The simple pendulum is one of the most studied systems in classical mechanics. By measuring how the period of oscillation changes with pendulum length, you can determine the local gravitational field strength g with remarkable precision from a school laboratory. The relationship T² ∝ L produces a clean linear graph whose gradient directly yields g, making this an ideal IB Physics IA for demonstrating rigorous uncertainty analysis.
This practical is suitable for IB Diploma Physics HL and SL and Edexcel IGCSE Physics.
Background Theory
For small angles of oscillation (less than ~15°), the period T of a simple pendulum is given by:
T = 2π√(L/g)
Squaring both sides:
T² = (4π²/g) × L
This is a linear equation: a graph of T² (y-axis) against L (x-axis) passes through the origin with gradient = 4π²/g. Rearranging: g = 4π² / gradient. The accepted value of g in Hong Kong is 9.788 m s⁻², which provides an excellent comparison point.
Variables
- Independent variable (IV): Length of pendulum L (m) — measured from pivot to centre of mass of bob, e.g. 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 1.00 m
- Dependent variable (DV): Period of oscillation T (s) — measured by timing 20 complete oscillations and dividing by 20
- Controlled variables (CV): Amplitude of swing (less than 15° throughout), same pendulum bob mass, same pivot point, same location (g is constant)
Equipment
- Pendulum bob (dense, small sphere e.g. steel ball bearing)
- Inextensible string or thread
- Clamp stand with G-clamp to fix pivot
- Ruler or metre stick (±0.5 mm)
- Digital stopwatch (±0.01 s)
- Protractor (to measure amplitude angle)
Safety
Safety
⚠️ Ensure the clamp stand is secure and cannot topple. Keep clear of the swinging bob. No chemical hazards — no waste disposal required.
Method
- Set up the pendulum with L = 0.10 m, measured from the pivot point to the centre of the bob. Use a ruler to measure precisely.
- Displace the bob to an angle of less than 15° (verify with protractor). Release without pushing.
- Time 20 complete oscillations (one oscillation = swing there and back). Record the total time t₂₀.
- Calculate T = t₂₀ / 20 and T².
- Repeat the timing three times for each length and calculate a mean T and mean T².
- Increase L in steps and repeat until L = 1.00 m.
- Record the uncertainty in L (±0.001 m from ruler) and in T (±0.01/20 = ±0.0005 s per oscillation from stopwatch resolution, plus reaction time ≈ ±0.1 s total, so δT ≈ ±0.005 s after dividing by 20).
Results Table
| L (m) | t₂₀ run 1 (s) | t₂₀ run 2 (s) | t₂₀ run 3 (s) | Mean t₂₀ (s) | T = t₂₀/20 (s) | T² (s²) |
|---|---|---|---|---|---|---|
| 0.10 | ||||||
| 0.20 | ||||||
| 0.30 | ||||||
| 0.40 | ||||||
| 0.50 | ||||||
| 0.60 | ||||||
| 0.70 | ||||||
| 0.80 | ||||||
| 0.90 | ||||||
| 1.00 |
Analysis
1. Plot T² (y-axis, s²) against L (x-axis, m). The graph should be linear and pass through the origin.
2. Draw a best-fit straight line. Calculate the gradient: gradient = ΔT² / ΔL
3. Calculate g: g = 4π² / gradient
4. Draw worst-case lines (maximum and minimum gradient) to find the uncertainty in the gradient δ(gradient), then propagate to find δg using: δg/g = δ(gradient)/gradient
5. Compare your value of g ± δg to the accepted value for your location (9.788 m s⁻² in Hong Kong). Does the accepted value fall within your uncertainty range?
Discussion Points
- Why does the period depend on length but not on the mass of the bob or the amplitude (for small angles)?
- Why is it better to time 20 oscillations rather than just 1?
- Why must the amplitude be kept below 15°? What happens to the period at larger angles?
- Why does the graph of T² vs L pass through the origin?
- What sources of systematic error might cause your value of g to differ from the accepted value?
Guidance
IA Guidance
This is a classic IB Physics IA that, despite its simplicity, rewards careful technique and sophisticated uncertainty analysis. To score highly:
- Research Design: Justify your length range and number of oscillations timed. Explain why you measure L to the centre of the bob and why this introduces uncertainty. Justify the small angle approximation with reference to the derivation.
- Data Analysis: Include uncertainty bars on both axes of your T² vs L graph. Use worst-case lines to find δ(gradient) and propagate to δg. Compare your result to the accepted local value of g.
- Conclusion: State g ± δg and comment on whether your result is consistent with the accepted value. Discuss whether any discrepancy is within experimental uncertainty.
- Evaluation: Discuss the dominant source of uncertainty — likely the measurement of L (estimating centre of mass of bob) or reaction time in starting/stopping the stopwatch. Suggest using a light gate and data logger to eliminate reaction time error entirely.
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