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

A high-voltage power line hanging between two transmission towers spans a 40-meter gap with a significant 10-meter vertical sag due to thermal expansion.The objective is to find the exact length of the power line required to connect these points safely.Engineers model this using a catenary, the natural curve of a hanging cable described mathematically by the hyperbolic cosine function.This curve is controlled by a parameter a, which relates the cable&#8217;s weight per meter to the horizontal tension. For this specific installation, the value of parameter a is assumed to be 20 meters.To find the exact length, the calculation uses the arc length function. Here, the derivative of the catenary gives the hyperbolic sine.A standard identity involving hyperbolic functions simplifies the expression further by removing the square root.The integral of the resulting hyperbolic cosine is simply the hyperbolic sine. Evaluating the hyperbolic sine function at the limits from negative twenty to positive twenty gives a total length of approximately 47 meters.

booglengte van een kromme: probleemoplossing

Arc Length of a Curve: Problem Solving

Een hoogspanningsleiding overspant een horizontale afstand van 40 m tussen twee hoogspanningsmasten, wat resulteert in een verticale doorhang van 10 m als gevolg van zwaartekracht en thermische uitzetting. De kromme die door de hangende kabel wordt gevormd, is een kettinglijn, die het gedrag van een uniforme, flexibele kabel onder invloed van zijn eigen gewicht nauwkeurig modelleert. In tegenstelling tot een parabolische vorm wordt de kettinglijn beschreven door de hyperbolische cosinusfunctie en biedt zij een precieze weergave van de vorm van de kabel.In deze opstelling kennen ingenieurs de parameter a een waarde van 20 m toe. Deze parameter komt overeen met de verhouding tussen de horizontale spankracht in de kabel en het gewicht per lengte-eenheid. Door de oorsprong van het co&#xF6;rdinatenstelsel in het laagste punt van de kromme te plaatsen, bevinden de masten zich symmetrisch op posities van 20 m aan weerszijden van de kromme.Om de exacte lengte van de kabel te bepalen die voor de installatie nodig is, wordt de booglengte van de kettinglijn tussen de twee masten berekend. De gebruikte formule is:\begin{equation*}L = \int_{-20}^{20} \cosh\!\liparens{\jfrac{x}{20}}\, dx\end{equation*}Het resultaat van deze integratie is:\begin{equation*}L = 40 \sinh(1) \approx 47.01 \,\text{meters}\end{equation*}Deze waarde geeft de totale lengte van de hoogspanningsleiding weer en houdt rekening met de doorhang die wordt veroorzaakt door het eigen gewicht van de kabel en thermische effecten.

Booglengte van een kromme: probleemoplossing

Een hoogspanningsleiding overspant een horizontale afstand van 40 m tussen twee hoogspanningsmasten, wat resulteert in een verticale doorhang van 10 m als ...

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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

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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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