Two Students, One Pendulum, and a Value of g That Is Too Low
Two students measure g with a simple pendulum. They hang a 100 g brass bob from a clamp stand on thin string and measure the length from the point where the string leaves the clamp to the centre of the bob with a metre rule: L = 1.000 m. They set the bob swinging with an amplitude of about 15° and time 20 complete oscillations three times, starting and stopping the stopwatch as the bob passes through the lowest point: 40.2 s, 40.6 s and 40.0 s. Using the mean period and T = 2π√(L/g), they obtain 9.74 m/s². Their teacher gives the accepted value as 9.81 m/s² and asks them to account for the discrepancy. The students' reaction times are about 0.2 s each, the metre rule is marked in millimetres, and in one unchecked measurement they took the length to the top of the bob rather than its centre.
1. Reproduce the calculation, estimate the uncertainty in g from the timing spread, and decide whether the discrepancy with 9.81 m/s² is significant.
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Mean time: (40.2 + 40.6 + 40.0) / 3 = 40.267 s for 20 oscillations, so T = 40.267 / 20 = 2.013 s.
Value of g: g = 4π²L / T² = (39.478 × 1.000) / (2.013)² = 39.478 / 4.053 = 9.74 m/s².
Uncertainty from the timing spread. The three times span 0.6 s; taking half the range as a crude standard uncertainty gives 0.3 s on 20 oscillations, i.e. 0.015 s on T. Since g ∝ T⁻², the fractional uncertainty is twice that of T: 2 × (0.015 / 2.013) ≈ 1.5%, so ±0.15 m/s².
Verdict. The result is 9.74 ± 0.15 m/s². The accepted value 9.81 lies well inside one standard uncertainty (the gap is 0.07). Statistically, there is no discrepancy to explain. The teacher's question is therefore a trap: it invites the students to invent a systematic error for a difference that the random scatter already accounts for. The honest answer is that the precision is too poor to distinguish the two values.
2. Evaluate three candidate explanations - the 15° amplitude, reaction time, and measuring to the top of the bob - and say which is consistent in sign and magnitude with a result that is too low.
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1. Finite amplitude (accounts for the sign and the size). The formula used is the small-angle approximation. The exact period is larger: T ≈ T₀(1 + θ₀²/16) with θ₀ in radians. Here θ₀ = 15° = 0.262 rad, so θ₀²/16 = 0.0043, a 0.43% lengthening of the period. Using a period that is 0.43% too long in g ∝ T⁻² makes g too small by about 0.85%, i.e. 0.08 m/s². Corrected: 9.74 / 0.9915 ≈ 9.82 m/s² - essentially the accepted value. This is the explanation.
2. Reaction time (wrong sign or negligible). The watch is started and stopped at the same point in the cycle, so a consistent lag cancels between start and stop. Random variation in reaction time widens the spread, which is already in the uncertainty, but it does not systematically depress g.
3. Length to the top of the bob (wrong sign). Taking the length to the top overstates L by about the bob's radius, ~1 cm, i.e. 1%. That would make g too high (about 9.84), not too low. It also refers to a measurement not used in the calculation.
Only the amplitude effect is both the right sign and the right size.
3. Recommend one change that would remove the systematic error you identified and one that would reduce the random uncertainty, explaining the reasoning behind each.
Read the worked answer
Remove the systematic error - reduce the amplitude and/or apply the correction. Keep the amplitude below about 5°, where θ₀²/16 < 0.05%, or measure the amplitude and correct T using T₀ = T / (1 + θ₀²/16). Reducing the amplitude alone is not free: at very small amplitudes the bob is more affected by air resistance and the timing becomes harder, so about 5° is a sensible compromise. The point is that a systematic error cannot be averaged away by repeating the experiment at the same amplitude - it must be changed or corrected.
Reduce random uncertainty - time more oscillations and automate the timing. Timing 50 oscillations rather than 20 divides the timing uncertainty per period by 2.5. Better still, use a light gate at the equilibrium position: it removes reaction time entirely and triggers at a well-defined phase, giving both a smaller spread and no operator-dependent phase error. Repeating and averaging then genuinely improves precision.
Bonus improvement. Measure L to the centre of the bob with vernier callipers or measure the bob diameter and correct for it, and take several readings of L at different points around the bob to check that the string is vertical. Also check that the clamp stand does not flex, since a moving pivot changes the effective length.