M057 · Precession and the sleeping top

Spinning Top: Precession & Nutation

When does a nearly upright spinning top stay upright? When it spins faster than a threshold set by its mass, shape and moments of inertia. A heavy top with its mass far from the tip needs more spin. Above the threshold, a small tilt just makes the top nod around upright; below it, the same tilt grows and the top falls away from vertical.

Spin a uniform solid cone in three dimensions with its tip held in a bearing. Tell apart its spin, precession and nutation, compare the two steady precession rates, and nudge an upright 'sleeping' top on either side of its stability threshold. The force at the pivot and an independent check of the numerical error update alongside.

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 $\text{Pending}$

Set up, measure, then verify

Set the initial angular momentum

Show physical vectorsEach kind of arrow is normalized on its own; the readouts give the actual magnitudes.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. The initialization setting chooses the start: 0 is a free release, 1 the slow steady-precession branch and 2 the fast branch. A nutation kick disturbs the slow branch; the fast branch always has zero kick and follows its exact solution. If no real branch exists, the free-release settings apply and the status line says so.

The top is a uniform solid cone: an ideal axisymmetric rigid body in uniform gravity, with its apex fixed by an ideal bearing. The same radius and height set the centre of mass and the moments of inertia about the pivot; the handle, marker and support have no mass. The equations of motion are integrated without any small-tilt or fast-spin approximation. Spin means the component of angular velocity along the body axis, which differs from the rate of the Euler spin angle. The bearing can pull as well as push, so this is not a tip that slides or bounces freely on a table. Friction, air drag, deformation, collisions with the floor or support, and spin decay are left out. Plots of the full run include samples already computed for later times. Fast rotation can alias at the screen's frame rate; pause and use the fine-step button to check the phase. The upright stability threshold applies only to infinitesimal disturbances of vertical spinning.

Physics tutorial

Spinning Top: Precession & Nutation

BackgroundA uniform solid cone spins in three dimensions with its tip held in a bearing. You can tell apart its spin, precession and nutation, compare the two steady precession rates, and nudge an upright 'sleeping' top on either side of its stability threshold. The force at the pivot and an independent check of the numerical error are shown alongside.

Why it mattersThe same rigid-body law explains a flywheel in an instrument and a nutating top. If you watch only the spinning surface pattern, you miss what matters: how the axis moves.

Start with the essentials

Focus question: When does a nearly upright spinning top stay upright?
One-sentence intuition: Gravity's sideways torque changes the angular momentum, but the energy, the vertical component of angular momentum and its component along the body axis all stay constant.

Core mathematical model

Cone geometry and inertia about the pivot

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

The top is a uniform solid cone whose apex is held by an ideal bearing. The same radius and height set the centre of mass and the moments of inertia about the pivot. The handle, marker and support have no mass.

Full vector equations of motion

\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 are taken about the fixed pivot. The symmetry axis and the angular momentum evolve together, and a quaternion keeps track of the spin angle.

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

The actual spin about the axis is not the rate of the Euler spin angle. Mixing the two up gives the wrong condition for exact steady precession.

Two rates of steady precession

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

Each real root gives a constant tilt, but only if the top starts with no nutation. A kick disturbs that motion. When the axis is horizontal, the equation becomes linear.

Slow-precession approximation

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

This estimate leaves out the transverse-inertia term. It is accurate when the precession is slow compared with the spin, and it 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}

The potential energy is zero when the centre of mass is level with the pivot. The largest errors are divided by nonzero physical scales of energy and angular momentum, so they still mean something when a conserved quantity is itself zero.

Upright stability threshold

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

This is a local, linear stability condition. Below the threshold, perfectly vertical spinning is still an exact solution; tilt the top slightly to reveal the instability.

Reaction force at the held pivot

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

The ideal bearing can pull the tip down as well as push it up. A tip resting freely on a table would need a separate model of one-sided contact and collisions.

Common difficulties

The angular momentum always points along the shaft

Typical misconceptionAt any spin rate, the angular momentum points exactly along the shaft.

Better mental modelWhenever the axis moves, the transverse inertia adds a part that points off the axis. Compare the blue arrow with the shaft in the low-spin or fast-branch preset.

Staying vertical proves stability

Typical misconceptionA vertical top that remains vertical must be stable.

Better mental modelBelow the threshold, the exact undisturbed solution still exists. Compare the small-tilt presets to see how the top responds when it is disturbed.

The pivot works like a tip on a table

Typical misconceptionAn ordinary table surface can pull the tip down too.

Better mental modelHere a bearing holds the tip in place. A tabletop cannot pull on the tip, and sliding, lifting off and collisions are outside the model.

Run the experiment

01
Set the tilt and look around
Select a free release, look at the overview, top and side views, then drag the orange tilt handle.
What to observe: The new starting axis changes both the three-dimensional scene and the recorded data.
02
Compare the two steady precessions
Select the slow and fast branches with the nutation kick at zero, then reverse the spin.
What to observe: The tilt stays constant only because the top starts with exactly the right precession. Released without it, the top generally nutates.
03
Test the approximation and the sleeping top
Lower the spin and compare with the approximation. Then compare the stable and unstable small-tilt presets for the conical top.
What to observe: The slow-precession approximation can fail. The local stability threshold shows up only when the top is disturbed.
04
Check the numerical accuracy
Replay the whole run and look at the conservation errors and the difference between two step sizes. Use the fine phase step to follow the rotor markings.
What to observe: Small numerical errors show that the model's own equations are solved accurately. They cannot vouch for friction or tabletop contact, which the model leaves out.