Revise Paper 5 Planning, Analysis & Evaluation for Cambridge 9702 (P5)

The 15-mark planning question requires a logical, detailed, and safe experimental plan. Begin by deconstructing the prompt to identify the independent variable (IV), the dependent variable (DV), and the relationship to be tested. Your response must be structured into clear sections. Start with a labelled diagram of the apparatus. Follow this with a step-by-step method for collecting data, specifying the instruments used (e.g., 'digital callipers' not 'a tool to measure width') and how you will vary the IV and measure the DV. Crucially, detail how you will keep other variables constant. The analysis section must explain how to process the data, typically by plotting a linearised graph, and how to derive the required constant from the gradient or intercept. Finally, include at least one specific and relevant safety precaution.

Question 2 tests your ability to process data, calculate uncertainties, and evaluate the result. You will be given a table of raw data to complete by calculating derived quantities and their absolute uncertainties. Remember the rules: for $y = a ± b$, $Δy = Δa + Δb$; for $y = ab$ or $y = a/b$, $Δy/y = Δa/a + Δb/b$. For $y = a^n$, $Δy/y = n(Δa/a)$. After processing, you must plot a graph, including error bars for the calculated DV. The length of each error bar is twice the absolute uncertainty of that point. Your final task is to determine a value from the graph, typically from the gradient, and find its uncertainty using the 'worst acceptable line' method. Always present your final answer as value ± uncertainty with correct units and significant figures.

The Worst Acceptable Line (WAL) is the key to finding the uncertainty in your gradient and y-intercept. It is the steepest or shallowest straight line that is still a plausible fit for the data, given the error bars. By definition, the WAL must pass through all of the error bars. To draw it, pivot your ruler on the very top of the first point's error bar and rotate it until it just touches the very bottom of the last point's error bar (this gives the shallowest WAL). Calculate the gradient of this line ($m_WAL$). The uncertainty in the gradient is then simply $Δm = |m_LOBF - m_WAL|$. A similar method gives the uncertainty in the y-intercept, $Δc = |c_LOBF - c_WAL|$. This technique is non-negotiable for top marks.

Consider an experiment to find $g$ using $T = 2π√(L/g)$. The data is linearised by plotting T² against $L$. The relationship is $T² = (4π²/g)L$. This is in the form $y = mx$, where $y = T²$, $x = L$, and the gradient $m = 4π²/g$. Suppose your LOBF gives a gradient $m_LOBF = 4.02 s²m⁻¹$. Your WAL gives a gradient $m_WAL = 4.18 s²m⁻¹$. The best value for $g$ is $g_best = 4π² / m_LOBF = 4π² / 4.02 = 9.82 ms⁻²$. The 'worst' value for $g$ is $g_worst = 4π² / m_WAL = 4π² / 4.18 = 9.45 ms⁻²$. The uncertainty in $g$ is $Δg = |g_best - g_worst| = |9.82 - 9.45| = 0.37 ms⁻²$. So, the final result is $g = (9.8 ± 0.4) ms⁻²$.

Every measurement you take is subject to errors. These are not 'mistakes' but inherent variations in the measurement process. They are broadly classified into two types: systematic errors and random errors.

These two terms are often used interchangeably in everyday language, but in physics, they have distinct meanings.

• Accuracy is how close a measured value is to the true or accepted value.
• Precision is how close repeated measurements are to each other.

Imagine an archer shooting at a target:

• Accurate and Precise: All arrows are in the bullseye.
• Precise but not Accurate: All arrows are clustered together, but far from the bullseye. This suggests a systematic error (e.g., the bow's sight is misaligned).
• Accurate but not Precise: The arrows are scattered all over the target, but their average position is the bullseye. This suggests significant random errors.
• Neither Accurate nor Precise: The arrows are scattered all over the target, and their average is not in the bullseye.

When you use measured quantities to calculate a new quantity, the uncertainties in your measurements 'propagate' to the final result. You need to know the rules for combining them.

In Paper 5, you'll often need to calculate uncertainties for values in a table, especially for quantities that are reciprocals or logarithms of your measurements.

For a reciprocal, $y = 1/x$: The fractional uncertainty rule gives $\frac{\Delta y}{y} = \frac{\Delta x}{x}$. Rearranging for the absolute uncertainty $\Delta y$ gives $\Delta y = y \frac{\Delta x}{x} = \frac{1}{x} \frac{\Delta x}{x} = \frac{\Delta x}{x^2}$.

For a logarithm, $y = \lg(x)$: The simplest method is to find the range: $\Delta y = \lg(x_{max}) - \lg(x) = \lg(x + \Delta x) - \lg(x)$. Alternatively, you can use $\Delta y = \lg(x) - \lg(x - \Delta x)$. Both are acceptable.

Question 1 (15 marks) asks you to plan a laboratory experiment from scratch. Examiners expect a labelled diagram, clear procedure, controlled variables, data analysis strategy, and safety notes. Structure your answer in short bullet points — not an essay. Always state which quantity is independent and which is dependent, and explain how a graph (often a log–log plot) will yield the constants in the suggested relationship.

A core task in Paper 5 is to determine the gradient and y-intercept of a graph and find the uncertainties in these values. This is done by drawing a line of best fit (LOBF) and a worst acceptable line (WAL).

1. Plot error bars: For each point, draw a vertical bar representing its absolute uncertainty.
2. Draw the LOBF: Draw a single straight line that passes as close as possible to all data points, with a balance of points above and below the line.
3. Draw the WAL: This is the line with the steepest or shallowest gradient that still passes through all the error bars. You only need to draw one WAL (either steepest or shallowest). To draw the steepest WAL, pivot the line around the lowest error bar until it touches the highest error bar. For the shallowest, pivot around the highest error bar until it touches the lowest.

Question 2 is where Senpai Corner’s Worst Acceptable Line (WAL) method matters. After calculating derived columns (e.g. $1/I$, $\lg T$) with uncertainties, you plot the graph with error bars, draw the line of best fit (LOBF), then draw one WAL — the steepest or shallowest straight line that still passes through every error bar. Gradient uncertainty is $|m_{\text{LOBF}} - m_{\text{WAL}}|$. Intercept uncertainty uses the same idea with the WAL intercept. This is stricter than “eyeballing” a spread — every error bar must be touched.