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The Tippe Top

Why a Spinning Top Can Turn Upside Down and Lift Its Centre of Mass

Big Question: How can a dissipative contact force drive a spinning eccentric body from a low-centre-of-mass orientation into a higher one?

Quick Answer

A tippe top is approximately an axisymmetric rounded body whose centre of mass does not coincide with the geometric centre of its spherical contact surface. When it spins on a rough horizontal surface, the contact point usually slips. Friction opposes that local slip and therefore exerts a torque about the centre of mass.

Because the mass centre is offset, that torque couples spin, precession, nutation and orientation. Above a sufficient initial spin and for suitable geometry and moments of inertia, the original upright spinning state becomes dynamically unstable while the inverted state is stable. The axis then evolves toward inversion.

Frictional dissipation reduces total mechanical energy throughout the process, but some rotational kinetic energy is converted into gravitational potential energy as the centre of mass rises. The exact trajectory depends on how sliding and rolling resistance are modelled, so a responsible explanation does not pretend there is one universal friction law for every real tippe top.

SIAM Journal on Applied Dynamical Systems — Tippe Top Inversion as a Dissipation-Induced Instability →

What You Will Learn

• Why the centre of mass is offset from the geometric centre.
• Why a frictionless tippe top does not perform the familiar inversion.
• How contact-point slip creates frictional torque.
• Why friction can change orientation while still dissipating energy.
• How spin, precession and nutation couple.
• Why inversion requires sufficient initial spin.
• How an unstable spinning state can evolve toward a stable inverted state.
• Why the centre of mass can rise while total energy falls.
• What a Jellett-type integral means in common idealised models.
• Why changing the friction model can change mathematical invariants and even inversion behaviour.
• How the tippe top differs from a rattleback and the intermediate-axis flip.
• How to test competing models with high-speed motion tracking.

Part 1 — The Object Is Not Balanced Around Its Geometric Centre

A useful idealisation is a nearly spherical axisymmetric body with a short stem and a centre of mass displaced along its symmetry axis.

When the stem initially points upward, the centre of mass lies relatively low. After complete inversion, the body spins with the stem toward the surface and the centre of mass is higher.

The eccentricity matters because the contact force does not generally act through the centre of mass.

Part 2 — A Smooth Floor Removes the Essential Mechanism

On a perfectly frictionless horizontal plane, the contact force is purely normal.

The familiar inversion is not produced because the tangential contact interaction needed to redirect the rotational state is absent.

Classic analyses of the tippe top therefore identify friction as essential to the rise.

Physica — On the Influence of Friction on the Motion of a Top →

Part 3 — Friction Opposes Slip at the Contact Point

A spinning rigid body can have complicated motion at the point touching the table.

If that contact point is sliding relative to the surface, kinetic friction acts opposite the slip velocity.

This statement is more precise than saying “friction opposes the top’s motion.” The top’s centre, axis and contact point do not all move in the same direction.

friction opposes local relative motion at contact; the torque produced by that friction can make the global orientation change in a surprising direction.

Part 4 — The Contact Force Produces a Torque About the Centre of Mass

The torque from a force depends on the lever arm from the centre of mass to the point where the force acts.

Because the centre of mass and geometric centre are offset, the normal force and tangential friction at the contact point create torques that couple the body’s orientation to its rotational velocity.

The resulting motion cannot be reduced to “spin slows down.” Spin, precession and nutation exchange angular-momentum components while friction steadily removes mechanical energy.

Part 5 — Fast Spin Creates a Different Stability Landscape

The upright and inverted configurations can both correspond to steady spinning solutions in idealised models.

Their stability depends on initial spin, mass-centre offset, body geometry and principal moments of inertia.

For an invertible tippe top spun above a critical range, the initial state becomes unstable to small tilts while the inverted state can act as an attractor under dissipation.

This is why merely placing a stationary tippe top on a table does not make it flip. Rotation changes the stability problem.

Part 6 — Dissipation Can Destabilise One Motion and Select Another

We often associate dissipation with “everything simply settles downward.” That intuition is incomplete in rotating constrained systems.

Friction removes energy, but the allowed motion is constrained by contact geometry and angular momentum structure. Energy loss can therefore drive the system away from one rotating state and toward a different state with higher gravitational potential but lower total mechanical energy.

This is why modern mathematical work describes tippe-top inversion as a dissipation-induced instability.

Part 7 — Energy Bookkeeping Removes the “Climbing Against Friction” Paradox

Suppose the centre of mass rises by height Δh. Its gravitational potential energy increases by

ΔU = MgΔh.

At the same time, rotational kinetic energy falls and friction converts some mechanical energy into heat and small vibrations.

The energy ledger can therefore be

large loss of rotational kinetic energy = smaller gain in gravitational potential energy + dissipated energy + changes in other kinetic components.

Total mechanical energy still decreases.

Part 8 — Angular Momentum Is More Subtle Than Energy

Friction with the table provides an external torque about the centre of mass, so the top’s angular momentum vector is not simply fixed in space.

However, the contact force has special geometry and some idealised models possess conserved combinations of angular velocity and orientation.

One famous example is a Jellett-type integral. It constrains the slow evolution while energy decreases.

The exact conserved quantities depend on the contact and friction model, which is why they should not be promoted to universal laws of every physical toy.

Part 9 — Why the Friction Law Is Not a Minor Technical Detail

Real contact can involve sliding friction, rolling resistance, spinning friction, deformation and small impacts.

Different mathematical models idealise these processes differently. Research has shown that changing the assumed rolling-resistance law can change which first integrals survive and whether inversion occurs in the model.

Nonlinear Dynamics — Friction Model and Tippe Top Inversion →

Part 10 — Why Some Tippe Tops Never Fully Invert

Inversion is not guaranteed for every rounded eccentric top.

Geometry and inertia determine whether the inverted steady state is dynamically accessible and stable.

Some parameter combinations do not invert even at high spin; others rise only to an intermediate angle.

This is an important counterexample to the oversimplified rule “spin fast + friction = flip.”

Part 11 — The Motion Passes Through Precession and Nutation

As the symmetry axis tilts, it generally precesses around the vertical and may nutate—its inclination angle oscillates as it evolves.

Near successful inversion, rapidly spinning tops can approach a gyroscopic balance regime where spin and precession are strongly coupled.

The detailed time history can look complicated even when the long-term trend is clear: the axis moves toward the inverted stable state.

Part 12 — Why the Spin Direction Appears to Change

A viewer may say the top “reverses its spin” after inversion.

Be careful about reference axes. When the body turns upside down, the direction of its symmetry axis relative to the laboratory vertical reverses. A body-fixed angular-velocity component and the apparent rotation viewed from above can therefore be described with different signs depending on convention.

Always define the axis before saying a rotational direction reversed.

Part 13 — This Is Not the Rattleback

A rattleback is an asymmetric object that can spin readily in one direction but convert spin into rocking and reverse direction in the other.

Its key physics involves chirality or asymmetry coupling rocking and spinning modes.

The tippe top instead owns friction-driven inversion of an approximately axisymmetric body with an offset centre of mass.

Part 14 — This Is Not the Intermediate-Axis Flip

A freely rotating rigid body in space can be unstable when spinning around its intermediate principal axis and repeatedly flip orientation.

That phenomenon requires no contact surface and no friction.

Tippe-top inversion is a different scientific job because surface contact and dissipation are essential.

Part 15 — Follow One Inversion

1. The top begins spinning rapidly with its stem upward.
2. The contact point has a small slip velocity.
3. Friction acts opposite that slip.
4. Because contact is offset from the centre of mass, friction generates torque.
5. A small tilt begins to grow if the upright state is unstable at that spin.
6. Spin, precession and nutation reorganise.
7. The centre of mass rises.
8. Rotational kinetic energy falls by more than gravitational potential energy rises.
9. Friction dissipates the energy difference.
10. The axis approaches the inverted orientation.
11. The top spins temporarily on or near its stem.
12. Continued dissipation eventually destroys the inverted spin and the top settles.

Failed Model → Better Model

Naive model	Why it fails	Better model
Friction only slows rotation.	Friction at an offset contact point also produces torque that changes orientation.	Track contact slip, torque and full rigid-body motion.
Energy loss means the centre of mass must fall.	Rotational kinetic energy can decrease by more than gravitational potential rises.	Use the complete energy ledger.
Angular momentum must point in one fixed direction.	The table exerts external torque.	Track angular momentum together with model-specific integral constraints.
One friction coefficient explains every inversion.	Sliding, rolling and spinning resistance can change the dynamics.	State and test the contact model explicitly.
Every eccentric spinning sphere flips.	Stability depends on geometry, inertia and initial spin.	Calculate the allowed stable states and thresholds.

How Do We Know?

• Track the top’s symmetry-axis angle with high-speed video.
• Measure spin and precession rates through the inversion.
• Track the contact-point motion and estimate slip direction.
• Compare rough and very smooth surfaces.
• Change initial spin and determine the inversion threshold.
• Use tops with different mass offsets and moments of inertia.
• Measure total mechanical energy versus time.
• Compare trajectories with several friction laws rather than fitting one model only.
• Check whether predicted conserved combinations remain approximately constant.

Observation vs Inference

• Observation: a sufficiently fast tippe top can invert on a rough surface.
• Observation: its centre of mass rises during inversion.
• Measurement: mechanical energy decreases while the top is slipping and spinning.
• Inference: contact friction supplies the torque and dissipation that destabilise the initial rotating state.
• Model: axisymmetric rigid-body equations with specified sliding/rolling friction law.
• Boundary: exact inversion trajectory, threshold and surviving integrals depend on the physical and mathematical contact model.

Common Misconceptions and Better Models

Misconception	Better model
Friction gives the top extra energy to climb.	Friction dissipates energy while redirecting the rotational dynamics; spin energy pays for the rise.
The top flips because angular momentum is conserved exactly.	External contact torques act; only particular combinations may be conserved in specific models.
Friction always acts opposite the centre-of-mass velocity.	It acts opposite relative slip at the contact point.
The inversion is the same as the tennis-racket theorem.	Intermediate-axis flipping is torque-free; tippe inversion is contact-driven and dissipative.
All tippe tops flip if spun fast enough.	Some geometry and inertia combinations do not admit complete inversion.

Checkpoint Questions

1. What geometric feature makes a tippe top different from a concentric sphere?
2. Why is friction essential?
3. What velocity does kinetic friction directly oppose?
4. How does contact force create a torque about the centre of mass?
5. Why can the centre of mass rise while mechanical energy falls?
6. Why does initial spin matter for stability?
7. What is meant by dissipation-induced instability?
8. Why must a Jellett integral be described as model-dependent?
9. How is a tippe top different from a rattleback?
10. How is it different from an intermediate-axis flip?

Answer Key

Open after attempting the questions

1. Its centre of mass is offset from the geometric centre along the symmetry axis.
2. Tangential contact force generates the torque and dissipation needed for the familiar inversion.
3. The slip velocity of the contact point relative to the table.
4. The force acts at a point displaced from the centre of mass, giving a non-zero lever arm.
5. Rotational kinetic energy decreases by more than gravitational potential energy increases, with the remainder dissipated.
6. Rotation changes which steady orientations are stable or unstable.
7. Dissipative forces make one rotating state unstable and drive the system toward another attractor.
8. Different friction/contact models preserve different first integrals or none of the same form.
9. Rattleback reversal depends on asymmetric rocking–spinning coupling rather than axisymmetric inversion.
10. Intermediate-axis flipping is free rigid-body instability without surface friction.

Primary Science Bridge

• friction changes motion;
• forces acting away from the centre can turn objects;
• spinning objects can move in complicated ways;
• energy can change form while some becomes heat;
• an object can move upward without gaining total mechanical energy.

Secondary and JC Bridge

Core idea	Higher-resolution route
Friction	Contact-point slip law
Turning effect	Torque about centre of mass
Rotation	Euler rigid-body dynamics
Energy	Dissipative mechanical systems
Stability	Dissipation-induced instability and attractors
Modelling	Sliding versus rolling-resistance laws

Unfamiliar Transfer Challenge

An eccentric spinning rotor touches a rough guide ring. As it loses rotational speed, its axis moves into a higher-potential orientation rather than simply falling to the lowest position.

What would you inspect before calling the motion impossible? Map the contact-point slip, contact torque, rotational-energy loss, potential-energy gain and stability of the initial and final rotating states.

Deep Science Window — Stability Is About Response to Small Disturbances

A steady spin can satisfy the equations perfectly and still be unstable. Give it a tiny tilt: if the disturbance decays, the state is stable; if the disturbance grows, it is unstable. Tippe-top inversion is therefore not explained merely by listing the possible vertical orientations. We must ask how small perturbations evolve while dissipation acts.

Deep Science Window — Why Dissipation Does Not Always Select Lowest Potential Energy

In constrained rotating systems, the accessible state is restricted by angular momentum structure and contact geometry. The system seeks lower total mechanical energy subject to its evolving constraints, not necessarily minimum gravitational potential at every instant. Rotational energy can therefore fund a climb while the overall ledger decreases.

Evidence Boundaries

• Friction-driven inversion ≠ friction supplies energy.
• Centre-of-mass rise ≠ total mechanical-energy increase.
• One ideal friction law ≠ every physical tippe top.
• Jellett-type conservation ≠ universal exact invariant under all rolling-resistance models.
• Fast spin ≠ guaranteed inversion for arbitrary geometry.
• Tippe top ≠ rattleback or intermediate-axis theorem.

Manual Summary — KNOW → CONNECT → EXPLAIN → APPLY → CHECK

KNOW: offset centre of mass, contact slip, friction torque, precession, stability, dissipation.

CONNECT: contact slip to friction, friction to torque, torque to orientation change, and lost rotational energy to both gravitational rise and dissipation.

EXPLAIN: why a spinning top can turn upside down while friction continuously removes energy.

APPLY: diagnose another rotating system whose higher-potential state becomes dynamically selected.

CHECK: state the contact/friction model explicitly and verify its predicted threshold and trajectory.

Research Sources and Further Reading

• SIAM Journal on Applied Dynamical Systems — Tippe Top Inversion as a Dissipation-Induced Instability
• SIAM Journal on Applied Dynamical Systems — Motion of the Tippe Top: Gyroscopic Balance Condition and Stability
• Nonlinear Dynamics — The Influence of First Integrals and Rolling Resistance on Tippe Top Inversion
• Physica — On the Influence of Friction on the Motion of a Top