Learning objectives
Recall the SI base quantities and units used in the syllabus, use the prefixes from pico to tera, and make sensible order-of-magnitude estimates
Express derived units in base units and test an equation for homogeneity, explaining the limits of the test
Distinguish random and systematic errors (including zero errors) and precision from accuracy, and combine absolute and percentage uncertainties
Distinguish scalars from vectors, and add, subtract and resolve coplanar vectors
Key definitions
SI base unit
One of the seven internationally agreed units (kg, m, s, A, K, mol, cd) from which all other units are built.
Homogeneous equation
An equation in which every term has the same base units.
Absolute uncertainty
The estimated range on either side of a measured value, quoted to one significant figure with the same units as the value.
Vector
A quantity with both magnitude and direction, e.g. displacement, velocity, force, momentum.
Zero error
A systematic error in which an instrument gives a non-zero reading when the true value is zero, so every reading is shifted by the same amount.
Precision
How close repeated measurements of the same quantity are to one another; a small spread of readings means high precision.
Accuracy
How close a measured value, or the mean of a set of readings, is to the true value.
Explanation
Every physical quantity is a number multiplied by a unit. The SI system defines seven base units, and all other units are derived from them. You need five of them — the kilogram (mass), metre (length), second (time), ampere (current) and kelvin (temperature) — plus the mole (amount of substance) in the A Level work on gases; the seventh, the candela, is not examined. To convert, start from a defining equation: the newton comes from F = ma, so 1 N = 1 kg m s⁻²; the joule from W = Fs, so 1 J = kg m² s⁻²; the pascal from p = F/A, giving kg m⁻¹ s⁻²; the volt from V = W/Q, giving kg m² s⁻³ A⁻¹. Examiners expect you to write these fluently, so practise chaining the definitions rather than memorising each result.
Prefixes and estimates. Very large and very small values are written with prefixes: pico (p) 10⁻¹², nano (n) 10⁻⁹, micro (μ) 10⁻⁶, milli (m) 10⁻³, centi (c) 10⁻², deci (d) 10⁻¹, kilo (k) 10³, mega (M) 10⁶, giga (G) 10⁹ and tera (T) 10¹². Convert to base units before substituting into an equation: 250 μA = 2.5 × 10⁻⁴ A and 3.0 GHz = 3.0 × 10⁹ Hz. For areas and volumes the prefix is squared or cubed as well, so 1 mm² = (10⁻³ m)² = 10⁻⁶ m² and 1 cm³ = 10⁻⁶ m³. You should also be able to make sensible order-of-magnitude estimates, to one significant figure or the nearest power of ten: the mass of an adult is about 70 kg, the height of a door about 2 m, the speed of a car on a motorway about 30 m s⁻¹, the power of a kettle about 2 kW, the wavelength of visible light about 5 × 10⁻⁷ m and the diameter of an atom about 10⁻¹⁰ m. An estimate is also a quick check on a calculated answer: a 70 kg person jogging at about 5 m s⁻¹ has a kinetic energy of roughly 10³ J, so an answer of 10⁶ J must contain an error.
An equation is homogeneous when the base units of every term agree. Checking homogeneity is a quick way to spot an error: in v² = u² + 2as each term is m² s⁻², so the equation passes. The test has a limit — it cannot detect a missing dimensionless constant or a wrong numerical factor. E = ½mv² and E = mv² are both homogeneous, but only one is correct. Homogeneity is therefore a necessary condition for a correct equation, not a sufficient one.
Every measurement carries an uncertainty. For a single reading the absolute uncertainty is usually half the smallest scale division, or the full division for an instrument read at two ends. When quantities are added or subtracted, add the absolute uncertainties; when multiplied or divided, add the percentage uncertainties; for a power, multiply the percentage uncertainty by the index. If R = V/I with V = 6.0 ± 0.1 V and I = 1.20 ± 0.02 A, the percentage uncertainties are 1.67% and 1.67%, so R = 5.0 Ω ± 3.3% = 5.0 ± 0.2 Ω.
Random and systematic errors; precision and accuracy. Random errors scatter repeated readings unpredictably on either side of the true value — reaction time when starting and stopping a stopwatch is an example. They are reduced by repeating readings and averaging, or by plotting a graph and drawing a line of best fit. Systematic errors shift every reading by the same amount in the same direction, so repeating and averaging does not remove them. A common systematic error is a zero error: an instrument that does not read zero when it should, such as a micrometer whose closed jaws read +0.02 mm or an ammeter that reads 0.1 A with no current. Correct for it by subtracting the zero reading from every measurement (which adds on the size of a negative zero reading). Precision describes how close repeated readings are to one another; accuracy describes how close a reading, or the mean of a set, is to the true value. Random errors limit precision and systematic errors limit accuracy, so readings can be precise but inaccurate — tightly grouped but all shifted by a zero error — or accurate but imprecise, widely scattered about the true value.
A scalar has magnitude only (mass, energy, temperature, speed, distance) while a vector also has direction (displacement, velocity, acceleration, force, momentum). Vectors add by drawing them tip-to-tail, or by resolving into perpendicular components: a force F at angle θ to the horizontal has components F cos θ horizontally and F sin θ vertically. Two perpendicular components combine back into a resultant of magnitude √(Fₓ² + Fᵧ²) at angle tan⁻¹(Fᵧ/Fₓ). Always state the direction of a vector answer — a magnitude alone loses the mark.
Subtracting vectors. A change in a vector quantity is found by subtraction, for example the change in velocity Δv = v − u. Subtracting a vector is the same as adding its negative, so draw v and then −u (u reversed) tip-to-tail; the arrow from the start of v to the end of −u is Δv. If a ball moving at 4.0 m s⁻¹ due north is deflected so that it moves at 3.0 m s⁻¹ due east, then Δv = 3.0 m s⁻¹ east + 4.0 m s⁻¹ south, so |Δv| = √(3.0² + 4.0²) = 5.0 m s⁻¹, directed 53° south of east. For motion along a line, give the two velocities signs: a ball that hits a wall at 5 m s⁻¹ and rebounds at 3 m s⁻¹ has a change of velocity of 8 m s⁻¹, not 2 m s⁻¹. The acceleration, and the resultant force that causes it, are in the direction of Δv.
Formulae
Prefixes: p 10⁻¹², n 10⁻⁹, μ 10⁻⁶, m 10⁻³, c 10⁻², d 10⁻¹, k 10³, M 10⁶, G 10⁹, T 10¹²
1 N = 1 kg m s⁻²; 1 J = 1 kg m² s⁻²; 1 Pa = 1 kg m⁻¹ s⁻²
Δ(A + B) = ΔA + ΔB
%(A × B) = %A + %B; %(Aⁿ) = n × %A
Resultant = √(Fₓ² + Fᵧ²); tan θ = Fᵧ/Fₓ
Change of a vector: Δv = v − u = v + (−u)
Worked examples
q: The period of a pendulum is measured as T = 1.80 ± 0.02 s and its length as L = 0.805 ± 0.005 m. Calculate g = 4π²L/T² with its absolute uncertainty.
steps: g = 4π² × 0.805 ÷ 1.80² = 31.78 ÷ 3.24 = 9.81 m s⁻²
% uncertainty in L = 0.005/0.805 = 0.62%; in T = 0.02/1.80 = 1.11%
% uncertainty in g = 0.62% + 2 × 1.11% = 2.84%
Δg = 0.0284 × 9.81 = 0.28 → g = 9.8 ± 0.3 m s⁻²
In context
The 1999 Mars Climate Orbiter was lost because one team used pound-force seconds and another used newton seconds — a units failure worth $125 million.
Engineers quote tolerances on every component: a shaft of 20.00 ± 0.02 mm must fit a bearing of 20.05 ± 0.02 mm.
Common misconceptions
Treating the kilogram-derived unit as "grams" — the SI base unit of mass is the kilogram.
Adding absolute uncertainties when quantities are multiplied — use percentage uncertainties instead.
Believing a homogeneous equation must be correct — homogeneity is necessary but not sufficient, because a numerical factor can still be wrong.
Adding vector magnitudes directly when they are not in the same direction — add them tip-to-tail or by perpendicular components.
Students often think precise readings must also be accurate. In fact readings can agree closely with one another yet all be shifted from the true value by a systematic error such as a zero error.
Study tips
Quote an uncertainty to 1 significant figure and round the value to the same decimal place.
When a question says "show that the equation is homogeneous", write the base units of each term explicitly and then state that they agree.
Summary
Base units kg, m, s, A and K (plus mol); prefixes run from pico (10⁻¹²) to tera (10¹²); derived units come from defining equations.
Homogeneity is necessary but not sufficient for a correct equation.
Zero errors are systematic; random errors limit precision and systematic errors limit accuracy.
Add absolute uncertainties for ±, percentage uncertainties for × and ÷, and multiply % by the power.
Vectors have direction: add tip-to-tail, subtract by adding the reversed vector, and resolve into perpendicular components.
Check your understanding
Question 1: The base units of the pascal are…
1. kg m s⁻²
2. kg m² s⁻²
3. kg m⁻¹ s⁻²
Question 2: If x = a³ and a has 2% uncertainty, the uncertainty in x is…
1. 8%
2. 2%
3. 6%
Question 3: 1 cm³ expressed in m³ is…
1. 10⁻⁶ m³
2. 10⁻² m³
3. 10⁻³ m³