Dimensional Analysis Balance

Left pan (LHS)

Quantity A

Right pan (RHS)

Quantity B

Multiply by a second quantity

Beam sensitivity 1.0×

Verification

—

M⁰L⁰T⁰ = M⁰L⁰T⁰

LHS dimension

—

RHS dimension

—

Mismatch |ΔD|

0.00

Beam angle

0.0°

How it works

Every physical quantity has a dimension built from the SI base quantities mass M, length L and time T, written as an exponent vector. Force, for example, is [M¹L¹T⁻²]. An equation is dimensionally homogeneous — a necessary (not sufficient) condition for it to be physically valid — only if both sides reduce to the exact same exponent vector:

[LHS] = [RHS]
D = (a, b, c)  such that  quantity ~ M^a L^b T^c
mismatch |ΔD| = ‖D_LHS − D_RHS‖₂

This scene turns that check into a real balance scale. Each pan's load is the Euclidean norm ‖D‖ of the quantities placed in it — a visualisation weight, not a literal mass — and the beam is a genuine damped rotational oscillator driven by the load difference:

τ = k·(W_R − W_L) − c·ω         (spring-like restoring drive + damping)
ω̇ = τ / I,   θ̇ = ω              (semi-implicit Euler, dt clamped)

The beam's tilt reflects how much "load" is on each side and settles smoothly like a real scale. The dimensional verdict (green/red) is computed independently and exactly: it is true quality control for an equation — it flags F = m·v (force set equal to momentum) as dimensionally inconsistent even though both are everyday mechanics quantities, and confirms F = m·a is valid. This is exactly the first checkpoint described in the article: before trusting any simulation's numerical output, verify its equations are dimensionally consistent.

Quantity A / B / C — pick from 14 common mechanical quantities, each with a known [M,L,T] exponent vector.
Multiply by a second quantity — combine two RHS quantities (their exponents add), e.g. mass × acceleration.
Random equation — samples a fresh trio, sometimes landing on a valid identity, sometimes not.

Left pan · LHS Right pan · RHS

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