Partial fraction decomposition and Catenary: Difference between pages
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''' | [[File:Catenary-pm.svg|right|thumb|350x350px|Plots of <math>y = a \cosh \left(\frac{x}{a}\right)</math> with <math>a = 0.5, 1, 2</math>. The variable <math>x</math> is on the horizontal axis and <math>y</math> is on the vertical axis.]] | ||
[[File:Kette_Kettenkurve_Catenary_2008_PD.JPG|right|thumb|290x290px|A chain hanging like this forms the shape of a catenary approximately]] | |||
A '''catenary''' is a type of [[curve]]. An ideal [[chain]] hanging between two supports and acted on by a uniform [[Gravity|gravitational force]] makes the shape of a catenary.<ref name="wolfram">{{cite web|url=http://mathworld.wolfram.com/Catenary.html|title=Catenary|publisher=Wolfram Research|accessdate=2016-10-30}}</ref> (An ideal chain is one that can bend perfectly, cannot be stretched and has the same [[density]] throughout.<ref name="csu">{{cite web|url=http://curvebank.calstatela.edu/catenary/catenary.htm|title=The Catenary - The "Chain" Curve|publisher=California State University|accessdate=2016-10-30}}</ref>) The supports can be at different heights and the shape will still be a catenary.<ref>{{cite web|url=http://people.math.aau.dk/~br/catenary.pdf|title=Catenary|first=Bo|last=Rosbjerg|publisher=Aalborg University|accessdate=2016-10-30}}</ref> A catenary looks a bit like a [[parabola]], but they are different.<ref>{{cite web|url=http://mathforum.org/library/drmath/view/65729.html|title=Catenary and Parabola Comparison|publisher=Drexel University|accessdate=2016-11-05}}</ref> | |||
The [[equation]] for a catenary in [[Cartesian coordinate system|Cartesian coordinates]] is<ref name="csu" /><ref name="math24">{{cite web|url=http://www.math24.net/equation-of-catenary.html|title=Equation of Catenary|publisher=Math24.net|accessdate=2016-10-30}}</ref> | |||
: <math>y = a \cosh \left(\frac{x}{a}\right)</math> | |||
where <math>a</math> is a [[parameter]] that determines the shape of the catenary<ref name="math24" /> and {{math|cosh}} is the [[Hyperbolic function|hyperbolic cosine]] function, which is defined as<ref name="stroud" /> | |||
: <math> \cosh x = \frac {e^x + e^{-x}} {2}</math>. | |||
Hence, we can also write the catenary equation as | |||
: <math> y = \frac{a\left(e^\frac{x}{a} + e^{-\frac{x}{a}}\right)}{2}</math>. | |||
< | The word "catenary" comes from the [[Latin]] word ''catena'', which means "chain".<ref name="stroud">{{cite book|first1=K. A.|last1=Stroud|first2=Dexter J.|last2=Booth|title=Engineering Mathematics|edition=7th|publisher=Palgrave Macmillan|date=2013|isbn=978-1-137-03120-4|page=438}}</ref> A catenary is also called called an alysoid and a chainette.<ref name="wolfram" /> | ||
== References == | |||
{{reflist}} | |||
[[Category:Geometry]] | |||
{{math-stub}} | |||
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Latest revision as of 21:53, 6 November 2016
A catenary is a type of curve. An ideal chain hanging between two supports and acted on by a uniform gravitational force makes the shape of a catenary.[1] (An ideal chain is one that can bend perfectly, cannot be stretched and has the same density throughout.[2]) The supports can be at different heights and the shape will still be a catenary.[3] A catenary looks a bit like a parabola, but they are different.[4]
The equation for a catenary in Cartesian coordinates is[2][5]
- [math]\displaystyle{ y = a \cosh \left(\frac{x}{a}\right) }[/math]
where [math]\displaystyle{ a }[/math] is a parameter that determines the shape of the catenary[5] and Template:Math is the hyperbolic cosine function, which is defined as[6]
- [math]\displaystyle{ \cosh x = \frac {e^x + e^{-x}} {2} }[/math].
Hence, we can also write the catenary equation as
- [math]\displaystyle{ y = \frac{a\left(e^\frac{x}{a} + e^{-\frac{x}{a}}\right)}{2} }[/math].
The word "catenary" comes from the Latin word catena, which means "chain".[6] A catenary is also called called an alysoid and a chainette.[1]