Quantities and measurement
Study H2 Physics units, estimates, measurement errors, uncertainty propagation, graphs and vectors, with lessons and topic practice.
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Before you begin
Every result in physics is a number, a unit and an uncertainty, and some also need a direction. The lessons start with SI units and how to check an equation with them, move on to making estimates, then deal with errors and uncertainties in measured and calculated quantities, and finish with adding vectors.
Be comfortable with: O-Level measurement, because this topic builds on physical quantities, units and reading instruments, and with powers of ten and basic trigonometry.
Learning goals
- Use SI quantities, units, prefixes and dimensional analysis.
- Estimate physical quantities and check the reasonableness of results.
- Assess random, systematic and propagated uncertainties.
- Resolve, add and subtract coplanar vectors.
Syllabus statements covered
- recall and use the following SI base quantities and their units: mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol)
- recall and use the following prefixes and their symbols to indicate decimal sub-multiples or multiples of both base and derived units: pico (p), nano (n), micro (μ), milli (m), centi (c), deci (d), kilo (k), mega (M), giga (G), tera (T)
- express derived units as products or quotients of the SI base units and use the named units listed in ‘Summary of Key Quantities, Symbols and Units’ as appropriate
- use SI base units to check the homogeneity of physical equations
- make reasonable estimates of physical quantities included within the syllabus
- show an understanding of the distinction between random errors and systematic errors (including zero error) which limit precision and accuracy
- assess the uncertainty in derived quantities by adding absolute or relative (i.e. fractional or percentage) uncertainties or by numerical substitution (rigorous statistical treatment is not required)
- distinguish between scalar and vector quantities, and give examples of each
- add and subtract coplanar vectors
- represent a vector as two perpendicular components
Lessons
Work through them in order.
Core lessons
- SI units, prefixes and homogeneityConvert prefixed units, express derived units in SI base units and use base units to check whether an equation is homogeneous.
- Making physical estimatesBuild an estimate from a simple model and sensible values, state it to a suitable precision and check that it makes sense.
- Errors, precision and accuracyDistinguish random from systematic errors, correct a zero error and state a measurement with its uncertainty.
- Uncertainty in derived quantitiesFind the uncertainty in a calculated result by adding absolute or fractional uncertainties, or by numerical substitution.
- Uncertainty in a graph gradientDraw the steepest and shallowest lines through the error bars to find the uncertainty in a graph's gradient.
- Scalars and vectorsAdd and subtract coplanar vectors with diagrams and with perpendicular components.
Practise and check
Or choose
Topic reference
Quick reference
| Idea | Rule | Example |
|---|---|---|
| SI base units | kg, m, s, A, K, mol | |
| derived units | write the defining equation in base units | N = kg m s⁻², J = kg m² s⁻² |
| 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 mm² = 10⁻⁶ m² |
| homogeneity | every term in an equation has the same base units | ρ gh and p are both kg m⁻¹ s⁻² |
| estimate | simple model, stated values, answer to 1–2 significant figures | bath of water: 2 × 10² kg |
| sum or difference | Δ q = Δ a + Δ b | |
| product or quotient | (Δ q)/q = (Δ a)/a + (Δ b)/b | |
| power q = aⁿ | (Δ q)/q = |n|(Δ a)/a | A = π r²: double the fractional uncertainty in r |
| gradient uncertainty | half the difference between the steepest and shallowest lines | |
| components | Fₓ = F cos θ, Fy = F sin θ | θ measured from the x-axis |
| resultant | R = square root of (Rₓ² + Ry²), tan θ = Ry/Rₓ |
Core knowledge to remember
- The SI base quantities and units are mass (kg), length (m), time (s), current (A), temperature (K) and amount of substance (mol).
- A homogeneous equation has the same base units in every term. A homogeneity check can show that an equation is wrong, but not that it is right.
- Random errors scatter readings unpredictably about the true value; they limit precision and are reduced by repeating and averaging. Systematic errors, including zero errors, shift every reading the same way; they limit accuracy and are not reduced by averaging.
- Precision is how close repeated readings are to each other. Accuracy is how close they are to the true value.
- The uncertainty in a derived quantity is found by adding absolute uncertainties (sums and differences), adding fractional uncertainties (products, quotients and powers), or by numerical substitution.
- A scalar has magnitude only; a vector has magnitude and direction. Vectors are added head to tail; subtracting a vector means adding its reverse.
Common mistakes
- Treating the gram as the base unit of mass, or converting cm² with a factor of 10⁻².
- Saying an equation is correct because it is homogeneous.
- Giving an estimate to more significant figures than the assumptions support.
- Thinking repeated readings remove a systematic error.
- Subtracting uncertainties when quantities are subtracted.
- Adding absolute uncertainties for a product or quotient, or forgetting to multiply by a power.
- Quoting a value to more decimal places than its uncertainty.
- Adding the magnitudes of vectors that point in different directions.