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

İki iletim kulesi arasında asılı olan yüksek voltajlı bir elektrik hattı, termal genişleme nedeniyle 10 metrelik ciddi bir dikey çökme ile 40 metrelik bir boşluk oluşturur.Amaç, bu noktaları güvenli bir şekilde bağlamak için gereken elektrik hattının tam uzunluğunu bulmaktır.Mühendisler bunu, hiperbolik kosinüs fonksiyonuyla matematiksel olarak tanımlanan asılı bir kablonun doğal eğrisi olan katener kullanarak modeller.Bu eğri, kablonun metre başına ağırlığını yatay gerilimle ilişkilendiren bir a parametresi tarafından kontrol edilir. Bu özel kurulum için, a parametresinin değeri 20 metre olarak kabul edilir.Tam uzunluğu bulmak için hesaplama yay uzunluğu fonksiyonunu kullanır. Burada, katenerin türevi hiperbolik sinüs verir.Hiperbolik fonksiyonları içeren standart bir özdeşlik, karekökü kaldırarak ifadeyi daha da basitleştirir.Ortaya çıkan hiperbolik kosinüsün integrali basitçe hiperbolik sinüstür. Hiperbolik sinüs fonksiyonunun eksik yirmiden pozitif yirmiye kadar sınırlarda değerlendirilmesi yaklaşık 47 metre toplam uzunluk elde eder.

bir eğrinin ark uzunluğu: problem Çözümü

Arc Length of a Curve: Problem Solving

Y&#xFC;ksek gerilim hatt&#x131;, iki iletim kulesi aras&#x131;ndaki 40 m&#x2019;lik yatay mesafeyi kapsar ve yer&#xE7;ekimi ile termal genle&#x15F;menin etkileri nedeniyle 10 m&#x2019;lik dikey bir sarkma olu&#x15F;turur. Ask&#x131;da duran kablonun olu&#x15F;turdu&#x11F;u e&#x11F;ri, kendi a&#x11F;&#x131;rl&#x131;&#x11F;&#x131; alt&#x131;nda d&#xFC;zg&#xFC;n ve esnek bir kablonun davran&#x131;&#x15F;&#x131;n&#x131; do&#x11F;ru bi&#xE7;imde modelleyen bir katenar e&#x11F;risidir. Parabolik bir &#x15F;ekilden farkl&#x131; olarak katenary, hiperbolik kosin&#xFC;s fonksiyonu ile tan&#x131;mlan&#x131;r ve kablonun &#x15F;eklini hassas bi&#xE7;imde temsil eder.Bu d&#xFC;zende m&#xFC;hendisler a parametresine 20 m de&#x11F;erini atar. Bu parametre, kablodaki yatay gerilmenin birim uzunluk ba&#x15F;&#x131;na a&#x11F;&#x131;rl&#x131;&#x11F;a oran&#x131;na kar&#x15F;&#x131;l&#x131;k gelir. Orijin e&#x11F;rinin en d&#xFC;&#x15F;&#xFC;k noktas&#x131;na yerle&#x15F;tirildi&#x11F;inde, kuleler orijinin her iki yan&#x131;nda 20 m uzakl&#x131;kta simetrik konumlarda bulunur.Kurulum i&#xE7;in gerekli kesin kablo uzunlu&#x11F;unu belirlemek amac&#x131;yla, iki kule aras&#x131;ndaki katenar e&#x11F;risinin ark uzunlu&#x11F;u hesaplan&#x131;r. Kullan&#x131;lan form&#xFC;l &#x15F;&#xF6;yledir:\begin{equation*}L = \int_{-20}^{20} \cosh\!\liparens{\jfrac{x}{20}}\, dx\end{equation*}Bu integralin sonucu &#x15F;u &#x15F;ekildedir:\begin{equation*}L = 40 \sinh(1) \approx 47.01 \,\text{meters}\end{equation*}Bu de&#x11F;er, kablonun kendi a&#x11F;&#x131;rl&#x131;&#x11F;&#x131; ve termal genle&#x15F;menin etkilerinden kaynaklanan sarkmay&#x131; dikkate alarak do&#x11F;ru kurulumu sa&#x11F;lamak i&#xE7;in gereken toplam y&#xFC;ksek gerilim hatt&#x131; uzunlu&#x11F;unu verir.

Bir Eğrinin Ark Uzunluğu: Problem Çözümü

Y&#xFC;ksek gerilim hatt&#x131;, iki iletim kulesi aras&#x131;ndaki 40 m&#x2019;lik yatay mesafeyi kapsar ve yer&#xE7;ekimi ile termal genle&#x15F;menin ...

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JoVE Core

Calculus

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Birim

Application of Techniques of Integration

İntegral Alma Tekniklerinin Uygulanması

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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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Pappus Theorem connects centroid motion to the volume and surface area of solids of revolution. It is also called the Pappus–Guldinus Theorem. The method uses geometry to find these values when a plane region or a plane curve is rotated about an external axis. The theorem has two related parts. The first part gives the volume of a solid formed by rotating a plane region around an axis outside the region. In this case, the volume equals the area of the region multiplied by the distance traveled...

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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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