General Physics 1.3 — Unit Conversion and the Metric System
Physics is only as useful as its numbers are comparable. The metric system provides a coherent, base-10 structure for measurement; unit conversion is the discipline of moving between scales without losing meaning.
Why Units Are Not Optional
A number without a unit is almost always meaningless. “The distance is 42” tells you nothing — 42 what? Meters, kilometers, miles, light-years? Units carry the physical meaning; the number only carries the magnitude within that meaning. Every calculation in physics is simultaneously a calculation on units, and getting units wrong produces answers that are numerically precise and physically nonsensical.
The most expensive unit error on record: NASA’s Mars Climate Orbiter was lost in 1999 because one team used pound-force seconds and another used newton-seconds for thruster impulse. The spacecraft approached Mars on the wrong trajectory and was destroyed. The calculation was done correctly — in the wrong units.
The Metric System (SI)
The International System of Units (SI, from the French Système International) is the global standard for science and most of engineering. It is built on seven base units, each measuring a fundamentally independent physical quantity:
| 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 derived from these seven. Speed is m/s, force is kg·m/s² (the newton), energy is kg·m²/s² (the joule), pressure is kg/(m·s²) (the pascal). There are no arbitrary conversion factors within the SI system — it is fully coherent.
Metric Prefixes
The metric system handles scale through prefixes — multipliers that are powers of ten, applied to any base unit:
| Prefix | Symbol | Factor |
|---|---|---|
| exa | E | 10¹⁸ |
| peta | P | 10¹⁵ |
| tera | T | 10¹² |
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| (base) | — | 10⁰ |
| centi | c | 10⁻² |
| milli | m | 10⁻³ |
| micro | μ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
| pico | p | 10⁻¹² |
| femto | f | 10⁻¹⁵ |
| atto | a | 10⁻¹⁸ |
Examples: 1 km = 1000 m, 1 ms = 0.001 s, 1 μA = 10⁻⁶ A. The prefix replaces the power-of-ten — you never write “1 kilo × 10³ metres.”
Unit Conversion: Dimensional Analysis
The reliable method for unit conversion is dimensional analysis (also called the factor-label method). The key insight: multiplying by a conversion factor equal to 1 changes the unit without changing the physical quantity.
Since 1 km = 1000 m, the fraction 1 km / 1000 m equals exactly 1. Multiplying any distance by this fraction leaves the distance unchanged — only the unit changes.
Example: Convert 65 miles/hour to metres/second.
Known conversion factors: 1 mile = 1609 m, 1 hour = 3600 s.
65 miles 1609 m 1 hour 65 × 1609
-------- × ------ × -------- = -------------- m/s ≈ 29.1 m/s
hour 1 mile 3600 s 3600
The miles cancel, the hours cancel, what remains is m/s. If you set up the fractions so that the units you want to eliminate are in opposite positions (one in numerator, one in denominator), they cancel algebraically. If your final unit is wrong, the setup was wrong — the units tell you.
Temperature Scales
Temperature is a common source of conversion errors because the three main scales have different zeros and different step sizes:
- Kelvin (K) — SI base unit. Absolute scale: 0 K is absolute zero, the lowest possible temperature. No degree symbol.
- Celsius (°C) — offset from Kelvin by 273.15. Water freezes at 0 °C (273.15 K), boils at 100 °C (373.15 K).
- Fahrenheit (°F) — different zero and different scale. Water freezes at 32 °F, boils at 212 °F.
Conversions:
- K = °C + 273.15
- °C = (°F − 32) × 5/9
- °F = °C × 9/5 + 32
Physics calculations must use Kelvin — equations involving temperature (ideal gas law, thermodynamics) require an absolute scale. Using Celsius or Fahrenheit directly in these equations gives wrong answers.
Order-of-Magnitude Check on Conversions
After any conversion, check the result makes intuitive sense at the order-of-magnitude level. Going from miles to kilometres, the number should get bigger (1 mile ≈ 1.6 km). Going from metres to kilometres, the number should get smaller. A 70 kg person should not convert to 70,000 g and then be reported as 700 g — the direction of the power-of-ten matters, and it is the most common error.
What stuck: Dimensional analysis is not a trick — it is a complete, self-checking method. If you carry units through every step of a calculation, the units in the final answer will either confirm correctness or reveal exactly where the logic broke down.