A fast-spinning gyroscope doesn't fall — it
precesses. Gravity's torque
τ = r×mg rotates the angular momentum vector,
making the spin axis sweep a cone at rate
Ω = mgr / (I·ω). Slow the spin and watch precession
speed up.
Spin ω: 20.0 rad/s
Precession Ω: 0.00 rad/s
Period T: —
L = Iω: —
τ = mgr·sinθ: —
🔭 Physics of Precession
When gravity pulls the gyroscope's centre of mass downward, it
exerts a horizontal torque τ = r × F perpendicular to
both the spin axis and vertical. Instead of falling, angular
momentum changes direction: dL/dt = τ, so the
spin axis sweeps a cone — a phenomenon called
steady precession.
Ω_prec = mgr·sinθ / (I·ω·sinθ) = mgr / (I·ω)
Key insight: precession rate Ω is
inversely proportional
to spin ω — a slower gyroscope precesses faster. This is why
well-thrown tops stay upright, and explains the 26,000-year wobble
of Earth's axis.
🌀 Gyroscope Precession
About this simulation
A spinning gyroscope tilted away from vertical does not topple over — instead its
spin axis slowly sweeps a cone, a motion called precession. The same physics
keeps spinning tops and bicycle wheels upright, stabilises spacecraft and satellites,
and drives the 26,000-year wobble of Earth's rotation axis. It is one of the most
counter-intuitive results in classical mechanics: pushing on a spinning object makes
it move sideways, not in the direction you pushed.
How it works
Gravity pulls down on the centre of mass, creating a torque about the pivot.
That torque is perpendicular to the angular momentum vector L of the spinning disk.
Because dL/dt = τ, the torque changes L's direction rather than its size.
The spin axis therefore rotates steadily, tracing a cone around the vertical.
Key equations
Omega = m·g·r / (I·omega) — precession rate Omega, where m is mass, g is
gravity, r is the distance from pivot to centre of mass, I is the moment of inertia, and
omega is the spin rate. Note that the tilt angle cancels out, and slower spin gives faster precession.
Controls
Spin rate ω — how fast the disk spins; lower ω makes precession visibly faster.
Tilt angle θ — lean of the axis from vertical.
CM distance r — pivot-to-centre-of-mass distance; larger r increases the torque.
Speed — time-scaling of the whole animation.
Presets, Pause, Vectors, Trace — load scenarios, freeze, show force arrows, draw the tip path.
Did you know?
The precession of Earth's axis means the "North Star" changes over millennia — around
3000 BC the pole star was Thuban, today it is Polaris, and in roughly 12,000 years it
will be the bright star Vega.
About this simulation
This simulation models an idealised symmetric top: a uniform disk of fixed radius 0.12 m and mass 0.3 kg, spinning at rate ω about an axle tilted θ degrees from vertical and pivoted at one end so gravity acts at lever arm r. Each frame the engine integrates the rigid-body law dL/dt = τ, where torque τ = mgr·sinθ comes from gravity acting on the disk's centre of mass and angular momentum L = Iω uses the solid-disk moment of inertia I = ½mR². Because τ is always perpendicular to L, it can only rotate L's direction, never its size — so the axle sweeps a cone at the steady precession rate Ω = mgr/(Iω), a relation in which the tilt angle cancels out entirely.
What it shows
A 3D-projected spinning disk on a pivoted axle, with colour-coded vector arrows for angular momentum L, torque τ, precession Ω, and gravity mg, plus an optional dashed theoretical precession circle and a live-coloured trace of the actual simulated tip path.
How to use it
Drag the spin rate ω, tilt θ, CM distance r, and speed sliders, or jump to a preset (Fast Spin, Slow Spin, Earth Tilt, Tilted Top); toggle Vectors or Trace on/off, and pause the animation at any moment to read the live fact chips for ω, Ω, period T, L, and τ.
Did you know?
The exact same Ω = τ/(Iω) relationship that spins this simulated disk also governs why a rolling bicycle wheel resists tipping, why gyrocompasses hold a fixed heading at sea, and why Earth's own rotation axis slowly traces a 26,000-year precession cone.
Frequently Asked Questions
Why does a spinning gyroscope not simply fall over?
Gravity pulling on the tilted disk's centre of mass creates a torque τ = mgr·sinθ about the pivot, and the simulation integrates the rigid-body law dL/dt = τ. Because this torque is always perpendicular to the angular momentum vector L = Iω, it can only rotate L's direction, never shrink it — so instead of toppling, the spin axis sweeps a steady cone around the vertical, which is precession.
What is the formula for precession rate, and why does the tilt angle cancel out?
The simulator computes precession rate as Ω = mgr / (Iω), where m is the disk's mass, g is 9.81 m/s², r is the pivot-to-centre-of-mass distance, I is the moment of inertia, and ω is the spin rate. Both the torque τ = mgr·sinθ and the horizontal component of angular momentum that must rotate scale with sinθ in the same way, so the sinθ factors cancel and the resulting precession rate is independent of tilt angle.
What determines the moment of inertia used in this simulation?
The spinning disk has a fixed radius of 0.12 m and mass 0.3 kg, and the code computes its moment of inertia about the spin axis with the standard solid-disk formula I = ½mR². This fixed I, together with the spin rate slider, sets the angular momentum L = Iω shown in the fact chips, and it is this same I that appears in the denominator of the precession-rate formula.
How does this relate to the 26,000-year wobble of Earth's rotation axis?
Earth is itself a giant, slightly tilted spinning top: the combined gravitational pull of the Sun and Moon on its equatorial bulge creates a torque analogous to mgr·sinθ here, and Earth's rotation axis responds exactly the way this simulator's disk does — by precessing around a cone instead of tipping over. Earth's enormous moment of inertia and comparatively slow torque stretch that same Ω = torque/(Iω) relationship out to a period of about 26,000 years, compared with the few seconds to minutes seen on screen.
What is the difference between the dashed circle and the coloured trace when Trace is enabled?
The dashed circle is the theoretical precession path: the fixed circle of radius 2r·sinθ that the axle tip would follow under perfectly steady precession. The coloured line is the actual simulated tip path, recorded frame by frame as the physics integrates φ += Ω·dt and spin phase += ω·dt. At the fast-spin, small-tilt settings used in most presets the two curves overlap almost exactly, which is a visual check that the numerical integration tracks the analytic precession formula correctly.