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Students can link things, such as a GeoGebra creation.

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Free online apps bundle from GeoGebra: get graphing, geometry, algebra, 3D, statistics, probability, all in one tool!

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Handwritten

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When students find a really cool webpage explaining a topic, they can post it here.

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It has often been pondered whether the shape of a suspension bridge cable is a catenary or a parabola.

However, what differentiates the bridge cable from a chain is that the bridge cable is not only holding up its own weight but also that of the bridge itself. So, let's consider two examples:

where the weight of the bridge is negligible compared to the cable;

where the weight of the support cable is negligible compared to that of the bridge.

By the way, if you ever assumed that the curve of a dangling chain is parabolic, you aren't alone, as it is often said that Galileo also assumed the shape to be parabolic. This is not surprising as this figure shows the comparison between the two curves:

Now consider the shape of the suspension cable assuming that the weight of the cable is negligible compared to the weight of the bridge. This is quite a bit easier to solve since the weight of the bridge is now uniform in the direction, whereas above it varied with the slope of the chain.

Find the shape of the cable of a suspension bridge (pictured above), assuming the weight of the cable is negligible compared to the weight of the bridge.

Lets define the same variables as before:

F = force along the cable whose direction will represent the slope of the cable at point P

T = horizontal component of the cable tension

W = the weight of the cable.

As in the example above,

But this time, the mass density doesn't vary with the slope, and we'll assume that the density of the bridge is constant for its entire span. So,

Once again, if we assume the low point of the cable is at the origin, then

So, in conclusion, we see that if the cable supports only its own weight, like a rope being held at two points, it forms the shape of a catenary. However, if it has negligible weight relative to the bridge it supports, it assumes the shape of a parabola.

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Hyperbolic Trigonometric Functions | Brilliant Math & Science Wiki

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It has often been pondered whether the shape of a suspension bridge cable is a catenary or a parabola. However, what differentiates the bridge cable from a chain is that the bridge cable is not only holding up its own weight but also that of the bridge itself.