Modelling & Approximations (IPhO Mechanics)

IPhO mechanics lesson on modelling choices, approximations, scaling, and sanity checks.

International Physics Olympiad preparation

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In olympiad mechanics, the hardest step is often deciding what to ignore without throwing away the effect the question is testing. This lesson gives you a repeatable workflow for choosing models, making approximations, and defending them with scaling and sanity checks.

Learning objectives

By the end, you should be able to:

IPHO-M-MOD-01: state the system boundary, coordinates, interactions and constraints before writing the governing equation;
IPHO-M-MOD-02: identify a dimensionless small parameter, keep a consistent leading order and estimate the first neglected correction; and
present at least one units check and one physical limit or sign check.

These are Mini Physics competency codes. They unpack the official IPhO requirements to model real situations, make appropriate approximations and apply ideas creatively.

Scroll diagram horizontally to read all labels.

An approximation is defensible only when a dimensionless ratio identifies the neglected scale and the final result survives unit, limit, and sign checks.

Dimensional Analysis (units and scaling checks)
Work, Energy & Power Hub (A Level)

1. Definitions

Model: a simplified description of the system that specifies degrees of freedom, forces/energies, constraints, and the equations you will solve.
Idealisation: a structural simplification (point mass, rigid rod, massless string, frictionless pivot). It changes the model class.
Approximation: a controlled neglect of smaller terms inside a chosen model (small-angle, h/R_⊕≪ 1, m_string/m≪ 1). It changes the equations inside the model.
Small parameter: a dimensionless number ε with ε≪ 1 that justifies dropping O(ε) or O(ε²) terms.
Leading order: the first nonzero term in an expansion in ε (what you keep). Next order: the first correction (what you can estimate to bound your error).
Scaling / order-of-magnitude: replace quantities by typical scales (e.g. x∼ L, t∼ τ) to compare terms quickly.
Dominant balance: identify which terms must balance to satisfy the equation in a regime (often the key modelling insight).
Sanity checks: dimensions, limits (small/large parameter), and physics (signs, conservation, monotonicity).

2. Key ideas

IPhO marking rewards explicit modelling. You often get a large fraction of the marks for:

Stating the system and coordinates clearly (with a sketch and sign convention).
Writing the assumptions/idealisations as sentences, not just thinking them.
Justifying each approximation by showing a small dimensionless ratio (or a numerical estimate).
Making one approximation at a time and keeping track of what changed.
Checking your final answer by units and by at least one limiting case.

System + coordinates: “Treat the body as a point mass at x(t) on a smooth track; take + x down the slope.”
Assumptions + why: “Neglect air drag because F_d/(mg)∼ 10⁻²; use small-angle since θ₀ = 0.2 rad so the error is O(θ₀²) ≈ 4%.”
Checks: “I will verify units and the μ → 0 and I → 0 limits.”

3. Detailed explanation

3.1 A modelling workflow you can reuse

Read for structure: what is asked (value, scaling, condition), what is given, what you may assume.
Define the system boundary: what is inside, what is external, what interactions cross the boundary.
Choose coordinates that respect constraints: if there is a constraint, build it into q instead of carrying extra variables.
List candidate “big tools”: momentum/impulse, energy, angular momentum, centre of mass, Lagrangians.
Pick scales and form ratios: write down typical L, T, V, m, v, and compare terms with dimensionless numbers.
Commit to approximations: state them, and note what they imply you are not modelling.
Solve and then check: units, signs, and at least one extreme regime.

3.2 How to justify “ignore this term”

You can drop a term only after you compare it to what remains.

If an equation contains two forces (or energies) A and B, you need a reason to claim A is negligible:

|A|/|B| = ε≪ 1.

Typical sources of small parameters in mechanics:

Geometry: a/L≪ 1 (object size vs curvature/length scale).
Gravity variation: h/R_⊕≪ 1 (uniform g).
Material stiffness: Δ L/L≪ 1 (rigid body or inextensible string).
Timescales: τ_fast/τ_slow≪ 1 (separation of dynamics).
Inertial vs potential: v²/(gL)≪ 1 or v²/cₛ²≪ 1 (quasistatic or “rigid enough”).

3.3 Common modelling atoms in olympiad mechanics

These show up constantly. You are allowed to use them, but you should know the condition they hide.

Point mass: internal rotation/deformation is irrelevant. A clean condition is a/L≪ 1, where a is object size and L is the length scale of variation of the external field/constraint.
Massless, inextensible string: tension is almost uniform and string kinetic/potential energy is negligible. A good small parameter is mₛ/(m₁ + m₂)≪ 1.
Frictionless pulley / pivot: dissipated energy is negligible compared to mechanical energy, or the axle friction torque is negligible compared to the torque required for the motion.
Ignore pulley inertia: the pulley contributes an effective translational mass I/R² to the moving coordinate (worked example below). Neglect when I/R²≪ the masses being accelerated.
Small-angle / small oscillations: expand about equilibrium and keep terms up to quadratic order in the displacement. If θ₀≪ 1, then sin θ ≈ θ with fractional error about θ₀²/6.

3.4 Nondimensionalisation (the clean way to see what matters)

If you can rewrite the dynamics in terms of dimensionless variables, the small parameter usually appears automatically.

Example template (one degree of freedom): suppose

mx double dot + kx + α x³ = 0.

Pick a typical amplitude A and time scale τ = square root of (m/k), and write x = A y and t = τ s. Then

(d² y)/ds² + y + ε y³ = 0, ε = (α A²)/k.

Now the statement “it is approximately a simple harmonic oscillator” becomes “ε≪ 1”.

4. Common mistakes

Dropping the dominant term: a tiny-looking symbol can still dominate if it multiplies a huge scale (or if other terms cancel).
Inconsistent approximations: using sin θ ≈ θ but keeping cos θ exact, or mixing “small angle” with large-angle geometry.
Approximating too early: simplify after you have a correct expression; otherwise you can lose a constraint or a sign.
Forgetting what is being held fixed: taking a limit without stating what stays constant (e.g. m → 0 at fixed I vs at fixed density).
Not checking units: wrong powers and missing factors show up immediately with dimensional analysis.
Treating vectors as scalars: dropping directions and sign conventions (especially in angular momentum and work).

5. Problem-solving tips

Put your modelling statement near the start. Even if the algebra later is messy, the marker can award method marks.
When you make an approximation, write the ratio explicitly (even just one line) and label it ≪ 1.
Prefer dimensionless errors: “fractional error ∼ 2%” is more persuasive than “seems small”.
Use limits as a checksum: does your result behave sensibly as μ → 0, g → 0, I → 0, or θ₀ → 0?
If you are unsure which terms dominate, do a quick scaling estimate before committing.

6. Worked examples

1) Flat-Earth approximation: when can you ignore Earth curvature?

Suppose a projectile travels a horizontal distance L from a launch point while the reference line remains tangent to Earth at launch. Over that distance, the surface drops below the tangent by

s = R_⊕- square root of (R_⊕²-L²) ≈ L²/2R_⊕ (L≪ R_⊕).

A clean modelling condition for “flat Earth” is s≪ H, where H is the vertical scale relevant to the problem (peak height, clearance, etc).

Numerical check: take R_⊕ ≈ 6.4 × 10⁶ m and L = 10 km = 10⁴ m:

s ≈ (10⁴)²/(2(6.4 × 10⁶)) ≈ 7.8 m.

Whether this is negligible depends on the required vertical precision and on the other vertical scales. For a metre-level prediction over 10 km, it is not negligible.

The familiar L²/(8R_⊕) sagitta belongs to a different geometry: the maximum gap between an arc and the chord joining two endpoints separated by chord length L.

2) Small-angle: how small is “small”? (error estimate)

The Taylor expansion is

sin θ = θ-θ³/6 + O(θ⁵).

If you replace sin θ by θ, the fractional error is roughly

(| sin θ-θ|)/|θ| ≈ θ²/6.

Example: if the amplitude is θ₀ = 0.2 rad, the fractional error is about 0.2²/6 ≈ 0.007 (below 1%). If θ₀ = 0.5 rad, the error is about 0.5²/6 ≈ 0.04 (a few percent).

This is exactly the kind of one-line justification that earns marks.

3) When is pulley inertia negligible? (effective mass idea)

Consider an Atwood-like system where the string does not slip on a pulley of radius R and moment of inertia I.

If the linear coordinate of the string is x(t), then the pulley’s angular speed is ω = x dot/R, so its rotational kinetic energy is

Tᵣₒₜ = 1/2 Iω² = (1/2)(I/R²)x dot ².

Compare this to translational kinetic energy terms ∼ 1/2 (m₁ + m₂)x dot ²: the pulley behaves like an extra mass

m_eff = I/R².

Modelling condition: ignore pulley inertia when I/R²≪ m₁ + m₂.

Quick special case: for a solid disk pulley of mass M,

I = (1/2)MR² ⇒ I/R² = M/2.

So “massless pulley” means M≪ 2(m₁ + m₂).

7. Extensions

1) Two drag laws in one equation (find regimes by scaling)

A falling object sometimes obeys

mv dot = mg - cv - dv²

with both linear and quadratic drag. Define the crossover speed v* by comparing the drag terms:

cv*∼ dv*² ⇒ v* = c/d.

Regimes: for v≪ v*, the dv² term is negligible (linear drag dominates). For v≫ v*, the cv term is negligible (quadratic drag dominates).

Terminal speed estimate: set v dot = 0 to get mg = cvₜ + dvₜ². If you already know which regime applies, you can estimate

vₜ ≈ mg/c (vₜ≪ v*), vₜ ≈ square root of (mg/d) (vₜ≫ v*).

The modelling step is deciding which estimate is self-consistent.

2) Period of a pendulum beyond small angle (estimate the correction)

The exact period of a simple pendulum of length ℓ and amplitude θ₀ can be written using an elliptic integral. For olympiad work, the key is the first correction:

T ≈ 2π square root of (ℓ/g) (1 + θ₀²/16 + O(θ₀⁴)).

Task: take θ₀ = 0.5 rad and estimate the fractional increase in period.

Answer: θ₀²/16 ≈ 0.25/16 ≈ 0.016, so the period is about 1.6% longer than the small-angle value.

8. Practice and evidence

Pick two IPhO mechanics problems with different contexts. Before doing algebra, write the system, coordinate, retained interactions and small parameter. Treat IPHO-M-MOD-01/02 as passed only if both solutions include a justified approximation and a successful limit or scale check. Then continue via the IPhO Mechanics Hub.

Syllabus and review details

No official syllabus alignment is listed for this lesson.

Last reviewed: 2026-07-20