Name: catenary cable length in power lines
Uploaded: 2026-01-12T12:24:12.000Z
Duration: 1 min 21 s
Channel: JoVE

Uma linha de alta tensão suspensa entre duas torres de transmissão atravessa uma distância de 40 metros com uma significativa queda vertical de 10 metros devido à expansão térmica.O objetivo é encontrar o comprimento exato da linha de energia necessária para conectar esses pontos com segurança.Engenheiros modelam isso usando uma catenária, a curva natural de um cabo suspenso descrita matematicamente pela função cosseno hiperbólico.Essa curva é controlada por um parâmetro a, que relaciona o peso do cabo por metro com a tensão horizontal. Para essa instalação específica, o valor do parâmetro a é assumido como 20 metros.Para encontrar o comprimento exato, o cálculo usa a função comprimento do arco. Aqui, a derivada da catenária dá o seno hiperbólico.Uma identidade padrão envolvendo funções hiperbólicas simplifica ainda mais a expressão ao remover a raiz quadrada.A integral do cosseno hiperbólico resultante é simplesmente o seno hiperbólico. Avaliar a função seno hiperbólica nos limites de menos vinte a vinte positivo resulta em um comprimento total de aproximadamente 47 metros.

comprimento do arco de uma curva: resolução de problemas

Arc Length of a Curve: Problem Solving

Uma linha de transmiss&#xE3;o de alta tens&#xE3;o se estende por uma dist&#xE2;ncia horizontal de 40 m entre duas torres de transmiss&#xE3;o, resultando em uma curvatura de 10 m devido aos efeitos da gravidade e da dilata&#xE7;&#xE3;o t&#xE9;rmica. A curva formada pelo cabo suspenso &#xE9; uma caten&#xE1;ria, que modela com precis&#xE3;o o comportamento de um cabo uniforme e flex&#xED;vel sob seu pr&#xF3;prio peso. Ao contr&#xE1;rio de uma forma parab&#xF3;lica, a caten&#xE1;ria &#xE9; descrita pela fun&#xE7;&#xE3;o cosseno hiperb&#xF3;lico e oferece uma representa&#xE7;&#xE3;o precisa da forma do cabo.Nesta configura&#xE7;&#xE3;o, os engenheiros atribuem ao par&#xE2;metro a o valor de 20 m. Este par&#xE2;metro corresponde &#xE0; raz&#xE3;o entre a tens&#xE3;o horizontal no cabo e seu peso por unidade de comprimento. Ao posicionar a origem no ponto mais baixo da curva, as torres est&#xE3;o localizadas simetricamente a 20 m de cada lado da curva.Para determinar o comprimento exato do cabo necess&#xE1;rio para a instala&#xE7;&#xE3;o, calcula-se o comprimento do arco da caten&#xE1;ria entre as duas torres. A f&#xF3;rmula utilizada &#xE9;:\begin{equation*}L = \int_{-20}^{20} \cosh\!\liparens{\jfrac{x}{20}}\, dx\end{equation*}O resultado desta integra&#xE7;&#xE3;o &#xE9;:\begin{equation*}L = 40 \sinh(1) \approx 47.01 \,\text{meters}\end{equation*}Este valor fornece o comprimento total necess&#xE1;rio da linha de transmiss&#xE3;o, garantindo a instala&#xE7;&#xE3;o adequada e considerando a curvatura causada pelo pr&#xF3;prio peso do cabo e pelos efeitos t&#xE9;rmicos.

Comprimento do arco de uma curva: resolução de problemas

Uma linha de transmiss&#xE3;o de alta tens&#xE3;o se estende por uma dist&#xE2;ncia horizontal de 40 m entre duas torres de transmiss&#xE3;o, resultando ...

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catenary cable length in power lines

Catenary cable length helps engineers measure the full span of a hanging power line. A high-voltage line stretches 40 meters between two transmission towers and sags 10 meters because of gravity and thermal expansion. The cable forms a catenary, which is the curve of a uniform, flexible cable under its own weight.

A catenary is not the same as a parabola. It is described by the hyperbolic cosine function, which gives a more accurate shape for a suspended cable. In this setup, engineers use the parameter a = 20 meters. This value represents the ratio between the horizontal tension in the cable and its weight per unit length.

To make the calculation easier, the origin is placed at the lowest point of the curve. That puts the two towers 20 meters to either side of the center. The arc length of the catenary between those towers is then found with an integration formula. The result gives the total cable length needed for installation, including the sag caused by the cable’s own weight and thermal effects.

Catenary Cable Length in Power Lines

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arc length of a curve: problem solving

Catenary cable length helps engineers measure the full span of a hanging power line. A high-voltage line stretches 40 meters between two transmission towers and sags 10 meters because of gravity and thermal expansion. The cable forms a catenary, which is the curve of a uniform, flexible cable under its own weight. A catenary is not the same as a parabola. It is described by the hyperbolic cosine function, which gives a more accurate shape for a suspended cable. In this setup, engineers use the...

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key terms and concepts

Find the cable length of a sagging power line by modeling it as a catenary. Learn how arc length, tension, and sag work together in this example.

Catenary Cable Length in Power Lines | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Catenary Cable Length in Power Lines

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