Tippe Top

The trick everyone has seen

You spin a little mushroom-shaped top — a rounded sphere with a short stem poking up. For a moment it whirls happily on its round belly. Then, with no help from you, the whole top tips, rolls up onto its stem, and keeps spinning upside-down. The rounded part that was on the table is now in the air, and the stem that pointed at the ceiling is now planted on the table.

The unsettling part is that the center of mass climbs upward during the flip. Things are supposed to fall, not lift themselves. So where does the energy come from, and what force does the lifting? The answer is the same in every case: the contact point is slipping, and sliding friction is doing the steering. No friction, no flip. Spin a tippe top on a sheet of glass with oil on it and it just sits there and precesses until it falls over the normal way.

How the flip actually happens

A tippe top is built so its center of mass sits below the center of the sphere. Call the sphere radius R and the distance the center of mass sits below the geometric center a. When the top is "upright" (belly down), the center of mass is low; when it stands on its stem, the center of mass is high — lifted by about 2a (the COM swings from a below the sphere's center to a above it), plus the stem length on top of that.

Spin it fast and the bottom of the sphere doesn't roll cleanly — it skids. At the contact point the surface of the top is moving sideways relative to the table, so kinetic friction f acts in the opposite direction. Crucially, that friction force acts at the contact point, which is offset from the symmetry axis once the top tilts even slightly. An offset force is a torque.

That frictional torque has a component that rotates the symmetry axis. Step by step, the spin axis tips further and further from vertical, swings through horizontal, and continues over until the stem points down. The motion is a slow, monotonic climb superimposed on the fast spin and a precession wobble — the axis spirals over the top of an imaginary dome. When the stem finally touches down, the contact point lands back on the symmetry axis, the offset goes to zero, the slipping torque vanishes, and the top is stable in its new inverted state.

The governing physics

Treat the top as a rigid body with mass m, principal moments of inertia I₁ = I₂ (about axes through the center of mass perpendicular to the symmetry axis) and I₃ (about the symmetry axis). The two governing equations are Newton's law for the center of mass and Euler's torque equation about the center of mass:

m (d²r/dt²) = N + f + m g          (N = normal force, f = friction)
dL/dt       = s × (N + f)          (torque of contact forces about COM)

Here s is the vector from the center of mass to the contact point, and L = I·ω is the angular momentum about the center of mass. Gravity exerts no torque about the center of mass; only the contact forces do. The normal force N and gravity govern the bouncing/holding; the friction force f governs the flip.

The single most useful result is Jellett's invariant (Jellett, 1872). Because the friction force passes through the contact point, the torque it produces has no component along the direction (a − R n̂) — the line joining the center of mass and the contact point. That means the projection

J = −L · (a − R n̂)   ≈ constant during the flip

is conserved (n̂ is the unit symmetry axis). Energy is not conserved (friction dissipates it) and the vertical angular momentum L_z is not conserved (the contact-force torque has a vertical component). But J stays fixed, which is what lets you predict the final spin. The spin about the symmetry axis after inversion is related to the initial spin by the change in geometry through J — the top comes out spinning the same sense in space but appears reversed relative to its own stem.

Whether the top inverts at all is a stability question. Linearizing about the spinning states, complete inversion is possible only when the upright (lowest-COM) state is unstable while the inverted (highest-COM) state is stable. The classic result (Ebenfeld & Scheck; Cohen) is a condition on the geometry that the inertia ratio must straddle unity. Writing the eccentricity ε = a / R and γ = I₁ / I₃, both stable-inverted and unstable-upright hold simultaneously when

1 − ε  <  γ = I₁ / I₃  <  1 + ε        (geometry must place γ in this band)
and    ω²  >  ω_crit²  ≈  m g R (I₁ − I₃ + m a R) / I₃²   (enough spin to invert)

The clean physical statement: I₃ and I₁ must be close — the inertia ratio I₁/I₃ has to fall within ε = a/R of one, the center of mass must sit below center, there must be enough friction to keep the contact slipping, and the spin must clear the threshold. Miss any one and there's no full inversion. (For a real toy top I₁/I₃ is only a little above 1, which sits inside this band.)

Where the lifting energy comes from

The flip raises the center of mass, so gravitational potential energy goes up. That energy is withdrawn from the rotational kinetic energy, ½ I₃ ω₃² plus the transverse terms. Friction slows the spin; some of the lost spin energy becomes the rise in potential energy, the rest becomes heat at the slipping contact.

Quantity	Before flip (belly down)	After flip (on stem)	Conserved?
Center-of-mass height	Low (≈ R − a)	High (raised by ≈ 2a + stem length)	No — rises
Gravitational PE	Lower	Higher	No — increases
Rotational KE	High	Lower	No — decreases
Vertical angular momentum L_z	Large positive	Smaller (sense flips relative to body)	No — torque has vertical part
Total energy	—	—	No — friction dissipates heat
Jellett invariant J = −L·(a − Rn̂)	J₀	≈ J₀	Yes — nearly constant

So the bookkeeping is honest: the top does not get energy for free. It trades a fast spin for a higher center of mass, and pays a friction tax in heat along the way. That is exactly why a low-friction surface kills the effect — without slipping friction there is no torque to do the steering and no dissipation channel to drive the system toward the lower-energy (inverted, slower-spinning) configuration.

Numbers for a real toy top

A typical commercial tippe top is a plastic hemisphere about 2–3 cm across with a short stem. Rough but representative figures:

Parameter	Symbol	Typical value
Sphere radius	R	1.0–1.5 cm
COM offset below center	a	0.1–0.3 R (a few mm)
Inertia ratio	γ = I₁ / I₃	≈ 1.0–1.1 (must lie in 1±ε for inversion)
Sliding friction coefficient	μ	≈ 0.1–0.3 (plastic on wood/laminate)
Critical spin to invert	ω_crit	~ 30–100 rad/s (≈ 300–1000 rpm)
Time to complete the flip	t_flip	≈ 0.3–1.0 s
Rise in COM height	Δh	≈ 2a, on the order of a few mm to ~1 cm

Plug in numbers and the energy ledger is small but real. A 10 g top with a center of mass that rises Δh ≈ 6 mm gains m g Δh = 0.01 × 9.8 × 0.006 ≈ 6 × 10⁻⁴ J of potential energy. The spin energy at, say, ω₃ = 60 rad/s with I₃ ≈ 5 × 10⁻⁷ kg·m² is ½ I₃ ω₃² ≈ 9 × 10⁻⁴ J — comparable in size, which is why the spin visibly slows as the top stands up, and why a slightly under-spun top runs out of rotational energy before completing the climb.

Conditions and regimes

• Spin too slow. Below the critical spin the upright state is stable. The top just precesses, wobbles, and topples over the ordinary way without inverting.
• Spin above critical. The upright state becomes unstable and the inverted state stable. The symmetry axis climbs smoothly over the top of the dome and locks onto the stem.
• Too little friction. On a near-frictionless surface there is no steering torque and no dissipation. The axis cannot turn over — no inversion regardless of spin.
• Too much friction / gripping. If the contact rolls without slipping, the slip-driven torque disappears. Real tops keep slipping because the sideways skid velocity stays nonzero for most of the climb.
• Wrong inertia ratio. If the inertia ratio I₁/I₃ falls outside the band 1±ε (ε = a/R) — for example a flat, disk-like (oblate) mass distribution spread out in the equatorial plane, which raises I₃ and pushes I₁/I₃ well below 1 — the inverted state is not stable and the top won't fully flip. Geometry, not effort, decides this.
• Intermediate states. A top can hang at a tilted "rising" angle for a while if friction and spin are marginal, spinning on its equator before either completing the flip or sliding back.

Where it shows up

• The spinning hard-boiled egg. Same mechanism: spin a cooked egg flat and it rises to stand on its end as friction turns the spin axis upward. A raw egg can't — the liquid sloshes and dissipates the spin first. A quick way to tell boiled from raw.
• The "rising" or "wobbling" coin / ring. A coin spun on a table or an Euler's disk settles by a related slip-and-precess interplay, though those dissipate rather than invert.
• Rattlebacks (celts). A cousin curiosity — also friction-driven, but it reverses spin direction instead of inverting. Both show that contact friction can produce motion that looks like it defies intuition.
• Spacecraft and the "tennis-racket" / Dzhanibekov effect. The intermediate-axis instability and the role of small dissipation in driving a spinning body toward its minimum-energy spin state are conceptual relatives — energy dissipation reorients spinning bodies, which is exactly why satellites are spin-stabilized about their maximum-inertia axis.
• Teaching nonholonomic dynamics. The tippe top is a standard case study in rigid-body mechanics for sliding-contact (frictional, non-conservative) systems — it's a clean demonstration that an "invariant" can survive even when energy and angular momentum don't.

A famous footnote

The toy has been sold since at least the 1890s, but its physics drew serious attention in the mid-20th century. There's a well-loved (and possibly embellished) story that Wolfgang Pauli and Niels Bohr were photographed in 1954 crouched at the floor in Lund, utterly absorbed in a tippe top — two architects of quantum mechanics stumped by a child's toy. The first satisfying rigorous treatments came from work by C. M. Braams, N. M. Hugenholtz, and later detailed analyses establishing the slipping-friction mechanism and the Jellett invariant as the right tools. It remains a favorite test problem precisely because the naive "tops can't lift their own weight" intuition is so strong and so wrong.

Common misconceptions and edge cases

• "It defies energy conservation." No. The center of mass rises, but the spin slows — rotational kinetic energy pays for the lift plus the friction heat.
• "Gravity flips it." Gravity exerts no torque about the center of mass. The flip torque comes entirely from friction at the slipping contact; gravity only sets which configuration is lower-energy.
• "It works on any surface." It needs a gripping surface. On ice or oiled glass the contact rolls or slides freely with negligible friction and the top won't invert.
• "The spin direction reverses." In space the angular momentum keeps roughly the same direction; what reverses is the orientation of the body relative to that spin — so the stem that pointed up now points down while spinning the same way in the lab frame.
• "Any rounded top will do it." You need an inertia ratio I₁/I₃ inside the band 1±ε, a center of mass below center, and enough spin. A symmetric solid sphere (no COM offset) or a top with the wrong inertia ratio will not flip.
• "It's the same as a gyroscope precessing." Precession is a conservative, torque-driven wobble. The tippe top's inversion is a dissipative, friction-driven reorientation — related, but not the same phenomenon.