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

Una linea elettrica ad alta tensione sospesa tra due torri di trasmissione si estende su un gap di 40 metri con un significativo abbassamento verticale di 10 metri dovuto all'espansione termica.L'obiettivo è trovare la lunghezza esatta della linea elettrica necessaria per collegare in sicurezza questi punti.Gli ingegneri modellano questo usando una catenaria, la curva naturale di un cavo sospeso descritta matematicamente dalla funzione coseno iperbolica.Questa curva è controllata da un parametro a, che mette in relazione il peso del cavo per metro alla tensione orizzontale. Per questa installazione specifica, si assume che il valore del parametro a sia di 20 metri.Per trovare la lunghezza esatta, il calcolo utilizza la funzione lunghezza d'arco. Qui, la derivata della catenaria dà il seno iperbolico.Un'identità standard che coinvolge funzioni iperboliche semplifica ulteriormente l'espressione rimuovendo la radice quadrata.L'integrale del coseno iperbolico risultante è semplicemente il seno iperbolico. Valutando la funzione sinusoidale iperbolica ai limiti da -venti a venti positivi si ottiene una lunghezza totale di circa 47 metri.

lunghezza dell’arco di una curva: risoluzione di problemi

Arc Length of a Curve: Problem Solving

Una linea elettrica ad alta tensione si estende per 40 m in orizzontale tra due tralicci di trasmissione, con una freccia verticale di 10 m dovuta agli effetti della gravit&#xE0; e della dilatazione termica. La curva formata dal cavo sospeso &#xE8; una catenaria, che modella con accuratezza il comportamento di un cavo uniforme e flessibile soggetto al proprio peso. A differenza di una forma parabolica, la catenaria &#xE8; descritta dalla funzione coseno iperbolico, cosh, e fornisce una rappresentazione precisa dell&#x2019;andamento del cavo.In questa configurazione, gli ingegneri assegnano al parametro a il valore di 20 m. Tale parametro corrisponde al rapporto tra la tensione orizzontale del cavo e il suo peso per unit&#xE0; di lunghezza. Ponendo l&#x2019;origine nel punto pi&#xF9; basso della curva, i tralicci risultano collocati simmetricamente a 20 m da ciascun lato della curva.Per determinare la lunghezza esatta del cavo necessaria per l&#x2019;installazione, si calcola la lunghezza dell&#x2019;arco della catenaria tra i due tralicci. La formula utilizzata &#xE8;:\begin{equation*}L = \int_{-20}^{20} \cosh\!\liparens{\jfrac{x}{20}}\, dx\end{equation*}Il risultato di questa integrazione &#xE8;:\begin{equation*}L = 40 \sinh(1) \approx 47.01 \,\text{meters}\end{equation*}Questo valore fornisce la lunghezza totale del cavo necessaria, garantendo una corretta installazione e tenendo conto della freccia dovuta al peso del cavo e agli effetti termici.

Lunghezza dell’arco di una curva: risoluzione di problemi

Una linea elettrica ad alta tensione si estende per 40 m in orizzontale tra due tralicci di trasmissione, con una freccia verticale di 10 m dovuta agli ...

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Calculus

Analisi matematica

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Application of Techniques of Integration

Applicazione delle tecniche di integrazione

KEY TERMS AND CONCEPTS

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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

arc length of a curve

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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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