/s}

Initial axis tilt

\theta_0

25.00\,\mathrm{{}^\circ}

Free-release precession

\dot\phi_0

0.00\,\mathrm{rad/s}

Initial nutation rate

\dot\theta_0

0.00\,\mathrm{rad/s}

Initial state preparation

s

0.00\,

Record duration

T

8.00\,\mathrm{s}

Uniform gravity

g

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

Reviewed fraction

t/T

0.00\,\%

Drag the orange tilt handle below. Initialization: 0 free release, 1 slow branch, 2 fast branch. A nutation kick perturbs the slow branch. The fast branch keeps zero kick and uses its exact solution. If no real branch exists, the free-release controls apply and the status says so.

Uniform solid cone with its apex fixed by an ideal captured bearing. Cone radius, height, centre of mass and pivot inertias share one geometry. The handle, marker and support are massless. Ideal axisymmetric rigid body with one point fixed in uniform gravity. No small-tilt or fast-spin approximation is used in the integrated dynamics. Spin means the angular-velocity component along the body axis, which differs from the Euler spin-angle rate. The bearing may pull as well as push: this is not a freely sliding or bouncing tip on a table. Friction, air drag, deformation, floor/support collisions and spin decay are omitted. Full-record plots include future calculated samples. Fast rotation can alias at screen frame rates; pause and use the fine-step button to inspect phase. The upright stability threshold applies only to infinitesimal perturbations around vertical rotation.

Physics tutorial

Spinning Top: Precession & Nutation

BackgroundInspect a uniform conical top with a captured apex in three dimensions. Separate axial spin, precession and nutation, compare two steady branches, and perturb a sleeping top on either side of its stability threshold. Read the pivot reaction and an independent numerical error ledger.

Why it mattersA shared rigid-body law explains both an instrument flywheel and a nutating top. Looking only at the spinning texture hides the important motion of its axis.

Start with the essentials

Focus question: When does a nearly upright spinning top stay upright?
One-sentence intuition: Gravity changes world angular momentum through a sideways torque; energy, its vertical component and its body-axis projection remain conserved.

Core mathematical model

Geometry and pivot inertia

\ell=\frac34h,\qquad I_3=\frac3{10}mR^2,\qquad I_\perp=\frac3{20}mR^2+\frac35mh^2

Uniform solid cone with its apex fixed by an ideal captured bearing. Cone radius, height, centre of mass and pivot inertias share one geometry. The handle, marker and support are massless.

Full vector dynamics

\dot{\mathbf L}=mg\ell\,\mathbf e_z\times\mathbf n,\qquad\dot{\mathbf n}=\frac{\mathbf L\times\mathbf n}{I_\perp}

All quantities refer to the fixed pivot. The symmetry axis and world angular momentum evolve together; the spin angle is reconstructed with a quaternion.

Angular velocity and axial spin

\boldsymbol\omega=\frac{\mathbf L}{I_\perp}+\left(\frac1{I_3}-\frac1{I_\perp}\right)(\mathbf L\cdot\mathbf n)\mathbf n,\qquad\omega_3=\dot\psi+\dot\phi\cos\theta

Physical axial spin is not the Euler spin-angle rate. Confusing them changes the exact steady-precession condition.

Two steady-precession branches

I_\perp\cos\theta\,\Omega^2-I_3\omega_3\Omega+mg\ell=0

A real root prepares constant tilt only with zero initial nutation. A kick perturbs that solution. At a horizontal axis the equation becomes linear.

Slow-precession approximation

\Omega_{\mathrm{slow}}\simeq\frac{mg\ell}{I_3\omega_3}

This estimate drops the transverse-inertia term. It becomes accurate when precession is small relative to axial spin, and is undefined at zero spin.

Three conserved quantities

E=\frac{I_\perp}{2}(\dot\theta^2+\dot\phi^2\sin^2\theta)+\frac{I_3\omega_3^2}{2}+mg\ell\cos\theta,\qquad L_z=\mathrm{const},\quad\mathbf L\cdot\mathbf n=\mathrm{const}

Potential energy is zero when the centre is at pivot height. Maximum errors are scaled by nonzero physical energy and momentum scales; they remain meaningful when an invariant itself vanishes.

Upright stability threshold

|I_3\omega_3|>2\sqrt{I_\perp mg\ell}

This is a local linear stability condition. Exactly vertical rotation remains an exact solution even below threshold; use a small tilt to reveal its instability.

Captured-pivot reaction

\mathbf R=m\ell\ddot{\mathbf n}+mg\mathbf e_z

The ideal bearing can supply a negative vertical reaction. A freely resting tip would need a separate unilateral contact and collision model.

Common difficulties

The momentum always follows the shaft

Typical misconceptionThe shaft direction and angular momentum are identical at every spin.

Better mental modelTransverse inertia contributes whenever the axis moves. Compare the blue arrow with the shaft in the low-spin or fast-branch preset.

Exactly sleeping proves stability

Typical misconceptionA vertical top that remains vertical must be stable.

Better mental modelThe exact unperturbed solution survives below threshold. Compare small-tilt presets to test the response to a perturbation.

A pivot is a free tabletop tip

Typical misconceptionA negative vertical reaction can be supplied by an ordinary surface.

Better mental modelThis bearing captures the point. A tabletop cannot pull the tip; sliding, detachment and collisions are outside the model.

Run the experiment

01
Prepare and orbit
Select free release, inspect overview, top and side, then drag the orange tilt handle.
What to observe: The new initial axis drives both the physical scene and its quantitative record.
02
Compare steady branches
Select slow and fast branches; leave the nutation kick at zero. Reverse the spin.
What to observe: Constant tilt is a prepared solution; a release without the required precession generally nutates.
03
Test the approximation or sleeping state
Reduce spin and inspect the approximation; for the conical top compare stable and unstable small-tilt presets.
What to observe: A slow-precession approximation can fail, and a local stability threshold needs a perturbation to become visible.
04
Check the numerical record
Review the full record and inspect conservation defects and the difference between two step sizes. Use fine phase steps for rotor markings.
What to observe: A small numerical discrepancy supports the retained model; it does not certify omitted friction or tabletop contact physics.