Maths IA · Calculus · topic 29 of 97
Hanging chains: catenary or parabola? · IB Maths IA idea: guide, sample graph and step-by-step plan
looks alike: but how alike?A hanging chain looks like a parabola, and Galileo thought it was one. It is not. You will measure how big the difference really is, and when it matters.
- ✓ you like fitting a curve to a photo you took yourself
- ✓ you want a reason to use a Maclaurin series that you can check
- ✓ you are curious about a mistake Galileo made
The big idea
A chain hanging under its own weight takes the shape y = a cosh(x/a), which is called a catenary. A parabola with the same span and sag looks almost the same. How close are they? The Maclaurin series of cosh is a polynomial that starts with the parabola, so the first correction is a fourth-power term. That lets you estimate the gap before you compute it. The comparison is the heart of the IA: fit both curves to a photographed chain, measure the largest gap against the sag-to-span ratio, and decide which model the data can tell apart. The marks come from showing where the parabola is good enough, and where it is not, with numbers.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
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Photograph a hanging chain or cable against a grid, read off coordinates and fit y = a cosh(x/a) and a parabola.
→ You compare two models on real data, with residuals. A good evaluation asks whether the data can tell them apart at all.Write cosh as a Maclaurin series and keep two or three terms, then compare the truncated series with the exact curve.
→ You find the first term that separates the catenary from a parabola and use it to estimate the gap for any sag.Fix the span and change the sag, then measure the largest vertical gap as a percentage of the sag.
→ A graph of the gap against one ratio. You can say how deep a sag must be before the parabola is a poor model.Compare the arc length of the catenary and the parabola for the same span and sag, using integration.
→ An arc-length integral for each, one with an exact answer. You explain how the lengths compare and why it matters to a builder.Compare a free-hanging chain with a suspension cable that carries a level deck, which loads the cable evenly across the span.
→ A different loading gives a different curve. You can say why bridge cables are close to parabolic and a chain is not.Hang the chain between supports at different heights and fit the curve.
→ The lowest point is no longer in the middle, so you need two parameters and a shifted curve.Your Maths IA research question: run the check
same span, same sag!Write your own question first. Then make sure it ticks every box:
- Names the chain or cable, the span and the sag, and how they were measured.
- Names the two models, the catenary and the parabola, and says how each is fitted.
- Says what will be compared: the largest gap, the cable length, or the residuals of a fit.
- Says how the Maclaurin series is used and what it is checked against.
- Is narrow enough for 12 to 20 pages and does not just plot two curves and say they look alike.
The model
A chain hangs between two supports at the same height. Compare the true shape with a parabola of the same span and sag. Draw yours, with your own measurements.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Why this problem, the history in two lines, and what you expect before calculating: how different will they look?
Every symbol defined, the catenary and the parabola written with the same span and sag, and how a is found.
The Maclaurin series of cosh, where it starts to differ from a parabola, and the estimate for the gap.
The largest gap against the sag-to-span ratio, exact against estimated, with a graph.
Your photographed chain, both fits, residuals and what the data can and cannot distinguish.
The two arc lengths and how they compare, or the bridge cable comparison.
Back to your aim: how different the curves are, when the parabola is good enough, and why.
Chain stiffness, measurement and photo distortion, uniform weight, and which assumption matters most.
Where every formula, history point and measurement came from.
What your Maths IA graph could look like
where does the estimate fail?Made-up numbers to show the form of a strong graph, not the answer: the largest gap between catenary and parabola against the sag-to-span ratio. Yours will look different. example only
- The gap against the ratio, with the series estimate on the same axes so you can see where it stops working.
- Axis labels give units: the ratio has none, and the gap is a percentage of the sag.
- Say where the estimate and the exact gap part company, and by how much.
- Plot your measured chains as points, with error bars from your reading of the photo.
- Another graph to consider: both curves fitted to one photo, with the residuals underneath.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
When does the parabola stop being good enough?
Rank five hangs on one number: the largest gap as a percentage of the sag. Work it out for your own chain and explain why the gap grows faster than the sag.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind out what a catenary is. Before any calculation, guess how big the gap to a parabola is, as a percentage of the sag.
- Choose the chainPick a chain or cable and a way to hang it so you can control the sag. Say how you will photograph it without distortion.
- Define both curvesWrite the catenary and the parabola with the same span and sag. Show how the parameter a is found from the sag, numerically if needed.
- Trial run (1 lesson)Photograph one chain against a grid and read ten points. Fit both curves. Is the photo good enough to tell them apart? Fix it now.
- Use the seriesWrite the Maclaurin series of cosh, find the first term beyond the parabola and use it to estimate the largest gap.
- Measure the gapCompute the exact gap for several sag-to-span ratios and compare it with the estimate on one graph.
- Test on real dataFit both models to your chains for several sags. Report the residuals and say what the data can and cannot distinguish.
- Go further, with a reasonCompare cable lengths, load the cable like a bridge deck or use unequal supports. Say what question the extra step answers. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it means for a builder and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck units everywhere, every graph labelled, every symbol defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Exponential functions – Define and sketch the catenary for several values of .
- Maclaurin series – Expand and use the fourth-power term to estimate the gap to a parabola.
- Solving for a parameter – Solve for numerically on a calculator, given the span and sag.
- Optimisation – Maximise the difference between the two curves over to find the largest gap and where it occurs.
- Arc length – Compute the length of each curve by integration, , and compare.
- Curve fitting and residuals – Fit both models to points read from a photograph and compare the residuals with the reading uncertainty.
Key equations
Syllabus links
- 2.9Exponential and logarithmic functions
- 2.3Graphs of functions
- 5.6Standard derivatives; chain, product and quotient rules
- 5.17Areas with the y-axis; volumes of revolutionHL
- 5.19Maclaurin seriesHL
- 5.13Limits with l'Hopital's rule or Maclaurin seriesHL
- 2.5Linear, quadratic, exponential, cubic and sinusoidal models
- 5.8Trapezoidal rule
- 5.9Further derivatives; chain, product, quotient rules; related ratesHL
- 2.9Logarithmic, logistic, piecewise and half-life modelsHL
Read up first
Data and sources to start from
Photograph a chain or cable hung with several different sags against a grid, and read coordinates from the photos with a ruler or graphing software.
References to cite
- Stewart, J. (2015). Calculus: Early transcendentals (8th ed.). Cengage Learning.
- Galilei, G. (1638). Discorsi e dimostrazioni matematiche intorno a due nuove scienze. Elsevier (Leiden).
- Maor, E. (1994). e: The story of a number. Princeton University Press.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
R² cannot tell them apart- Fitting a parabola and a catenary to a photo and calling the one with the larger R² the true shape, when both are nearly the same.
- Comparing curves that do not have the same span and sag, so the gap includes an avoidable scale difference.
- Quoting cosh without defining it, or mixing up its definition with cos.
- Using photos taken from an angle, so the grid and the chain are both distorted.
- Calling a bridge cable a catenary without saying how it is loaded.
- Reading a very small gap off a graph by eye instead of computing it.
More Maths IA ideas like this
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This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.