Gyroscope
3D precession, nutation, and torque-free motion with calculator
About this tool
A symmetric rigid top rotates about a fixed frictionless pivot. Gravity acts at a centre of mass d along body axis 3. I is each transverse principal moment about the pivot; I₃ is the axial moment. The transverse moment about the centre of mass is I − m d². Physical moments require I ≥ m d² + I₃/2. The drawn rotor is schematic, not a mass-density reconstruction.
Select an example and Play, or change Initial parameters. ω₁, ω₂ and ω₃ are initial body-frame angular velocity components. Tilt θ is measured from vertical +Y; positive ω₃ follows the right-hand rule about the body axis. The torque-free example retains its transverse components and precesses about its fixed angular momentum; a symmetric free top has a constant cone angle about that vector. Gravity examples can also nutate.
Playback advances the physical clock at the chosen time factor. Step always adds 0.05 s, independently of that factor. Reset returns to the current initial parameters and pauses. Changes restart the initial state; invalid fields hide obsolete values and geometry. A hidden view or browser tab pauses. Returning within the app retains a paused motion; reloading restores only valid parameters and requires Play again. Historical unvalidated states are ignored.
The dynamics use coupled Euler equations and quaternion attitude, with torque τ = r × mg. Numerical RK4 substeps bound the angular phase by 0.015 rad. Long display frames contribute at most 50 ms before the time factor; lag is not caught up. Paused views redraw only on interaction or resize. Fast spinning markers can appear to reverse because of display sampling: use slow motion.
The fast, nearly axial spin reference is Ω ≈ mgd/(I₃ω₃), signed about +Y, and T ≈ 2π/|Ω|. The sine of tilt cancels between torque and the precession cone. These are reference estimates, not measured instantaneous precession. They are hidden for zero/vertical spin or when |Ω| ≥ 0.1|ω₃| or transverse speed ≥ 0.2|ω₃|; these thresholds are a display heuristic, not a rigorous validity guarantee.
Physical quantities show K, total E = K + mgd·nY, vertical angular momentum Lᵧ and axial I₃ω₃. E and both components are conserved in this ideal gravity model; all lab-frame angular momentum components are conserved without torque. Floating-point integration has finite error. Direction arrows have fixed display lengths; compare the numerical values, not the arrow lengths. The trail stores at most 400 recent axis-tip positions.
Moments range from 10⁻⁴ to 0.1 kg·m², mass from 0.01 to 2 kg, d from 0 to 0.3 m, spin from −600 to 600 rad/s, transverse rates from −200 to 200 rad/s and tilt from 0 to 180°. No friction, driving, bearing compliance or collisions are modeled. All calculations stay in the browser.