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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.
The student is asked to investigate experimentally:
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.
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.
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.
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.
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.
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 |
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.
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.
For the practical-endorsement element of the A-Level, students must satisfy five CPAC criteria:
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.
Once the basic experiment is complete, the following extensions demonstrate Grade A* practical thinking:
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
Specimen question modelled on the AQA paper format. Nine marks.
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