Paper 4 Measurement Practical-Style Answers

Question 1: Spring data, graph and gradient [22]

(a) Independent variable: mass $m$ [1]. Dependent variable: extension $x$ [1].

(b) The table headings state quantities [1] and units using the form quantity/unit, so units are not repeated in every cell [1].

(c) Award independently:

• axes labelled $m/\mathrm{kg}$ and $x/\mathrm{m}$, with $x$ vertical [1];
• simple scales using at least half the grid in each direction [1];
• all six points plotted accurately as small crosses [2];
• one straight line of best fit representing the overall trend [1].

An indicative completed graph is shown below. Accept reasonable differences in best-fit placement.

(d) Use two well-separated points on the candidate’s best-fit line. An acceptable result is

$$\text{gradient}\approx 0.40\mathrm{m}\mathrm{k}{\mathrm{g}}^{-1}.$$

Large triangle using points on the line [1]; value consistent with the candidate’s line, typically $0.39$ to $0.41$ [1]; unit $\mathrm{m}\mathrm{k}{\mathrm{g}}^{-1}$ [1].

(e) Since the gradient is $g/k$,

$$k=\frac{g}{\text{gradient}}\approx \frac{9.81}{0.40}=24.5\mathrm{N}{\mathrm{m}}^{-1}.$$

Correct rearrangement and follow-through [1]; value with unit [1].

(f) It could indicate a zero error in the extension measurement, such as using an incorrect unloaded reference length [1], or that the spring/model does not behave proportionally over the chosen range [1].

(g) Random uncertainty: for example, the load may still oscillate or the ruler reading may vary because alignment/parallax varies; this causes scatter in $x$ [2]. Systematic error: for example, all extensions may be found from an incorrectly recorded unloaded reference position, shifting every $x$ value in one direction [2]. Accept other experiment-specific sources with a correct mechanism. Do not award unrelated generic errors such as reaction time.

(h) Any one well-explained improvement: take more data over a wider range of $x$; repeat readings and average; use an instrument with suitable finer resolution; ensure consistent alignment. Improvement [1]; link to gradient uncertainty [1].

Question 2: Planning a diameter and density measurement [20]

(a) Measure mass $m$ [1], length $L$ [1], and diameter $d$ or radius $r$ [1].

(b)

$$\rho =\frac{m}{V},V=\pi r^{2}L=\frac{\pi d^{2}L}{4}.$$

Correct density relation [1]; cylindrical volume [1]; expression in measured quantities [1].

(c) Measure the diameter at several positions along the wire [1] and in different orientations if the cross-section may not be perfectly circular [1]. Record all readings to micrometer precision [1] and calculate a mean diameter [1].

(d) The wire diameter is small [1], so the micrometer’s finer resolution gives a smaller percentage uncertainty than a metre rule [1].

(e) A zero error gives a systematic bias in every diameter reading [1]. Since volume depends on $d^2$, this biases the calculated density in the opposite direction to the volume error [1].

(f) Use columns with quantity and unit in headings [1], record repeated diameter readings consistently to the same decimal place [1], and include calculated mean diameter and measured mass/length clearly without units repeated in cells [1].

(g) Accept a genuine safety precaution, such as covering or bending back sharp wire ends, wearing eye protection while cutting, or directing cut ends away from people [1]. “Do not overtighten the micrometer” is a handling/accuracy precaution, not by itself a safety precaution.

(h) Any two: check units converted to SI if using $\mathrm{kg}{\mathrm{m}}^{-3}$; compare with order of magnitude for metals; check that a larger measured diameter would increase volume and reduce density; check significant figures/uncertainty; verify $d$ was not used as $r$ [2].

Review Figures

Figure: Practical graph marks usually separate table headings, axis labels, sensible scales, plotted points, best-fit line and gradient interpretation. Question 1 uses the same skills with a new original dataset.

Figure: Instrument resolution sets a default reading uncertainty only when no better uncertainty is stated. For Question 2, the same principle explains why a micrometer is more suitable than a metre rule for measuring a thin wire diameter.