Two Measurements of g: Precision Without Accuracy
In a first-year physics lab, two groups determine g. Group A uses a simple pendulum with a clamp-to-bob-center length of 0.950 m and times 20 complete oscillations in 40.0 s. Group B drops a steel ball from rest through 1.50 m and records photogate times of 0.552 s, 0.548 s, 0.560 s, and 0.556 s. The accepted local value is 9.79 m/s². Group A reports 9.37 m/s² and notes that repeated 20-swing timings varied by only ±0.2 s. Group B reports 9.78 m/s². A classmate argues that Group B is better because it agrees with the accepted value; another says Group A is more trustworthy because pendulums are less affected by air resistance. The teacher asks the class to separate precision from accuracy and to decide whether each discrepancy is random or systematic. The pendulum bob is 2.0 cm in diameter, and the string is tied to a clamp at the top.
1. Calculate g from each group's data. For Group A, what effective pendulum length would be required to produce the accepted value at the measured period? What does that imply about the likely error?
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Group A: T = 40.0/20 = 2.00 s, so g = 4π²L/T² = 4π²(0.950)/(2.00)² ≈ 9.38 m/s². Group B: mean t = (0.552 + 0.548 + 0.560 + 0.556)/4 = 0.554 s, so g = 2(1.50)/(0.554)² ≈ 9.78 m/s². For Group A to get 9.79 m/s² at T = 2.00 s, the effective length would need to be L = gT²/(4π²) = 9.79(4.00)/(39.48) ≈ 0.992 m, about 4.2 cm longer than the stated 0.950 m. The bob radius is only 1.0 cm, so measuring to the bob center cannot explain the whole difference. A likely systematic error is that the actual pivot is above the clamp point, the string stretches, or the length was not measured from the true pivot to the bob center. That length error makes the calculated g too low even though the timing is precise.
2. Design a test that would distinguish a systematic length error from random timing error in Group A's pendulum experiment. Include the measurements and the pattern you would expect.
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Repeat the experiment at five or more lengths, measuring each length with a plumb line from the actual pivot to the bob center and timing at least 20 swings per trial. Plot T² versus L. For a simple pendulum, T² = (4π²/g)L, so the points should lie on a straight line with slope 4π²/g. If the points are linear but the slope consistently gives g ≈ 9.38, the error is systematic-most likely an effective-length offset or a timing calibration issue. If the points scatter randomly about a line whose slope gives the accepted g, the error is random. A nonzero intercept on the T² axis is direct evidence of a length offset. Also vary the bob mass: if g is unchanged, air resistance and mass effects are not the explanation.
3. Group B's four times scatter. Estimate the standard deviation of the mean time and the resulting uncertainty in g. Is Group B's result statistically consistent with 9.79 m/s²? What does agreement with the accepted value not prove?
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The mean is 0.554 s. Deviations are −0.002, −0.006, +0.006, and +0.002 s; the sample standard deviation is sqrt[(4 + 36 + 36 + 4)×10⁻⁶/3] ≈ 0.00516 s, and the standard error of the mean is about 0.00258 s. Since g = 2h/t², δg/g ≈ 2δt/t = 2(0.00258)/0.554 ≈ 0.0093, so δg ≈ 0.09 m/s². Group B therefore reports 9.78 ± 0.09 m/s², consistent with 9.79. Agreement with the accepted value does not prove the method is unbiased: a systematic error could be hidden, the uncertainty could be underestimated, or the accepted value could share the same calibration. Independent checks are still needed.