Catenary

In mathematics, the catenary is the shape of a hanging flexible chain or cable when supported at its ends and acted upon by a uniform gravitational force (its own weight). The chain is almost vertical near the points of suspension because this part of the chain has the most weight pulling down on it. Toward the bottom, the slope of the chain decreases because the chain is supporting less and less weight.

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The word catenary is derived from the Latin word for "chain." The curve is also called the alysoid, funicular, and chainette. Galileo claimed that the curve of a chain hanging under gravity would be a parabola, but this was disproved by Jungius in 1669. In 1691, Leibniz, Christiaan Huygens, and Johann Bernoulli derived the equation in response to a challenge by Jakob Bernoulli. Huygens first used the term 'catenary' in a letter to Leibniz in 1690, and David Gregory wrote a treatise on the catenary in 1690.

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If you roll a parabola along a straight line, its focus traces out a catenary (see roulette). As proved by Euler in 1744, the catenary is also the curve which, when rotated about the x axis, gives the surface of minimum surface area (the catenoid) for the given bounding circle. Square wheels can roll perfectly smoothly if the road has evenly spaced bumps in the shape of a series of inverted catenary curves.

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The intrinsic equation of the shape of the catenary is given by the hyperbolic function

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:y = a cdot cosh left ({x over a} ight ).

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Shape: In geometry, two objects are of the same shape if one can be transformed to another (ignoring color) by dilating (that is, by multiplying all distances by the same factor) and then, if necessary, rotating and translating. Dilation changes the size but not the shape; rotation and translation preserv...

Chain: A chain can be any of the following:...

Cable: A cable is two or more wires bound together which may be bare or covered or insulated. It may be protected by a jacket sheath which protects all. Electrical cables can also act as carriers for other media, including optical fibers....