Question
(a) Explain what is meant by the accuracy of a measured value. (1 mark)
(b) Two solid cubes, A and B, are measured to determine the density of their materials.
Table 1.1 shows the measurements for cube A.
Table 1.1
| quantity | measurement |
| length of side | \( (1.53 \pm 0.01)\,\mathrm{cm} \) |
| mass | \( (31.3 \pm 0.5)\,\mathrm{g} \) |
(i) Show that the calculated density of the material of cube A is \(8.7 \times 10^3\,\mathrm{kg\,m^{-3}}\). (2 marks)
(ii) Calculate the percentage uncertainty in the density of the material of cube A. (2 marks)
percentage uncertainty = ____________________ %
(iii) The density of the material of cube B is determined to be \(9.2 \times 10^3\,\mathrm{kg\,m^{-3}} \pm 6\%\).
State and explain whether cube A and cube B could be made from the same material. (2 marks)
Syllabus Topic Codes (A Level Physics P2 syllabus):
• 4.3: Density and pressure — parts (b)(i) and (b)(iii)
▶️ Answer/Explanation
(a) Accuracy [1 mark]
Accuracy describes how close a measured value is to the true value of the quantity.
(b)(i) Density [2 marks]
For a cube,
\(V=l^3\)
\(V=(1.53\times10^{-2})^3\,\mathrm{m^3}\)
Density is given by
\(\rho=\dfrac{m}{V}\)
\(\rho=\dfrac{31.3\times10^{-3}}{(1.53\times10^{-2})^3}\)
\(\rho=8.7\times10^3\,\mathrm{kg\,m^{-3}}\)
Answer: \( \boxed{8.7\times10^3\,\mathrm{kg\,m^{-3}}} \)
(b)(ii) Percentage uncertainty [2 marks]
The percentage uncertainty in the mass is
\(\dfrac{0.5}{31.3}\times100=1.6\%\)
Since \(V=l^3\), the percentage uncertainty in the volume is
\(3\times\dfrac{0.01}{1.53}\times100=2.0\%\)
Therefore, the percentage uncertainty in density is
\(\%\text{ uncertainty}=1.6+2.0\)
\(=3.6\%\approx4\%\)
Answer: \( \boxed{4\%} \)
(b)(iii) Comparison of densities [2 marks]
For cube A, the density is approximately \(8.7\times10^3\,\mathrm{kg\,m^{-3}}\) with an uncertainty of \(4\%\).
The range for cube A is therefore approximately
\(8.7\times10^3\pm4\%\)
which gives a range of approximately \(8.35\times10^3\) to \(9.05\times10^3\,\mathrm{kg\,m^{-3}}\).
For cube B, the density range is
\(9.2\times10^3\pm6\%\)
which gives approximately \(8.65\times10^3\) to \(9.75\times10^3\,\mathrm{kg\,m^{-3}}\).
The two ranges overlap, so the difference between the measured densities is within the uncertainty.
Answer: \( \boxed{\text{Yes, they could be made from the same material.}} \)
The density ranges overlap, so the measurements are consistent with the cubes being made from the same material.