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the difference between a parabola and a catenary

In: Mathematics

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In terms of engineering forms:

A parabola is the deflected shape that forms when the load is uniform along the *span* of a member (assuming uniform stiffness and pinned ends).

A catenary is when the load is uniform along the *length* or *slope* of the cable. The difference is that the cable has a slope to it, so if you take a 1 foot span (horizontal) length near the ends there will be maybe 1.2 feet of cable sloping down, while at the mid span a 1 foot length of span will be basically 1 foot as the cable is essentially flat. This means there’s somewhat more load towards the ends of the span than there is at the middle.

So things like chains and power cables hang as catenaries, while something like a suspension bridge will be hang as a catenary during construction (prior to hanging the deck from the cables) but will be *mostly* a parabola once complete as the deck is much heavier than the cables and applies a mostly uniform load to the span.

I’ll try to make this an explanation for a five year old.

Both are curves that looks similar but aren’t exactly the same.

A parabola is the shape a baseball follows when you toss it at up an angle and gravity slows it down and pulls it back to earth.

A catenary is the shape a rope takes when it hangs tied at both ends and sags under its own weight.

The baseball has only the force of gravity acting on it. It is pulled down evenly the whole time.

The rope has the force of gravity but also the tension of the two ends of the rope. Gravity pulls down but also pulls “out” because rope is pulling on the anchors it’s tied to.

So the equations that are required to describe each type of curve are very different. You can’t describe a catenary with a parabola. It doesn’t account for the pulling back of the rope. It’s more complex and needs different kinds of equations to describe them.

Fundamentally, it’s a different formula. Parabolas rely on a square to create the curve ( y=x^(2) ) while catenaries use the hyperbolic cosine function “cosh” ( y=cosh(x) ).

One of the simpler differences is that if you extended the “arms” of the curve to infinity, a parabola’s arms will trend toward parallel, while a catenary’s will not.