OBSERVATION CHAMBER / ACTIVE
Gyroscope Precession Observatory
Change spin, tilt, and gravity. Observe a fixed-pivot top precess and nutate while tracking its angular momentum and energy.
ROUTE 93 / DYNAMICS / PRECESSION
A spinning axis under gravity
A top can turn around the vertical while its tilt oscillates. Follow the axis tip, then switch the lens to see angular momentum and gravitational torque.
The calculation and all numerical readings remain available. PNG is disabled.
Drag left or right to turn the view. Focus the field: Space or P = PLAY / PAUSE, S = STEP, R = RESET, L = LENS. Touch scrolling stays available.
Calculations stay in this browser.
- TIME
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- TILT θ
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- AZIMUTH φ
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- PRECESSION φ̇
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- AXIAL SPIN s / I∥
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- FAST LIMIT k / s
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INVARIANTS
Watch the numerical drift
- | |n| − 1 |
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- NORMALIZED ΔE
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- NORMALIZED ΔLz
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- NORMALIZED Δ(L · n)
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- FINITE
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- STEPS
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Each conserved-value drift is divided by max(1, |initial value|). The axis is never renormalized: the norm residual is the actual integration error. All readings are dimensionless except the displayed angles.
k / s is only a fast-spin, small-nutation approximation. It is not an exact prediction for every run. Axial spin s / I∥ and azimuth precession φ̇ describe different motions. Azimuth is undefined on the vertical axis.
DETERMINISTIC ACCEPTANCE
Numerical checks —
| CHECK | ERROR | LIMIT | RESULT |
|---|
Independent exact vector solutions test regular precession and torque-free rotation. Boundary runs test normalized invariant drift ≤ 10⁻⁵ over 30 time units. Halving the step must reduce the chosen nutation residual to at most 20%.
Current observation report
OBSERVATION GUIDE
Compare precession and nutation
- Start with FAST SPIN. Press PLAY and follow the axis tip around the vertical. Compare φ̇ with the fast-limit estimate.
- Choose REGULAR PRECESSION. The initial momentum is chosen for constant tilt when a real regular solution exists. NUTATION adds transverse momentum so the tilt also changes.
- Choose TORQUE-FREE. L stays fixed while the axis rotates around L. Use the vector lens and drag to inspect the direction.
- REPORT and PNG pause the run. Any state or view change clears the old output. RESET also clears the trail. Hidden tabs stop without catch-up; reduced motion starts paused and never restarts automatically.
A finite rigid-body observation model, not a physical measurement or stability certification.
Runs inside the browser with no upload and no registration.
MODEL NOTE
Fixed-pivot symmetric heavy top
n points from the pivot toward the centre of mass; ez points upward. L and n are world-frame vectors. Moments about the pivot are I⊥ = 1 and I∥ = 0.5; k = mgr ≥ 0.
ω = L / I⊥ + (1/I∥ − 1/I⊥)(L · n)n
n′ = ω × n
L′ = −k n × ez
E = ½ L · ω + k nz
Exact regular-precession fixture
θ = 0.4, s = 10, k = 1
Ω = 2 / (10 + √(100 − 4 cos θ))
L₀ = Ω ez + (10 − Ω cos θ)n₀
n(t) = Rz(Ωt)n₀
L(t) = Ω ez + (10 − Ω cos θ)n(t)
This solves (s − I⊥Ω cos θ)Ω = k. The free general solution rotates n about the fixed L at |L| / I⊥. Neither fixture relies only on conserved scalars.
OBSERVATION POLICY
This fixed-pivot axisymmetric rigid-body model runs in your browser. Inputs, calculations, reports, and images stay local. It does not measure a real gyroscope or visitors.
Route
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