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

Pappo e Guldino svilupparono teoremi per trovare la superficie e il volume di qualsiasi corpo di rivoluzione.
Per generare l'area della superficie, ruotare una curva piana di lunghezza nota attorno all'asse x.
Si consideri un elemento di linea differenziale. Quando viene ruotato, genera un anello, che viene integrato per ottenere l'intera superficie. 
Il primo teorema afferma che l'area di una superficie di rivoluzione è il prodotto della lunghezza della curva generatrice e della distanza percorsa dal baricentro della curva durante la generazione della superficie.
Allo stesso modo, per trovare il volume di rivoluzione, un'area piana viene ruotata attorno a un asse non intersecante.
Si consideri un'area differenziale, che genera un elemento di volume quando viene ruotata. Il volume differenziale viene quindi integrato per determinare l'intero volume. 
Il secondo teorema afferma che il volume di un corpo di rivoluzione è il prodotto dell'area generatrice e della distanza percorsa dal baricentro dell'area durante la generazione del volume.
In entrambi i casi, la formula cambia di conseguenza se la rivoluzione avviene solo attraverso un angolo.

<ol>
	<li>Hibbeler, R.C. (2016). Ingegneria Meccanica: Statica. Quattordicesima edizione, New Jersey: Pearson. Sezione 9.3, pp. 502-503.</li>
	<li>Meriam, J.L., Kraige, L.G. e Bolton, J.N. (2016). Meccanica ingegneristica: statica e dinamica. Ottava edizione, Singapore: John Wiley & Sons. Sezione 5.5 pp. 261-262.</li>
</ol>

i teoremi di pappo e guldino

I due teoremi sviluppati da Pappo e Guldino sono ampiamente utilizzati in matematica, ingegneria e fisica per trovare l'area superficiale e il volume di qualsiasi corpo di rivoluzione. Ciò viene eseguito facendo ruotare una curva piana attorno a un asse che non interseca la curva per trovare la sua area superficiale o facendo ruotare un'area piana attorno a un asse non intersecante per calcolarne il volume.

Per trovare l'area superficiale, considera un elemento di linea differenziale che genera un anello con area superficiale dA quando viene ruotato. Integrando questa area differenziale si determina l'area superficiale della rivoluzione, che è descritta dal primo teorema. Il primo teorema afferma che l'area superficiale della rivoluzione è uguale al prodotto della lunghezza della curva generatrice per la distanza percorsa dal suo baricentro durante la generazione dell'area superficiale.

Analogamente, per calcolare il volume di rivoluzione, considera un elemento di area differenziale che genera un anello con volume differenziale quando viene ruotato. Integrando questo volume differenziale si determina il volume di rivoluzione, secondo il secondo teorema. Il secondo teorema di Pappo e Guldino afferma che il volume di rivoluzione è uguale al prodotto dell'area generatrice per la distanza percorsa dal suo baricentro nella generazione di tale volume. Comprensibilmente, se sia la curva che l'area vengono ruotate attraverso un angolo invece di una rotazione completa di 360°, le rispettive formule dovrebbero essere modificate di conseguenza.

I teoremi di Pappo e Guldino

I due teoremi sviluppati da Pappo e Guldino sono ampiamente utilizzati in matematica, ingegneria e fisica per trovare l'area superficiale e il volume di qualsiasi corpo ...

Education

Didattica

JoVE Core

Mechanical Engineering

Ingegneria Meccanica

Unit 1

Unità 1

Center of Gravity and Centroid

Centro di gravità e centroide

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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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