1.2 Estimation and the limitations of measurement — Practise

Work through these on your own. For an estimate, round each quantity to one significant figure first, and say where the estimate is weak.

The questions and the answers below come from a set that is checked by computer every time the book is built.

A classroom is $10\,\text{m}$ long, $6.0\,\text{m}$ wide and $2.5\,\text{m}$ high. The density of air is $1.2\,\text{kg}\,\text{m}^{-3}$ . Estimate the mass of air in the room to one significant figure.
Estimate the number of times a person heart beats in 70 years. Take the rate as 70 beats per minute, and give your answer to one significant figure.
A stack of 100 sheets of paper is $12\,\text{mm}$ thick. Find the thickness of one sheet, and say why a direct measurement of one sheet would be unreliable.
A metre rule has a smallest division of $1\,\text{mm}$ . State the uncertainty in a single reading and explain why it is not zero.
Two students each measure the diameter of the same wire five times. Student A gets readings that are all close together but about $0.20\,\text{mm}$ too large. Student B gets readings spread over $0.10\,\text{mm}$ that average the true value. Which student is more precise, and which is more accurate? Give a reason for each answer.
Classify each error as systematic or random, and state one way to reduce it. (a) A top-pan balance reads $0.05\,\text{g}$ when the pan is empty. (b) A student reaction time when a stopwatch is started and stopped. (c) A thermometer is calibrated $0.5\,^\circ\text{C}$ too high.
A stopwatch can be started and stopped with an uncertainty of about $0.2\,\text{s}$ . A student times 20 oscillations in one measurement and divides by 20 to find the period. Work out the uncertainty in the period.
You must measure the diameter of a wire that is about $0.30\,\text{mm}$ across. The instruments available are a metre rule (smallest division $1\,\text{mm}$ ), a vernier caliper (smallest division $0.1\,\text{mm}$ ) and a micrometer (smallest division $0.01\,\text{mm}$ ). Choose the instrument you would use, give a reason, and state the uncertainty in one reading.
Estimate how many Earth diameters would fit across the Sun. The radius of the Earth is $6.4 \times 10^3\,\text{km}$ and the radius of the Sun is $7.0 \times 10^5\,\text{km}$ . Give your answer to one significant figure.
You want to estimate the mass of water in a full bath. State the measurements you would take, estimate each one, and say which estimate is likely to give the largest percentage uncertainty in the mass.
A student calculates a speed as $2.876\,18\,\text{m}\,\text{s}^{-1}$ from measurements that are each known to two significant figures. Write the speed to the correct number of significant figures.
Explain why a measurement can never be exact. Use the words resolution, systematic error and random error.

Answers and concise solutions

1. The estimated mass of air

Answer: Volume $= 10 \times 6.0 \times 2.5 = 150\,\text{m}^3$ . Mass $= \rho V = 1.2 \times 150 = 180\,\text{kg}$ , which is $2 \times 10^2\,\text{kg}$ to one significant figure.

2. The estimated number of heartbeats

Answer: $70 \times 60 \times 24 \times 365 \times 70 = 2.6 \times 10^9$ , so about $3 \times 10^9$ heartbeats to one significant figure. The estimate is only as good as the rate and the length of a year, so one figure is enough.

3. The thickness of one sheet

Answer: The thickness is $\dfrac{12}{100} = 0.12\,\text{mm}$ . One sheet is thinner than the smallest division of a metre rule, so a direct reading would be dominated by the resolution of the rule. Measuring 100 sheets divides the reading uncertainty by 100.

4. The uncertainty in one reading

The resolution of the rule is the smallest division, 1 mm.
The uncertainty in a single reading is half the smallest division, so 0.5 mm.
The true value could lie on either side of the mark, within half a division of it, so the reading has a limit.

5. Precision and accuracy

Student B is more precise, because the repeated readings are closer together than Student A readings.
Student B is also more accurate, because the mean is close to the true value. Student A has a systematic error of about 0.20 mm, so every reading is too large.

6. The type of each error

The empty balance error is systematic. Set the balance to read zero before the measurement, or subtract 0.05 g from every reading.
The reaction time is random. Repeat the timing several times and use the mean.
The thermometer calibration error is systematic. Check the thermometer against a known fixed point and correct every reading.

7. The uncertainty in the period

Answer: The uncertainty in the total time is $0.2\,\text{s}$ . Dividing the total time by 20 also divides its uncertainty by 20, so the uncertainty in the period is $\dfrac{0.2}{20} = 0.010\,\text{s}$ .

8. The instrument and its uncertainty

Use the micrometer, because its smallest division is much smaller than the diameter of the wire.
The metre rule and caliper have a resolution that is too coarse for a 0.30 mm wire, so the percentage uncertainty would be large.
The uncertainty in one micrometer reading is half the smallest division, 0.005 mm.

9. The estimate to one significant figure

Answer: A diameter is twice a radius, and the factor of 2 cancels in the ratio. So $\dfrac{7.0 \times 10^5}{6.4 \times 10^3} = 1.09 \times 10^2$ , which is about $1 \times 10^2$ to one significant figure.

10. The measurements and the weakest estimate

Estimate the length, the width and the depth of the water, and multiply them to get the volume.
Multiply the volume by the density of water, 1000 kg m^-3, to get the mass.
The depth is the hardest to estimate, because the bath is not a simple rectangular box and the water level is not a clean edge, so it carries the largest percentage uncertainty.

11. The speed to the correct figures

Answer: The result can only be as precise as the least precise measurement, so the speed is $2.9\,\text{m}\,\text{s}^{-1}$ to two significant figures.

12. Why no measurement is exact

Every instrument has a resolution, so a reading can only be located to within part of the smallest division.
A systematic error shifts every reading in the same direction, so it is not removed by repeating the measurement.
A random error scatters the readings, so a single reading may be too high or too low. Repeating and averaging reduces its effect but does not remove it.