IPhO mechanics
Modelling, approximations and high-leverage methods for olympiad mechanics beyond the standard syllabus.
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Before you begin
IPhO mechanics problems reward clean modelling, controlled approximations and early method selection. Start by learning to state a model and test its approximations. The lessons below develop force, energy and momentum methods for unfamiliar constraints and regimes; Lagrangian mechanics offers an optional alternative method.
Be comfortable with: A-Level kinematics; forces and dynamics; circular motion; work, energy and power; and the IPhO problem-solving framework.
Learning goals
- Apply mechanics models and problem-solving methods in competition settings.
Lessons
Work through them in order.
Build the bridge
Build the official core
- Rotational dynamics and rollingSolve rolling problems from translation, rotation and the no-slip condition, and find the friction instead of guessing it.
- Fluids: Bernoulli, continuity and scalingApply continuity, Bernoulli's equation and the momentum flux, and use the Reynolds number to decide whether viscosity matters.
Transfer to unfamiliar problems
- Small oscillations and normal modesExpand about a stable equilibrium to find small-oscillation frequencies and the normal modes of coupled systems.
- Central forces and the effective potentialReduce central-force orbits to radial motion in an effective potential, and find turning points and stable orbits.
Optional: extend your toolkit
Practise and check
Topic reference
What you should be able to do
- Define a mechanics model by naming the system, coordinates, interactions, constraints and approximation regime.
- Choose and combine force, energy, momentum, angular-momentum or fluid-balance methods without contradicting the constraints.
- Check a result using units, limiting cases, stability, contact feasibility or the size of a neglected term.
Official scope: Covers selected routes through official mechanics and hydrodynamics content; the optional Lagrangian lesson is a training extension.
Official coverage and competencies
The official IPhO syllabus supplies the coverage boundary. Its Mechanics section names kinematics and rigid-body motion, statics, dynamics and conservation laws, celestial mechanics, and hydrodynamics. Its general theoretical skills also require appropriate modelling, approximation and use of symmetry; the wider syllabus covers single oscillators and approximate mathematical methods.
The official document deliberately gives broad topics rather than lesson-sized outcomes and says that past IPhO problems indicate the expected depth. Mini Physics therefore adds the competency codes below as an instructional decomposition. These codes are not issued or endorsed by IPhO.
| Mini Physics competency | Observable evidence | Official anchor |
|---|---|---|
| IPHO-M-MOD-01 Define the model | State the system, coordinates, interactions and constraints before solving. | General theoretical skills: modelling and symmetry |
| IPHO-M-MOD-02 Control approximations | Produce a dimensionless small parameter and check the neglected order. | General skills; approximate mathematical methods |
| IPHO-M-ROT-01 Couple rigid-body motion | Use centre-of-mass translation, rotation and the rolling constraint consistently. | Mechanics: kinematics and dynamics |
| IPHO-M-ROT-02 Test contact feasibility | Determine the friction direction, solve its magnitude and verify the static-friction bound. | Mechanics: statics and dynamics |
| IPHO-M-OSC-01 Linearise near equilibrium | Find equilibrium, expand to the required order and obtain the small-oscillation frequency. | Statics: stable equilibrium; single oscillator |
| IPHO-M-OSC-02 Resolve coupled modes | Obtain mode frequencies and amplitude ratios, then check symmetry or decoupling limits. | Training extension connecting oscillators, waves and linear algebra |
| IPHO-M-CEN-01 Reduce central-force motion | Use angular momentum and energy to obtain a radial equation. | Dynamics and celestial mechanics |
| IPHO-M-CEN-02 Classify radial motion | Read allowed regions, turning points and stability from an effective-potential graph. | Training method for dynamics and celestial mechanics |
| IPHO-M-FLU-01 Select fluid balances | State the regime and decide between continuity, Bernoulli and momentum. | Mechanics: hydrodynamics |
| IPHO-M-FLU-02 Analyse a control volume | Include pressure forces and momentum flux with a declared sign convention. | Training extension for hydrodynamic force problems |
| IPHO-M-LAG-01 Choose generalized coordinates | Encode holonomic constraints in a minimal coordinate set. | Optional method; not named in the official syllabus |
| IPHO-M-LAG-02 Exploit a Lagrangian | Derive an equation of motion or conserved momentum and verify it by a standard method. | Optional method; not named in the official syllabus |
These six lessons build on one another. They do not cover all of official IPhO mechanics: projectile and relative kinematics, centre-of-mass statics, elasticity, non-inertial frames, surface tension and capillarity still need practice elsewhere or future lessons.
Revision
Method selector
| Clue in the problem | Start with | Mandatory check |
|---|---|---|
| Constraints hide forces you do not need | Energy or a minimal coordinate | Every constraint is satisfied by the coordinates |
| Collision or short interaction | Momentum, impulse or angular momentum | Define the system and external impulse/torque |
| Motion near equilibrium | Potential curvature or linearised equation | Expand about the actual equilibrium |
| Orbit or radial force | Angular momentum plus energy | Allowed radii have E ≥ V_eff |
| Pipe speeds and pressures | Continuity, then Bernoulli if its assumptions hold | State streamline, viscosity and compressibility assumptions |
| Fluid force on a bend, nozzle or plate | Control-volume momentum | Include pressure forces and vector directions |
Coverage gaps to train separately
- Vector kinematics in moving frames, including when Coriolis acceleration vanishes.
- Centre-of-mass calculation by integration and one- or two-dimensional equilibrium.
- Stress, strain, Young modulus and elastic-energy models.
- Rotating frames, centrifugal potential and inertial forces.
- Kepler laws and energy on elliptical orbits.
- Buoyancy, surface tension, surface energy and capillary pressure.
Training habits
- Work official problems in timed blocks.
- Tag every lost mark with a competency code, or with coverage gap if this unit has no matching objective.
- Write the missing model or check step, then re-solve the problem after 48–72 hours without notes.
- Reading or revealing a solution is preparation, not evidence of independent problem solving. Re-solve without hints and test the method in a problem with a changed assumption, representation or constraint.
- Keep one mechanics problem in your weekly cycle after you move on.