Designing a Fair Test for Pendulum Period
A student investigates whether pendulum period depends on amplitude. They have a 1.00 m string, brass and plastic bobs of similar size but different mass, a protractor, a stopwatch, and a retort stand. They plan to release the bob from 5°, 15°, 30°, and 45° and time 10 oscillations each, using the brass bob throughout. Their teacher asks them to justify the choice of variables, control the confounders, and decide whether their planned data will let them distinguish the small-angle approximation from the exact behaviour.
1. Identify the independent, dependent, and controlled variables and explain one control that is essential for a fair test.
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Independent variable: release angle (amplitude). Dependent variable: period, obtained by timing 10 oscillations and dividing by 10. Controlled variables: string length (1.00 m), bob mass and type, release method (no push), and the same pivot. An essential control is string length, because period depends strongly on length; if length changed between trials, any angle effect would be confounded. Using the same bob throughout also controls mass. Releasing from rest without a push ensures the amplitude is exactly the protractor angle.
2. The student uses only the brass bob. Explain why this is acceptable for this investigation and when it would become a flaw.
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The research question concerns amplitude, not mass, so holding mass constant is correct control. Using one bob removes mass as a confounder. It would become a flaw only if the student wanted to generalise the amplitude result to all bobs, or if they claimed mass independence without testing it. Since the small-angle approximation predicts period independent of mass, a single bob cannot test that prediction, but it can still test the amplitude prediction. The design is fit for the stated aim but not for a broader claim.
3. Predict whether the planned angles will distinguish the small-angle model from exact behaviour, and justify.
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The small-angle approximation gives T = 2π√(L/g) = 2π√(1.00/9.8) = 2.01 s, independent of angle. The exact period grows with amplitude: at 5° the correction is negligible (~0.05%), at 15° about 0.4%, at 30° about 1.7%, at 45° about 4%. Timing 10 oscillations at 45° gives about 20.9 s versus 20.1 s, a 0.8 s difference, which a stopwatch can resolve. At 5° and 15° the differences (0.01–0.08 s over 10 swings) are within human reaction time (~0.2 s), so those points cannot distinguish the models. The design works only at the larger angles, and only if timing uncertainty is reduced by more oscillations.