|Real Life Parabolas|
Real Life Parabolas
- Parabolas are very well-known and are seen frequently in the field of mathematics. Their applications are varied and are apparent in our every day lives. For example, the main image on the right is of the Golden Gate Bridge in San Francisco, California. It has main suspension cables in the shape of a parabola.
For a detailed overview of parabolas, see the page, Parabola. However, we will provide a brief summary and description of parabolas below before explaining its applications to suspension bridges.
You may informally know parabolas as curves in the shape of a "u" which can be oriented to open upwards, downwards, sideways, or diagonally. But to be more mathematical, a parabola is a conic section formed by the intersection of a cone and a plane. Below is an image illustrating this.
When you were first introduced to parabolas, you learned that the quadratic equation, is its algebraic representation (where and are the coordinates of the vertex and and are the coordinates of an arbitrary point on the parabola.
Suspension Bridges are the most commonly built bridges. Known for their long spans, these bridges feature a deck with vertical supports, from which long wire cables hang above. These cables are made up of hangers that run vertically downwards to hold the cable up. The suspension cables hang over the towers until they are anchored on land by the ends of the bridges. Notably, the way these cables are hung resemble the shape of a parabola.
Usefulness of Suspension Bridges
Due to their elegant structure, suspension bridges are used to transport loads over long distances, whether it be between two distant cities or between two ends of a river. Suspension bridges are able to work efficiently because of their cables, which are interesting from a mathematical perspective.
Since the bridge’s deck spans a long distance, it must be very heavy in weight by its own, not to mention all the weight of the heavy load of traffic that it must carry. Because of all this weight, this results in two active forces: compression and tension. The cable’s parabolic shape results in order for it to effectively address these forces acting upon the bridge. For instance, the deck sags from all the weight of the traffic because of compression forces, which travels upwards the cables. The cables then transfer those compression forces downwards the vertical towers, down into the foundations buried deep within the earth. However, the cables receive the brunt of the tension forces, as they are supporting the bridge’s weight and its load of traffic, being stretched by the anchors' ends on-land.
Overall, the suspension bridge does its job with minimal material (as most of the work is accomplished by the suspension cables), which means that it is economical from a construction cost perspective.
A conceptual explanation
This links to other page, Catenary. But we shall explain the differences between parabola and catenary with more emphasis on the parabola.
Why is that the main suspension cables hang in a shape of a parabola, and not in a catenary, a similar ‘u-shaped’ curve?
Despite their visual similarities, catenaries and parabolas are two very different curves, both conceptually and mathematically.
A catenary curve is created by its own weight, pulling down because of gravity. The parabolic curves of the suspension cable are not created by gravity alone, but also by other forces: compression and tension acting on it. Also the weight of the suspension cable is negligible compared to that of the deck, but it is also supporting the weight of the deck. This is also another conceptual reason why the suspension cables hang in a parabolic curve.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Calculus, Physics
Some basic differential calculus is needed to derive an equation for the suspension cables, which giv [...]
Some basic differential calculus is needed to derive an equation for the suspension cables, which gives us a parabola
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