What it does
The simple pendulum is the classic way to measure the strength of gravity in a school or university lab. For small swings its period is T = 2π√(L/g), so a graph of T² against L is a straight line through the origin with slope 4π²/g. This analyser fits that line to your measurements by least squares and gives g with its standard uncertainty and a 95 % interval, the residuals, and how your value compares with the one you expect.
It also handles the two things that most often spoil a pendulum result:
- Large swings. The small-angle formula is only approximate: at 15° the true period is 0.43 % longer, at 30° 1.74 %. If you give the amplitude, each period is corrected with the exact formula (the complete elliptic integral) before fitting.
- An unknown length offset. If you measured to the top of the bob, or the pivot point is uncertain, choose the fit with an extra length δ. It fits T² = 4π²(L + δ)/g, so the offset no longer biases g, and it tells you δ.
How to use it
- Paste or type one line per length: length, period and optionally the amplitude in degrees. A header line is fine, and you can open a CSV or Excel file.
- Choose the length unit.
- If you timed several swings at once (much more accurate), type how many in full swings and paste the total times.
- Give an amplitude for rows that have none, or 0 for no correction.
- Choose the fit: through the origin, or with the unknown extra length.
- The expected g is only for comparison (about 9.78 at the equator to 9.83 at the poles, rounded).
The second card is a large-angle calculator: for any length and amplitude it shows the small-angle period, the exact period and the series approximation, and a table of the error at amplitudes from 2° to 120°.
Worked example
The made-up example has six lengths from 0.2 m to 1.2 m, each swung at 15°. Fitting through the origin, the page shows g = 9.825 ± 0.017 m/s² (95 % interval 9.7819 to 9.8685), R² = 0.999924 and an rms period scatter of 5.3 ms. It is +0.15 % from 9.81, within two uncertainties. Without the amplitude correction the same data give 9.7411: the 15° swing alone moves g by 0.86 %. Switching to the fit with an extra length gives g = 9.827 ± 0.043 m/s² and δ = 0.157 ± 3.4 mm, consistent with zero, so the lengths were measured correctly. In the correction card, a 1 m pendulum at 30° has a small-angle period of 2.0061 s and an exact period of 2.0410 s, 1.74 % longer.
Tips
- Time 10 or 20 swings and start counting from “zero” as the bob passes the lowest point, where it moves fastest and the moment is sharpest.
- Measure the length to the centre of the bob. If you cannot, use the offset fit.
- Keep the amplitude below about 10°, or record it, and use a dense, small bob so air drag is small.
- If the residuals curve rather than scatter, look for a length-dependent error such as a stretchy string.
FAQ
Why fit T² and not T? T² is proportional to L, so the graph is a straight line whose slope gives g directly.
How big is the large-angle effect? About θ₀²/16 (θ₀ in radians): 0.05 % at 5°, 0.19 % at 10°, 1.7 % at 30° and 18 % at 90°.
What about a real (physical) pendulum? A large bob or a heavy rod changes the effective length. For a sphere of radius r on a light string, the length becomes L + 2r²/5L; this tool treats the bob as a point.
Where the numbers come from
- Simple pendulum period, the exact large-amplitude period 4√(L/g) K(sin(θ₀/2)) and its series 1 + θ₀²/16 + 11θ₀⁴/3072: standard classical mechanics results. Complete elliptic integral by the arithmetic-geometric mean (Abramowitz and Stegun, 17.6).
- Least squares with and without an intercept, standard errors and Student t intervals: standard statistics.
- Checked against SciPy (ellipk, linregress, Student t quantiles), NumPy least squares with the full covariance matrix, and a SymPy series expansion.
- The example measurements are made up.
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Last updated 2026-11-09.