General Physics 1.1 — SI Units & Dimensional Analysis
Foundations of measurement: the seven SI base units, derived units, and using dimensional analysis to verify equations and convert between unit systems.
First session of a general physics study track. This chapter is usually treated as bookkeeping, but dimensional analysis in particular is a genuine problem-solving tool — worth treating carefully.
The Seven SI Base Units
All physical quantities are built from seven independent base units:
| Quantity | Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Temperature | kelvin | K |
| Amount of substance | mole | mol |
| Luminous intensity | candela | cd |
Every other unit in physics is a combination of these. Force (newton) is kg·m·s⁻². Energy (joule) is kg·m²·s⁻². Voltage (volt) is kg·m²·s⁻³·A⁻¹.
Derived Units and Prefixes
Derived units are shorthand names for combinations that appear frequently. It is worth staying comfortable converting them back to base units — that is where dimensional analysis operates.
Standard SI prefixes:
| Prefix | Symbol | Factor |
|---|---|---|
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| milli | m | 10⁻³ |
| micro | μ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
Dimensional Analysis
Every valid physical equation must be dimensionally homogeneous — the dimensions on both sides must match. Dimensions are written using the base quantities: L (length), M (mass), T (time), I (current), Θ (temperature), etc.
Checking an equation
Kinetic energy: KE = ½mv²
- Left side: [energy] = M·L²·T⁻²
- Right side: [mass] × [velocity]² = M × (L·T⁻¹)² = M·L²·T⁻²
Dimensions match — the equation is dimensionally consistent.
Deriving relationships
If you know which quantities a physical result depends on, dimensional analysis can suggest the form of the equation. Example: the period of a simple pendulum depends on length L and gravitational acceleration g (dimensions L·T⁻²).
Set up: [T] = [L]ᵃ · [L·T⁻²]ᵇ
Matching exponents: a + b = 0 and -2b = 1 → b = -½, a = ½
This gives T ∝ √(L/g) — the correct functional form, without solving the differential equation.
Unit conversion
Dimensional analysis makes unit conversion mechanical. Multiply by conversion factors written as fractions equal to 1:
60 mph → m/s: 60 mi/hr × (1609 m / 1 mi) × (1 hr / 3600 s) ≈ 26.8 m/s
Each fraction is dimensionally 1; only the units change.
What Stuck
Dimensional analysis is a consistency check and a derivation tool, not just bookkeeping. If an equation fails a dimensional check it is definitely wrong. If it passes, it is not necessarily right — but dimensional analysis narrows the form substantially. Building the habit of checking dimensions before trusting a result catches algebra errors early.