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Dimensional Analysis

Intuition

You can learn a shocking amount about a physical system before solving any equation — just by insisting that the units balance. G. I. Taylor famously estimated the (classified!) yield of the Trinity nuclear test from a published photograph, using nothing but dimensions. The method: guess which quantities matter, demand the units match, and the physics has nowhere to hide.

The principle

Every physical quantity has dimensions built from a small basis — mass \([M]\), length \([L]\), time \([T]\), temperature \([\Theta]\), current \([I]\). Any valid physical equation must be dimensionally homogeneous: units on the left = units on the right.

Recipe:

  1. List the parameters that can plausibly matter: \(\{A_1, A_2, \dots\}\)
  2. Reduce to an independent set \(\{B_1, B_2, \dots\}\)
  3. Write down the dimensions of each
  4. Posit a power law: \(C \propto B_1^{\alpha} B_2^{\beta}\cdots\)
  5. Match exponents of \([M], [L], [T], \dots\) and solve

Worked example: the Trinity blast wave

Assume the fireball radius \(R\) depends on released energy \(E\), time \(t\), and ambient density \(\rho\). Posit \(E \propto R^\alpha t^\beta \rho^\gamma\):

\[[E] = ML^2T^{-2} = L^\alpha\, T^\beta\, (ML^{-3})^\gamma\]

Matching: \(\gamma = 1\) (mass), \(\beta = -2\) (time), \(\alpha - 3\gamma = 2 \Rightarrow \alpha = 5\):

\[\boxed{\,E \sim \frac{\rho R^5}{t^2}\,}\]

One photo (radius + timestamp) then yields the yield. The full solution is the Sedov–Taylor blast wave, and the same scaling reappears in supernova remnants.

Worked example: boiling eggs

Boiling time \(t\) vs egg weight \(W\). Heat conduction (\(\partial_t \theta = k\nabla^2\theta\), \([k] = L^2 T^{-1}\)) sets the clock: \(t \sim L^2/k\), and geometric similarity plus equal density gives \(W \propto L^3\). Hence

\[t \propto W^{2/3}\]

A 60 g chicken egg takes 7 min ⇒ a 10 g quail egg takes \(7\times(10/60)^{2/3} \approx 2.1\) min; a 1.6 kg ostrich egg takes \(7\times(1600/60)^{2/3} \approx 62\) min.

Worked example: animal locomotion

  • Walking (pendulum law): legs swing like pendulums, \(T = 2\pi\sqrt{L/g}\), so stepping frequency \(f \propto L^{-1/2}\) — bigger animals stroll slower.
  • Running (strength-limited law, A. V. Hill): muscle stress \(\sigma\) is universal; force \(\propto L^2\), torque \(\propto L^3\), moment of inertia \(\propto L^5\) ⇒ angular acceleration \(\propto L^{-2}\) and \(f \propto L^{-1}\).

Log–log plots of real animals show both slopes — physics visible in zoology.

Scale models

Fluids have no intrinsic length scale: the governing equations are scale-invariant once expressed in dimensionless groups. Match the groups (chiefly the Reynolds number) between a wind-tunnel model and the full-scale aircraft, and the flows are dynamically similar — same physics, cheaper experiment.

Common mistakes

  • Omitting a relevant parameter (the method can't warn you) or including a redundant one (you get an undetermined exponent — actually a hint that a dimensionless group is free).
  • Expecting the dimensionless prefactor. Dimensional analysis gives \(\sim\), never the constant (Sedov's full calculation gives the O(1) factor).
  • Treating angles/counts as dimensional — they're dimensionless; they ride along in the unknown function.

Knowledge graph position

Prerequisites: none beyond units — accessible anytime. Leads to: Buckingham Pi, Reynolds number, Kolmogorov scaling.

Quiz

Q1 (computational). A pendulum's period may depend on \(L\), \(m\), \(g\). Find the dependence.

Answer

\(T \propto L^a m^b g^c\): time needs \(c = -1/2\), mass forces \(b = 0\), length gives \(a = 1/2\). \(T \propto \sqrt{L/g}\) — mass drops out automatically.

Q2 (conceptual). In the egg problem, why does the boiling time scale as \(W^{2/3}\) rather than \(W\)?

Answer

Heat must diffuse to the centre: \(t \sim L^2/k\), quadratic in size, while \(W \sim L^3\). So \(t \sim W^{2/3}\) — cooking time is set by conduction depth, not by total mass.

Q3 (multiple choice). Dimensional analysis alone can determine:

  • (a) exact numerical answers (b) functional forms up to dimensionless factors
  • (c) which parameters are physically relevant (d) boundary conditions
Answer

(b). Relevance (c) must be assumed going in; constants and boundary effects need real analysis or experiment.