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

Stellen Sie sich einen zylindrischen Vorratsbehälter mit konischem Deckel vor. 
Der Tank stellt einen Verbundkörper dar, der in eine endliche Anzahl von Teilen mit einfacheren Formen unterteilt werden kann.
Die Theoreme von Pappus und Guldinus können angewendet werden, um die Oberfläche und das Volumen solcher zusammengesetzten Formen zu finden.
Um die Oberfläche zu ermitteln, werden die Generierungskurven für jede Form identifiziert und um eine sich nicht schneidende Achse gedreht. Dann wird der Schwerpunkt für jede Kurve lokalisiert.
Die Länge der Erzeugungskurve und die vom Schwerpunkt der Kurve zurückgelegte Strecke werden dann verwendet, um die Gesamtoberfläche des Tanks zu erhalten.
Auf ähnliche Weise wird zur Schätzung des Volumens der erzeugende Bereich jeder Form identifiziert und um dieselbe Achse gedreht.
Das Volumen eines Umdrehungskörpers ist gleich dem Produkt aus der erzeugenden Fläche und der Strecke, die der Schwerpunkt der Form zurücklegt.
Durch das Ermitteln der Fläche jedes Teils und die Identifizierung der jeweiligen Schwerpunkte werden die Werte im Satz eingesetzt, um das Volumen des Tanks zu erhalten.

<ol>
	<li>Hibbeler, R.C. (2016). Technische Mechanik: Statik. Vierzehnte Auflage, New Jersey: Pearson. Abschnitt 9.3, S. 503, 507.</li>
</ol>

theoreme von pappus und guldinus: problemlösung

Die Theoreme von Pappus und Guldinus sind leistungsstarke mathematische Prinzipien, die zur Ermittlung der Oberfläche und des Volumens von zusammengesetzten Formen verwendet werden. Betrachten wir zum Beispiel einen zylindrischen Lagertank mit einer konischen Spitze. Die Bestimmung der Oberfläche oder des Volumens kann für solche komplexen Formen eine Herausforderung darstellen. Diese Theoreme sind besonders nützlich für die Berechnung des Volumens und der Oberfläche solcher Systeme. Hier kann der zylindrische Lagertank mit einer konischen Spitze in zwei einfache Formen aufgeteilt werden: ein Zylinder und ein Kegel. Um dieses Problem zu lösen, wird das erste Theorem angewendet, um die Oberfläche des durch das Umdrehen jeder der erzeugenden Kurven um eine nicht-schneidende Achse entstehenden Festkörpers zu finden. Dies beinhaltet das Einsetzen relevanter Werte, wie z.B. die Länge der erzeugenden Kurve und die von ihrem Schwerpunkt zurückgelegte Strecke, das Erhalten separater Werte für jeden Teil und deren Addition, um ein Gesamtergebnis zu erhalten. Zur Berechnung des Volumens drehen wir eine Fläche um dieselbe Achse wie zuvor und verwenden Informationen aus dem zweiten Theorem, das das Multiplizieren der erzeugten Flächen mit den von ihren jeweiligen Schwerpunkten zurückgelegten Strecken beinhaltet. Durch die Durchführung von Berechnungen mit diesen beiden Prinzipien wird es einfacher, genaue Werte für sowohl die Oberfläche als auch das Volumen von komplexen Formen wie diesem Lagertank zu finden.

Theoreme von Pappus und Guldinus: Problemlösung

Die Theoreme von Pappus und Guldinus sind mächtige mathematische Prinzipien, die zur Bestimmung der Oberfläche und des Volumens von zusammengesetzten Formen verwendet werden. Für...

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

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Center of Gravity and Centroid

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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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