Units, Conversions & Dimensional Analysis – White to Black Belt Mastery

by | Aug 11, 2025

🥋 Units, Conversions, and Dimensional Analysis – White to Black Belt Mastery

Intro Quote or Motto

“A measurement without units is like a punch without aim — it has no real meaning.”

Key Concept (White Belt Level)

Units give meaning to numbers. Conversions change a measurement from one unit to another
without changing its value. Dimensional analysis checks whether equations make physical sense.
Throughout this guide, final answers are shown to 3 significant figures (3SF).

Core Principles

  1. SI base units: meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), candela (cd).
  2. Common derived units: newton (N = kg·m/s²), joule (J = N·m), watt (W = J/s), pascal (Pa = N/m²).
  3. Prefixes: milli (m, 10⁻³), micro (µ, 10⁻⁶), kilo (k, 10³), mega (M, 10⁶), giga (G, 10⁹).
  4. Factor–label method: multiply by conversion factors that equal 1 so units cancel cleanly.
  5. Dimensional analysis: both sides of an equation must match in dimensions (e.g., [M], [L], [T]).

Common Mistakes and Pitfalls

  • Mixing unit systems (e.g., m with km) without converting first.
  • Dropping units during steps; always carry and cancel them visibly.
  • Rounding too early — keep guard digits, then round the final answer to 3SF.
  • Using formulas that aren’t dimensionally consistent.

Sensei’s Shortcuts

  • Write units on every line; let them guide each operation.
  • Convert prefixes first (e.g., km → m) before other calculations.
  • If dimensions don’t match, fix the algebra before plugging in numbers.
  • Sanity-check magnitudes: does the result look reasonable?

Worked Example – Step by Step (White Belt)

Problem. Convert 65.0 km/h to m/s, and convert 2.50 h to seconds. Report 3SF.

Solution.

Speed: 65.0 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 18.1 m/s.

Time: 2.50 h × (3600 s / 1 h) = 9.00 × 10³ s.

Final Answer: 18.1 m/s; 9.00 × 10³ s.

Practice Drill

  1. Convert 3.60 km/min to m/s.
  2. Convert 45.0 m/s to km/h.
  3. Convert 3.50 g/cm³ to kg/m³.

Answers

  1. 60.0 m/s
  2. 162 km/h
  3. 3.50 × 10³ kg/m³

Yellow Belt Extension – Deeper Skills

Dimensional check: s = u t + 0.5 a t²

  • u t ⇒ (L·T⁻¹)·T = L
  • a t² ⇒ (L·T⁻²)·T² = L
  • Sum gives L, matching s (length).

Pendulum scaling via dimensions. Suppose period T depends on length L and g:

Assume T ∝ L^α g^β. Dimensions: [T] = [L]^α [L·T⁻²]^β = L^{α+β} T^{−2β}.
Match exponents ⇒ −2β = 1 ⇒ β = −1/2; α + β = 0 ⇒ α = 1/2 ⇒
T ∝ √(L/g).

Black Belt Mastery – Exam Strategy and Challenge

Challenge.

  1. A cyclist rides at 27.8 m/s. Convert to mph.
  2. Pressure is 1.02 × 10⁵ Pa. Convert to kPa and to atm (1 atm = 1.013 × 10⁵ Pa).
  3. Someone claims power = force × distance. Use dimensions to accept or reject this.

Sensei Strategy Notes

  • Chain factors for (1): m→km→mi and s→h.
  • For (2), Pa → kPa (÷1000) and compare to atm.
  • Power should be force × velocity; check dimensions.

Suggested Answers (3SF)

  1. 27.8 m/s × 3.60 km/h per m/s ÷ 1.61 km/mi = 62.2 mph (approx).
  2. 1.02 × 10⁵ Pa = 102 kPa; (1.02 × 10⁵) / (1.013 × 10⁵) = 1.01 atm.
  3. [Force × distance] = (M·L·T⁻²)·L = M·L²·T⁻² (work), not power (M·L²·T⁻³). Correct relation: power = force × velocity.

Sensei’s Final Words

“Units are your stance, conversions your footwork, and dimensional analysis your guard. Master them, and every calculation lands clean.”