Measuring g with a simple pendulum
PhysicsMeasurement & UncertaintyAges 15–16
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Sign in to playTime 1, 10 or 20 oscillations of a simple pendulum with a stopwatch, either by hand or with a simulated experimenter whose reaction time varies randomly. Each trial gives T and T² with the uncertainty caused by reaction time; the graph of T² against L gets a best-fit line, and its gradient k gives g = 4π²/k with an uncertainty. Increase the angular amplitude to see a systematic error.
Lesson: Required practical: measuring g with a pendulum – measurement uncertainty
What it shows
For small swings the period of a simple pendulum is T = 2π√(L/g), so T² = (4π²/g)·L: a graph of T² against L is a straight line through the origin with gradient 4π²/g. Timing by hand adds a random error at the start and at the stop of the stopwatch, and that error is the same whether you time one swing or twenty, so timing many oscillations and dividing makes the period far more precise. Plotting several lengths and using the gradient averages out random scatter. Large amplitudes lengthen the period slightly, a systematic error that repetition cannot remove.
How to use
Set the Length L and choose how many oscillations to Time. Press Start timing as the bob passes the marker, count the passes and press Stop timing after N oscillations, or press Auto experimenter to collect trials quickly. Repeat at several lengths until the best-fit line and g appear. Compare 1, 10 and 20 oscillations, raise Reaction σ, then tick Compare with true g and try a large Amplitude.
Parameters you can change
- String length 0.2–2 m
- Oscillations per timing 1 oscillation, 10 oscillations, 20 oscillations
- Reaction-time spread (σ) 0–0.3 s
- Angular amplitude 5–40 degrees
- True gravitational acceleration (hidden from students) 1.6–25 m/s²
Questions to explore
- Why does timing 20 oscillations and dividing by 20 give a more precise period than timing one?
- Why is it better to find g from the gradient of T² against L than from a single measurement?
- With a 40° amplitude, is the measured g larger or smaller than the true value, and why?
