Inquiry skills: measurement, uncertainty and graphs

1. From a physical question to a testable relationship

An investigation starts with a relationship that could be contradicted by measurements. “Study circular motion” is a topic, whereas “How does rotation period depend on radius when inward force and rotating mass are held constant?” identifies an independent variable, a dependent variable and two controls. For uniform circular motion, F = 4π²mr/T², so T² should be proportional to r when F and m remain constant. Predicting the straight-line form before collecting data makes the test more discriminating than looking for any convenient pattern afterwards.

A practical method must explain how each quantity is measured and how the controls are maintained. If a suspended mass supplies string tension, mark the string at a fixed distance below the tube and keep that mark stationary while timing. A rising or falling suspended mass indicates that the assumed tension is not simply its weight. Time ten revolutions and divide by ten to reduce the percentage effect of reaction time. Take repeated readings at each radius, use a sufficiently broad radius range, and describe precautions against a rotating object striking someone. These details connect the method to the model's assumptions.

Do not call a variable controlled merely because its name appears in a list. The rotating mass must actually remain unchanged, and the effective radius must be measured to its centre of mass rather than to an arbitrary edge. In the conclusion, distinguish the observed proportionality from the stronger claim that the model is universally true. Agreement within measurement uncertainty supports the relationship under the tested conditions; it does not establish behaviour outside the measured range or remove possible shared systematic effects.

2. Resolution, random variation and systematic effects

A measured value is an estimate accompanied by uncertainty. A ruler marked every millimetre does not justify reporting a length to a micrometre. For a simple analogue reading, half the smallest division may be a reasonable reading uncertainty, but the physical setup can dominate: a blurred edge, parallax or an inaccessible centre can make the uncertainty larger. Digital resolution is likewise only one contribution. A display with many digits is not evidence that calibration, alignment or timing is equally accurate.

Random effects produce scatter when measurements are repeated. For example, a hand-timed set of ten revolutions might take 12.4, 12.6 and 12.5 s. The mean is 12.5 s and a simple half-range estimate is 0.1 s; a stated method may require another estimate. Dividing the timing interval by ten gives T = 1.25 s with timing uncertainty 0.01 s, before including other relevant effects. Repetition helps estimate scatter and can improve the precision of the mean, but it does not guarantee that the mean lies close to the true value.

A systematic effect shifts readings consistently or distorts their scale. A balance that reads 0.8 g with an empty pan, a ruler with a damaged zero or a sensor whose calibration factor is wrong can produce very repeatable but inaccurate results. Repeating the same procedure does not remove the offset. Check a zero, compare with a known reference, reverse an arrangement where appropriate, or calibrate over the range used. Explain the direction of the resulting bias when possible: measuring radius to the outer edge rather than the centre overestimates r and changes any quantity calculated from it. “Human error” alone names neither a mechanism nor a correction.

3. Absolute and percentage uncertainty

Write an absolute uncertainty in the same unit as the measured quantity: L = (0.840 ± 0.005) m. Its percentage uncertainty is 100 × 0.005/0.840 = 0.595%, normally reported as about 0.6%. The absolute interval is 0.835–0.845 m. A percentage is a relative comparison, not another unit of length. Reporting both forms helps distinguish a coarse absolute reading of a large quantity from the same absolute reading of a small one.

For the syllabus's conservative combination rules, add absolute uncertainties when adding or subtracting measured quantities. If two positions are x₁ = (0.120 ± 0.002) m and x₂ = (0.780 ± 0.002) m, the displacement is 0.660 m with uncertainty 0.004 m. The uncertainty grows even though the values are subtracted: the possible errors can act in opposite directions. Treat the measurements as independent for this rule; a common zero offset can cancel in a difference, but that cancellation must be justified from the apparatus rather than assumed.

For multiplication or division, add percentage uncertainties. A speed calculated from s = (2.00 ± 0.01) m and t = (0.80 ± 0.02) s is 2.50 m s⁻¹. The relative contributions are 0.5% and 2.5%, giving 3.0% overall, or 0.075 m s⁻¹. A sensible final form is (2.50 ± 0.08) m s⁻¹. Keep extra digits in intermediate steps and round the final value consistently with the uncertainty. This conservative estimate is not a statistical confidence interval. Do not replace it with a root-sum-square rule unless the question explicitly introduces that convention.

4. Linearising data and interpreting a gradient

A graph is a physical argument. Put the independent quantity on the horizontal axis and the dependent quantity on the vertical axis, label both with units, choose scales that use the available space, and show uncertainty bars where available. Plot measured points accurately. A line of best fit represents an overall relationship; connecting each neighbouring point with a zigzag treats random scatter as a physical feature. A curve should be used when the model predicts curvature and no transformation has been made.

For a circular-motion experiment at constant tension F and mass m, plot T² vertically against r horizontally. Rearranging F = 4π²mr/T² gives T² = (4π²m/F)r. The gradient therefore has units s² m⁻¹ and equals 4π²m/F. If m = 0.0500 kg and the fitted gradient is 2.00 s² m⁻¹, F = 4π²(0.0500)/2.00 = 0.987 N. This conclusion depends on the axes: swapping them would make the gradient the reciprocal relationship, with units m s⁻².

Calculate the gradient using two widely separated points on the fitted line, not automatically two adjacent measured points. A wide triangle reduces the relative effect of reading the graph. If the chosen line points are (0.20 m, 0.42 s²) and (0.80 m, 1.62 s²), the gradient is (1.62 − 0.42)/(0.80 − 0.20) = 2.00 s² m⁻¹. The intercept is 0.02 s², which may indicate an offset or may be consistent with zero once uncertainty is considered. Forcing a line through the origin conceals that diagnostic information unless the evidence and method justify the constraint.

5. Uncertainty bars, anomalous points and model limits

Uncertainty bars show the plausible interval assigned to a measured coordinate. They do not necessarily represent the full variation of the underlying population or a 95% confidence interval. State what the bars mean. When drawing a best-fit line, consider whether the line is compatible with the intervals as a group, and inspect residual patterns. Points distributed randomly about the line suggest a different issue from a smooth curve in the residuals, which may reveal a model that is inadequate over the chosen range.

An anomalous point is one that does not follow the surrounding pattern to an extent that the stated uncertainty cannot readily explain. It is not permission to delete an inconvenient result. Check the original reading, repeat that condition if possible, inspect the apparatus and record a reason for excluding a value. A timing error because only nine revolutions were counted instead of ten is a specific methodological explanation. A high-radius point deviating repeatedly may instead reveal that tension was no longer constant, so discarding it would hide the boundary of the experiment.

Where a task asks for extreme acceptable gradients, draw the steepest and shallowest lines reasonably compatible with the uncertainty bars, following the task's stated convention. Their spread estimates gradient uncertainty and can be propagated to the inferred constant. Percentage difference from a reference is a separate calculation: if an experiment gives 9.65 m s⁻² against a reference 9.80 m s⁻², the magnitude of the percentage difference is 100 × 0.15/9.80 = 1.53%. Whether that is acceptable depends on the uncertainty and validity of the setup; a small difference alone cannot prove a sound method.

6. Writing a justified scientific conclusion

A useful conclusion answers the original question at the strength supported by the evidence. State the measured relationship, include a quantitative result where appropriate, and connect it to the model. “T² increased approximately linearly with radius, with gradient 2.00 s² m⁻¹ over 0.20–0.80 m” is more informative than “the hypothesis was correct”. If the result supports F = 4π²mr/T², explain which quantities were held constant and whether the intercept and scatter are consistent with the expected proportionality.

Evaluate reliability and validity separately. Repeat measurements address repeatability; an apparatus that measures the wrong radius repeatedly can still be invalid for the intended test. An improvement should target a named limitation. An optical gate reduces timing reaction uncertainty. Measuring from the rotation axis to the mass centre corrects the geometrical definition of radius. A wider range can test curvature more clearly, but it may also introduce a new regime where the constant-tension assumption fails. Explain that trade-off rather than claiming every larger data set is automatically better.

Dimensional analysis provides a final reasonableness check. Since force is measured in kg m s⁻², the expression mv²/r has the correct units, whereas mv/r does not. Correct dimensions are necessary but not sufficient: both mv²/r and 2mv²/r have force units, so only evidence and derivation distinguish their factors. Check physical direction, limiting cases and scale as well. In an examination response, link each limitation to its effect on the conclusion and then to a feasible improvement. Clear causal explanation earns more defensible credit than a list of unexplained labels such as accuracy, precision and reliability.