What an A2 planning or evaluation question asks for

Specifying apparatus: range, resolution and the relevant dimension (for example a pendulum length measured to the centre of the bob)

When to use a data logger, and how to choose its sampling rate

Calibration and zero checks, including background count and fixed-point calibration

Techniques that improve a measurement: set squares, fiducial marks, data loggers, video

Control variables, repeats, and the number and range of readings; how the data will be used

Safety with radioactive sources, high voltages, hot liquids and charged capacitors

Recording results, anomalous readings, and the words precision, accuracy, resolution and sensitivity

1. A good A2 plan specifies every instrument with its range and resolution and says exactly which dimension is measured and where from, such as "to the centre of the bob" or "from the front of the source holder to the window of the tube".

2. Use a data logger when the event is too fast (or too slow) for hand timing; choose a sampling rate that gives many readings within each time constant or cycle.

3. Check zeros and calibrate against known values; subtract the background count from every count rate.

4. Plan at least six values over the widest sensible range, spaced evenly in the quantity you will plot; repeat where readings scatter, count for longer where they are random.

5. Safety must name the hazard of this experiment: tongs and distance for sources, current-limited supplies for high voltages, stable mounting and care with hot liquids.

6. Precise = little spread; accurate = close to the true value; sensitive = large output change per unit input change; resolution = smallest change shown.

Unit 6 asks the same planning questions as Unit 3, but with experiments from Units 4 and 5 and sharper detail. The paper is written, but every question is about a real experiment, often one you have not done: write a plan, criticise one, process readings or evaluate a result.

Choose each instrument so its range covers all the readings and its resolution is small compared with the smallest reading.

Use a data logger when the change is over in a second or two, when many readings are needed at precise times, when two quantities must be read at the same instant, or when a slow change needs readings over a long time.

Sampling rate: choose it so there are many readings, at least 10 to 20, within each time constant, period or event. Too slow a rate misses the peak of a short pulse.

A data logger removes reaction time, but its sensor still needs a suitable range and resolution, and zeroing or calibration.

WORKED EXAMPLE · QUESTION

A student plans to measure how the p.d. across a 220 µF capacitor, charged to 6.00 V, changes as it discharges through a 2.2 kΩ resistor. She intends to read a voltmeter every few seconds while timing with a stopwatch. (a) Calculate the time constant. (b) Explain why her method is unsuitable. (c) Suggest suitable apparatus and settings.

SOLUTION

(a) RC=2.2×10³×220×10⁻⁶= 0.484 s

(b) A reaction-time error of about 0.2 s is 0.2÷0.484×100=41% of the time constant, and the discharge is nearly complete in about 5RC=2.42 s, so there is no time to read the meter more than once or twice.

(c) A data logger with a voltage sensor across the capacitor, sampling at 50 readings per second (every 0.020 s): about 24 readings in each time constant and about 121 over 5RC.

Alternatively, use a 100 kΩ resistor: RC=22 s and the 0.2 s error is under 1%.

Changing the apparatus to suit the instrument is a good planning answer too.

Many equations contain a length measured between particular points; saying which points is part of specifying the apparatus.

Techniques and extra apparatus worth naming

Set square: stand it on the bench against a rule to make the rule vertical, or line up the bottom of a bob or pointer with the scale.

Fiducial mark: a pin, or a line drawn with a marker pen, at the centre of an oscillation, where the object moves fastest and its passing is sharpest. Count passes in the same direction and start the stopwatch on "zero".

Many cycles: time 10 or 20 oscillations so the reaction-time error is a small fraction of the total.

Small amplitudes for pendulums and springs (under about 10°) so the motion is simple harmonic and the period does not depend on amplitude.

Data logger, light gates or video against a scale remove reaction time.

Clamp blocks at the top of a pendulum give a fixed, sharply defined point of suspension.

WORKED EXAMPLE · QUESTION

A student measures the string of a pendulum from the bottom of the clamp blocks to the top of the bob as 0.842 m, and the bob diameter with a vernier caliper as 24.6 mm. Three timings of 20 oscillations give 37.0 s, 37.2 s and 37.1 s. (a) Determine g. (b) A second student uses 0.842 m as the length. Compare the error this causes with the uncertainty from the timing.

SOLUTION

(a) Relevant length: l=0.842+0.0123=0.8543 m. Mean of 20 oscillations = 37.1 s, so T=37.1÷20=1.855 s. Rearrange T=2π√(l/g): g=4π²l/T²=4π²×0.8543÷1.855²= 9.80 m s⁻²

(b) Leaving out the radius makes l 0.0123÷0.8543×100=1.4% too small, giving g=9.66 m s⁻²: every reading would be low by the same proportion, a systematic error. The timing half-range, 0.1 s on 37.1 s, is 0.27% in T, so about 0.5% in g (T is squared). The missing radius is bigger than the timing uncertainty. (Combining percentage uncertainties for powers and products is on 6.2.)

Zero checks: tare a balance with the apparatus in place; zero a Hall probe away from magnets and currents; check a sensor reads zero with no input; close the jaws of a caliper.

Background count: with the source locked away, count for a long time (several minutes) and work out the background count rate. Subtract it from every count rate before analysing.

Calibration compares an instrument with known values. Two fixed points give a straight calibration line: melting ice (0 °C) and steam above boiling water (100 °C at standard pressure) for a thermometer; a known field inside a long solenoid for a Hall probe; known masses for a force sensor.

A zero error moves every reading by the same amount and cancels when you take a difference. A scale (gradient) error changes readings in proportion and does not cancel in a difference; it must be corrected using the calibration line.

WORKED EXAMPLE · QUESTION

A thermometer reads 2.0 °C in melting ice and 96.0 °C in steam at standard pressure. During an experiment it reads 20.0 °C at the start and 60.0 °C at the end. (a) Determine the true final temperature. (b) Determine the true temperature rise and the percentage error if the readings were used uncorrected.

SOLUTION

(a) Gradient of the calibration line: (96.0-2.0)÷(100-0)=0.94 (reading per °C). Invert: true = (reading − 2.0) ÷ 0.94, so (60.0-2.0)÷0.94= 61.7 °C

(b) True start: (20.0-2.0)÷0.94=19.1 °C, so the rise is 58.0÷0.94-18.0÷0.94= 42.6 °C. The uncorrected rise is 40.0 °C: (42.6-40.0)÷42.6×100= 6% too small. The offset of 2.0 cancels in the difference, but the scale error does not.

A safety answer names the hazard in this experiment and the precaution that deals with it.

Number: at least six (better seven or eight) values, so the shape is clear and an odd point stands out.

Range: as wide as the apparatus and safety allow. For a decay or discharge, cover at least two or three time constants or half-lives.

Spacing: aim for points spread evenly along the axis you will plot. If you plot 1/d², readings evenly spaced in d crowd together at the small-1/d² end of the axis, so take more readings at small d, where the quantity changes fastest.

Repeats: repeat where readings scatter (hand timing, rebound heights); for random processes such as radioactive decay, count for longer so each count contains several hundred events.

Control variables: name each one, say why it matters and how it is fixed: "same source and same GM tube, at the same voltage setting, throughout".

Use of the data: rearrange into a straight-line form; say what is plotted, the expected shape, and what the gradient and intercept give.

WORKED EXAMPLE · QUESTION

A sealed gamma source sits a few millimetres inside its holder, so the true source-to-detector distance is d+d₀, where d is the distance measured from the front of the holder to the window of a GM tube and d₀ is unknown. Theory predicts a corrected count rate C=k/(d+d₀)². (a) Plan an experiment to test this relationship and find d₀. (b) Use the graph below to determine d₀.

SOLUTION

(a) Background: with the source in its box, count for 300 s: here 108 counts → 0.36 s⁻¹, subtracted from every count rate.

Measurements and technique: clamp the source holder and the tube on a line along a bench rule (1 mm); vary d from 0.050 m to 0.200 m in seven steps, using a set square to line up the front of the holder and the tube window with the rule; count for 100 s at each distance so every count is several hundred or more.

Controls and safety: same source, tube and counter voltage; nothing moved between source and tube. Source handled with tongs, pointed away from people, returned to its lead-lined box as soon as the readings are complete.

Use of the data: rearrange to C^(-1/2)=(d+d₀)/√k. Plot C^(-1/2) against d: a straight line confirms the law; it crosses the d axis at -d₀, and d₀ = y-intercept ÷ gradient.

(b) At d=0: 0.0275 s^½; at d=0.200 m: 0.483 s^½. Gradient =(0.483-0.0275)÷0.200=2.28 s^(1/2) m⁻¹. d₀=0.0275÷2.28= 0.012 m (about 12 mm)

The graph turns a systematic error you cannot measure directly into an intercept you can read.

That is why a plan should say how the data will be used, not just which readings to take.

Every column heading is quantity / unit, for example d / m or C / s⁻¹. Units go in the heading, not after each value.

Raw readings are recorded to the resolution of the instrument and to the same number of decimal places down a column: 0.050, 0.075, 0.100 m, never 0.05, 0.075, 0.1.

A calculated column has the same number of significant figures as the least precise raw data used, or one more.

The unit of a derived quantity is worked out, not guessed: the unit of T² is s²; of a count rate, s⁻¹; of C^(-1/2), s^½.

Put repeats and the mean in the same table, so an anomalous repeat can be seen.

WORKED EXAMPLE · QUESTION

A student records the first four rows of the experiment in Worked example 4 like this (background 0.36 s⁻¹):

Identify the faults and write out a corrected table, adding a column for C^(-1/2).

SOLUTION

Headings: "Distance" has no unit → d / m; "Count rate / s" has the wrong unit → / s⁻¹; "Corrected rate" has no unit → C / s⁻¹.

Decimal places: distances measured to 1 mm must all show three decimal places; 10.6 should be 10.60 and 50.0 should be 50.00, to match the rest of their columns.

Anomalous readings

An anomalous reading is one that does not fit the pattern of the others: a repeat far from the other repeats, or a point well off the best-fit line.

Spot it while the experiment is running (plot as you go, compare repeats) so it can be measured again. If it cannot be repeated, leave it out of the mean and the best-fit line, but keep it in the table, marked, with the reason.

Suggest a physical cause where you can: a balance not re-zeroed, a wire that had warmed up, a timing started late, a source moved.

Do not call a point anomalous just because it is inconvenient. A curve in the data is a result, not an anomaly.

WORKED EXAMPLE · QUESTION

In the experiment in the figures, the length of wire in the field is 45.0 mm. The balance has a resolution of 0.01 g. (a) Identify the anomalous reading and describe what the student should do. (b) Use the graph to determine the magnetic flux density B. (c) Explain why the smallest current gives the least reliable point.

SOLUTION

(a) I=2.50 A, Δm=0.95 g: the line predicts about 0.80 g. Re-zero the balance and repeat the reading at 2.50 A; if the repeat lies on the line, use it. If not, exclude 0.95 g from the line and note it in the table.

(b) Gradient from a large triangle: (1.28-0.16) g÷(4.00-0.50) A=0.32 g A⁻¹=3.2×10⁻⁴ kg A⁻¹. Since Δm g=BIl, gradient =Bl/g, so B=3.2×10⁻⁴×9.81÷0.0450= 0.070 T

(c) ±0.005 g on 0.16 g is 3.1%, but on 1.27 g only 0.39%: small readings carry the largest percentage uncertainty.

A set of results can be precise but inaccurate (a systematic error moves them all the same way), or accurate on average but imprecise (large random scatter).

Random errors limit precision; systematic errors limit accuracy. Repeating improves the mean of a scattered set, but it cannot fix a systematic error.

"More accurate" does not mean "more decimal places", and many significant figures do not make a value "precise".

WORKED EXAMPLE · QUESTION

A temperature sensor gives an output p.d. that falls as the temperature rises: 3.92 V at 10 °C, 3.40 V at 20 °C, 1.78 V at 50 °C and 1.36 V at 60 °C. Its output is read with a voltmeter of resolution 0.01 V. (a) Calculate the sensitivity between 10 °C and 20 °C and between 50 °C and 60 °C. (b) Determine the smallest temperature change the system can detect in each range. (c) A student says the sensor is "more accurate at low temperatures". Comment.

SOLUTION

(a) Sensitivity = change in output ÷ change in input: (3.92-3.40)÷10= 0.052 V K⁻¹ (52 mV K⁻¹); (1.78-1.36)÷10= 0.042 V K⁻¹ (42 mV K⁻¹).

(b) Smallest detectable change = resolution ÷ sensitivity: 0.01÷0.052= 0.19 K; 0.01÷0.042= 0.24 K.

(c) The sensor is more sensitive at low temperatures, so it can detect smaller changes there;

whether its readings are accurate depends on how well it has been calibrated against known temperatures, not on its sensitivity.

Evaluating and improving

Find the largest source of uncertainty by comparing percentage uncertainties (a squared quantity counts twice; see 6.2), and look for systematic errors: a non-zero intercept where theory says zero, results all on one side of the accepted value, a hidden offset such as d₀.

Suggest a specific improvement tied to that source and say what it improves; comment in context: does the result agree with the accepted value within its uncertainty, and do the assumptions hold (small amplitude, point source, constant temperature)?

2 Uncertainties and logarithmic graphs, 6.3 A2 core practicals

After the readings: combining uncertainties, error bars, worst-fit lines and log graphs. Every A2 core practical uses the planning skills on this page.

1 Planning at IAS

Verniers, micrometers, zero errors, parallax and the basic plan checklist.

Units 4 and 5

The physics behind the contexts: F=BIl and capacitor discharge (Unit 4); simple harmonic motion, gas laws, thermal energy and radioactivity (Unit 5).

1. A 470 µF capacitor discharges through 47 kΩ. Is a stopwatch suitable for timing the discharge?

2. A GM counter records 4.40 counts per second near a source; the background is 0.20 s⁻¹. What count rate is due to the source?

3. A sensor's output changes by 25 mV for each kelvin. It is read with a voltmeter of resolution 0.01 V. What is the smallest temperature change it can detect?

4. Five values of the speed of sound are 331, 332, 331, 330, 332 m s⁻¹; the accepted value at that temperature is 343 m s⁻¹. Describe the results.

5. Why is a fiducial mark placed at the centre of an oscillation rather than at one end?

1. RC=470×10⁻⁶×47×10³=22.1 s. A 0.2 s reaction time is under 1% of this, so hand timing with readings every 5 or 10 s is reasonable.

2. 4.40-0.20= 4.20 s⁻¹. Always subtract the background before analysing.

3. 0.01 V÷0.025 V K⁻¹= 0.40 K.

4. Precise (spread only ±1 m s⁻¹) but not accurate (about 12 m s⁻¹ low): there is a systematic error, which repeating will not remove.

5. The object moves fastest at the centre, so the instant it passes is sharply defined; at the ends it is momentarily at rest and the turning point is hard to judge.