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

Considérons un réservoir de stockage cylindrique avec un sommet conique. 
Le réservoir représente un corps composite, qui peut être divisé en un nombre fini de pièces aux formes plus simples.
Les théorèmes de Pappus et de Guldinus peuvent être appliqués pour déterminer la surface et le volume de telles formes composites.
Pour trouver l’aire de surface, les courbes génératrices de chaque forme sont identifiées et tournent autour d’un axe non sécant. Ensuite, le centroïde est localisé pour chaque courbe.
La longueur de la courbe génératrice et la distance parcourue par le centroïde de la courbe sont ensuite utilisées pour obtenir la surface totale du réservoir.
De même, pour estimer le volume, la zone génératrice de chaque forme est identifiée et tournée autour du même axe.
Le volume d’un corps de révolution est égal au produit de l’aire génératrice et de la distance parcourue par le centroïde de la forme.
En trouvant l'aire de chaque pièce et en identifiant leurs centroïdes respectifs, les valeurs sont substituées dans le théorème pour obtenir le volume du réservoir.

<ol>
	<li>Hibbeler, R.C. (2016). Mécanique du génie : Statique. Quatorzième édition, New Jersey : Pearson. Section 9.3, p. 503 et 507.</li>
</ol>

les théorèmes de pappus et de guldinus : résolution de problèmes

Les théorèmes de Pappus et de Guldinus sont de puissants principes mathématiques utilisés pour trouver l'aire de surface et le volume de formes composites. Par exemple, considérez un réservoir de stockage cylindrique avec un dessus conique. Trouver l'aire de surface ou le volume peut être difficile pour de telles formes complexes. Ces théorèmes sont particulièrement utiles pour calculer le volume et l'aire de surface de tels systèmes. Ici, le réservoir de stockage cylindrique avec un dessus conique peut être décomposé en deux formes simples : un cylindre et un cône. Pour résoudre ce problème, le premier théorème est appliqué pour trouver l'aire de surface du solide formé en faisant tourner chacune des courbes génératrices autour d'un axe non intersectant. Cela implique de substituer des valeurs pertinentes telles que la longueur de la courbe génératrice et la distance parcourue par son centroïde, d'obtenir des valeurs séparées pour chaque partie, puis de les additionner pour obtenir un total global. Pour calculer le volume, nous faisons tourner une aire autour du même axe qu'auparavant et utilisons des informations provenant du deuxième théorème, qui multiplie les aires générées par les distances parcourues par leurs respectifs centroïdes. En effectuant des calculs avec ces deux principes, il devient plus facile de trouver des valeurs précises pour à la fois l'aire de surface et le volume de formes complexes telles que ce réservoir de stockage.

Les théorèmes de Pappus et de Guldinus : Résolution de problèmes

Les théorèmes de Pappus et de Guldinus sont de puissants principes mathématiques qui sont utilisés pour déterminer la surface et le volume des formes composites. Pour...

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

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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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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'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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