Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

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

AA SLAI SLAA HLAI HL Accessible Also in: Statistics

Research question

Is the relationship between a pendulum's length and its period a power law with exponent ½, and at what swing angle does the simple model stop working?

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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

Modelling the data, step by step

This idea usually needs a power model. See it worked step by step, with a criterion tip at every step: Power y = a·xⁿ (log–log).

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for 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.

A possible outline

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

Pitfalls that cost marks

Timing single swings (reaction-time error dominates).
Measuring length to the top of the bob instead of its centre.
Reporting g without an uncertainty.

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.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Power models y = axⁿ; Log-log linearisation and regression
AA HL	Fits, but add an HL technique	Power models y = axⁿ; Log-log linearisation and regression
AI SL	Good fit	Power models y = axⁿ; Log-log linearisation and regression
AI HL	Fits, but add an HL technique	Power models y = axⁿ; Log-log linearisation and regression

Level: Accessible. A good first extended piece of maths, with room to go deeper. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: timing single swings (reaction-time error dominates); and measuring length to the top of the bob instead of its centre. Say how each one affects your answer.

Use of mathematics (E)

SL: power models y = axⁿ; log-log linearisation and regression; percentage error and uncertainty; estimating g from the fitted constant, each explained in your own words and carried out correctly.

How modelling explorations reach the top bands in general: Modelling with functions IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

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

Extending it for HL

Add an HL technique the question genuinely needs: what an HL modelling exploration adds.

See a complete IA, marked

Should I lease or buy my first car? (AI SL) is a complete modelling exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AI SL, AA HL and AI HL. We rate it Accessible: a good first extended piece of maths, with room to go deeper.

What mathematics does this IA use?

Power models y = axⁿ, log-log linearisation and regression, percentage error and uncertainty and estimating g from the fitted constant.

Where can I get data for this IA?

Build a pendulum with 8+ lengths, time 10 swings three times each; use phyphox's pendulum or acceleration tools for precise timing. Checked sources and tools: phyphox (RWTH Aachen) and Tracker video analysis.

Turn this idea into your IA

The planner opens with “Does a pendulum's period really depend on the square root of its length?” filled in: the question, the maths and the data sources.

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