🎯 Hook – the $327 million unit mistake
In 1999, NASA's Mars Climate Orbiter burned up in the Martian atmosphere because one engineering team's software reported thruster impulse in pound-force seconds while the navigation software expected newton seconds. No equation was wrong — the units were. A coherent system of units isn't bureaucratic fussiness; it's what stops a $327 million spacecraft from being lost to a label.
🧱 SI base quantities and units
| Base quantity | Unit | Symbol |
|---|---|---|
| Mass | kilogram | kg |
| Length | metre | m |
| Time | second | s |
| Electric current | ampere | A |
| Thermodynamic temperature | kelvin | K |
| Amount of substance | mole | mol |
Derived units are products or quotients of base units, many with special names built from defining equations:
| Quantity | Defining equation | Unit | Base units |
|---|---|---|---|
| Force | F = ma | newton, N | kg m s⁻² |
| Energy | W = Fs | joule, J | kg m² s⁻² |
| Power | P = W/t | watt, W | kg m² s⁻³ |
| Pressure | p = F/A | pascal, Pa | kg m⁻¹ s⁻² |
| Charge | Q = It | coulomb, C | A s |
| Potential difference | V = W/Q | volt, V | kg m² s⁻³ A⁻¹ |
| Resistance | R = V/I | ohm, Ω | kg m² s⁻³ A⁻² |
🔢 Prefixes, conversions and IB notation
Converting units: speed km h⁻¹ → m s⁻¹ divide by 3.6; area/volume squares or cubes the prefix (\(1\ \text{cm}^2=(10^{-2}\,\text{m})^2=10^{-4}\,\text{m}^2\); \(1\ \text{cm}^3=10^{-6}\,\text{m}^3\)); density \(1\ \text{g cm}^{-3}=1000\ \text{kg m}^{-3}\); temperature \(T(\text{K})=\theta(°\text{C})+273\) (a change of 1 K equals a change of 1 °C); energy \(1\ \text{eV}=1.60\times10^{-19}\ \text{J}\), \(1\ \text{kWh}=3.6\times10^{6}\ \text{J}\).
Accepted non-SI units in IB physics: eV, u (atomic mass unit), kWh, light year, parsec, AU, °C, minute/hour/day/year, degree (angle).
IB notation: use negative indices (m s⁻¹, not m/s); a space separates number and unit and separates unit symbols (9.81 m s⁻²); symbols are case-sensitive (mW ≠ MW); no plurals, no full stops (5 kg, not 5 kgs); table headings and graph axes use quantity/unit, e.g. "v / m s⁻¹".
🌐 Deeper notes — the 2019 redefinition and dimensionless quantities
Since 20 May 2019, all seven SI base units are defined by fixing the numerical values of constants of nature — the caesium hyperfine frequency (second), the speed of light (metre), the Planck constant (kilogram), the elementary charge (ampere). The kilogram is no longer defined by a metal cylinder kept in Paris, which could be damaged or drift in mass over time; a constant of nature is reproducible anywhere.
Dimensionless quantities — ratios of the same kind of quantity have no unit: strain, refractive index, efficiency, coefficient of friction. Angles in radians are formally dimensionless (m/m) but the "rad" label is kept for clarity.
Coherence: combining base units in an equation gives a derived unit with a factor of exactly 1 (\(1\ \text{N}=1\ \text{kg}\times1\ \text{m s}^{-2}\)); mixing units (g with m, say) breaks this and gives a wrong number by a power of ten.
📊 Graph mastery — temperature and prefix scales
| Graph type | Axes (X, Y) | Shape | What to read |
|---|---|---|---|
| Temperature in K vs °C | °C · K | Straight line, gradient 1 | Intercept at 273 K (0 °C) |
| Log-scale prefix chart | power of ten · example object | Evenly spaced decades, femto to peta | Where a given quantity sits among the prefixes |
✍️ Worked example 1 — deriving units from an equation
✍️ Worked example 2 — conversions in practice
🌍 Real-world: unit mistakes cost money
The Mars Climate Orbiter's loss is the textbook case, but unit slips happen constantly in engineering, medicine (dosage units) and finance. A written "/ unit" column on every table and graph axis is not a formality — it is the single habit that catches this class of error before it becomes expensive.
🚀 Going further — why fix constants, not artefacts?
A physical artefact (like the old kilogram cylinder) can be damaged, contaminated, or simply drift in mass with handling over decades — and only one copy exists to compare against. Defining units from constants of nature (the Planck constant, the speed of light) gives a definition any suitably equipped lab anywhere can realise independently, with no risk of loss or drift.
Challenge: strain (extension/original length) has no unit. What does this tell you about combining it in an equation like stress = (Young modulus) × strain? Since strain is dimensionless, the Young modulus must carry the same units as stress (Pa) — nothing is "hidden" in a dimensionless factor.
📋 Quick practice — answers in italics
(a) Convert 108 km h⁻¹ to m s⁻¹. 30 m s⁻¹.
(b) Convert 450 nm to m. 4.50×10⁻⁷ m.
(c) Convert 37 °C to kelvin. 310 K.
(d) A 60 W lamp runs for 3.0 h — find the energy used in J and kWh. 6.48×10⁵ J = 0.18 kWh.
(e) Identify the errors in "the car's speed was 20 M/S and its mass 1.2 Kgs." "M" should be lowercase "m" (M means mega); units take no plural "s"; the slash form is non-standard IB style (m s⁻¹, kg).
⚠️ Trap corner – common mistakes
| Misconception | Correct view |
|---|---|
| The gram is the SI base unit of mass | The kilogram is the base unit — the only base unit that already carries a prefix |
| Prefixes apply linearly to squared/cubed units | \(\text{cm}^2=10^{-4}\ \text{m}^2\), not \(10^{-2}\) — the prefix is squared (or cubed) too |
| mW and MW are easy to mix up | m = milli (10⁻³), M = mega (10⁶) — a million-fold difference; read case carefully |
| Kelvin takes a degree sign | 300 K, never "300 °K" — kelvin is an absolute scale with no degree symbol |
| km h⁻¹ → m s⁻¹ by multiplying by 3.6 | Divide by 3.6 to convert km h⁻¹ to m s⁻¹ (multiply to go the other way) |
| kWh and kW measure the same thing | kWh is energy; kW is power — don't confuse a rate with a total |
Complete the statements:
✓ (b) \(1.1\times10^8/1.01\times10^5\approx1.1\times10^3\) → about 3 orders of magnitude.
✓ (c) The atmosphere (atm), the bar, or mmHg.
✓ (b) \(k=540/30^2=\mathbf{0.60\ kg\,m^{-1}}\).
✓ (c) \(90\ \text{km h}^{-1}=25\ \text{m s}^{-1}\); \(F=0.60\times25^2=\mathbf{375\ N}\).
✓ (b) \(t=E/P=5.14\times10^{5}/2200\approx234\ \text{s} \approx\mathbf{3.9\ min}\).
✓ (b) \(V=\pi r^2h=\pi(0.0125)^2(0.0320)=1.57\times10^{-5}\ \text{m}^3\); density \(=0.0426/1.57\times10^{-5}\approx\mathbf{2710\ kg\,m^{-3}}\).
✓ (c) %unc: mass 0.24%, diameter 0.4% (doubled to 0.8% for r²), height 0.31% — total ≈ 1.3%, so absolute uncertainty ≈ ±35 kg m⁻³.
✓ (d) 2710 ± 35 kg m⁻³ agrees with aluminium (2700 kg m⁻³) well within uncertainty, and is far from zinc.
✓ (e) The diameter dominates (doubled for r²) — use a micrometer instead of a vernier caliper, or average several measurements at different points.
✓ The Mars Climate Orbiter was lost because one software team used pound-force seconds and another expected newton seconds for the same impulse value — a non-SI/SI mismatch that coherent, consistent SI units would have prevented.
✓ A constant of nature (e.g. the Planck constant) is the same everywhere and at all times, so any sufficiently equipped laboratory can realise the unit independently, with no risk of loss or drift.
✓ Either could, in principle, have been chosen as fundamental — the choice is a practical convention for reproducible measurement, not a deep physical necessity.
✓ Risk: forgetting to convert back to SI before combining with SI-based equations gives an answer wrong by a large, easily unnoticed factor.
✓ Example: treating 100 cm³ as \(100\times10^{-2}=1\ \text{m}^3\) instead of the correct \(100\times10^{-6}=10^{-4}\ \text{m}^3\) is wrong by a factor of 10⁴.