Name: pappus and guldinus theorems for surface area
Uploaded: 2023-09-22T11:49:32.000Z
Duration: 1 min 12 s
Channel: JoVE

Konik bir tepeye sahip silindirik bir depolama tankı düşünün. 
Tank, daha basit şekillere sahip sonlu sayıda parçaya bölünebilen kompozit bir gövdeyi temsil eder.
Pappus ve Guldinus'un teoremleri, bu tür bileşik şekillerin yüzey alanını ve hacmini bulmak için uygulanabilir.
Yüzey alanını bulmak için, her şekil için oluşturan eğriler tanımlanır ve kesişmeyen bir eksen etrafında döndürülür. Ardından, her eğri için centroid bulunur.
Üretme eğrisinin uzunluğu ve eğrinin merkezi tarafından kat edilen mesafe daha sonra tankın toplam yüzey alanını elde etmek için kullanılır.
Benzer şekilde, hacmi tahmin etmek için, her şeklin üretim alanı tanımlanır ve aynı eksen etrafında döndürülür.
Bir devir cisminin hacmi, üreten alanın çarpımına ve şeklin merkez noktasının kat ettiği mesafeye eşittir.
Her bir parçanın alanını bularak ve ilgili merkezlerini tanımlayarak, tankın hacmini elde etmek için değerler teoremde ikame edilir.

<ol>
	<li>Hibbeler, R.C. (2016). Mühendislik Mekaniği: Statik. On Dördüncü Baskı, New Jersey: Pearson. Bölüm 9.3, s. 503, 507.</li>
</ol>

pappus ve guldinus teoremleri: problem Çözme

Pappus ve Guldinus teoremleri, birleşik şekillerin yüzey alanını ve hacmini bulmak için kullanılan güçlü matematik ilkesidirler. Örneğin, koni şeklinde bir üst kapağı olan bir silindirik depo düşünelim. Bu gibi karmaşık şekillerin yüzey alanını veya hacmini bulmak zor olabilir. Bu teoremler, özellikle bu sistemlerin hacmini ve yüzey alanını hesaplamada kullanışlıdır. Burada, koni şeklinde bir üst kapağı olan silindirik depo, iki basit şekle, bir silindir ve bir kona şekline ayrılabilir.

Bu problemi çözmek için, ilk teorem, üretilen eğrilerin her birini birbirine kesişmeyen bir eksen etrafında dönerek oluşan katı yüzey alanını bulmak için uygulanır. Bu, üretilen eğrinin uzunluğu ve ağırlık merkezinin kat ettiği mesafe gibi ilgili değerlerin yerine konması, her parça için ayrı ayrı değerlerin elde edilmesi ve bunların toplanarak genel bir toplamın elde edilmesini içerir. Hacmi hesaplamak için, önceki teoremde olduğu gibi bir alanı aynı eksene göre döndürürüz ve ikinci teoremden elde edilen bilgileri kullanırız, bu da üretilen alanları ilgili ağırlık merkezinin kat ettiği mesafelerle çarpmayı içerir.

Bu iki prensiple yapılan hesaplamalarla, bu depo gibi karmaşık şekillerin hem yüzey alanı hem de hacmi için doğru değerleri bulmak daha kolay hale gelir.

Pappus ve Guldinus teoremleri: Problem Çözme

Pappus ve Guldinus'un teoremleri, bileşik şekillerin yüzey alanını ve hacmini bulmak için kullanılan güçlü matematiksel ilkelerdir. İçin...

theorems-of-pappus-and-guldinus-problem-solving

Education

JoVE Core

Mechanical Engineering

Makine Mühendisliği

Unit 1

Ünite 1

Center of Gravity and Centroid

Ağırlık Merkezi ve Geometrik Merkez

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

Theorems of Pappus and Guldinus: Problem Solving

pappus and guldinus theorems for surface area

Pappus and Guldinus's theorems help find the surface area and volume of composite shapes. These rules are useful when a solid is made from simple parts that are hard to measure directly. A cylindrical storage tank with a conical top is one example.

The tank can be split into two simple shapes: a cylinder and a cone. To find surface area, the first theorem is used for the solid formed by revolving each generating curve around a non-intersecting axis. The calculation uses the length of the generating curve and the distance traveled by its centroid, or center of mass.

The surface area is found separately for each part and then added together. To find volume, the second theorem is used. It works by revolving an area around the same axis and multiplying the generated area by the distance traveled by its centroid. This makes it easier to calculate accurate surface area and volume for complex shapes like the storage tank.

Pappus and Guldinus Theorems for Surface Area

pappus-and-guldinus-theorems-for-surface-area

center of gravity

Center of Gravity in Balance and Design

The center of gravity is the point where an object’s weight appears to act. It is the spot used to balance the object most evenly. In mechanics, this point helps explain stability and moments of inertia. The center of gravity does not always sit inside the object’s shape. In some cases, it can lie outside the body. That is why the idea is useful for both simple and more complex shapes. Its location can be found with the principle of moments. The object is divided into very small parts, and...

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Center of Mass in Motion

The center of mass helps explain how an object moves through space. It is the point where the total mass of an object can be treated as concentrated. This idea applies to objects of different shapes and sizes. The center of mass is related to the center of gravity, but they are not the same. The center of gravity is the point where an object’s weight seems to be concentrated. It can be used to balance an object perfectly. Knowing the center of mass also helps predict motion. When a ball is...

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The centroid is the geometric center of a body. It is an important idea in engineering, physics, and mechanics. For many objects, it lies inside the body. A body with holes or cavities can be an exception. For a homogeneous body, the centroid matches the center of mass or the center of gravity. Homogeneous means the material has a uniform density. In that case, the centroid can often be found by balancing the moments of the body’s volume. Symmetry can also help locate the centroid. If a...

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composite bodies

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theorems of pappus and guldinus

Surface Area and Volume by Centroid Paths

Pappus and Guldinus theorems help find the surface area and volume of bodies of revolution. A body of revolution is formed when a plane curve or plane area is turned around an axis that does not intersect it. For surface area, a curve is revolved around a non-intersecting axis. A small line element on the curve creates a thin ring as it moves. Adding these ring areas gives the total surface area of the solid. The first theorem says that this surface area equals the length of the generating...

theorems of pappus and guldinus: problem solving

Pappus and Guldinus's theorems help find the surface area and volume of composite shapes. These rules are useful when a solid is made from simple parts that are hard to measure directly. A cylindrical storage tank with a conical top is one example. The tank can be split into two simple shapes: a cylinder and a cone. To find surface area, the first theorem is used for the solid formed by revolving each generating curve around a non-intersecting axis. The calculation uses the length of the...

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key terms and concepts

Learn how Pappus and Guldinus's theorems help calculate surface area and volume for composite shapes like a tank with a conical top.

Pappus and Guldinus Theorems for Surface Area | JoVE K12

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mechanical engineering

Pappus and Guldinus Theorems for Surface Area

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