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

Pappus i Guldinus opracowali twierdzenia, aby znaleźć powierzchnię i objętość dowolnego ciała obrotowego.
Aby wygenerować pole powierzchni, należy obrócić krzywą płaską o znanej długości wokół osi x.
Rozważmy element linii różnicowej. Po obróceniu generuje pierścień, który jest zintegrowany w celu uzyskania całej powierzchni. 
Pierwsze twierdzenie mówi, że pole powierzchni obrotu jest iloczynem długości krzywej generującej i odległości, jaką pokonuje środek ciężkości krzywej podczas generowania powierzchni.
Podobnie, aby znaleźć objętość obrotu, obszar płaszczyzny jest obracany wokół osi, która się nie przecina.
Rozważmy obszar różnicowy, generujący element objętości po obróceniu. Objętość różnicowa jest następnie integrowana w celu określenia całej objętości. 
Drugie twierdzenie mówi, że objętość ciała obrotowego jest iloczynem obszaru generującego i odległości, jaką pokonuje środek ciężkości obszaru podczas generowania objętości.
W obu przypadkach wzór zmienia się odpowiednio, jeśli obrót odbywa się tylko pod kątem.

<ol>
	<li>Hibbeler, R.C. (2016). Mechanika inżynierska: Statyka. Czternasta edycja, New Jersey: Pearson. Sekcja 9.3, s. 502-503.</li>
	<li>Meriam, J.L., Kraige, L.G. i Bolton, J.N. (2016). Mechanika inżynierska: statyka i dynamika. Ósma edycja, Singapur: John Wiley & Sons. Sekcja 5.5 str. 261-262.</li>
</ol>

twierdzenia pappusa i guldinusa

Dwa twierdzenia opracowane przez Pappusa i Guldinusa są szeroko stosowane w matematyce, inżynierii i fizyce do obliczania powierzchni i objętości dowolnego bryła obrotowej. Dokonuje się tego przez obrót krzywej płaszczyzny wokół osi, która nie przecina krzywej, w celu znalezienia jej powierzchni lub obrót obszaru płaskiego wokół nieprzecinającej osi w celu obliczenia objętości.

Aby obliczyć powierzchnię, należy wziąć pod uwagę różniczkowy element linii, który generuje pierścień o powierzchni dA po obróceniu. Zintegrowanie tego różniczkowego obszaru określa powierzchnię obrotu, którą opisuje pierwsze twierdzenie. Pierwsze twierdzenie mówi, że powierzchnia obrotu jest równa iloczynowi długości krzywej generującej i odległości pokonywanej przez jej centroid podczas generowania powierzchni.

Podobnie, aby obliczyć objętość obrotową, należy wziąć pod uwagę różniczkowy element obszaru, który generuje pierścień o różniczkowej objętości po obróceniu. Zintegrowanie tej różniczkowej objętości wyznacza objętość obrotu zgodnie z drugim twierdzeniem. Drugie twierdzenie Pappusa i Guldinusa mówi, że objętość obrotu jest równa iloczynowi generującego obszaru i odległości pokonywanej przez jego centroid podczas generowania takiej objętości. Oczywiście, jeśli zarówno krzywa, jak i obszar są obracane o pewien kąt, a nie pełny obrót o 360°, ich odpowiednie wzory muszą być odpowiednio zmienione.

Twierdzenia Pappusa i Guldinusa

Dwa twierdzenia opracowane przez Pappusa i Guldinusa są szeroko stosowane w matematyce, inżynierii i fizyce do określania powierzchni i objętości dowolnego ciała ...

Education

JoVE Core

Mechanical Engineering

Inżynieria mechaniczna

Unit 1

Jednostka 1

Center of Gravity and Centroid

Środek ciężkości i środek geometryczny

KEY TERMS AND CONCEPTS

Kluczowe terminy i pojęcia

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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Surface Area and Volume by Centroid Paths

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