This practice outline investigates string resonance. It supports development of a locally approved method and risk assessment; it is not a complete validated procedure. Check current Pearson documents for course-specific requirements.
Aim and scientific principle
Test how resonant frequency depends on vibrating length, tension or linear mass density, keeping the other model quantities and the identified mode fixed. The model assumes a uniform flexible string, small transverse displacements, approximately constant tension and approximate nodes at both ends of the measured vibrating segment.
Begin by translating the title into a testable question. Identify what is deliberately changed, what is measured directly, what is calculated and what must remain controlled. Predict a relationship only after naming the mechanism. A prediction without a reason is weak; a reason without measurable variables cannot guide a practical design.
Apparatus and setup
Use vibration generator, signal generator, string, pulley, masses, balance, metre rule and supports. Consider measuring range, resolution, uncertainty, calibration and suitability for the sample, not the instrument name alone. Draw a labelled scientific setup that shows the working geometry and exactly where readings are taken. Include a control or blank when it distinguishes the target effect from drift, contamination or background response.
Before collecting results, check zero readings, calibration, leaks, alignment or sterility as relevant. Record the instrument resolution beside the first raw-data table. If a digital device displays more digits than its underlying accuracy supports, do not treat every digit as equally certain.
Method
Identify the effective vibrating segment and approximate end nodes. Measure its length L between those boundaries, not the entire string. Estimate linear mass density from the mass and length of a representative uniform sample, noting any change in length or density under working tension.
Include the hanger in the total suspended mass M. With the mass stationary and pulley friction negligible, estimate string tension as T=Mg
Check this topic from memory
Attempt the matching topic bank before reopening the notes. Use each missed idea to decide what to review next.
. Otherwise the hanging weight need not equal the tension in the vibrating segment.
Identify a resonance by its node and antinode pattern, recording the mode number n as well as frequency. Keep n fixed while changing one of L, T or linear density; monitor the remaining quantities and use small amplitudes. Choose a justified range and repeat resonance identification rather than copying a fixed number of settings.
For fixed mode and linear density, test frequency against reciprocal length at fixed tension, or against square root tension at fixed length. For a density investigation, use inverse square root density at fixed length and tension. Record changed strings and reassess uniformity, stiffness and boundary conditions.
Plan the analysis before starting. This reveals whether an additional measurement is needed for a derived quantity and prevents a completed experiment from lacking a mass, length, concentration, background or calibration value. A method should be detailed enough for another trained student to reproduce, while avoiding decorative steps that do not affect validity.
Variables and controls
Independent variable: length, tension or mass per unit length.
Dependent variable: resonant frequency.
Important controls: mode number and the model quantities not being investigated, approximate nodal boundaries, small amplitude and pulley behaviour. Changing strings to vary linear density may also change stiffness or elasticity; record these limitations instead of claiming every material property remains fixed.
For each control, state how it is maintained or monitored. Simply writing "keep temperature constant" is incomplete if the method has no bath, probe, equilibration time or other means of doing so. Choose a range wide enough to reveal the relationship but within safe and scientifically valid limits. Choose the number, range and spacing of values for the expected response and analysis. Concentrating values near an uncertain threshold or optimum can be more useful than uniform spacing; no fixed count establishes an adequate design.
What is distinctive about this practical
For approximate nodes at both ends, L=nλ/2 and fn=(n/(2L))T/μ, where n is a positive integer and the linear mass density is μ=ms/ℓs for a representative sample. There are n antinodes and n + 1 nodes including both ends. Adjacent nodes are separated by half a wavelength, not a whole wavelength. A vibrating driver is not an exact fixed endpoint; the actual boundary pattern and effective length must be checked.
At fixed n, the inverse-length, square-root-tension and inverse-square-root-density relations each require the other model quantities to remain fixed. String stretching can change both vibrating length and linear density. Pulley friction, stiffness, large amplitude or uncertain mode identification can invalidate the simple interpretation.
Recording and processing results
Record L, total suspended mass, the tension estimate and its assumptions, sample mass and length, mode pattern and repeated resonance frequencies. At fixed n, T and linear density, a graph of frequency against reciprocal length has gradient nv/2, so v=2k/n if k denotes that gradient. A frequency versus square-root-tension gradient is a different quantity and must not be interpreted as wave speed. A straight trend alone does not establish proportionality: inspect the intercept and uncertainty, and do not force a zero intercept to hide an end correction. Put units in column headings, record raw readings consistently with instrument resolution and the method and distinguish repeats from calculated means. Do not delete an anomaly merely because it is inconvenient. Recheck the condition, look for a procedural explanation and state any exclusion rule.
Choose a fitted line, curve or ruled joins according to the data, scientific relationship and question; do not connect or smooth points mechanically. A gradient should use a large triangle and values read from the fitted line. When a relationship is linearised, explain the transformation and connect gradient or intercept to the requested physical quantity. When observations are categorical, define the category rule before seeing the result.
Safety, ethics and responsible practice
Wear eye protection, secure masses and keep clear of a snapping string. A useful safety statement names the hazard, identifies who or what could be harmed and gives a control that reduces likelihood or severity. "Take care" does not specify an action that reduces the identified risk.
Worked application: from raw evidence to a defensible conclusion
This is an ideal-model counting check, not measured data. For a segment containing three loops, n = 3: there are three antinodes and four nodes including both ends. The wavelength is λ=2L/3. If the pattern is wrongly labelled as the fundamental, using twice the vibrating length as the wavelength makes the inferred wave speed three times the correctly mode-labelled value at the same frequency. This check does not validate the tension estimate or nodal boundary approximation.
In an actual investigation, preserve the observed pattern and repeat resonance identification before averaging frequencies. Check whether changing tension stretches the string or shifts its effective boundary. Report deviations and investigate these assumptions rather than treating any linearised trend as proof of the ideal model.
This chain is stronger than a memorised recipe because every step answers the practical question. If the graph contradicts the prediction, report the evidence honestly and investigate method, range, controls and model assumptions before claiming a new scientific law.
Evaluation: limitation, effect and improvement
Limitation: end corrections, pulley friction and subjective resonance judgement alter effective tension or length.
Likely effect: pulley friction makes string tension differ from the hanging weight, while uncertain bridge position changes effective length. Both can bias the expected frequency relation, and subjective resonance judgement adds scatter.
Targeted improvement: use well-defined approximate nodal boundaries, check pulley behaviour and string extension, keep amplitude small, and record the mode pattern. A sensor-defined amplitude peak may reduce subjective judgement, but it does not establish the mode number, correct pulley friction or validate the ideal string model.
Separate random variation, systematic bias and model limitation. Repeats help reveal random variation but do not remove a zero error. Calibration establishes the relation between reference values and indications; appropriate adjustment or correction can address an identified offset. It does not make biased sampling representative. Finer resolution and greater repeatability are distinct properties, and finer resolution may improve an estimate only when the reading is a material source of uncertainty.
Common misconceptions and corrections
Repeats automatically make a result accurate. Repeats estimate random variation; they do not correct a biased method or calibration error.
A control variable is just a list item. Explain the apparatus or procedure that keeps it stable.
The line must pass through every point. Use the overall pattern and a best-fit relationship, then investigate justified anomalies.
Percentage error and percentage uncertainty are interchangeable. Error compares a measured value with a suitable reference. Uncertainty characterises dispersion of values attributed to the measurand; it may be expressed as a standard uncertainty or an interval with stated coverage, and includes more than resolution alone. State the reference, denominator and uncertainty convention used.
A successful endpoint proves the theory. The observation supports a conclusion only within the method, range and assumptions tested.
More sophisticated apparatus is always better. The improvement must address the dominant limitation and be practical in the stated context.
Assessment guidance
Pearson assesses practical understanding throughout the science papers and concentrates it in written Units 3 and 6. Planning answers need an operational independent variable, a measured dependent variable, a justified range, controls, repeats, safety and a stated analysis. Data answers need units, consistent precision, sensible significant figures and transparent calculations. Graph answers need labelled axes, a scale using the grid well and line treatment appropriate to the data and question. Evaluation answers should use a limitation-effect-improvement chain specific to the apparatus. Do not claim that a memorised core-practical recipe is the only acceptable method when the question supplies a different valid context.
Retrieval practice
Extend or adapt this practical to answer a related question. Where the design uses a continuous variable, justify its planned values. List apparatus with range and resolution where relevant, control methods, a raw record layout, the expected analysis and one hazard-control pair. Then identify one random uncertainty, one systematic bias and one model limitation. Explain which can be reduced by repeats, which needs calibration followed by appropriate correction or adjustment, or a redesign, and how your conclusion would change if the anomaly were confirmed.
