Interpreting a Discrepant g Measurement
A student measures g with a simple pendulum, timing 20 oscillations with a stopwatch for five lengths between 0.4 m and 1.0 m. She plots T^2 against L and obtains a gradient of 4.12 s^2/m with a stated uncertainty of ±0.06 s^2/m. The accepted local value is 9.79 m/s^2. Separately, a spring-mass oscillator in the same lab gives 9.6 ± 0.3 m/s^2. The demonstrator says the pendulum result 'must be wrong because it is more than 1% off' and asks her to repeat everything. The student notices she timed from the first swing after release rather than waiting for steady oscillation, and that the bob was a large sphere on a long thread.
1. Extract g from the pendulum gradient with its uncertainty and assess whether the discrepancy is statistically significant.
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Extraction. For a simple pendulum T^2 = (4π^2/g)L, so the gradient m = 4π^2/g and g = 4π^2/m.
With m = 4.12 s^2/m: g = 39.48/4.12 ≈ 9.58 m/s^2.
Propagating uncertainty. Since g ∝ 1/m, the fractional uncertainty in g equals that in m: 0.06/4.12 ≈ 1.46%. So Δg ≈ 0.0146 × 9.58 ≈ 0.14 m/s^2. Result: g = 9.58 ± 0.14 m/s^2.
Significance test. Difference from accepted value: 9.79 − 9.58 = 0.21 m/s^2. In units of the uncertainty: 0.21/0.14 ≈ 1.5σ. A 1.5σ deviation is entirely ordinary - roughly 13% of honest measurements would fall at least this far out. It is not evidence of a blunder.
The demonstrator's '1% rule' is not a statistical criterion; the correct comparison uses the measurement uncertainty, and here the result is consistent with the accepted value at the 1.5σ level.
2. Identify the systematic errors suggested by the setup and explain how each would bias g.
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Timing from release. Starting the clock at release includes the transient during which the amplitude is larger and the period slightly longer. Timing 20 swings averages this in, inflating T and therefore T^2 for each length. Since g = 4π^2L/T^2, an inflated T^2 lowers the extracted g - consistent with the low value of 9.58.
Finite bob size. A large sphere on a long thread is not a point mass; the effective length is to the centre of mass, and the moment of inertia adds a correction. The physical pendulum period is T = 2π sqrt(I/(mgd)), which for a sphere of radius r gives a slightly longer period than the point-mass formula. Using thread length only (not centre of mass) would systematically mis-scale L. If she measured to the top of the bob, L is underestimated, which also lowers g.
Amplitude. Large-amplitude swings lengthen the period by a factor (1 + θ0^2/16). At 10° this is about 0.2%, small but in the same direction.
All three biases push g low, which explains the sign of the discrepancy rather than random scatter.
3. Recommend a concrete revision and state what evidence would justify accepting the corrected result.
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Revision plan.
- Measure L to the centre of the bob and record bob radius so the finite-size correction can be applied or shown negligible.
- Release and allow several swings to settle, then time from a marked equilibrium crossing to avoid reaction-time bias at the extremes.
- Keep amplitude below about 5° and record it, applying the amplitude correction if needed.
- Time more oscillations (e.g. 50) to reduce the stopwatch reaction-time contribution, and have the same person start and stop.
- Repeat each length and use the scatter to check the quoted gradient uncertainty.
Acceptance criterion. After correction, recompute g and its uncertainty. The corrected value should be quoted as g ± Δg, and the result is acceptable if it lies within about 2σ of 9.79 m/s^2. Agreement with the spring-mass value (9.6 ± 0.3) within combined uncertainty would be supporting but not decisive evidence, since both could share a calibration error. The key test is whether the corrected pendulum value moves toward 9.79 and whether the residual is consistent with the stated uncertainty.