Scaling & Dimensional Analysis (IPhO Math Tools)

Scaling is your fastest “sanity-check and first-answer” tool. In IPhO problems, a clean scaling argument often earns method marks even before you do any detailed calculus.

1. Definitions (Must Know)

A. Dimensions and units

• A unit is a human convention (meter, second, kilogram).
• A dimension describes what a quantity is made of in terms of base dimensions:
  [Q] = L^a M^b T^c (and sometimes I,Θ,…).

Common physics dimensions:

• [v] = LT⁻¹, [a] = LT⁻², [F] = MLT⁻²
• [E] = ML²T⁻², [P] = ML²T⁻³
• [ρ] = ML⁻³, [p] = ML⁻¹T⁻²

B. Dimensional homogeneity

An equation is dimensionally homogeneous if every term has the same dimensions.

Example: x = x₀ + vt + (1/2)at² is valid because each term has dimension L.

C. Scaling laws

To say “y scales with x” usually means a power law of the form

where C is a dimensionful or dimensionless constant that may depend on other parameters.

D. Dimensionless groups (similarity parameters)

A dimensionless quantity has [Π] = 1. Examples:

• Reynolds number: Re = (ρ v L)/μ
• Mach number: Ma = v/c
• A “small parameter” ε used for approximations

If two physical situations have the same set of relevant dimensionless groups, they are often dynamically similar (same shape of solution after rescaling).

E. Nondimensionalization

Pick characteristic scales and rewrite variables as

so your equations become dimensionless. The coefficients that remain are the dimensionless numbers that control regimes.

F. Order-of-magnitude estimates

An order-of-magnitude statement keeps only the scale (powers of ten and simple factors):

In olympiad physics, you often treat unknown geometric factors (like π or 2) as “order 1” unless the question explicitly needs high accuracy.

2. Key Ideas (What Earns Marks)

• Start by listing the variables you think matter. If you cannot justify a variable, do not include it.
• Use dimensional homogeneity as a fast error detector.
• If a problem statement has no intrinsic length/time scale, expect a power law.
• Turn messy expressions into dimensionless form to spot the controlling parameter.
• Use limiting cases and scaling checks:
  • If you double a length scale, how should time/energy scale?
  • What happens as a parameter becomes very small or very large?
• State the symmetry or regime assumption explicitly (laminar versus turbulent, small angle, thin layer, and so on).

3. Detailed Explanations

A. Dimensional checks (the fastest points)

If you derive a formula under time pressure, do one explicit check:

1. Write dimensions for each symbol.
2. Confirm the final expression matches the required dimension.

This catches missing factors of g, R, ρ, and similar “silent” errors.

B. Power-law scaling from dimensions

If you expect

then write dimensions:

and solve for a,b,c by matching exponents of L,M,T.

This works best when:

• there is only one physically relevant way to combine the variables, and
• you are not mixing additive contributions from different physics.

C. Buckingham Pi (quick version)

If you have n variables built from k base dimensions, you can form n-k independent dimensionless groups Π₁,Π₂,….

Practical workflow:

1. Choose a set of repeating variables that cover all base dimensions (often one length scale, one time scale, one mass scale).
2. Multiply the remaining variables by powers of the repeating variables to make dimensionless groups.
3. The physics becomes a relation like f(Π₁,Π₂,…) = 0.

You usually do not need the full theorem statement in an IPhO script. What matters is producing the right dimensionless combinations and interpreting them.

D. Nondimensionalize to find the regime

Example pattern (without committing to a specific system):

• Suppose your equation contains two terms:
  Term 1 + Term 2 = 0.
• After rescaling, it becomes
  1 + ε = 0,
  where ε is dimensionless.

Then:

• if ε ≪ 1, Term 2 is a small correction,
• if ε ≫ 1, Term 2 dominates and you should simplify using the opposite limit.

E. Scaling for integrals and derivatives (rule of thumb)

If a function changes by an amount Δ f over a scale Δ x, then

If an integrand is roughly size A over a width w, then

These crude estimates become powerful when you combine them with a physical picture of where the integral “gets its area”.

4. Common Mistakes

• Mixing units (centimeters in one place, meters elsewhere) and then trusting a numerical answer.
• Forgetting that angles in series expansions use radians, not degrees.
• Missing a hidden variable (for fluids: ρ, μ; for gravity: G; for EM: ε₀, μ₀).
• Using dimensional analysis when two different physical mechanisms add (then you need a dimensionless parameter to decide which dominates).
• Dropping dimensionless geometric factors that are not actually order 1 (for example, logarithms can be large).

5. Exam Tips

• Write your variable list before you do algebra. It signals method and avoids missing parameters.
• If you use dimensional analysis, write the dimension-matching step explicitly (one line is enough).
• Always do a “factor-of-two” sanity check:
  • What changes if you double the system size?
  • What changes if you double the density?
• If the question asks for an estimate, do not waste time chasing exact constants.

6. Worked Examples

A. Pendulum period (classic dimensional analysis)

A simple pendulum of length L swings with small amplitude in gravity g. Estimate the period T.

B. Diffusion time scale (scaling shortcut)

A dye blob spreads by diffusion with diffusion constant D (dimensions L²T⁻¹). Estimate the time t to spread a distance L.

C. Capillary rise (dimensions plus physics choice)

Liquid rises in a thin tube of radius r due to surface tension γ (dimensions MT⁻²). The liquid has density ρ and gravity is g. Estimate the height h.

7. Mind Stretchers

A. Blast wave radius (Sedov-Taylor scaling)

An explosion releases energy E into uniform air of density ρ. Find how the shock radius R scales with time t.

8. Practice