Experimental Design

Experimental Design

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Board Coverage AQA Paper 3 | Edexcel CP6 | OCR (A) Paper 3 | CIE P5

1. Key Principles of Experimental Design

Independent, Dependent, and Control Variables

Definition. An independent variable is the variable that is deliberately manipulated or changed by the experimenter to observe its effect on the dependent variable.

Definition. A dependent variable is the variable that is measured or observed as a response to changes in the independent variable.

Definition. A control variable is a variable that is kept constant throughout an experiment to ensure that any observed changes in the dependent variable are due only to changes in the independent variable.

• Independent variable: the variable you deliberately change (e.g., length of wire)
• Dependent variable: the variable you measure (e.g., resistance)
• Control variables: variables you keep constant (e.g., temperature, material, cross-sectional area)

Identifying Variables: Worked Examples

Example 1 — Investigating g with a pendulum:

Type	Variable	How it is controlled/changed
Independent	Length of pendulum, $L$	Vary with a metre rule from 0.2 m to 1.0 m
Dependent	Period, $T$	Measure with a stopwatch (time 20 oscillations)
Control	Mass of bob	Use the same bob throughout
Control	Amplitude (angle of release)	Keep below $5^\circ$
Control	String material and thickness	Use inextensible string

Example 2 — Investigating the effect of temperature on the resistance of a thermistor:

Type	Variable	How it is controlled/changed
Independent	Temperature, $T$	Water bath, thermometer
Dependent	Resistance, $R$	Ohmmeter or voltmeter/ammeter method
Control	Type of thermistor	Same component throughout
Control	Immersion depth	Thermistor fully submerged at fixed depth
Control	Heating rate	Heat slowly, allow thermal equilibrium

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Common Pitfall Students often confuse the independent and dependent variables. A reliable mnemonic: "I change the Independent variable, and I measure the Dependent variable."

Control Variables in Depth

Control variables are often the most neglected part of experimental design, yet failing to control them is the most common reason experiments produce invalid results.

Why control variables matter: If a control variable is not held constant, it becomes a confounding variable — you cannot determine whether the change in the dependent variable is due to the independent variable or the uncontrolled factor.

Strategies for controlling variables:

1. Physical isolation — e.g., placing the experiment in a water bath to control temperature
2. Standardised procedure — e.g., always measuring from the same reference point
3. Matching conditions — e.g., using the same equipment and setup for every trial
4. Monitoring and recording — if a variable cannot be perfectly controlled, at least measure it so its effect can be assessed

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Board Coverage AQA Paper 3 Section A | Edexcel CP6 (Core Practical 6) | OCR (A) PAG 1 | CIE Paper 5 Q1 (Planning)

• AQA and Edexcel tend to ask students to identify variables from a given method. Practice reading a method description and extracting all three types.
• OCR (A) PAG activities require students to write a full risk assessment, so control variables must be explicitly listed.
• CIE Paper 5 explicitly marks the "control of variables" section, and examiners look for methods of control, not just listing the variable name.

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Reliability, Validity, and Accuracy

Definition. Accuracy is the closeness of a measured value to the true (accepted) value of the quantity being measured.

Definition. Precision is the closeness of agreement between repeated measurements of the same quantity, reflecting the reproducibility of the measurements.

Definition. Repeatability is the precision of measurements made by the same experimenter using the same equipment and method over a short period of time.

Definition. Reproducibility is the precision of measurements made by different experimenters using different equipment and/or methods but following the same experimental protocol.

Definition. A systematic error is a consistent, repeatable deviation of measured values from the true value, caused by a flaw in the experimental setup or instrument calibration. It affects accuracy but not precision.

Definition. A random error is an unpredictable variation in measured values caused by limitations of measurement or environmental fluctuations. It affects precision but not accuracy on average.

Definition. Uncertainty is an estimate of the range within which the true value of a measured quantity is likely to lie, expressed as an absolute value $\pm \Delta x$ or as a percentage.

Reliability: Can the experiment be repeated and give consistent results? Improved by:

• Repeating measurements and calculating a mean
• Using the same equipment and method
• Controlling all variables carefully

Validity: Does the experiment actually test what it claims to test? Improved by:

• Controlling all confounding variables
• Using an appropriate method
• Ensuring the measurement directly relates to the hypothesis

Accuracy: How close is the result to the true value? Improved by:

• Using calibrated instruments
• Eliminating systematic errors
• Reducing random errors

Reliability in Depth

Reliability is fundamentally about repeatability and reproducibility. An experiment is reliable if:

• Repeatability: the same person, using the same equipment, under the same conditions, obtains the same results on repeated trials.
• Reproducibility: a different person, using different (but equivalent) equipment, can reproduce the same results.

Quantifying reliability: Calculate the spread of repeated measurements. If the standard deviation is small relative to the mean, the results are reliable. For a small number of repeats, the range (maximum − minimum) is often used as a simpler indicator.

Example. A student measures the period of a pendulum five times: 1.98 s, 2.01 s, 2.00 s, 1.99 s, 2.00 s. The mean is 2.00 s and the range is 0.03 s, giving a spread of $\pm 0.015$ s. This is reliable — the variation is less than 1% of the mean.

Validity in Depth

Validity has two aspects that examiners distinguish:

1. Internal validity: Does the experiment isolate the effect of the independent variable on the dependent variable? This requires that all confounding variables are controlled. For example, in an experiment on the resistance of a wire, if the wire heats up as current flows, the changing temperature becomes a confounding variable that threatens internal validity.

2. External validity: Can the results be generalised beyond the specific experimental conditions? For example, results obtained with a 1 mm copper wire may not apply to a 5 mm aluminium wire.

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Board Coverage AQA Paper 3 | Edexcel CP6 | OCR (A) Paper 3 | CIE P5

• AQA emphasises the distinction between validity and reliability in 6-mark practical questions.
• Edexcel Core Practicals require students to comment on the validity of their conclusions.
• OCR (A) PAG reports ask students to evaluate the validity of their method and results.
• CIE Paper 5 marks validity through the "justification of the method" requirement — students must explain why their approach is valid, not just state that it is.

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Accuracy in Depth

Accuracy depends on minimising both systematic and random errors:

$\mathrm{Accuracy} \propto \frac◆LB◆1◆RB◆◆LB◆|\mathrm{systematic error}| + \mathrm{random error}◆RB◆$

Systematic errors shift all readings by a constant amount. They cannot be reduced by averaging. Examples include:

• A zero error on a balance (always reads 0.02 g too high)
• A thermometer calibrated against the wrong scale
• Consistently starting a stopwatch too late due to a delayed reflex

Random errors cause scatter around the true value. They can be reduced by averaging. Examples include:

• Fluctuations in the mains voltage affecting electrical readings
• Reading the last digit of an analogue scale differently each time
• Air currents affecting a pendulum swing

$\boxed{\mathrm{True value} = \mathrm{measured value} \pm \mathrm{uncertainty}}$

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Common Pitfall Do not confuse these three terms. An experiment can be reliable (consistent) but invalid (measuring the wrong thing), or valid but inaccurate (systematic error).

2. Planning an Experiment

When asked to design an experiment, address:

1. Aim: What are you investigating?
2. Variables: Independent, dependent, and control variables
3. Method: Step-by-step procedure
4. Measurements: What instruments, what range, what intervals
5. Safety: Identify hazards and precautions
6. Reliability: How many repeats, how to handle outliers
7. Analysis: How to process the data (typically linearise and plot)
8. Uncertainty: Sources of error and how to minimise them

Worked Example: Planning an Experiment to Determine g

Aim: To determine the acceleration of free fall, $g$, using a simple pendulum.

Variables:

• Independent: length of pendulum, $L$ (varied from 0.20 m to 1.00 m in 0.10 m steps)
• Dependent: period of oscillation, $T$ (measured by timing 20 oscillations with a digital stopwatch)
• Control: mass of bob, amplitude of swing ($\lt{} 5^\circ$), string material

Method:

1. Set up a retort stand with a clamp. Attach a piece of inextensible string and a pendulum bob.
2. Measure the length $L$ from the point of suspension to the centre of the bob using a metre rule.
3. Displace the bob by a small angle ($\lt{} 5^\circ$) and release it.
4. Time 20 complete oscillations using a digital stopwatch. Divide by 20 to find $T$.
5. Repeat for each length. Take three readings at each length and calculate a mean $T$.
6. Plot $T^2$ against $L$. The gradient is $4\pi^2/g$.

Safety: Ensure the clamp stand is stable. Do not swing the bob at head height.

Reliability: Three repeats at each length. Discard anomalous results and retake.

Uncertainty: The dominant uncertainty is the timing measurement. The absolute uncertainty in $T$ is $\pm 0.2$ s / 20 $= \pm 0.01$ s. The length measurement has uncertainty $\pm 0.005$ m from the metre rule resolution.

Analysis: Since $T = 2\pi\sqrt{L/g}$, plotting $T^2$ vs $L$ gives a straight line through the origin with gradient $4\pi^2/g$. Calculate $g$ from the gradient and compare with the accepted value $g = 9.81 \mathrm{ m s}^{-2}$.

Choosing Measurement Ranges and Intervals

When designing an experiment, choosing the correct range and interval for the independent variable is critical:

• Range: Should be as wide as practical to maximise the spread of data points. A wider range produces a more reliable gradient. However, the range must stay within the valid region of the model (e.g., keep pendulum angles small).
• Intervals: Should be roughly evenly spaced. Aim for at least 6–8 data points. Smaller intervals near regions of rapid change can be useful (e.g., near a resonance peak).
• Number of repeats: At least 3 repeats at each value. If time allows, more repeats improve the reliability of the mean.

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Board Coverage AQA Paper 3 Section A | Edexcel CP6 | OCR (A) PAG | CIE P5

• AQA often provides a method and asks students to identify improvements — focus on range, intervals, and repeats.
• Edexcel Core Practicals have prescribed ranges; students should justify why the chosen range is appropriate.
• OCR (A) PAG reports require a detailed method with explicit ranges and intervals.
• CIE Paper 5 Q1 requires students to specify a range with at least 5 values and justify the choice of measuring instruments.

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3. Common Apparatus and Techniques

Measuring Instruments

Instrument	Typical Range	Resolution	Common Use
Metre rule	0–100 cm	1 mm	Lengths
Vernier calipers	0–15 cm	0.01 mm	Internal/external diameters
Micrometer	0–25 mm	0.001 mm	Wire diameters
Digital balance	0–200 g	0.01 g	Mass
Stopwatch	0–60 min	0.01 s	Time intervals
Thermometer	$-10$ to $110^\circ\mathrm{C}$	1°C or 0.1°C	Temperature
Voltmeter	0–20 V	0.01 V or 0.1 V	Potential difference
Ammeter	0–10 A	0.01 A or 0.1 A	Current
Signal generator	1 Hz – 1 MHz	1 Hz	AC frequency
Data logger	Various	Depends on sensor	Continuous recording

Choosing the Right Instrument

The choice of instrument depends on the quantity being measured, the required precision, and the context of the experiment:

• Lengths > 10 cm: metre rule (resolution 1 mm)
• Lengths 1–10 cm: vernier calipers (resolution 0.01 mm)
• Lengths < 1 cm: micrometer screw gauge (resolution 0.001 mm)
• Masses > 10 g: digital balance (resolution 0.01 g)
• Masses < 10 g: more sensitive balance or use the difference method

The difference method: When measuring small masses (e.g., a single paper clip), measure the mass of 10 clips together and divide by 10. This reduces the percentage uncertainty by a factor of 10.

Improving Precision

• Use instruments with finer resolution
• Measure larger quantities (e.g., time 10 oscillations, not 1)
• Use appropriate measuring ranges
• Reduce parallax errors by reading scales at eye level

Improving Accuracy

• Calibrate instruments before use
• Correct for zero errors
• Use methods that minimise systematic errors
• Compare with accepted values

4. Risk Assessment

Risk assessment is a required part of practical work at A Level. You must identify hazards, assess the risk, and describe precautions.

Risk Assessment Methodology

A risk assessment follows four steps:

1. Identify the hazard — what could cause harm? (e.g., hot water, mains electricity, laser)
2. Identify who is at risk — usually the experimenter and nearby students
3. Evaluate the level of risk — consider both the likelihood and the severity of harm
4. Implement control measures — what precautions reduce the risk to an acceptable level?

Risk Assessment Matrix

Likelihood / Severity	Minor (e.g., small burn)	Moderate (e.g., electric shock)	Severe (e.g., eye damage)
Unlikely	Low	Medium	High
Possible	Medium	High	Very High
Likely	High	Very High	Unacceptable

Common Hazards in Physics Practicals

Hazard	Risk	Control Measures
Hot water / water bath	Scalding	Use heat-resistant gloves, warning signs
Mains electricity	Electric shock, electrocution	Check insulation, use low-voltage supplies
Laser (Class 2)	Eye damage	Never look directly at beam, wear laser goggles
Radioactive sources	Radiation exposure	Use tongs, minimise exposure time, store in lead-lined box
Heavy objects (e.g., masses)	Crushing injury	Secure clamp stands, keep feet clear
Sharp wire ends	Cuts	File ends smooth, handle with care
Oscilloscope / CRO	Electric shock	Check connections before powering on

Writing a Risk Assessment for an Exam

When asked to write a risk assessment, structure your answer as follows:

1. State the hazard clearly — "The mains electricity supply poses a risk of electric shock."
2. State the severity — "This could cause serious injury or death."
3. Describe the control measure — "Use a low-voltage (e.g., 12 V) power supply instead of mains, and ensure all connections are insulated."

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Board Coverage AQA Paper 3 | Edexcel CP1–CP12 | OCR (A) PAG (all) | CIE P5

• AQA requires a risk assessment as part of the 12 required practicals. Risk assessments are assessed in written papers.
• Edexcel Core Practicals include risk assessment in the student lab book. Exams may ask students to complete a risk assessment table.
• OCR (A) PAG reports must include a risk assessment section. Examiners check that hazards are realistic and control measures are specific.
• CIE Paper 5 Q1 typically includes a "safety" sub-question worth 1–2 marks. Students must name a specific hazard and a corresponding precaution.

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warning

Common Pitfall Do not list trivial risks (e.g., "tripping over a bag") when more serious hazards exist. Examiners award marks for identifying the most significant hazards and providing specific, practical control measures. "Be careful" is never an acceptable control measure.

5. Evaluating Experiments

When asked to evaluate an experiment:

1. Identify the largest source of uncertainty — this limits the overall precision.
2. Suggest specific improvements — not vague statements like "be more careful".
3. Discuss systematic errors — are there unaccounted biases?
4. Assess whether the method is valid — does it actually measure what is intended?

Systematic vs Random Errors in Evaluation

When evaluating, you must distinguish between the two types of error:

• Random errors cause scatter in data points. They are reduced by repeating measurements and averaging. Evidence of random error: data points do not lie exactly on the line of best fit.
• Systematic errors shift all readings in the same direction. They are not reduced by averaging. Evidence of systematic error: the line of best fit does not pass through the origin when it should, or the gradient differs from the accepted value.

Example. In a resistivity experiment, a student plots $R$ vs $L$ and finds the line of best fit has a non-zero y-intercept. This indicates a systematic error — likely contact resistance at the crocodile clips. The intercept represents this fixed resistance.

Evaluating Graphs

When a graph is plotted, check:

1. Does the line pass through the origin? If the theory predicts it should, a non-zero intercept indicates a systematic error.
2. Are all points close to the line of best fit? Large deviations suggest random errors or anomalous results.
3. Does the gradient match the expected value? A discrepancy suggests either a systematic error or an uncontrolled variable.
4. Is there evidence of curvature? This may indicate that the assumed relationship is not valid over the chosen range.

Common Experimental Contexts

Determining $g$ with a pendulum:

• Measure period $T$ for various lengths $L$
• Plot $T^2$ vs $L$; gradient $= 4\pi^2/g$
• Largest uncertainty: reaction time when timing

Determining resistivity:

• Measure $R$ for various lengths $L$ of wire
• Plot $R$ vs $L$; gradient $= \rho/A$
• Largest uncertainty: wire diameter (use micrometer, measure at multiple positions and average)

Determining the Young modulus:

• Measure extension $\Delta L$ for various loads
• Plot stress vs strain; gradient $= E$
• Largest uncertainty: cross-sectional area (measure diameter with micrometer)

Investigating g by free fall (light gates):

• Release a card through a light gate; measure the velocity at different fall distances
• Plot $v^2$ vs $h$; gradient $= 2g$
• Largest uncertainty: the height measurement and ensuring the card is released without an initial velocity

Investigating the emf and internal resistance of a cell:

• Vary the external resistance using a variable resistor
• Plot $V$ vs $I$; the y-intercept is emf and the gradient (negative) is $-r$
• Largest uncertainty: voltmeter and ammeter readings, and ensuring the cell does not heat up

Investigating wave properties (ripple tank):

• Measure wavelength for different frequencies using a strobe light
• Plot $\lambda$ vs $1/f$; gradient $= v$ (wave speed)
• Largest uncertainty: measuring wavelength from a frozen ripple pattern

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Board Coverage AQA Paper 3 | Edexcel CP6, CP9 | OCR (A) PAG 2, PAG 3 | CIE P5

• AQA Required Practical 5 uses a falling object to determine g; Required Practical 8 uses a potentiometer or voltmeter/ammeter method for internal resistance.
• Edexcel Core Practical 6 uses a pendulum for g; Core Practical 9 investigates waves on a string.
• OCR (A) PAG 2 covers mechanics experiments (including g); PAG 3 covers electrical experiments.
• CIE Paper 5 Q2 often asks students to analyse data from a provided experiment — knowing common experimental contexts helps you spot the relationship between variables quickly.

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tip

Exam Technique When asked "how could you improve this experiment?", always be specific. Instead of "use a more accurate instrument", say "use a micrometer instead of a ruler to measure the wire diameter, reducing the uncertainty from $\pm 0.5$ mm to $\pm 0.005$ mm."

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Common Pitfalls

• Stating "repeat and average" without explaining why: Simply saying you will repeat readings is insufficient. You must explain that repeating and averaging REDUCES THE EFFECT OF RANDOM ERRORS. It does NOT reduce systematic errors (which affect all readings equally). Distinguish between these two types of error in your answer.

• Confusing precision with accuracy: Precision refers to the consistency of repeated measurements (small spread). Accuracy refers to how close the mean value is to the true value. You can have high precision but low accuracy (consistent but wrong readings due to a systematic error like a zero offset).

• Forgetting to discuss control variables: When designing an experiment, you must identify and explain how you will keep all variables constant EXCEPT the independent variable. Simply listing them is not enough -- explain HOW you control each one (e.g., "use the same wire throughout to keep material constant").

• Writing an insufficient evaluation: A good evaluation does not just list errors -- it identifies SPECIFIC sources of error in the particular experiment, estimates their MAGNITUDE and DIRECTION (does each error make the result too high or too low?), and suggests specific IMPROVEMENTS. Vague statements like "human error" score few marks.

Problem Set

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Problem 1

Design an experiment to determine the resistivity of a metal wire. Identify the independent, dependent, and control variables.

Answer. Independent variable: length $L$ of the wire. Dependent variable: resistance $R$ (measured using a voltmeter and ammeter). Control variables: material, cross-sectional area, temperature.

Method: Measure the diameter with a micrometer at several points and average. Set up the wire in a circuit with an ammeter in series and a voltmeter in parallel. Vary the length from 0.10 m to 1.00 m in 0.10 m steps. Record $V$ and $I$ at each length, calculate $R = V/I$. Plot $R$ vs $L$. Gradient $= \rho/A$, so $\rho = \mathrm{gradient} \times A$.

Improvements: Use a low current to avoid heating. Ensure good contact at the crocodile clips. Measure diameter at multiple positions.

If you get this wrong, revise: Key Principles of Experimental Design

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Problem 2

In an experiment to determine $g$ using a simple pendulum, a student measures the time for 20 oscillations. Explain why this improves the reliability compared to timing a single oscillation.

Answer. The uncertainty in timing a single oscillation is dominated by human reaction time ($\sim 0.2$ s). For a period of $\sim 1$ s, this gives a percentage uncertainty of $\sim 20\%$. Timing 20 oscillations gives a total time of $\sim 20$ s, and the reaction time uncertainty is still $\sim 0.2$ s (at the start and end), giving a percentage uncertainty of $\sim 0.4/20 = 2\%$. The period uncertainty is reduced by a factor of 20.

If you get this wrong, revise: Improving Precision

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Problem 3

Identify the independent, dependent, and control variables in an experiment to investigate how the resistance of a thermistor varies with temperature.

Answer. Independent variable: temperature $T$ (varied using a water bath). Dependent variable: resistance $R$ (measured with an ohmmeter). Control variables: type of thermistor, immersion depth, heating rate, method of measurement.

If you get this wrong, revise: Independent, Dependent, and Control Variables

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Problem 4

A student measures the diameter of a wire using a ruler and obtains 1.0 mm. The actual diameter is 1.02 mm. Explain the systematic error and suggest a better instrument.

Answer. The ruler has a resolution of 1 mm, which is too coarse for measuring a wire diameter of $\sim 1$ mm. The reading of 1.0 mm is an approximation. A micrometer (resolution 0.001 mm) should be used, reducing the uncertainty from $\pm 0.5$ mm to $\pm 0.005$ mm.

If you get this wrong, revise: Measuring Instruments

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Problem 5

A student obtains data points that do not lie on a straight line when plotting $T^2$ against $L$ for a pendulum. Suggest three possible reasons and how to investigate each.

Answer. (1) Large angle oscillations: the formula $T = 2\pi\sqrt{L/g}$ assumes small angles. For angles $> 10^\circ$, the period increases. Solution: use smaller angles ($< 5^\circ$) and repeat. (2) String not ideal: the string may stretch or have mass. Solution: use a lighter, inextensible string. (3) Air resistance: damping affects the period at large amplitudes. Solution: use a denser bob to minimise air resistance effects.

If you get this wrong, revise: Evaluating Experiments

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Problem 6

Explain the difference between reliability and validity in the context of an experiment to determine the speed of sound in air using echo timing.

Answer. Reliability: repeating the experiment gives consistent values of the speed of sound. This is ensured by using the same apparatus, same distance, same temperature control, and taking multiple readings. Validity: the method actually measures the speed of sound, not some other quantity. This requires that the distance to the reflecting surface is accurately known, the time measured is truly the echo time (not other reflections), and temperature is controlled (since the speed of sound depends on temperature).

If you get this wrong, revise: Reliability, Validity, and Accuracy

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Problem 7

A student investigates how the resistance of a length of constantan wire changes with temperature. The wire is heated in a water bath and its resistance is measured at regular temperature intervals. The student's data shows that the resistance increases linearly with temperature, but the gradient is significantly different from the accepted value. Identify two possible systematic errors in this experiment.

Answer. (1) Contact resistance: The crocodile clips may add a fixed resistance that is not accounted for. This shifts the entire line up, changing the apparent gradient. Solution: measure the resistance at 0°C (ice-water mixture) and subtract this intercept, or use a four-wire (Kelvin) connection. (2) Temperature lag: The wire may not be at the same temperature as the water bath (thermal equilibrium not reached). If the wire lags behind the water temperature, the measured resistance corresponds to a lower temperature than recorded, distorting the gradient. Solution: allow sufficient time for thermal equilibrium before each reading, and stir the water bath continuously.

If you get this wrong, revise: Accuracy in Depth

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Problem 8

Write a risk assessment for an experiment using a Class 2 laser to investigate single-slit diffraction. The laser is directed at a slit, and the diffraction pattern is observed on a screen 1.5 m away.

Answer.

Hazard	Who is at risk	Risk level	Control measures
Laser beam	Experimenter, others	High	Never look directly at the beam or its reflection; wear laser safety goggles; ensure beam is below eye level; place a beam stop behind the screen; remove reflective surfaces
Electrical equipment	Experimenter	Low	Check mains connections are insulated; keep liquids away from the laser power supply

If you get this wrong, revise: Risk Assessment

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Problem 9

A student determines the internal resistance of a battery by plotting a graph of terminal potential difference $V$ against current $I$. The equation relating these quantities is $V = \mathcal{E} - Ir$, where $\mathcal{E}$ is the emf and $r$ is the internal resistance. The student obtains a y-intercept of 1.48 V and a gradient of $-1.25 \Omega$. The accepted emf is 1.50 V. Evaluate this result.

Answer. The measured emf ($\mathcal{E} = 1.48$ V) is close to the accepted value (1.50 V), giving a percentage difference of $(1.50 - 1.48)/1.50 \times 100 = 1.3\%$, which is within typical measurement uncertainty. The internal resistance $r = 1.25 \Omega$. The graph passes approximately through the origin (the x-intercept should be $\mathcal{E}/r = 1.50/1.25 = 1.20$ A). If the line does not pass through the origin on a $V$ vs $I$ plot this is expected (the y-intercept is the emf). To improve accuracy: use a more precise voltmeter, ensure the battery does not heat up (which would change $r$ during the experiment), and take readings quickly to minimise changes in the battery's internal state.

If you get this wrong, revise: Evaluating Graphs

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Problem 10

Design an experiment to determine the Young modulus of a copper wire. In your answer, identify all variables, describe the method, and explain how you would ensure the experiment is both reliable and valid.

Answer.

Variables:

• Independent: applied load (force $F$), varied by adding slotted masses
• Dependent: extension $\Delta L$, measured with a vernier scale or scribe mark on a reference scale
• Control: wire material and cross-sectional area, temperature, method of measuring extension

Method:

1. Measure the diameter of the wire at several points using a micrometer. Calculate the mean diameter $d$ and cross-sectional area $A = \pi d^2/4$.
2. Clamp the wire vertically and add a small initial load to straighten it (this load is not counted).
3. Measure the original length $L$ from the clamp to the reference point.
4. Add masses in 100 g increments. At each mass, wait 30 s for the wire to stabilise, then record the extension.
5. Continue until the elastic limit is approached (check by removing masses to see if the wire returns to its original length).
6. Plot stress ($F/A$) against strain ($\Delta L/L$). The gradient is the Young modulus $E$.

Reliability: Repeat the experiment twice with the same wire. Compare gradients.

Validity: Ensure the wire is within its elastic limit (check by removing masses). Ensure the extension is measured from a fixed reference, not the moving clamp. The experiment is valid only if the wire obeys Hooke's law throughout the range used.

If you get this wrong, revise: Worked Example: Planning an Experiment to Determine g

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Problem 11

In an experiment to determine the speed of sound in air, a student claps once and measures the time for the echo to return from a wall 50 m away. The student repeats the measurement 5 times and obtains the following times: 0.294 s, 0.301 s, 0.288 s, 0.310 s, 0.295 s. The accepted speed of sound at 20°C is 343 m s$^{-1}$.

(a) Calculate the mean time and the percentage uncertainty in the time measurement. (b) Calculate the speed of sound from the mean time and compare with the accepted value. (c) Identify the largest source of error and suggest an improvement.

Answer.

(a) Mean time $= (0.294 + 0.301 + 0.288 + 0.310 + 0.295) / 5 = 0.298$ s. Range $= 0.310 - 0.288 = 0.022$ s. Half-range uncertainty $= 0.011$ s. Percentage uncertainty $= 0.011 / 0.298 \times 100 = 3.7\%$.

(b) Speed $= 2d / t = 2 \times 50 / 0.298 = 336$ m s$^{-1}$. Percentage difference from accepted value $= (343 - 336) / 343 \times 100 = 2.0\%$.

(c) The largest source of error is the timing uncertainty (reaction time). The distance measurement (50 m) using a tape measure has a much smaller percentage uncertainty ($\sim 0.02/50 = 0.04\%$). Improvement: Use electronic timing (e.g., two microphones connected to a data logger) to eliminate human reaction time entirely.

If you get this wrong, revise: Reliability in Depth