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Science project idea

Hanging chain or hanging bridge? Catenary and parabola compared

A chain hanging under its own weight forms a catenary, while a cable carrying a load spread evenly along the horizontal, as in a suspension bridge, forms a parabola. The two curves look alike but have different equations.

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What it shows

A chain hanging under its own weight forms a catenary, while a cable carrying a load spread evenly along the horizontal, as in a suspension bridge, forms a parabola. The two curves look alike but have different equations.

The science

For a hanging chain of uniform mass per length, the curve is y = a cosh(x / a), where a is a constant set by the tension and the weight per length, and y is measured from the lowest point plus a. A parabola, y = k x squared, arises when the weight is spread uniformly over the horizontal distance, which is what a bridge deck does for its cables. Near the lowest point both curves are similar, because cosh(u) is about 1 + u squared / 2 for small u. They separate as one goes further from the middle, where the catenary rises faster than the parabola.

Materials

Cost: under Rs 50. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.

  • A light bead chain or a thin metal chain, 1 m
  • Two nails or drawing pins on a board
  • A pegboard or a notice board covered with graph paper
  • A string, 1 m, and 11 equal washers or nuts
  • Sticky tape, a ruler and a plumb line
  • A phone to photograph the curves square on
  • A spreadsheet or a calculator with cosh

Safety

For this project: An adult must help with nails and pins. Use a cork board and keep the board flat or upright so nothing falls on feet.

Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.

All the safety rules

Step by step

8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.

  1. Fix two nails 60 cm apart at the same height on the board with graph paper behind. Hang the chain between them so it sags about 25 cm.

  2. Photograph the chain square on, then print or project the photo over the grid. Read the coordinates of 8 points on the curve from the lowest point as the origin.

  3. Test the parabola idea by calculating y divided by x squared at each point. If it is a parabola this ratio is constant, otherwise it changes.

  4. Hang the loaded string between the same nails with a washer every 5 cm along the horizontal. Take 8 points the same way.

  5. Calculate y divided by x squared for the loaded string and compare with the chain.

  6. For the chain, choose a from the sag and half-span, then compute y = a (cosh(x/a) minus 1) for each x value and compare with the measured y.

  7. Plot both data sets and both theoretical curves on the same graph.

  8. Write which curve fits each set better and by how much, using the largest difference between measured and calculated heights.

What you should see

The loaded string follows y = k x squared with a nearly constant ratio y divided by x squared, while the chain is better fitted by the cosh curve, with its ratio increasing with x.

Viva questions

A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.

1What is a catenary?

The curve formed by a uniform chain hanging freely from two points.

2Why do suspension bridge cables form parabolas?

The weight of the deck is spread evenly along the horizontal, not along the cable.

3How are the two curves alike near the bottom?

cosh(u) is nearly 1 + u squared / 2, which is a parabola.

4Where is the catenary arch used?

Arches inverted from a hanging chain are very stable, as designers such as Gaudi found.

Free ebook: Viva Questions for Science Projects

Take it further

Invert a photo of the hanging chain and build the arch from card strips to show that it stays up under load.

Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.

Free ebooks for this project

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