OxfordAQA A Level Physics: Measurements and Their Errors — Practice Questions

These are original questions written for Marlbridge, in the style and at the standard of the examination. They are not reproduced past-paper questions — examination boards hold copyright in their own papers. Use these alongside the official past papers available free from your board.

Related: Measurements and Their Errors revision notes

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Section A

1. State the SI base units and the quantity each measures. [3]

2. Express the newton, the joule and the watt in SI base units. [3]

Section B

3. Explain the difference between random and systematic errors, and state how each is reduced. [4]

4. A wire of length 1.250 m ± 0.001 m has a diameter of 0.42 mm ± 0.01 mm.

(a) Calculate the percentage uncertainty in the length and in the diameter. [2] (b) The cross-sectional area is calculated from A = πd²/4. Calculate the percentage uncertainty in A. [2] (c) Explain why the diameter contributes far more to the overall uncertainty than the length. [2]

5. State the rules for combining uncertainties when quantities are:

(a) added or subtracted [1] (b) multiplied or divided [1] (c) raised to a power [1]

6. A student measures the period of a pendulum by timing 20 oscillations rather than one.

(a) Explain why this improves the result. [2] (b) The reaction time is about 0.2 s. Estimate the percentage uncertainty in the period if 20 oscillations take 30.0 s. [3]

7. Define resolution, and explain why a metre rule reading to 1 mm being finer resolution than one reading to 1 cm does not by itself make a measurement more accurate. [3]

8. Five repeated readings of a length give: 12.4, 12.6, 12.3, 12.5, 12.4 mm. Estimate the uncertainty using the range method. [2]

9. Describe how to find the uncertainty in a gradient from a graph with error bars. [3]

10. A calculation from data with 2 significant figures gives a calculator display of 4.8571429. State the result to an appropriate number of significant figures, and explain why. [2]

11. A cube of metal has a measured side length of 2.00 cm ± 0.02 cm and a mass of 43.0 g ± 0.5 g.

(a) Calculate the volume of the cube and the percentage uncertainty in the volume. [2] (b) Calculate the density of the metal and its absolute uncertainty, giving your final answer to an appropriate number of significant figures. [3]

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Answers

1. Any three: metre — length; kilogram — mass; second — time; ampere — current; kelvin — temperature; mole — amount of substance; candela — luminous intensity [1] [1] [1].

2. N = kg m s⁻² [1]. J = kg m² s⁻² [1]. W = kg m² s⁻³ [1].

3. Random errors vary unpredictably in size and direction between readings [1] and are reduced by taking repeated readings and calculating a mean [1]. Systematic errors shift every reading by the same amount in the same direction [1] and are reduced by checking the zero and calibrating the instrument — repeats do not help [1]. Precision and accuracy are independent: a set of readings can be precise (tightly clustered) but inaccurate (clustered around the wrong value) — the signature of a systematic error.

4. (a) Length: (0.001 ÷ 1.250) × 100 = 0.08% [1]. Diameter: (0.01 ÷ 0.42) × 100 = 2.4% [1]. (b) The diameter is squared, so its percentage uncertainty is doubled [1]: 2 × 2.4 = 4.8% [1]. (c) The absolute uncertainty in the diameter is a much larger fraction of a much smaller measured value [1]; it is also squared in the formula, which doubles its contribution [1].

5. (a) Add the absolute uncertainties [1]. (b) Add the percentage uncertainties [1]. (c) Multiply the percentage uncertainty by the power [1].

6. (a) The uncertainty from the timer and reaction time is spread over 20 periods [1], so the percentage uncertainty in a single period is about 20 times smaller [1]. (b) Percentage uncertainty in the total time = (0.2 ÷ 30.0) × 100 [1] = 0.67% [1]; dividing by 20 does not change the percentage uncertainty, so the uncertainty in T is also 0.67% [1].

7. Resolution is the smallest change an instrument can detect [1]. Finer resolution reduces one source of uncertainty, but it does not remove a systematic error [1]; a finely-resolved but miscalibrated instrument still gives precise, consistently wrong (inaccurate) readings [1].

8. Range = 12.6 − 12.3 = 0.3 mm [1]; uncertainty = 0.3 ÷ 2 = ±0.15 mm [1].

9. Draw the steepest and shallowest lines that still pass through all the error bars [1] [1]; the uncertainty in the gradient is half the difference between these two gradients [1].

10. 4.9 [1] — a calculated result cannot be more precise than the least precise measurement used to obtain it, so it should be quoted to 2 significant figures [1].

11. (a) Volume = (2.00)³ = 8.00 cm³ [1]. Percentage uncertainty in side = (0.02 ÷ 2.00) × 100 = 1% [1]; volume involves the side cubed, so percentage uncertainty in volume = 3 × 1% = 3% [1]. (b) Density = mass ÷ volume = 43.0 ÷ 8.00 = 5.375 g/cm³ [1]. Percentage uncertainty in mass = (0.5 ÷ 43.0) × 100 = 1.16% [1]; percentage uncertainty in density = 3% + 1.16% = 4.16% [1], giving an absolute uncertainty of 5.375 × 0.0416 = 0.22 g/cm³, so the final answer is 5.4 ± 0.2 g/cm³ (2 significant figures, matching the least precise input) [1].

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Where marks are usually lost

• Giving the newton as a base unit.
• Forgetting to double the percentage uncertainty for a squared quantity.
• Adding percentage uncertainties when quantities are added.
• Assuming finer resolution automatically means a more accurate measurement.
• Using the largest or smallest reading alone rather than half the range for repeated-reading uncertainty.
• Quoting a calculated result to more significant figures than the least precise measurement justifies.
• Forgetting that a cubed or squared quantity multiplies its percentage uncertainty by the power, then failing to add it to the other combined percentage uncertainties.