Name: surface area and volume by centroid paths
Uploaded: 2023-09-22T11:49:32.000Z
Duration: 1 min 10 s
Channel: JoVE

Pappus ve Guldinus, herhangi bir devrim cisminin yüzey alanını ve hacmini bulmak için teoremler geliştirdiler.
Yüzey alanını oluşturmak için, x ekseni etrafında bilinen uzunlukta bir düzlem eğrisi döndürün.
Bir diferansiyel çizgi elemanı düşünün. Döndürüldüğünde, tüm yüzey alanını elde etmek için entegre edilmiş bir halka oluşturur. 
İlk teorem, bir devrim yüzeyinin alanının, üreten eğrinin uzunluğunun ve eğrinin merkezinin yüzeyi oluştururken kat ettiği mesafenin çarpımı olduğunu belirtir.
Benzer şekilde, devir hacmini bulmak için, bir düzlem alanı kesişmeyen bir eksen etrafında döndürülür.
Döndürüldüğünde bir hacim elemanı oluşturan bir diferansiyel alan düşünün. Diferansiyel hacim daha sonra tüm hacmi belirlemek için entegre edilir. 
İkinci teorem, bir devir cisminin hacminin, üreten alanın ve alanın merkezinin hacmi üretirken kat ettiği mesafenin çarpımı olduğunu belirtir.
Her iki durumda da, devir sadece bir açıdan geçiyorsa formül buna göre değişir.

<ol>
	<li>Hibbeler, R.C. (2016). Mühendislik Mekaniği: Statik. On Dördüncü Baskı, New Jersey: Pearson. Bölüm 9.3, sayfa 502-503.</li>
	<li>Meriam, J.L., Kraige, L.G. ve Bolton, J.N. (2016). Mühendislik Mekaniği: Statik ve Dinamik. Sekizinci Baskı, Singapur: John Wiley & Sons. Bölüm 5.5, sayfa 261-262.</li>
</ol>

pappus ve guldinus teoremleri

Matematik, mühendislik ve fizikte Pappus ve Guldinus tarafından geliştirilen iki teorem, dönme cisimlerin yüzey alanı ve hacmini bulmak için yaygın olarak kullanılır. Bunun yapılması için eğriyi kesmeyen bir eksende bir düzlem eğrisini döndürerek yüzey alanını bulmak veya kesişmeyen bir eksende bir düzlem alanını döndürerek hacmini hesaplamak gerekmektedir.

Yüzey alanını bulmak için, döndürüldüğünde yüzey alanı dA olan bir halka üreten bir diferansiyel çizgi elemanını düşünelim. Bu diferansiyel alanın tümünü işleme alarak yüzey alanını belirleyebiliriz, ki bu da birinci teorem tarafından açıklanır. Birinci teorem, döndürme yüzey alanının üretilen eğri uzunluğu ile gerçekleştirilen yüzey alanı üretimi sırasında merkezine kat edilen mesafenin çarpımına eşit olduğunu belirtir.

Benzer şekilde, dönme hacmini hesaplamak için, döndürüldüğünde bir diferansiyel hacim ile bir halka üreten bir diferansiyel alan elemanını düşünelim. Bu diferansiyel hacimin tümünü işleme alarak dönme hacmini belirleyebiliriz, ikinci teoreme göre. Pappus ve Guldinus'un ikinci teoremi, dönme hacminin üretilen alan ile merkezinin bu hacim üretimi sırasındaki kat ettiği mesafenin çarpımına eşit olduğunu belirtir. Anlaşılır bir şekilde, eğri veya alanın tam bir 360Â° dönme yerine bir açı boyunca döndürülmesi durumunda, ilgili formüllerin buna göre değiştirilmesi gerekmektedir.

Pappus ve Guldinus teoremleri

Pappus ve Guldinus tarafından geliştirilen iki teorem, herhangi bir cismin yüzey alanını ve hacmini bulmak için matematik, mühendislik ve fizikte yaygın olarak kullanılmaktadır.

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

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 curve multiplied by the distance traveled by its centroid. The centroid is the geometric center of the curve as it moves.

For volume, a plane area is revolved around a non-intersecting axis. A small area element creates a ring with a tiny volume as it turns. Integrating these small volumes gives the total volume of revolution.

The second theorem says that this volume equals the generating area multiplied by the distance traveled by its centroid. If the curve or area turns through an angle instead of a full 360° rotation, the formula must be adjusted. These ideas are useful in mathematics, engineering, and physics.

Surface Area and Volume by Centroid Paths

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Center of Gravity in Balance and Design

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Centroid and Uses of a Paraboloid Shape

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Center of Gravity in Composite Bodies

Composite bodies are made of multiple connected parts that act as one larger object. Each part still has its own weight and center of gravity. To find the center of gravity of the whole body, engineers must account for every part. When the density or specific weight is constant, the center of gravity matches the centroid. The centroid is the geometric center of a shape. This idea is useful when analyzing parts that behave as a single unit. Composite bodies are common in mechanical engineering.

theorems of pappus and guldinus

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...

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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...

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key terms and concepts

Learn how Pappus and Guldinus theorems use centroid motion to find surface area and volume of bodies of revolution, with 360° and angled rotations.

Surface Area and Volume by Centroid Paths | JoVE K12

JoVE Core Makine Mühendisliği, denge, kuvvetler, kafes kiriş analizi ve yapısal mekanik gibi temel konuları kapsayan kısa, anlaşılması kolay animasyonlu videolar sunar. Gerçek dünyadaki uygulamalarını keşfedin ve bilim insanlarının önderlik ettiği deneylerle bu kavramların uygulanmasına tanık olun.

mechanical engineering

Surface Area and Volume by Centroid Paths

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