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

Il momento della forza è definito come il prodotto incrociato dei vettori di posizione e forza.
Il prodotto incrociato per il momento può essere espresso nella forma determinante, utilizzando vettori unitari e la forma cartesiana dei vettori di posizione e forza.
Il determinante viene espanso per determinare la formulazione vettoriale cartesiana del momento della forza.
Si consideri che una forza di 10 N viene applicata ad una porta girevole in un punto dato dal vettore di posizione nella forma cartesiana delle componenti. Qui, il momento della forza nella formulazione del vettore deve essere determinato.
In questo caso, il vettore di forza è perpendicolare al piano y-z , il che fa sì che le componenti y e z del vettore di forza siano zero. Un determinante viene costruito utilizzando il vettore di posizione e il vettore di forza.
Infine, espandendo il determinante, viene determinato il momento della porta girevole della forza attorno all'asse z.
Qui, il momento è nella direzione z positiva, il che implica che la porta ruota in senso antiorario.

<ol>
	<li>Hibbeler, R.C. (2016). Ingegneria Meccanica \u2012 Statica e Dinamica. Hoboken, New Jersey: Pearson Prentice Hall. pagine 129.</li>
</ol>

forma cartesiana per la formulazione vettoriale

La forma cartesiana per la formulazione vettoriale, è un processo per calcolare il momento della forza utilizzando i vettori posizione e forza. Il momento della forza è definito come il prodotto vettoriale di questi vettori, rendendolo una grandezza vettoriale. La forma cartesiana dei vettori posizione e forza, coinvolge i vettori unitari, che possono essere utilizzati per esprimere il prodotto vettoriale in forma di determinante.

<img alt="Equation 1" src="/files/ftp_upload/14254/14254_ Equation_1v2.svg" style="text-align: center; height: 60px;" />

Il determinante può quindi essere espanso per definire la formulazione vettoriale cartesiana del momento della forza.

<img alt="Equation 2" src="/files/ftp_upload/14254/14254_ Equation_2v2.svg" style="text-align: center; height: 15px;" />

Ad esempio, supponiamo che una forza di 10 N venga applicata a una porta girevole lungo la direzione x in un punto dato dal vettore posizione r = 0i - 4j + 0k nella forma delle componenti cartesiane. In questo caso, il vettore forza è perpendicolare al piano y-z sul quale si trova la porta, il che significa che le componenti y e z del vettore forza sono zero. Per calcolare il momento della forza, viene costruito un determinante utilizzando il vettore posizione e il vettore forza, F = 10i + 0j + 0k.

<img alt="Equation 3" src="/files/ftp_upload/14254/14254_ Equation_3v2.svg" style="text-align: center; height: 60px;" />

Infine, espandendo il determinante, possiamo determinare che il momento della forza della porta girevole è di 40 N.m sull'asse z. Il momento positivo implica che la porta gira in senso antiorario.

La forma cartesiana per la formulazione vettoriale ci permette di calcolare il momento della forza utilizzando i vettori posizione e forza in modo diretto.

Forma cartesiana per la formulazione vettoriale

La forma cartesiana per la formulazione vettoriale è un processo per calcolare il momento della forza utilizzando i vettori posizione e forza. Il momento della forza è ...

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Mechanical Engineering

Ingegneria Meccanica

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Force System Resultants

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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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key terms and concepts

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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