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

Rozważ cylindryczny zbiornik magazynowy ze stożkowym blatem. 
Zbiornik reprezentuje korpus kompozytowy, który można podzielić na skończoną liczbę części o prostszych kształtach.
Twierdzenia Pappusa i Guldinusa można zastosować do znalezienia pola powierzchni i objętości takich złożonych kształtów.
Aby znaleźć pole powierzchni, krzywe generujące dla każdego kształtu są identyfikowane i obracane wokół osi, która się nie przecina. Następnie środek ciężkości znajduje się dla każdej krzywej.
Długość krzywej generującej i odległość przebyta przez środek ciężkości krzywej są następnie wykorzystywane do uzyskania całkowitej powierzchni zbiornika.
Podobnie, aby oszacować objętość, obszar generowania każdego kształtu jest identyfikowany i obracany wokół tej samej osi.
Objętość ciała obrotowego jest równa iloczynowi powierzchni generującej i odległości przebytej przez środek ciężkości kształtu.
Znajdując obszar każdej części i identyfikując ich odpowiednie centroidy, wartości są podstawiane w twierdzeniu w celu uzyskania objętości zbiornika.

<ol>
	<li>Hibbeler, R.C. (2016). Mechanika inżynierska: Statyka. Czternasta edycja, New Jersey: Pearson. Sekcja 9.3, s. 503, 507.</li>
</ol>

twierdzenia pappusa i guldinusa: rozwiązywanie problemów

Twierdzenia Pappusa i Guldinusa to potężne zasady matematyczne, które są stosowane do obliczania powierzchni i objętości kształtów złożonych. Na przykład, rozważmy cylindryczny zbiornik magazynowy z wierzchołkiem stożkowym. Obliczenie powierzchni lub objętości może być trudne dla tak skomplikowanych kształtów. Te twierdzenia są szczególnie przydatne do obliczania objętości i powierzchni takich systemów. Tutaj, cylindryczny zbiornik magazynowy z wierzchołkiem stożkowym może zostać podzielony na dwa proste kształty: cylinder i stożek.

Aby rozwiązać ten problem, pierwsze twierdzenie jest stosowane do obliczenia powierzchni ciała utworzonego przez obrót każdej z krzywych generujących wokół nieprzecinającej się osi. Wymaga to podstawienia odpowiednich wartości, takich jak długość krzywej generującej i odległość przebytej przez jej środek ciężkości, uzyskania osobnych wartości dla każdej części, a następnie dodania ich, aby uzyskać ogólny wynik. Do obliczenia objętości obracamy obszar wokół tej samej osi co wcześniej i korzystamy z informacji z drugiego twierdzenia, które polega na pomnożeniu wygenerowanych obszarów przez odległości przebyte przez ich odpowiednie środki ciężkości.

Przeprowadzając obliczenia z tymi dwoma zasadami, łatwiej jest znaleźć dokładne wartości zarówno dla powierzchni, jak i objętości skomplikowanych kształtów, takich jak ten zbiornik magazynowy.

Twierdzenia Pappusa i Guldinusa: Rozwiązywanie problemów

Twierdzenia Pappusa i Guldinusa to potężne zasady matematyczne, które są używane do znajdowania pola powierzchni i objętości kształtów kompozytowych. Dla...

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

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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: 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

pappus-and-guldinus-theorems-for-surface-area

center of gravity

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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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