Two Pendulum Values for g: Whose Number Goes in the Class Table?
Two lab groups in the same classroom measure g with a simple pendulum during a 50-minute period. Group A hangs a brass bob on a string they measure as 1.000 m, releases it at about 5 degrees, and times 20 oscillations with a photogate, obtaining a period of 2.003 s. Group B uses the same support but a lighter aluminum bob, releases at 25 degrees because they want 'visible' swings, and times 10 oscillations with a stopwatch, reading 19.9 s total. The room's ventilation duct blows toward Group B's apparatus. Group A reports g = 9.84 m/s^2 with an estimated uncertainty of +/-0.02; Group B reports 9.97 +/- 0.06. The accepted local value is 9.81 m/s^2. In the post-lab discussion Group B argues their value is better because the bob was lighter and the stopwatch was 'read to hundredths,' and the teacher asks the class to decide which value belongs in the shared class data table and with what uncertainty.
1. Propagate Group B's timing uncertainty properly and compare it with Group A's. Is the 0.13 m/s^2 difference significant? For each error source (reaction time, 25-degree amplitude, ventilation draft, lighter bob), state whether it can explain Group B's value, taking the direction of the effect into account.
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Reaction time. Two human events, start and stop, contribute about +/-0.15 s each, so about +/-0.21 s on a 19.9 s total. That is a 1.06% uncertainty in T. Since g = 4(pi)^2 L / T^2, the relative uncertainty in g is twice that, about 2.1%, or +/-0.21 m/s^2. Group B's quoted +/-0.06 is understated by roughly a factor of three; they quoted the resolution of the display, not the uncertainty of the measurement.
Significance. The A-B difference is 0.13 m/s^2 against a combined uncertainty of sqrt(0.21^2 + 0.02^2) = 0.21 m/s^2. That is about 0.6 sigma, so the two results are statistically consistent once uncertainty is handled honestly.
Direction matters. For 25 degrees, theta = 0.436 rad and the finite-amplitude correction is (1 + theta^2/16) = 1.0119. The real period is 1.2% longer than the small-angle period, so the small-angle formula underestimates g by about 2.4%: corrected, Group B's value moves from 9.97 to about 10.21 m/s^2. That is the wrong direction to explain why B is already high, so amplitude cannot be the culprit; it actually deepens the discrepancy.
Ventilation and bob mass. Air movement adds damping, which lengthens the period only in second order and cannot produce 1.6%. The pendulum period is independent of bob mass, so a lighter bob is not 'more precise'; if anything, a light bob damps faster and may swing less cleanly.
2. What should be entered in the shared class table, and how should it be labelled? Evaluate the proposal to average 9.84 and 9.97 into a single class value.
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Averaging treats the two numbers as equally credible measurements of the same quantity, and neither condition holds.
First, the uncertainties differ by an order of magnitude. An inverse-variance weighted mean gives Group A a weight of 1/0.02^2 = 2500 and Group B a weight of 1/0.21^2 = 23. The weighted mean is (9.84 x 2500 + 9.97 x 23) / 2523, which is 9.84 m/s^2. The average is dominated entirely by Group A, so 'averaging' changes nothing while giving the false impression that two independent results were combined.
Second, Group B has an identified systematic (finite amplitude) that must be corrected before any combination, and correcting it moves the value further from the class consensus. Systematic errors do not cancel by averaging; random errors do.
Defensible table entry: Group A, g = 9.84 +/- 0.02 m/s^2, photogate, 20 oscillations, amplitude about 5 degrees. Group B, g = 9.97 +/- 0.21 m/s^2 raw, flagged as measured at 25 degrees with a stopwatch and therefore carrying a known amplitude bias plus an unquantified counting risk. A teacher should also note that Group A's value is 1.5 sigma above 9.81, which is acceptable but worth mentioning rather than hiding.
3. The teacher allows each group one repeat run. Design the repeat so that the class can distinguish a timing/counting error from a genuine difference in effective length, and state what outcome would support each explanation.
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The two hypotheses make different predictions about the same apparatus, so the decisive move is to hold the apparatus fixed and change only the measurement method.
Run design. Group B repeats on their own support, with their own bob and string, but times 20 oscillations with the photogate placed at the equilibrium point, released from 5 degrees. In parallel, a third student re-measures the pivot-to-bob-centre length with a steel tape, sighting squarely, because the effective length is the pivot-to-centre-of-mass distance, not the string length. The repeat is cheap and removes both the reaction-time and amplitude variables at once.
Outcome supporting a timing/counting error. If the photogate gives T about 2.006 s and g about 9.81 m/s^2 on the same apparatus, then B's 1.99 s was an artefact of starting or stopping the watch part-way through a swing, or of a fencepost miscount such as timing 9 intervals but dividing by 10. A 0.5% error in T is enough; that is only a tenth of one period, which matches the precision of hand timing.
Outcome supporting a length difference. If the photogate reproduces T about 1.99 s, then L must be genuinely short. From L = gT^2/(4 pi^2) with T = 1.99 s, L = 0.98 m, about 20 mm less than the 1.000 m recorded. That is exactly the size of error produced by measuring the string alone and forgetting the bob radius, or by a knot that slipped. The steel-tape check then either confirms or rules this out.