OxfordAQA International AS and A-level Physics: SI units and conversions, error treatment and order-of-magnitude estimation (9630) – Revision Notes

Quick-recall notes for section 3.1 (3.1.1 to 3.1.3) of the OxfordAQA International AS and A-level Physics (9630) specification, Version 4.4. Section 3.1 is shared content, assessed at International AS and at International A-level. For full explanations and longer worked examples, use the study guide. To test yourself afterwards, use the practice questions, the course hub, the checklist and the free diagnostics. An earlier set of measurements revision notes is also available.

3.1.1 Units and conversions

Base units to know

Quantity	Unit	Symbol
length	metre	m
time	second	s
current	ampere	A
mass	kilogram	kg
temperature	kelvin	K
amount of substance	mole	mol

Not required: the candela, definitions of the base quantities, dimensional analysis.

Derived units in base form:

Unit	Base form
newton	kg m s⁻²
joule	kg m² s⁻²
watt	kg m² s⁻³
pascal	kg m⁻¹ s⁻²

Prefix table

Prefix	T	G	M	k	c	m	μ	n	p	f
Power of ten	12	9	6	3	−2	−3	−6	−9	−12	−15

Standard form: A × 10ⁿ with 1 ≤ A < 10.

Method in steps

1. Write the prefix as a power of ten.
2. If the unit is squared or cubed, apply the same power to the whole factor: 1 cm² = 10⁻⁴ m², 1 mm³ = 10⁻⁹ m³.
3. Multiply or divide by the stated factor for any non-SI unit.
4. Check the size: a smaller unit means a bigger number.

Energy conversions

Conversion	Rule
eV to J	multiply by 1.60 × 10⁻¹⁹
J to eV	divide by 1.60 × 10⁻¹⁹
kW h to J	multiply by 3.6 × 10⁶
J to kW h	divide by 3.6 × 10⁶

The kW h figure comes from 1000 W × 3600 s. Use the value of the elementary charge that the question supplies.

Worked reminder (eV). A proton in an accelerator has 7.0 TeV. 7.0 × 10¹² × 1.60 × 10⁻¹⁹ = 1.12 × 10⁻⁶ J.

Worked reminder (kW h). An 85 W lamp is left on for 8 h. Energy = 0.085 kW × 8 h = 0.68 kW h = 0.68 × 3.6 × 10⁶ = 2.4 × 10⁶ J (2.448 × 10⁶ J before rounding).

3.1.2 Errors and uncertainty

Must-know distinctions

Pair	Difference
Random vs systematic	Random scatters both ways and is cut by averaging; systematic shifts everything the same way and is not cut by averaging
Accuracy vs precision	An accurate result sits near the true value; a precise set of readings sits close together
Repeatable vs reproducible	Repeatable: same operator and kit. Reproducible: different operator, kit or method
Resolution vs accuracy	Resolution is the smallest change displayed; a fine scale can still carry a zero error

Removing systematic error: zero the instrument, calibrate it, correct for the zero error, change the technique (for example read at eye level to avoid parallax). Reducing random error: repeat and average, time many cycles, use a finer-resolution instrument, or a larger measured quantity.

Forms of uncertainty

Form	Formula
absolute	Δx (with unit)
fractional	Δx / x
percentage	Δx / x × 100%

Repeat readings: mean ± half-range is a common estimate. Resolution of a single reading sets a minimum.

Combination rules

Calculation	Combine by
sum or difference	adding absolute uncertainties
product or quotient	adding percentage uncertainties
power n (including roots)	percentage uncertainty × n

No trigonometric or logarithmic combinations are required.

Worked reminder. R = V/I with V = 6.0 ± 0.1 V and I = 0.25 ± 0.01 A. R = 24 Ω. Percentage uncertainties: 1.67% and 4.0%, total 5.67%. Absolute = 0.0567 × 24 = 1.4 Ω, giving 24 ± 1 Ω.

Worked reminder (root). For T ∝ √l with l = 0.640 ± 0.004 m, the percentage uncertainty in l is 0.625%, so in T it is 0.625 ÷ 2 = 0.31%.

Significant figures

The uncertainty is normally quoted to 1 significant figure, and the value is given to the same decimal place. A value written with more digits than the uncertainty supports, such as 9.795 ± 0.2, is inconsistent. A recorded reading should show the resolution of the instrument.

Graphs

• Error bar: from value − uncertainty to value + uncertainty, vertical for y and horizontal for x.
• The two extreme lines should still pass through every error bar.
• Gradient uncertainty: subtract the shallow gradient from the steep one and halve it.
• Intercept uncertainty = (larger intercept − smaller intercept) ÷ 2, using the same two extreme lines.
• No error bars: use the extreme lines that still fit the scatter reasonably, then the same half-difference method.
• A line through the origin expected, but with an intercept that is much larger than its uncertainty, points to a systematic error.

Method in steps

1. Plot the points and draw the best-fit line.
2. Draw the steepest acceptable line and the shallowest acceptable line.
3. Read two well-separated points on each line and calculate its gradient.
4. Take half the difference for the gradient uncertainty.
5. Read each line’s intercept and take half the difference.
6. Convert to the required quantity (for example a = gradient ÷ 2) and carry the percentage uncertainty across.

Worked reminder (using a gradient). A graph of T² against l for a pendulum has gradient 4.0 ± 0.2 s² m⁻¹, and T² = (4π²/g) l. Then g = 4π² ÷ gradient = 9.87 m s⁻². The gradient’s percentage uncertainty is 5%, and g depends on its reciprocal, so g also has 5% uncertainty: 0.49 m s⁻². Result: g = 9.9 ± 0.5 m s⁻².

Whenever a quantity is found from a gradient or intercept, carry the percentage uncertainty into it using the product, quotient and power rules above.

3.1.3 Estimation

• Nearest order of magnitude means the nearest power of ten. A common convention: leading figure below about 3.2 rounds down, above rounds up.
• Reference sizes: person about 10² kg; atom about 10⁻¹⁰ m; Earth’s radius about 10⁷ m.
• State each estimate, combine with an equation, then round the final answer.
• Check an estimate by asking whether the answer is plausible.

Worked reminder. A sprinter of mass 70 kg runs at 8 m s⁻¹. Kinetic energy = ½ × 70 × 8² = 2240 J, so about 10³ J.

Quick self-test

1. Convert 450 cm³ to m³.
2. Write 2.5 ns in seconds, in standard form.
3. Convert 6.4 × 10⁻¹⁰ J into GeV (1 eV = 1.60 × 10⁻¹⁹ J).
4. A 2.5 kW oven runs for 3.0 h. State the energy in kW h and in J.
5. Readings of a length are all 0.5 cm too high. Name the type of error and give one remedy.
6. Two positions are 3.50 ± 0.07 m and 1.20 ± 0.05 m. Give the separation and its uncertainty.
7. A cube edge is 4.00 ± 0.08 cm. State the percentage uncertainty in the volume.
8. The steepest and shallowest lines have gradients 5.2 and 4.6, with the best-fit gradient 4.9. State the gradient with its uncertainty.
9. Write 0.3478 ± 0.0212 with a suitable number of significant figures.
10. State the order of magnitude of 2.9 × 10⁸ and of 3.4 × 10⁸.

Answers

1. 450 × 10⁻⁶ = 4.5 × 10⁻⁴ m³.
2. 2.5 × 10⁻⁹ s.
3. 6.4 × 10⁻¹⁰ ÷ 1.60 × 10⁻¹⁹ = 4.0 × 10⁹ eV = 4.0 GeV.
4. 2.5 × 3.0 = 7.5 kW h; 7.5 × 3.6 × 10⁶ = 2.7 × 10⁷ J.
5. Systematic (zero or offset); correct the zero, recalibrate, or subtract the offset.
6. 3.50 − 1.20 = 2.30 m; 0.07 + 0.05 = 0.12 m, so 2.30 ± 0.12 m (or 2.3 ± 0.1 m).
7. 0.08 ÷ 4.00 = 2%; cubed gives 6%.
8. (5.2 − 4.6) ÷ 2 = 0.3, so 4.9 ± 0.3.
9. 0.35 ± 0.02.
10. 10⁸ and 10⁹.

Where marks are usually lost

• Using 10⁻³ for mm² instead of 10⁻⁶.
• Putting grams into a formula that needs kilograms.
• Multiplying when dividing was needed between kW h and J, or between eV and J.
• Adding percentage uncertainties for a sum, or absolute ones for a product.
• Not doubling (or tripling) the percentage uncertainty for squared (or cubed) quantities.
• Saying that repeating readings removes a zero error.
• Mixing up accuracy and precision, or giving resolution as if it were accuracy.
• Quoting an answer to more decimal places than its uncertainty.
• Drawing extreme lines that miss some error bars.
• Giving an estimate as an exact figure with no stated assumptions.

Official syllabus

OxfordAQA International AS and A-level Physics (9630), Version 4.4, section 3.1 Measurements and their errors, for International AS and A-level exams from May/June 2020 onwards. Published by OxfordAQA.