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

Una línea de alta tensión colgando entre dos torres de transmisión abarca un espacio de 40 metros con una caída vertical significativa de 10 metros debido a la expansión térmica.El objetivo es encontrar la longitud exacta de la línea eléctrica necesaria para conectar estos puntos de forma segura.Los ingenieros modelan esto usando una catenaria, la curva natural de un cable colgante descrita matemáticamente por la función coseno hiperbólico.Esta curva se controla mediante un parámetro a, que relaciona el peso por metro del cable con la tensión horizontal. Para esta instalación específica, se asume que el valor del parámetro a es de 20 metros.Para encontrar la longitud exacta, el cálculo utiliza la función de longitud de arco. Aquí, la derivada de la catenaria da la sinusoidea hiperbólica.Una identidad estándar que involucra funciones hiperbólicas simplifica aún más la expresión eliminando la raíz cuadrada.La integral del coseno hiperbólico resultante es simplemente el seno hiperbólico. Evaluar la función seno hiperbólica en los límites de -veinte a veinte positivo da una longitud total de aproximadamente 47 metros.

longitud de arco de una curva: resolución de problemas

Arc Length of a Curve: Problem Solving

Una l&#xED;nea el&#xE9;ctrica de alta tensi&#xF3;n recorre una distancia horizontal de 40 m entre dos torres de transmisi&#xF3;n, lo que resulta en una flecha de 10 m debido a los efectos de la gravedad y la expansi&#xF3;n t&#xE9;rmica. La curva del cable suspendido es una catenaria, que modela con precisi&#xF3;n el comportamiento de un cable uniforme y flexible bajo su propio peso. A diferencia de una par&#xE1;bola, la catenaria se describe mediante la funci&#xF3;n coseno hiperb&#xF3;lico y ofrece una representaci&#xF3;n precisa de la forma del cable.En esta configuraci&#xF3;n, los ingenieros asignan al par&#xE1;metro a un valor de 20 m. Este par&#xE1;metro corresponde a la relaci&#xF3;n entre la tensi&#xF3;n horizontal del cable y su peso por unidad de longitud. Colocando el origen en el punto m&#xE1;s bajo de la curva, las torres se ubican sim&#xE9;tricamente a 20 m de cada lado de la curva.Para determinar la longitud exacta del cable necesaria para la instalaci&#xF3;n, se calcula la longitud de arco de la catenaria entre las dos torres. La f&#xF3;rmula utilizada es:\begin{equation*}L = \int_{-20}^{20} \cosh\!\liparens{\jfrac{x}{20}}\, dx\end{equation*}El resultado de esta integraci&#xF3;n es:\begin{equation*}L = 40 \sinh(1) \approx 47.01 \,\text{meters}\end{equation*}Este valor proporciona la longitud total necesaria de la l&#xED;nea el&#xE9;ctrica, considerando la flecha debida al peso propio del cable y los efectos t&#xE9;rmicos.

Longitud de arco de una curva: resolución de problemas

Una l&#xED;nea el&#xE9;ctrica de alta tensi&#xF3;n recorre una distancia horizontal de 40 m entre dos torres de transmisi&#xF3;n, lo que resulta en una ...

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Application of Techniques of Integration

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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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