A pendulum that will not keep time: separating amplitude error from length error
A Class 12 student is asked to measure the acceleration due to gravity with a simple pendulum and to state a realistic uncertainty. She clamps a fine steel wire between the jaws of a retort stand, hangs a steel sphere of diameter 2.00 cm on it, and uses a metre scale (least count 1 mm) to measure 1.000 m from the clamp jaws to the top of the sphere. A stopwatch is started as the bob crosses its mean position: 20 oscillations take 40.36 s at a swing of about 4 degrees and 40.53 s at a swing of about 15 degrees, each reproducible to plus or minus 0.05 s. Using the 15 degree timing and L = 1.000 m she reports g = 9.61 m/s^2, argues that the wire must stretch at large amplitude, and quotes plus or minus 0.3 percent. The accepted local value is 9.79 m/s^2. Her teacher replies that the two timings are consistent, that no stretching is needed, and that almost all of the remaining error is in the length, not the timing.
1. Show with numbers that the two timings are consistent with one another once the finite amplitude of the swing is accounted for. Is the stretching explanation needed?
Read the worked answer
For a simple pendulum the finite-amplitude correction is
T = T0 (1 + theta0^2/16), theta0 in radians.
- 15 degrees: theta0 = 0.2618 rad, theta0^2/16 = 0.0685/16 = 4.28e-3, so T0 = 2.0265/1.00428 = 2.0179 s (20 oscillations = 40.357 s).
- 4 degrees: theta0 = 0.0698 rad, correction factor 1.00030, so T0 = 2.0180/1.00030 = 2.0174 s (40.348 s).
The two corrected periods differ by 0.0005 s, i.e. 0.01 s over 20 oscillations, far below the plus or minus 0.05 s reading limit. The apparent 0.8 percent difference in the raw timings is entirely the standard amplitude correction; the period at small amplitude is independent of amplitude to first order, so there is no evidence for stretching, and a steel wire under a 100 g bob would stretch by well under a micrometre in any case. The stretching hypothesis fails on two counts: it is quantitatively unnecessary and physically negligible.
2. After the amplitude correction both runs give the same period, yet g is still about 1 percent low. Identify the dominant systematic, show that it accounts for the discrepancy, and give the corrected value.
Read the worked answer
Since g = 4pi^2L/T0^2, g scales linearly with L. A 1 percent deficit in g needs about a 1 percent (10 mm) overestimate of L, which is exactly the bob radius: 1.00 cm out of 1.000 m.
Effective length = distance from pivot to the centre of mass of the bob = 1.000 + 0.010 = 1.010 m.
Using T0 = 2.0179 s:
g = 4*pi^2 * 1.010 / (2.0179)^2 = 39.478 x 1.010 / 4.0719 = 9.79 m/s^2.
This coincides with the accepted value, so the whole residual discrepancy is the definition of the length. The timing contributes only about 0.05/40 = 0.13 percent in T (0.25 percent in g), and the small-amplitude correction is negligible. Note that if the wire mass were not negligible the effective length would be longer still, so the heavy-bob assumption should be checked separately.
3. Propose two concrete design changes that would let her quote plus or minus 0.5 percent with confidence, and say which error source each one attacks.
Read the worked answer
Budget first: delta(g)/g = 2*delta(T)/T + delta(L)/L.
Fix the length definition. Measure to the centre of the bob (vernier calipers on the sphere plus the wire length, or mark the wire and use a travelling microscope). This removes the 1 percent bias, which is 2-3 times larger than the random error, and is the change that matters most.
Reduce the relative timing error by timing 50 oscillations instead of 20 and starting/stopping at the same fiducial (a mark on the bob passing a reference line), or by using a light gate. This cuts the random contribution to about 0.05/100 = 0.05 percent in T, i.e. 0.1 percent in g, and averages out reaction-time bias.
Use a longer thread (say 2 m). The same 1 mm absolute error in L then costs only 0.05 percent, and the amplitude correction stays smaller because the angular amplitude for the same arc is reduced.
Keep the amplitude below about 5 degrees so the correction is under 0.05 percent, and repeat the whole measurement several times so the quoted uncertainty is a standard error rather than a guess.