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

Pappus e Guldinus desenvolveram teoremas para encontrar a área de superfície e o volume de qualquer corpo de revolução.
Para gerar a área de superfície, gire uma curva plana de comprimento conhecido em torno do eixo x.
Considere um elemento de linha diferencial. Quando girado, gera um anel, que é integrado para obter toda a área da superfície. 
O primeiro teorema afirma que a área de uma superfície de revolução é o produto do comprimento da curva geradora e a distância que o centróide da curva percorre enquanto gera a superfície.
Da mesma forma, para encontrar o volume de revolução, uma área plana é girada em torno de um eixo que não se cruza.
Considere uma área diferencial, gerando um elemento de volume quando revolucionado. O volume diferencial é então integrado para determinar todo o volume. 
O segundo teorema afirma que o volume de um corpo de revolução é o produto da área geradora e a distância que o centróide da área percorre enquanto gera o volume.
Em ambos os casos, a fórmula muda de acordo se a revolução for apenas através de um ângulo.

<ol>
	<li>Hibbeler, R.C. (2016). Engenharia Mecânica: Estática. Décima quarta edição, Nova Jersey: Pearson. Seção 9.3, pp. 502-503.</li>
	<li>Meriam, J.L., Kraige, L.G. e Bolton, J.N. (2016). Engenharia Mecânica: Estática e Dinâmica. Oitava Edição, Cingapura: John Wiley & Sons. Seção 5.5 pp. 261-262.</li>
</ol>

teoremas de pappus e guldinus

Os dois teoremas desenvolvidos por Pappus e Guldinus são amplamente utilizados em matemática, engenharia e física para encontrar a área de superfície e o volume de qualquer corpo de revolução. Isso é feito ao girar uma curva planar em torno de um eixo que não intersecta a curva para encontrar sua área de superfície, ou ao girar uma área planar em torno de um eixo que não intersecta para calcular seu volume.

Para encontrar a área de superfície, considere um elemento de linha diferencial que gera um anel com área de superfície dA quando girado. Integrar essa área diferencial determina a área de superfície da revolução, que é descrita pelo primeiro teorema. O primeiro teorema afirma que a área de superfície de revolução é igual ao produto do comprimento da curva geradora e a distância percorrida pelo seu centroide durante a geração da área de superfície.

Da mesma forma, para calcular o volume de revolução, considere um elemento de área diferencial que gera um anel com volume diferencial quando girado. Integrar esse volume diferencial determina o volume de revolução, de acordo com o segundo teorema. O segundo teorema de Pappus e Guldinus afirma que o volume de revolução é igual ao produto da área geradora e a distância percorrida pelo seu centroide na geração desse volume. Compreensivelmente, se tanto a curva quanto a área forem giradas através de um ângulo em vez de uma rotação completa de 360°, suas respectivas fórmulas precisariam ser alteradas conforme necessário.

Teoremas de Pappus e Guldinus

Os dois teoremas desenvolvidos por Pappus e Guldinus são amplamente utilizados em matemática, engenharia e física para encontrar a área de superfície e o volume de qualquer corpo ...

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

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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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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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

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mechanical engineering

Surface Area and Volume by Centroid Paths

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