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

Considera un serbatoio di stoccaggio cilindrico con una parte superiore conica. 
Il serbatoio rappresenta un corpo composito, che può essere suddiviso in un numero finito di parti con forme più semplici.
I teoremi di Pappo e Guldino possono essere applicati per trovare la superficie e il volume di tali forme composite.
Per trovare l'area della superficie, le curve generatrici per ogni forma vengono identificate e ruotate attorno a un asse non intersecante. Quindi, il baricentro viene individuato per ogni curva.
La lunghezza della curva generatrice e la distanza percorsa dal baricentro della curva vengono quindi utilizzate per ottenere la superficie totale del serbatoio.
Allo stesso modo, per stimare il volume, l'area generatrice di ciascuna forma viene identificata e ruotata attorno allo stesso asse.
Il volume di un corpo di rivoluzione è uguale al prodotto dell'area generatrice e della distanza percorsa dal baricentro della forma.
Trovando l'area di ciascuna parte e identificando i rispettivi centroidi, i valori vengono sostituiti nel teorema per ottenere il volume del serbatoio.

<ol>
	<li>Hibbeler, R.C. (2016). Ingegneria Meccanica: Statica. Quattordicesima edizione, New Jersey: Pearson. Sezione 9.3, pp. 503, 507.</li>
</ol>

i teoremi di pappo e guldino: risoluzione di problemi

I teoremi di Pappo e Guldino sono potenti principi matematici utilizzati per trovare l'area superficiale e il volume di figure composite. Ad esempio, consideriamo un serbatoio di stoccaggio cilindrico con una parte superiore conica. Trovare l'area superficiale o il volume può essere difficile per forme così complesse. Questi teoremi sono particolarmente utili nel calcolare il volume e l'area superficiale di tali sistemi. Qui, il serbatoio cilindrico con una parte superiore conica può essere scomposto in due figure semplici: un cilindro e un cono.

Per risolvere questo problema, viene applicato il primo teorema per trovare l'area superficiale del solido formato ruotando ciascuna delle curve generatrici attorno a un asse non intersecante. Ciò comporta la sostituzione dei valori pertinenti come la lunghezza della curva generatrice e la distanza percorsa dal suo baricentro, ottenendo valori separati per ciascuna parte e poi sommandoli per ottenere un totale complessivo. Per calcolare il volume, ruotiamo un'area attorno allo stesso asse di prima e utilizziamo informazioni dal secondo teorema, che prevede la moltiplicazione delle aree generate per le distanze percorse dai rispettivi baricentri.

Effettuando calcoli con questi due principi, diventa più facile trovare valori accurati sia per l'area superficiale sia per il volume di forme complesse come questo serbatoio di stoccaggio.

I teoremi di Pappo e Guldino: Risoluzione di problemi

I teoremi di Pappo e Guldino sono potenti principi matematici che vengono utilizzati per trovare la superficie, l'area e il volume delle forme composite. Per...

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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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

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'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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