Does a pendulum's period really depend on the square root of its length?

Why it makes a good exploration

The textbook formula T = 2π√(L/g) assumes small swings. Testing the exponent with a log-log plot and then pushing the angle until the model fails gives two linked questions with clear answers.

The mathematics you'll need

• Power models y = axⁿ
• Log-log linearisation and regression
• Percentage error and uncertainty
• Estimating g from the fitted constant

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Build a pendulum with 8+ lengths, time 10 swings three times each; use phyphox's pendulum or acceleration tools for precise timing.

• phyphox (RWTH Aachen) — Free app that turns your phone's accelerometer, microphone, barometer and gyroscope into data loggers with CSV export.
• Tracker video analysis — Free tool to track an object frame by frame in a video and export its x–y coordinates.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

1. Explain the model and its small-angle assumption.
2. Collect periods for a range of lengths.
3. Plot ln T against ln L; fit a line and interpret the gradient and intercept.
4. Estimate g and compare with 9.81 m s⁻².
5. Repeat at larger angles and find where the error exceeds 1%.

Showing personal engagement

• Predict the exponent before you collect data.
• Use a playground swing as a real-world test of your model.
• Explain the large-angle correction you observe.

See Criterion C: personal engagement for what examiners look for.

Taking it further

HL: compare the measured large-angle periods with the series correction T ≈ 2π√(L/g)(1 + θ₀²/16).