AQA A-Level Physics: Measurements and Their Errors

A student is investigating how the period of a simple pendulum depends on its length. She uses a stopwatch with a resolution of 0.01 s to time the swings of a pendulum bob, and a metre rule to measure the length of the string from the support to the centre of the bob.

When she first sets up the experiment she notices that the stopwatch reads 0.04 s before she has started timing, and that the metre rule has a few millimetres worn away from its zero end.

Describe and explain how she should carry out her timing and length measurements so as to reduce the effect of both random error and systematic error in her results. In your answer you should distinguish clearly between the two types of error and give a specific way of reducing each.

(6 marks)

A student determines the density of a small solid metal cylinder. She measures its mass on an electronic balance, its diameter with a micrometer and its length with a vernier caliper. Her readings, with their absolute uncertainties, are shown below.

Quantity	Reading	Absolute uncertainty
Mass, $m$m	56.3 g	± 0.1 g
Diameter, $d$d	12.70 mm	± 0.01 mm
Length, $L$L	45.0 mm	± 0.1 mm

The density is given by $\rho = \dfrac{4m}{\pi d^2 L}$ρ=πd2L4m​.

(a) Calculate the density of the metal in g cm⁻³. (2 marks)

(b) Calculate the percentage uncertainty in the density, and hence express the density as a value with an absolute uncertainty. (4 marks)

A class determines the acceleration of free fall, $g$g, using a free-fall timer. Six students each repeat the experiment and obtain the following values.

Trial	1	2	3	4	5	6
$g$g / m s⁻²	9.79	9.83	9.45	9.81	9.78	9.82

The accepted value of $g$g at this location is 9.81 m s⁻².

(a) Identify the anomalous result and state why it should be excluded. (1 mark)

(b) Using the remaining values, calculate the mean value of $g$g and estimate its absolute uncertainty as half the range. (2 marks)

(c) Comment on the precision and the accuracy of the class's results, justifying each comment with reference to your figures and the accepted value. (2 marks)

Two physical quantities that you may not have met before can each be written in terms of SI base units by rearranging a defining equation.

(a) Newton's law of gravitation is $F = \dfrac{G M m}{r^2}$F=r2GMm​, where $F$F is a force, $M$M and $m$m are masses and $r$r is a distance. Show that the SI base units of the gravitational constant $G$G are $\text{m}^3\,\text{kg}^{-1}\,\text{s}^{-2}$m3kg−1s−2. (3 marks)

(b) The capacitance $C$C of a capacitor is defined by $Q = C V$Q=CV, where $Q$Q is the charge stored and $V$V is the potential difference. Given that charge has base units A s and that potential difference (the volt) has base units $\text{kg}\,\text{m}^2\,\text{s}^{-3}\,\text{A}^{-1}$kgm2s−3A−1, determine the SI base units of capacitance. (2 marks)

Estimate, to an order of magnitude, the number of molecules of air in a typical school classroom.

In your answer you should make and state sensible estimates for any quantities you need, and show the chain of reasoning that leads to your final order of magnitude. (Take the molar volume of a gas at room conditions to be about $24\ \text{dm}^3\,\text{mol}^{-1}$24 dm3mol−1 and the Avogadro constant to be $6.0 \times 10^{23}\ \text{mol}^{-1}$6.0×1023 mol−1.)

(4 marks)

In the context of physical measurements, the terms precision, accuracy and resolution have distinct meanings.

(a) State what is meant by the precision of a set of measurements and by the accuracy of a measurement, making clear how the two differ. (2 marks)

(b) A digital ammeter reads to the nearest 0.01 A. State its resolution and explain how this limits the uncertainty in a single reading from the instrument. (1 mark)