Name: calculating moment with cartesian vectors
Uploaded: 2025-05-22T09:06:02.000Z
Duration: 1 min 26 s
Channel: JoVE

Das Kraftmoment ist definiert als das Kreuzprodukt von Orts- und Kraftvektoren.
Das Kreuzprodukt für den Moment kann in der Determinantenform ausgedrückt werden, unter Verwendung von Einheitsvektoren und der kartesischen Form von Orts- und Kraftvektoren.
Die Determinante wird erweitert, um die kartesische Vektorformulierung des Kraftmoments zu bestimmen.
Stellen Sie sich vor, dass eine Kraft von 10 N an einem Punkt auf eine Karusselltür ausgeübt wird, der durch den Positionsvektor in der Form der kartesischen Komponenten gegeben ist. Hier muss das Kraftmoment in der Vektorformulierung bestimmt werden.
In diesem Fall steht der Kraftvektor senkrecht zur y-z-Ebene , was dazu führt, dass die y- und z-Komponenten des Kraftvektors Null sind. Eine Determinante wird mit Hilfe des Ortsvektors und des Kraftvektors konstruiert.
Schließlich wird durch die Erweiterung der Determinante das Moment der Kraft um die z-Achse der Drehtür bestimmt.
Hier befindet sich der Moment in positiver z-Richtung, was bedeutet, dass sich die Tür gegen den Uhrzeigersinn dreht.

<ol>
	<li>Hibbeler, R.C. (2016). Technische Mechanik \u2012 Statik und Dynamik. Hoboken, New Jersey: Pearson Prentice Hall. Seite 129.</li>
</ol>

kartesische form für vektorformulierung

Die kartesische Form für die Vektorformulierung ist ein Verfahren zur Berechnung des Moments der Kraft unter Verwendung der Positions- und Kraftvektoren. Das Moment der Kraft wird als das Kreuzprodukt dieser Vektoren definiert, was es zu einer Vektorgröße macht. Die kartesische Form der Positions- und Kraftvektoren beinhaltet Einheitsvektoren, die verwendet werden können, um das Kreuzprodukt in Determinantenform auszudrücken. <img alt="Equation 1" src="/files/ftp_upload/14254/14254_ Equation_1v2.svg" style="text-align: center; height: 60px;"> Die Determinante kann dann erweitert werden, um die kartesische Vektorformulierung des Moments der Kraft zu bestimmen. <img alt="Equation 2" src="/files/ftp_upload/14254/14254_ Equation_2v2.svg" style="text-align: center; height: 15px;"> Angenommen, eine Kraft von 10 N wird in x-Richtung auf eine Drehtür an einem Punkt ausgeübt, der durch den Positionsvektor r = 0i - 4j + 0k in der kartesischen Komponentenform gegeben ist. In diesem Fall ist der Kraftvektor senkrecht zur y-z-Ebene, entlang der die Tür steht, was bedeutet, dass die y- und z-Komponenten des Kraftvektors null sind. Um das Moment der Kraft zu berechnen, wird eine Determinante mit dem Positionsvektor und dem Kraftvektor F = 10i + 0j + 0k konstruiert. <img alt="Equation 3" src="/files/ftp_upload/14254/14254_ Equation_3v2.svg" style="text-align: center; height: 60px;"> Schließlich können wir durch Erweitern der Determinante feststellen, dass das Moment der Kraft für die Drehtür 40 N.m um die z-Achse beträgt. Das positive Moment impliziert, dass sich die Tür gegen den Uhrzeigersinn dreht. Die kartesische Form für die Vektorformulierung ermöglicht es uns, das Moment der Kraft unter Verwendung von Positions- und Kraftvektoren auf einfache Weise zu berechnen.

Kartesische Form für Vektorformulierung

Die kartesische Form für die Vektorformulierung ist ein Verfahren zur Berechnung des Kraftmoments unter Verwendung der Orts- und Kraftvektoren. Das Moment der Kraft ist ...

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Cartesian Form for Vector Formulation

calculating moment with cartesian vectors

Calculating moment with Cartesian vectors shows how position and force vectors work together. The moment of force is the cross product of these vectors, so it is a vector quantity. In Cartesian form, unit vectors help write the vectors in component form.

The cross product can be arranged in determinant form. That determinant is then expanded to find the Cartesian vector form of the moment of force. This method gives a clear way to work through the calculation step by step.

A revolving door example shows the process in action. A force of 10 N is applied in the x-direction at the position vector r = 0 i - 4 j + 0 k. The force vector has no y or z components, so it can be written as F = 10 i + 0 j + 0 k.

Using these vectors, the determinant is built and expanded. The result is a moment of 40 N.m about the z-axis. The positive sign means the door rotates counterclockwise.

Calculating Moment with Cartesian Vectors

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Calculating moment with Cartesian vectors shows how position and force vectors work together. The moment of force is the cross product of these vectors, so it is a vector quantity. In Cartesian form, unit vectors help write the vectors in component form. The cross product can be arranged in determinant form. That determinant is then expanded to find the Cartesian vector form of the moment of force. This method gives a clear way to work through the calculation step by step. A revolving door...

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Learn how Cartesian vector form is used to calculate moment of force with unit vectors, determinants, and a revolving door example in a simple step-by-step method.

Calculating Moment with Cartesian Vectors | JoVE K12

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Calculating Moment with Cartesian Vectors

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