Timing a Pendulum to Determine g
Three learners in a school laboratory determine the acceleration of free fall using a simple pendulum. They suspend a steel bob of diameter 2.4 cm from a clamp stand and measure the length from the pivot to the centre of the bob as 0.850 m with a metre rule whose smallest division is 1 mm. Using a hand-held stopwatch, they time 20 complete oscillations three times and record 36.8 s, 37.4 s and 36.5 s. The amplitude is about 10 degrees. They calculate T = 1.85 s and g = 9.86 m/s^2, then write in their report: 'Our value is accurate because it is close to 9.8 m/s^2.' The teacher asks them to justify the uncertainty statement, to decide which measurement limits the experiment, and to consider whether the 10 degree release angle matters.
1. Calculate the percentage uncertainty in the period arising from the timing spread and compare it with the percentage uncertainty in the length. Which dominates?
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Mean time = (36.8 + 37.4 + 36.5)/3 = 36.9 s, so T = 1.845 s. The spread is 0.9 s; a simple half-range estimate is 0.45 s, or 0.0225 s per oscillation, giving 0.0225/1.845 = 1.2%. The length is 0.850 m, and even allowing plus or minus 2 mm for locating the centre of the bob gives 0.002/0.850 = 0.24%; the metre rule's 1 mm division is smaller still. Timing dominates by about a factor of five. Reaction time when starting and stopping the watch is also a timing effect, although timing 20 swings reduces its relative size.
2. The measured g is 9.86 m/s^2. Is the result precise? Is it accurate? Explain what the spread and the comparison with 9.81 m/s^2 tell you.
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Precision is indicated by the spread: about 1.2% in T, which propagates to about 2.4% in g because g is proportional to T^-2, so g = 9.86 plus or minus 0.24 m/s^2. Accuracy requires comparison with the accepted 9.81 m/s^2: the difference is 0.05, only about 0.2 of the stated uncertainty, so it is consistent. However, 'close to 9.8' is not a valid accuracy argument by itself. A finite amplitude of 10 degrees increases T by about theta^2/16 = 0.19%, lowering g by 0.38% (about 0.04 m/s^2), which is small. A larger systematic error would arise if the length were measured to the bottom of the bob rather than its centre; the report should state the measurement definition.
3. They propose reducing the timing uncertainty by timing only 5 oscillations but taking 10 repeats. Evaluate this plan.
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Timing 5 oscillations gives about 9.2 s per run, so a 0.2 s reaction error is about 2.2% per run. Ten repeats would reduce the random component by a factor of sqrt(10) to about 0.7%, but any systematic reaction-time bias remains, and the total timing work is similar (50 oscillations versus 60). A better plan is to keep at least 20 oscillations per run, use a fiducial mark at the equilibrium position so the eye judges the same point, and have the same learner operate the watch. Repeating the whole procedure and averaging periods is preferable to shortening each run.