📐Dimensional Analysis - Equation Checker, Buckingham π, Unit Simplifier

How to use Dimensional Analysis Tool

1. 1Under “Equation check”, type an equation or load an example such as s = v₀t + ½at²
2. 2Check the quantity assigned to each symbol and change it from the list or type a unit
3. 3Read the colours: green terms match the left side, red ones do not, and the problems are listed below
4. 4Under “Simplify units”, enter a unit expression to get its named derived unit and simpler forms
5. 5Under “π theorem”, list the relevant variables to get the dimension matrix, its rank and a set of π groups

FAQ

If an equation is dimensionally consistent, is it correct?

Not necessarily. Dimensional homogeneity is necessary, not sufficient: both sides and every term being added must share a dimension, so an inconsistent formula is certainly wrong. But pure-number factors such as ½, 2π or √2 and the form of functions are invisible to dimensional analysis; s = v₀t + at² passes the check yet is missing a ½. Use it to catch mistakes quickly, not to prove a result.

Why must the arguments of sin, exp and ln be dimensionless?

These functions are power series, e.g. eˣ = 1 + x + x²/2 + …, and if x carried a dimension each term would have a different one and could not be added. So in e^(−t/τ) the ratio t/τ is dimensionless, and in sin(ωt) the product ωt is an angle in radians. Likewise a dimensional quantity can only be raised to a fixed number; x^t has no physical meaning.

What is the Buckingham π theorem and how do I use it?

If a physical law involves n quantities whose dimension matrix has rank r (usually the number of base dimensions involved), it can always be written as a relation between n − r dimensionless groups, Π₁ = Φ(Π₂, …). Drag on a sphere involves F, ρ, v, D and μ with rank 3, giving F/(ρv²D²) and the Reynolds number ρvD/μ, so the drag coefficient can only depend on Re. With a single group it must be constant: for a pendulum T²g/L = constant, which gives T ∝ √(L/g) directly.

How should I choose the repeating variables, and why do different choices give different groups?

The repeating variables must be dimensionally independent, number exactly r and together contain every base dimension present; typical choices are a length, a density and a velocity, never the quantity you are solving for. Different choices give different sets of π groups, but each set can be obtained from another by multiplying groups and raising them to powers, so they describe the same relation. Some forms, like the Reynolds number, are simply more conventional.

What are the Planck and natural unit tables for?

Theorists often set ħ = c = 1 (particle physics) or G = ħ = c = k_B = 1 (Planck units) to simplify formulas, then have to convert numbers back to SI. The tables give the length (about 1.97 × 10⁻¹⁶ m) and time corresponding to 1 GeV⁻¹, the mass and temperature of 1 GeV, the Planck length and time, and atomic units such as the Bohr radius and hartree, all computed from CODATA 2022 constants.