SI units, prefixes and uncertainties
Every A-level physics paper expects you to handle units and uncertainties without being told how. A handful of rules covers it: break any unit into base units, and combine uncertainties according to how the quantities are combined.
Part 1 of 3: Learn it
In short
- Six SI base units come up at A-level: kg, m, s, A, K and mol.
- Adding or subtracting quantities: add their absolute uncertainties.
- Multiplying or dividing: add percentage uncertainties; raising to a power n: multiply the percentage by n.
Where this is in your specification
Spec points: AQA 7408 3.1.1 and 3.1.2, OCR A H556 2.2, Edexcel 9PH0 Topic 1
| Board | Topic: Measurements, errors and uncertainties |
|---|---|
| AQA 7408 | 3.1 |
| Edexcel 9PH0 | Topic 1 |
| OCR H556 | Module 2 (and 1.1) |
| Higher C857 76 | Units, prefixes and uncertainties |
Base units and derived units
| Quantity | Base unit |
|---|---|
| mass | kilogram, kg |
| length | metre, m |
| time | second, s |
| current | ampere, A |
| temperature | kelvin, K |
| amount of substance | mole, mol |
Every other unit can be written in base units by using an equation it appears in. From F = ma, the newton is kg m s⁻². From W = Fs, the joule is kg m² s⁻². Both sides of a correct equation have the same base units, which is a quick check on any formula.
Prefixes
| Prefix | Symbol | Factor |
|---|---|---|
| tera | T | 10¹² |
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| centi | c | 10⁻² |
| milli | m | 10⁻³ |
| micro | μ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
| pico | p | 10⁻¹² |
Kinds of uncertainty
- Absolute uncertainty: the ± value with the same units as the measurement, such as 2.40 ± 0.02 m.
- Fractional uncertainty: absolute uncertainty ÷ value.
- Percentage uncertainty: fractional uncertainty × 100.
For a single reading, the uncertainty is often half the smallest division (or the resolution for a digital meter). For repeated readings, half the range is a common estimate.
Combining uncertainties
| Calculation | Rule |
|---|---|
| a + b or a − b | add the absolute uncertainties |
| a × b or a ÷ b | add the percentage uncertainties |
| aⁿ | multiply the percentage uncertainty by n |
Write the newton in SI base units.
Show the answer
kg m s⁻².
Part 2 of 3: See it worked
Worked examples
Example 1
A sheet measures (2.40 ± 0.02) m by (1.20 ± 0.01) m. Find its area with an absolute uncertainty.
- Area = 2.40 × 1.20 = 2.88 m²
- Percentages: 0.02 ÷ 2.40 = 0.83% and 0.01 ÷ 1.20 = 0.83%, total 1.67%
- 1.67% of 2.88 = 0.048
Answer: (2.88 ± 0.05) m².
Example 2
A trolley travels (50.0 ± 0.5) cm in (8.0 ± 0.2) s. Find its mean speed with an uncertainty.
- Speed = 50.0 ÷ 8.0 = 6.25 cm/s
- Percentages: 0.5 ÷ 50.0 = 1% and 0.2 ÷ 8.0 = 2.5%, total 3.5%
- 3.5% of 6.25 = 0.22
Answer: (6.3 ± 0.2) cm/s.
Common mistakes
- Adding absolute uncertainties when two quantities are multiplied.
- Forgetting to double the percentage uncertainty when a quantity is squared.
- Quoting an uncertainty to more significant figures than the value justifies.
- Treating the gram as the SI base unit of mass. It is the kilogram.
A radius has a 2% uncertainty. What is the percentage uncertainty in r²?
Show the answer
4%.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Write the newton in SI base units.
kg m s⁻².
A radius has a 2% uncertainty. What is the percentage uncertainty in r²?
4%.
Write 450 nm in metres using standard form.
4.5 × 10⁻⁷ m.
Jobs that use this
- Physicist (opens a new tab)
- Laboratory technician (opens a new tab)
- Mechanical engineer (opens a new tab)
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses