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

Pappus und Guldinus entwickelten Theoreme, um die Oberfläche und das Volumen eines beliebigen Rotationskörpers zu bestimmen.
Um die Oberfläche zu erzeugen, drehen Sie eine ebene Kurve bekannter Länge um die x-Achse.
Betrachten Sie ein differenzielles Linienelement. Beim Drehen entsteht ein Ring, der integriert ist, um die gesamte Oberfläche zu erhalten. 
Der erste Satz besagt, dass die Fläche einer Rotationsfläche das Produkt aus der Länge der erzeugenden Kurve und der Strecke ist, die der Schwerpunkt der Kurve zurücklegt, während er die Oberfläche erzeugt.
Um das Umdrehungsvolumen zu ermitteln, wird ein ebener Bereich um eine sich nicht schneidende Achse gedreht.
Stellen Sie sich eine Differenzfläche vor, die beim Drehen ein Volumenelement erzeugt. Anschließend wird das Differenzvolumen integriert, um das gesamte Volumen zu bestimmen. 
Der zweite Satz besagt, dass das Volumen eines Umdrehungskörpers das Produkt aus der erzeugenden Fläche und der Entfernung ist, die der Schwerpunkt der Fläche zurücklegt, während er das Volumen erzeugt.
In beiden Fällen ändert sich die Formel entsprechend, wenn die Umdrehung nur durch einen Winkel erfolgt.

<ol>
	<li>Hibbeler, R.C. (2016). Technische Mechanik: Statik. Vierzehnte Auflage, New Jersey: Pearson. Abschnitt 9.3, S. 502-503.</li>
	<li>Meriam, J.L., Kraige, L.G. und Bolton, J.N. (2016). Technische Mechanik: Statik und Dynamik. Achte Auflage, Singapur: John Wiley & Sons. Abschnitt 5.5, S. 261-262.</li>
</ol>

die theoreme von pappus und guldinus

Die beiden von Pappus und Guldinus entwickelten Theoreme werden in der Mathematik, Ingenieurwissenschaften und Physik weit verbreitet verwendet, um die Oberfläche und das Volumen eines beliebigen Rotationskörpers zu berechnen. Dies wird erreicht, indem eine ebenen Kurve um eine Achse rotiert wird, die die Kurve nicht schneidet, um die Oberfläche zu berechnen oder eine ebene Fläche um eine nicht schneidende Achse rotiert wird, um das Volumen zu ermitteln. Um die Oberfläche zu berechnen, wird ein differentielles Linienelement betrachtet, das bei der Rotation einen Ring mit der Oberfläche dA erzeugt. Durch Integration dieses differentiellen Bereichs wird die Oberfläche der Rotation bestimmt, die durch das erste Theorem beschrieben wird. Das erste Theorem besagt, dass die Oberfläche der Rotation gleich dem Produkt aus der Länge der erzeugenden Kurve und der von ihrem Schwerpunkt zurückgelegten Entfernung während der Erzeugung der Oberfläche ist. Ähnlich wird zur Bestimmung des Volumens der Rotation ein differentielles Flächenelement betrachtet, das bei der Rotation einen Ring mit differentiellem Volumen erzeugt. Durch Integration dieses differentiellen Volumens wird das Volumen der Rotation gemäß dem zweiten Theorem ermittelt. Das zweite Theorem von Pappus und Guldinus besagt, dass das Volumen der Rotation gleich dem Produkt aus der erzeugenden Fläche und der von ihrem Schwerpunkt zurückgelegten Entfernung während der Erzeugung eines solchen Volumens ist. Verständlicherweise müssten die jeweiligen Formeln entsprechend angepasst werden, wenn entweder die Kurve oder die Fläche anstatt einer vollständigen 360° Drehung um einen Winkel rotiert wird.

Die Theoreme von Pappus und Guldinus

Die beiden von Pappus und Guldinus entwickelten Theoreme werden in der Mathematik, im Ingenieurwesen und in der Physik häufig verwendet, um die Oberfläche und das Volumen eines Körpers zu ermitteln ...

Education

Lehre

JoVE Core

Mechanical Engineering

Maschinenbau

Unit 1

Einheit 1

Center of Gravity and Centroid

Schwerpunkt und Flächenschwerpunkt

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

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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