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

Considere um tanque de armazenamento cilíndrico com um topo cônico. 
O tanque representa um corpo composto, que pode ser dividido em um número finito de peças com formas mais simples.
Os teoremas de Pappus e Guldinus podem ser aplicados para encontrar a área de superfície e o volume de tais formas compostas.
Para encontrar a área de superfície, as curvas geradoras para cada forma são identificadas e giradas em torno de um eixo sem interseção. Em seguida, o centróide está localizado para cada curva.
O comprimento da curva geradora e a distância percorrida pelo centróide da curva são então usados para obter a área total da superfície do tanque.
Da mesma forma, para estimar o volume, a área geradora de cada forma é identificada e girada em torno do mesmo eixo.
O volume de um corpo de revolução é igual ao produto da área geradora e à distância percorrida pelo centróide da forma.
Ao encontrar a área de cada parte e identificar seus respectivos centróides, os valores são substituídos no teorema para obter o volume do tanque.

<ol>
	<li>Hibbeler, R.C. (2016). Engenharia Mecânica: Estática. Décima quarta edição, Nova Jersey: Pearson. Seção 9.3, pp. 503, 507.</li>
</ol>

teoremas de pappus e guldinus: solução de problemas

Os teoremas de Pappus e Guldinus são princípios matemáticos poderosos que são utilizados para encontrar a área de superfície e o volume de formas compostas. Por exemplo, considere um tanque de armazenamento cilíndrico com uma parte superior cônica. Encontrar a área de superfície ou o volume pode ser desafiador para formas tão complexas. Esses teoremas são particularmente úteis no cálculo do volume e da área de superfície de tais sistemas. Aqui, o tanque de armazenamento cilíndrico com uma parte superior cônica pode ser dividido em duas formas simples: um cilindro e um cone.

Para resolver esse problema, o primeiro teorema é aplicado para encontrar a área de superfície do sólido formado pela rotação de cada uma das curvas geradoras em torno de um eixo que não se intersecta. Isso envolve a substituição de valores relevantes, como o comprimento da curva geradora e a distância percorrida pelo seu centroide, obtendo valores separados para cada parte e, em seguida, somando-os para obter um total geral. Para calcular o volume, giramos uma área em torno do mesmo eixo anterior e utilizamos informações do segundo teorema, que envolve multiplicar as áreas geradas pelas distâncias percorridas por seus respectivos centroides.

Ao realizar cálculos com esses dois princípios, torna-se mais fácil encontrar valores precisos tanto para a área de superfície quanto para o volume de formas complexas como esse tanque de armazenamento.

Teoremas de Pappus e Guldinus: Solução de Problemas

Os teoremas de Pappus e Guldinus são princípios matemáticos poderosos usados para encontrar a área de superfície e o volume de formas compostas. Durante...

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

Education

JoVE Core

Mechanical Engineering

Engenharia Mecânica

Unit 1

Unidade 1

Center of Gravity and Centroid

Centro de Gravidade e Centroide

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

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

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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

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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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Pappus and Guldinus Theorems for Surface Area

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