Required Practical 10: Investigating Simple Harmonic Motion

Required Practical 10 in AQA A-Level Physics is the experimental investigation of the period of a simple harmonic oscillator and its dependence on the system parameters. Students typically work with a vertical mass-spring system (varying the mass to verify T = 2π√(m/k)) or a simple pendulum (varying the length to verify T = 2π√(L/g)). The practical is examined directly in Paper 2 of AQA 7408 and is the basis for the CPAC (Common Practical Assessment Criteria) endorsement that accompanies the written exam grade. This lesson is the experimental anchor for the course: it ties the theoretical material of lessons 3–5 (SHM, mass-spring, pendulum) to laboratory measurement, uncertainty analysis, and graphical extraction of physical parameters.

Spec mapping: This lesson is the explicit treatment of AQA 7408 Required Practical 10: the investigation of factors affecting the period of a simple harmonic oscillator. It covers the experimental procedure for both the mass-spring (variant A) and the simple pendulum (variant B), uncertainty propagation, graphical linearisation, and the CPAC competencies assessed. The synoptic theory in lessons 3–5 supplies the analytical framework. (Refer to the official AQA specification document for exact wording.)

Synoptic links:

• Section 3.6.1.2 (SHM, lessons 3–5): the formulas T = 2π√(m/k) and T = 2π√(L/g) are derived from the SHM defining equation a = −ω²x; this practical tests them experimentally.
• Section 3.1 (use of SI units and limitations of physical measurements): uncertainty propagation, percentage uncertainty, and the choice of significant figures all anchor here.
• Section 3.7.2 (gravitational fields): the pendulum variant of RP10 is a direct measurement of g; the practical is the basis for the precision pendulum measurements that historically defined the metre and the second.

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Aims of the Practical

The student is asked to investigate experimentally:

1. Whether the oscillator obeys SHM — that is, whether the period is independent of amplitude (the isochronous prediction) within the small-amplitude regime.
2. The relationship between period and the system parameter — mass (mass-spring) or length (pendulum) — and to test the theoretical prediction T ∝ √(parameter).
3. The system's characteristic constant — the spring constant k (mass-spring) or local gravitational field strength g (pendulum) — extracted from a graphical fit.

A high-quality write-up demonstrates the data, the graph, the extracted constant, an explicit uncertainty estimate, and a critical comparison with the textbook value.

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Variant A — Mass-Spring System

Apparatus

• A helical (coil) spring of known approximate spring constant (typically 10–30 N m⁻¹ for school-lab use).
• A set of standard slotted masses (50 g or 100 g increments, ideally with a mass hanger; total range about 100 g to 800 g).
• A retort stand with clamp to suspend the spring vertically.
• A digital stopwatch (resolution at least 0.01 s).
• A ruler and a set square (or a fiducial marker) to identify a reference position for the start/end of each oscillation.
• A balance (resolution 0.1 g) to verify mass values.

Procedure

1. Determine the natural length of the spring. Suspend the spring (without any added mass) from the clamp and record its natural length L₀ using a metre rule.
2. Estimate the spring constant from a static measurement. Add a mass (say, 200 g = 0.20 kg) and measure the extension x. Compute k = mg/x and record this static value as a baseline.
3. Set up the oscillator. With a chosen mass on the hanger, pull the mass down by a small amount (about 2 cm — small compared with the static extension to ensure the coils do not touch on the upward swing) and release.
4. Measure the period. Time 20 complete oscillations with the stopwatch. To do this accurately, start the stopwatch as the mass passes through the equilibrium position (or any other clearly identifiable reference point, such as the lowest point of the swing) and stop it when the mass passes through the same reference position after 20 complete cycles. Divide by 20 to obtain T.
5. Repeat each timing three times at the same mass. Compute the mean.
6. Vary the mass. Repeat the timing procedure for at least six different mass values, evenly spaced across the available range (e.g. 100, 200, 300, 400, 500, 600, 700, 800 g).
7. Record the data in a table. Columns: mass m (kg), three timings of 20T (s), mean 20T (s), period T (s), T² (s²).

Data Analysis

Plot T² (on the y-axis) against m (on the x-axis). According to T² = (4π²/k)m, this should be a straight line through the origin with gradient 4π²/k.

• Test isochronism by varying the amplitude (e.g. release from 1 cm, 2 cm, 4 cm) at a single mass value and confirming that the period is independent of amplitude (within ~2%).
• Extract k from the gradient: k = 4π² / gradient.
• Compare with the static value of k from step 2 — they should agree to within a few per cent.

Worked Example — Spring Constant from RP10 Data

A student records the following data:

m (kg)	20T (s)	T (s)	T² (s²)
0.10	12.6	0.630	0.397
0.20	17.8	0.890	0.792
0.30	21.8	1.090	1.188
0.40	25.2	1.260	1.588
0.50	28.2	1.410	1.988
0.60	30.9	1.545	2.387

Plotting T² against m gives a straight line with gradient ≈ 4.0 s² kg⁻¹. From k = 4π² / gradient = 4π² / 4.0 = 39.48 / 4.0 = 9.87 N m⁻¹.

The theoretical spring constant from the static measurement might be, say, k_static = 10.2 N m⁻¹, an agreement to within 3% — a satisfactory result given typical measurement uncertainties.

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Variant B — Simple Pendulum

Apparatus

• A small dense bob (steel ball or brass) on a light inextensible string.
• A retort stand with clamp; the pivot must be sharp-edged and the string secured at a single point.
• A metre rule (or vernier calipers for short pendulums) to measure the length.
• A digital stopwatch.
• A protractor to measure the initial release angle (should be less than 10°).

Procedure

1. Set up the pendulum. Tie one end of the string to the clamp at the pivot. Attach the bob to the other end. Adjust the string length so that the centre of mass of the bob is at a chosen distance L from the pivot (typically 200–1000 mm). The centre of mass for a typical dense bob is at its geometric centre; for a string-tied bob, allow for the radius of the bob when measuring.
2. Measure L carefully. Use a metre rule, measuring from the pivot to the centre of the bob. Record L with appropriate uncertainty (typically ±2 mm on a metre rule, but ±0.5 mm on calipers).
3. Determine the period. Displace the bob through a small angle (≤ 10°) and release. Time 20 complete oscillations. Divide by 20 to obtain T.
4. Repeat three times at the same length, take the mean.
5. Vary L across at least six values (e.g. 200, 400, 600, 800, 1000 mm).
6. Test isochronism at one length by varying the initial amplitude (5°, 10°, 15°, 20°) and confirming that T is approximately independent of amplitude (it will be — but the largest amplitude may show a 1–2% increase).
7. Tabulate m (kg), L (m), three timings of 20T (s), mean 20T (s), T (s), T² (s²).

Data Analysis

Plot T² (y-axis) against L (x-axis). According to T² = (4π²/g)L, this should be a straight line through the origin with gradient 4π²/g.

• Extract g from the gradient: g = 4π² / gradient.
• Compare with the textbook value of g ≈ 9.81 m s⁻² — the agreement should be to within 1% for a well-conducted experiment.

Worked Example — g from a Pendulum

A student records:

L (m)	20T (s)	T (s)	T² (s²)
0.200	17.95	0.898	0.806
0.400	25.40	1.270	1.613
0.600	31.10	1.555	2.418
0.800	35.92	1.796	3.226
1.000	40.15	2.008	4.032

Plotting T² against L gives a straight line with gradient ≈ 4.0 s² m⁻¹. From g = 4π² / gradient = 39.48 / 4.0 = 9.87 m s⁻².

The textbook value of g is 9.81 m s⁻². The student's measurement agrees to within 0.6% — within typical experimental uncertainty.

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Uncertainty Analysis

A high-quality RP10 write-up includes a quantitative uncertainty estimate. The principal sources of uncertainty and their typical magnitudes are:

Source	Magnitude	Type	Mitigation
Stopwatch reaction time	±0.2 s on each start/stop	Random	Time 20 oscillations (divide reaction error by 20)
Length measurement (ruler)	±0.5 mm on metre rule	Random	Use vernier calipers (±0.05 mm)
Length measurement (to centre of mass)	±5 mm if string-only is measured	Systematic	Always measure to bob's centre of mass
Amplitude (pendulum)	±1° on protractor	Random	Repeat readings and average
Stopwatch resolution	±0.01 s (digital)	Negligible	—
Parallax (reading positions)	±1 mm	Random	Eye at same level as reference mark

Propagation of Uncertainty in T

If the stopwatch uncertainty is ±0.2 s on a 20-oscillation total of, say, 25 s, the percentage uncertainty in 20T is:

Δ(20T)/(20T) × 100% = 0.2/25 × 100% = 0.8%.

This is also the percentage uncertainty in T (since T = (20T)/20 — the 20 is exact).

If the same single oscillation were timed (T ≈ 1.25 s), the percentage uncertainty would be 0.2/1.25 × 100% = 16%. The 20-oscillation technique reduces the uncertainty by a factor of 20. This is the most important uncertainty-reduction technique in RP10.

Propagation of Uncertainty in g (Pendulum)

g = 4π² × L / T². The fractional uncertainty in g is the sum of fractional uncertainties:

ΔG/g = ΔL/L + 2 × ΔT/T.

If ΔL/L ≈ 0.5% and ΔT/T ≈ 0.8%, then Δg/g = 0.5% + 2 × 0.8% = 2.1%, so Δg ≈ 0.21 m s⁻². Quoting g = 9.81 ± 0.21 m s⁻² is more meaningful than just "g = 9.81 m s⁻²".

The factor of 2 on ΔT comes from squaring T in the formula — for any power n in the propagation formula, the fractional uncertainty is multiplied by n. This is examinable.

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CPAC (Common Practical Assessment Criteria)

For the practical-endorsement element of the A-Level, students must satisfy five CPAC criteria:

1. CPAC 1 — Apparatus and techniques. Follow practical procedures methodically; safely handle the apparatus; secure connections; use the timer correctly.
2. CPAC 2 — Plan and conduct experiments. Make a clear experimental plan; identify and use appropriate techniques; modify the procedure as needed.
3. CPAC 3 — Collect and analyse data. Make accurate measurements; record observations clearly and quantitatively; use units correctly.
4. CPAC 4 — Conclusions. Draw conclusions consistent with the observations; quantify uncertainties; compare with theory.
5. CPAC 5 — Health and safety. Identify hazards; assess and manage risk (e.g. ensure the mass is supported safely, avoid trip hazards from cables).

The endorsement is reported separately from the A-Level grade as "pass" or "fail"; almost all candidates pass with appropriate preparation. The CPAC competencies are integrated throughout RP10.

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Improvements and Extensions

Once the basic experiment is complete, the following extensions demonstrate Grade A* practical thinking:

1. Use a datalogger. A photogate (or motion sensor) placed at the equilibrium position can time the period to ±0.001 s — a 10× improvement on hand-stopwatch timing. The datalogger output is automatically recorded and can be averaged over hundreds of oscillations.
2. Plot a log-log graph. A log-log plot of T against m (or L) should give a straight line of gradient ½, confirming the T ∝ √(parameter) relationship. This is a more demanding test than the linear T²-vs-parameter plot.
3. Measure the period as a function of amplitude. Vary the release angle from 5° to 30° and plot T against the amplitude. The small-angle approximation should give a flat horizontal line up to about 10°; beyond that, T should slowly rise. The increase is the leading correction T(θ_max) ≈ T₀ × (1 + θ_max²/16).
4. Compare a spring's static and dynamic k values. The static k from kx = mg should match the dynamic k from 4π²m/T². Disagreement may indicate that the spring has internal mass (which would lower the dynamic k).
5. Repeat at a different latitude or altitude. A pendulum's measured g varies with location — at the equator g ≈ 9.78 m s⁻², at the poles g ≈ 9.83 m s⁻². The same pendulum at a different latitude gives a different T and hence a different inferred g.

flowchart TD
    A["RP10: Investigate factors<br/>affecting SHM period"] --> B{"Choose variant"}
    B -->|"A: Mass-spring"| C["Vary m, plot T² vs m<br/>Gradient → k"]
    B -->|"B: Pendulum"| D["Vary L, plot T² vs L<br/>Gradient → g"]
    C --> E["Critical technique:<br/>Time 20 oscillations,<br/>repeat 3 times, average"]
    D --> E
    E --> F["Uncertainty analysis"]
    F --> G["Δg/g = ΔL/L + 2ΔT/T<br/>or Δk/k = Δm/m + 2ΔT/T"]
    G --> H["Compare with textbook<br/>9.81 m/s² or static k"]
    H --> I{"Within 2%?"}
    I -->|"Yes"| J["Pass: report and improve"]
    I -->|"No"| K["Investigate sources<br/>of systematic error"]

    style J fill:#27ae60,color:#fff
    style K fill:#e67e22,color:#fff

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Specimen Exam Question

Specimen question modelled on the AQA paper format. Nine marks.