The 15-Degree Pendulum and the 'Weak Gravity' Claim
During a unit on measurement and uncertainty, a student team is asked to determine g with a simple pendulum and then to defend or retract its result. The bob is a steel sphere 2.54 cm in diameter; the string, measured with a meter stick from the pivot to the top of the sphere, is 0.837 m, so the team records L = 0.850 m to the center. They release the bob from a protractor reading of 15 degrees and start a stopwatch as it passes the equilibrium point, timing 20 complete oscillations. Five trials give 37.12 s, 37.05 s, 37.20 s, 37.09 s and 37.15 s. The stopwatch resolves to 0.01 s. Using T = mean time / 20 and g = 4π²L/T², the team computes 9.74 m/s² and writes in its report that gravity in the building appears to be about 0.7% weaker than accepted, probably because the lab is on the second floor. The instructor asks the team to support that claim with evidence or withdraw it.
1. Compute g from the team's data, then decide whether the trial-to-trial scatter can account for the gap between 9.74 and 9.81 m/s².
Read the worked answer
Mean time = (37.12 + 37.05 + 37.20 + 37.09 + 37.15)/5 = 185.61/5 = 37.122 s, so T = 37.122/20 = 1.8561 s. Then g = 4π²L/T² = 39.48 × 0.850/3.445 = 9.74 m/s², about 0.71% below 9.81.
Does scatter explain it? The five times have a standard deviation of about 0.057 s; the standard error of the mean is 0.057/√5 ≈ 0.026 s, which is 0.0013 s per oscillation, or 0.07% in T. Because g ∝ 1/T², that is only 0.14% in g, about ±0.014 m/s². Stopwatch resolution is smaller still (0.01 s in 37 s ≈ 0.03%). The observed deficit of 0.07 m/s² is roughly five standard errors, so random scatter does not explain it. Something shared by every trial is biasing the result. That does not mean gravity is anomalous: a systematic effect inside the apparatus or the procedure is far more likely, and the team must identify it before drawing any conclusion about the building.
2. The team argues that 15 degrees is a small angle. Test that assumption quantitatively and state what it implies for the reported value.
Read the worked answer
Fifteen degrees is not small for a pendulum. In radians, θ₀ = 15 × π/180 = 0.262, and the finite-amplitude result is T ≈ T₀(1 + θ₀²/16). Here θ₀²/16 = 0.0685/16 = 0.00428, so the measured period is 0.43% longer than the small-angle period.
Since g is computed from 1/T², a 0.43% period error becomes a 0.86% error in g, in the direction that makes g too small: g_corrected = 9.74 × (1.00428)² = 9.82 m/s². That single correction removes the entire reported deficit and slightly overshoots, which is expected because the correction is only first order in θ₀². The honest conclusion is that the team measured 9.74 m/s² because it released the bob at 15 degrees, not because gravity is weaker on the second floor. The two remaining suspects are timing bias (a student who consistently starts the watch late) and the effective length, which must be measured to the bob's center of mass and can change if the string stretches under load.
3. Recommend the changes that would most improve this experiment, and describe a test that separates a systematic timing bias from ordinary random error.
Read the worked answer
Replace the stopwatch with a photogate at the equilibrium point. Reaction time is a bias, not noise: a student who starts the watch a fixed fraction of a second after release adds the same offset to all five trials, so averaging never removes it. The photogate removes it entirely and reduces timing uncertainty by an order of magnitude.
Then repeat at release angles of 5, 10, 15 and 20 degrees and plot T against θ₀². Finite-amplitude theory predicts a straight line with a positive slope and an intercept T₀. If the intercept returns g = 9.81 within uncertainty, the amplitude effect is confirmed and the timing is clean. If the intercept is displaced, the cause is a bias: swap which student starts and stops the timer, or run stopwatch and photogate simultaneously on the same swings. A consistent difference of a few tenths of a second across all trials is bias; a difference that changes sign from trial to trial is noise. Measure L to the bob center with calipers and report g with a combined uncertainty from the fit rather than from one run.