Is a hanging chain a parabola? Catenary versus quadratic models

Why it makes a good exploration

Galileo thought a hanging chain formed a parabola; it does not, but the two curves are so close that the difference is surprisingly hard to see. Measuring a real chain and quantifying the gap is a lovely way to test a famous claim with your own data.

The mathematics you'll need

• Quadratic models through three points and by least squares (SL)
• Exponential functions and the definition cosh x = (eˣ + e⁻ˣ)/2 — new, explain it
• Sum of squared residuals to compare models
• Derivatives to find the angle the chain makes at its supports
• HL: arc length by integration, or solving for the parameter numerically

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

Hang a light chain or necklace against a grid (or photograph it square-on with a ruler in shot), read 15–25 points off the photo in GeoGebra, and repeat for two different span-to-sag ratios.

• GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.
• Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

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. Photograph the chain square-on and set up axes with the lowest point at the origin; state your scale.
2. Fit a parabola by least squares; record the residuals.
3. Introduce the catenary, explain where cosh comes from, and choose a by trial or technology.
4. Compare the models with residual plots and sums of squares; repeat for a deeper sag where the curves diverge more.
5. Discuss where the difference matters (arches, cables) and where it does not.

Showing personal engagement

• Predict before measuring which model will win and by how much.
• Link it to a structure you know (a local bridge, the Gateway Arch, a washing line) and check it.
• Explain why the difference grows as the chain sags more — and test that prediction.

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

Taking it further

HL students can derive the catenary from a force balance (a differential equation) or compute the chain's length by integration and compare it with the measured length.