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

Pappus et Guldinus ont développé des théorèmes pour déterminer la surface et le volume de tout corps de révolution.
Pour générer la surface, faites pivoter une courbe plane de longueur connue autour de l’axe x.
Considérons un élément de ligne différentielle. Lorsqu’il est tourné, il génère un anneau, qui est intégré pour obtenir toute la surface. 
Le premier théorème stipule que l'aire d'une surface de révolution est le produit de la longueur de la courbe génératrice et de la distance parcourue par le centroïde de la courbe lors de la génération de la surface.
De même, pour déterminer le volume de révolution, une aire plane est tournée autour d’un axe non sécant.
Considérons une aire différentielle, générant un élément de volume lorsqu’il est tourné. Le volume différentiel est ensuite intégré pour déterminer l’ensemble du volume. 
Le deuxième théorème stipule que le volume d’un corps de révolution est le produit de l’aire génératrice et de la distance parcourue par le centroïde de l’aire lors de la génération du volume.
Dans les deux cas, la formule change en conséquence si la révolution n’est que d’un angle.

<ol>
	<li>Hibbeler, R.C. (2016). Mécanique du génie : Statique. Quatorzième édition, New Jersey : Pearson. Section 9.3, p. 502 et 503.</li>
	<li>Meriam, J.L., Kraige, L.G. et Bolton, J.N. (2016). Mécanique de l’ingénierie : statique et dynamique. Huitième édition, Singapour : John Wiley & Sons. Section 5.5, p. 261 et 262.</li>
</ol>

les théorèmes de pappus et guldinus

Les deux théorèmes développés par Pappus et Guldinus sont largement utilisés en mathématiques, en ingénierie et en physique pour trouver l'aire de surface et le volume de tout corps de révolution. Cela est fait en faisant tourner une courbe plane autour d'un axe qui n'intersecte pas la courbe pour trouver son aire de surface ou en faisant tourner une surface plane autour d'un axe non intersectant pour calculer son volume. Pour trouver l'aire de surface, considérons un élément de ligne différentielle qui génère un anneau avec une aire de surface dA lorsqu'il est tourné. En intégrant cette aire différentielle, nous déterminons l'aire de surface de la révolution, qui est décrite par le premier théorème. Le premier théorème stipule que l'aire de surface de la révolution est égale au produit de la longueur de la courbe génératrice et de la distance parcourue par son centroïde pendant la génération de l'aire de surface. De même, pour calculer le volume de révolution, considérons un élément de surface différentielle qui génère un anneau avec un volume différentiel lorsqu'il est tourné. En intégrant ce volume différentiel, nous déterminons le volume de révolution, selon le deuxième théorème. Le deuxième théorème de Pappus et Guldinus stipule que le volume de révolution est égal au produit de la surface génératrice et de la distance parcourue par son centroïde lors de la génération de ce volume. De manière compréhensible, si la courbe ou la surface est tournée d'un angle au lieu d'une rotation complète de 360°, leurs formules respectives devraient être adaptées en conséquence.

Les théorèmes de Pappus et Guldinus

Les deux théorèmes développés par Pappus et Guldinus sont largement utilisés en mathématiques, en ingénierie et en physique pour déterminer l’aire et le volume de n’importe quel corps ...

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Center of Gravity and Centroid

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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 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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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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Surface Area and Volume by Centroid Paths

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