- #1
SAZAR
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When you take a paper strip, put it on the flat surface (table), and contract it so it stands up - is it parabola or something else?
Is a Paper Strip Contracted on a Table a Parabola?
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In summary, the conversation discusses the shape created when a paper strip is placed on a flat surface and contracted by pushing the ends towards each other. The resulting curve is not a parabola, but rather a complicated curve involving trigonometric and/or exponential factors. The conversation also discusses the possible properties of this curve, such as a single focal point.
- #2
tiny-tim
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Hi SAZAR! 
You mean you push the ends towards each other, so that the rest of the paper bends naturally?
I think that's a catenary … see http://en.wikipedia.org/wiki/Catenary#The_inverted_catenary_arch
You mean you push the ends towards each other, so that the rest of the paper bends naturally?
I think that's a catenary … see http://en.wikipedia.org/wiki/Catenary#The_inverted_catenary_arch
- #3
SAZAR
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no.
I meant when you place a paper strip on a desk, then press the ends of the paper strip with index fingers, and then slide them toward each other with paper strip ends stuck to the fingers, so the section of the paper strip between fingers lifts up from the desk and forms a curve.
I meant when you place a paper strip on a desk, then press the ends of the paper strip with index fingers, and then slide them toward each other with paper strip ends stuck to the fingers, so the section of the paper strip between fingers lifts up from the desk and forms a curve.
- #4
tiny-tim
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Isn't that what I said?
Anyway, still a catenary.
- #5
SAZAR
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In the example with paper strip I described the index fingers are not placed at the ends of the paper strip - they are placed some distance away from the ends of the paper strip, so the elastic forces of the paper strip are at work at the point where index fingers are.
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The catenary pictures (examples) on wikipedia show things hanging or bulging at sharp angle from ends of the structure bending.
In example I described, however, the structure continues beyond - the direction of the part beyond is horizontal - not in direction of the curve (it transits from horizontal to the curve)...
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The catenary pictures (examples) on wikipedia show things hanging or bulging at sharp angle from ends of the structure bending.
In example I described, however, the structure continues beyond - the direction of the part beyond is horizontal - not in direction of the curve (it transits from horizontal to the curve)...
- #6
Elucidus
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If I am understanding your description correctly, the resulting arching curve is flat underneath the fingers and then bugles upwards in the center. If this is correct, then the curve is definitely not a parabola.
I strongly suspect it is a very complicated curve that may involve trigonometric and/or exponential factors. Even worse still, if the ends are pushed sufficiently close, I believe the curve stops being a function as is balloons out past where the fingers are ( much like a light bulb shape or the letter [itex]\Omega[/itex]).
Am I way off base here?
--Elucidus
I strongly suspect it is a very complicated curve that may involve trigonometric and/or exponential factors. Even worse still, if the ends are pushed sufficiently close, I believe the curve stops being a function as is balloons out past where the fingers are ( much like a light bulb shape or the letter [itex]\Omega[/itex]).
Am I way off base here?
--Elucidus
- #7
SAZAR
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I'm interested only in the case when elastic forces of the paper are still at work, not when paper "breaks" and fold at sharp angle.
Imagine that finger-nails face each other, so the nails pin the paperstrip to the surface of the desk while paper slides e.i. contracts which make it bulge - still maintaining its elastic properties (not "breaking").
Try it yourself - take any paper you have there - do what I described. What is that curve you get when you watch it from its profile? ...
Imagine that finger-nails face each other, so the nails pin the paperstrip to the surface of the desk while paper slides e.i. contracts which make it bulge - still maintaining its elastic properties (not "breaking").
Try it yourself - take any paper you have there - do what I described. What is that curve you get when you watch it from its profile? ...
- #8
SAZAR
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Anyway - would it have at least some properties of parabola? (such as a single focal point; at least in some cases)
FAQ: Is a Paper Strip Contracted on a Table a Parabola?
What is a paper strip contracted on a table?
A paper strip contracted on a table refers to the act of bending or folding a long, thin strip of paper and then pressing it onto a flat surface, such as a table. This creates a curved shape that resembles a parabola.
How is a parabola formed with a paper strip?
When a paper strip is contracted on a table, the paper's fibers are compressed on one side and stretched on the other. This creates a curved shape with a concave side and a convex side, resembling a parabola.
What is the significance of creating a parabola with a paper strip?
Creating a parabola with a paper strip can be used as a visual demonstration of a mathematical concept. It can help to illustrate the properties of a parabola, such as its focus and directrix, and how it relates to quadratic equations.
Can a paper strip contracted on a table create a perfect parabola?
No, a paper strip contracted on a table will not create a perfect parabola. This is because the paper's fibers will not be perfectly compressed and stretched, and the curve may not be symmetrical. However, it can still serve as a useful approximation for learning about parabolas.
Are there any real-life applications of the concept of a paper strip contracted on a table as a parabola?
Yes, the concept of a paper strip contracted on a table as a parabola can be applied in various fields, such as engineering and architecture. It can be used to understand and design structures with curved surfaces, such as bridges and arches, which often use parabolic shapes for their strength and stability.