1.2 Estimation and the limitations of measurement — Practise with help

Work each problem first. Open a hint only if you are stuck, and the full working only to check.

Warm-up: remember

Write $2.5\,\text{mm}^2$ in $\text{m}^2$ .
Write $0.000\,30\,\text{m}$ in standard form.
Work out $\dfrac{0.2}{20}$ .

Answers

$2.5 \times 10^{-6}\,\text{m}^2$ , because the prefix is squared.
$3.0 \times 10^{-4}\,\text{m}$ .
$0.010$ .

If question 1 goes wrong, go back to the prefix table in Section 1.1 and square the factor. If question 2 goes wrong, count the places the decimal point moves. If question 3 goes wrong, remember that dividing by 20 is not the same as dividing by 2.

Practise with help

1. A room is 8.0 m long, 5.0 m wide and 3.0 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.

Hint

Find the volume of the room first. The mass is density multiplied by volume.

Next step

Write $V = 8.0 \times 5.0 \times 3.0$ and then $m = \rho V$ .

Full working

\begin{aligned} V &= 8.0 \times 5.0 \times 3.0 = 120\,\text{m}^3 \\ m &= 1.2 \times 120 = 144\,\text{kg} \\ &= 1 \times 10^2\,\text{kg} \quad (1\ \text{significant figure}) \end{aligned}

To one significant figure the estimate is $1 \times 10^2\,\text{kg}$ , which is about $10^2\,\text{kg}$ .

2. A micrometer reads to the nearest 0.01 mm. State its resolution, the uncertainty in one reading, and one reason it is a better choice than a metre rule for the diameter of a wire.

Hint

The resolution is the smallest change the instrument can show. The reading uncertainty is half of that.

Next step

The resolution is 0.01 mm, so the reading uncertainty is 0.005 mm. Now compare that with the 0.5 mm uncertainty of a metre rule.

Full working

The resolution is 0.01 mm. The uncertainty in one reading is half the smallest division, so 0.005 mm. A wire may be about 0.30 mm across, and the metre rule uncertainty of 0.5 mm is larger than the wire itself. The micrometer reads to a much smaller division, so the percentage uncertainty in the diameter is much smaller.

3. Classify each error as systematic or random, and say how to reduce it. (a) A ruler has a worn end that removes 2 mm from every reading. (b) A student varies the moment at which a stopwatch is pressed. (c) A thermometer reads 0.3 °C low at every temperature.

Hint

Ask whether the error shifts every reading the same way, or scatters them.

Next step

For each one, decide whether repeating the measurement would change the error. If it would not, the error is systematic.

Full working

(a) Systematic. Every reading is 2 mm too large, so repeating will not help. Start the measurement from a different mark on the rule, or allow for the 2 mm. (b) Random. The press time is sometimes early and sometimes late, so repeat the timing and take the mean. (c) Systematic. Every reading is 0.3 °C too low, so the thermometer must be checked against a known fixed point and the readings corrected.

4. A student times 25 oscillations in one measurement. The total time has an uncertainty of 0.3 s. Find the uncertainty in the period.

Hint

The period is the total time divided by the number of oscillations. The uncertainty is divided in the same way.

Next step

Divide 0.3 s by 25 to find the uncertainty in one period.

Full working

\text{uncertainty in } T = \frac{0.3\,\text{s}}{25} = 0.012\,\text{s}

Timing more oscillations reduces the uncertainty in the period, because the same uncertainty in the total time is shared over more oscillations.

5. A book has 500 sheets and is 5.0 cm thick. Estimate the thickness of one sheet in millimetres.

Hint

Divide the total thickness by the number of sheets. Keep the unit with the number.

Next step

Work out $\dfrac{5.0\,\text{cm}}{500}$ , then change centimetres to millimetres.

Full working

\text{thickness} = \frac{5.0\,\text{cm}}{500} = 0.010\,\text{cm} = 0.10\,\text{mm}

One sheet is 0.10 mm thick. Measuring 500 sheets divides the reading uncertainty of the ruler by 500.

6. You must measure the thickness of one page of this book. Choose between measuring one page directly with a metre rule and measuring a block of 500 pages. State which method you would use, and give two reasons.

Hint

Compare the thickness of one page with the smallest division of a metre rule.

Next step

Decide whether a single page can produce a reading larger than the smallest division of the rule.

Full working

Measure the block of 500 pages. One page is about 0.1 mm thick, which is much smaller than the 1 mm smallest division of a metre rule, so a direct reading would be all uncertainty. Measuring the block gives a reading many times larger than the smallest division, and dividing by 500 divides the reading uncertainty by 500. The method also averages out the small variations between pages.