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

Moment siły jest definiowany jako iloczyn poprzeczny wektorów położenia i siły.
Iloczyn wektorowy na chwilę można wyrazić w postaci wyznacznika, za pomocą wektorów jednostkowych oraz kartezjańskiej postaci wektorów położenia i siły.
Wyznacznik jest rozszerzony w celu wyznaczenia wektora kartezjańskiego sformułowania momentu siły.
Załóżmy, że siła 10 N jest przyłożona do drzwi obrotowych w punkcie wyznaczonym przez wektor położenia w postaci składowych kartezjańskich. W tym miejscu należy określić moment siły w sformułowaniu wektorowym.
W tym przypadku wektor siły jest prostopadły do płaszczyzny y-z , co powoduje, że składowe y i z wektora siły wynoszą zero. Wyznacznik jest konstruowany za pomocą wektora położenia i wektora siły.
Wreszcie, rozszerzając wyznacznik, określa się moment siły działającej na drzwi obrotowe względem osi z.
W tym przypadku moment jest w dodatnim kierunku z, co oznacza, że drzwi obracają się w kierunku przeciwnym do ruchu wskazówek zegara.

<ol>
	<li>Hibbeler, R.C. (2016). Mechanika inżynierska \u2012 Statyka i dynamika. Hoboken, New Jersey: Pearson Prentice Hall. str. 129.</li>
</ol>

postać kartezjańska dla formułowania wektorów

Postać kartezjańska dla formułowania wektorów polega na obliczaniu momentu siły za pomocą wektorów położenia i siły. Moment siły jest definiowany jako iloczyn wektorowy tych wektorów, co powoduje, że jest to wielkość wektorowa. Kartezjańska postać wektorów położenia i siły wykorzystuje wektory jednostkowe, które mogą być używane do wyrażenia iloczynu wektorowego w postaci wyznacznikowej.

<img alt="Equation 1" src="/files/ftp_upload/14254/14254_ Equation_1v2.svg" style="text-align: center; height: 60px;" />

Następnie wyznacznik może zostać rozwinięty, aby określić kartezjańską postać wektorową momentu siły.

<img alt="Equation 2" src="/files/ftp_upload/14254/14254_ Equation_2v2.svg" style="text-align: center; height: 15px;" />

Na przykład, załóżmy, że na obracające się drzwi działa siła o wartości 10 N wzdłuż kierunku x w punkcie określonym przez wektor położenia r = 0i - 4j + 0k w postaci kartezjańskiej. W tym przypadku wektor siły jest prostopadły do płaszczyzny y-z, wzdłuż której stoją drzwi, co oznacza, że składowe y i z wektora siły wynoszą zero. Aby obliczyć moment siły, konstruowany jest wyznacznik z użyciem wektora położenia i wektora siły, F = 10i + 0j + 0k.



<img alt="Equation 3" src="/files/ftp_upload/14254/14254_ Equation_3v2.svg" style="text-align: center; height: 60px;" />

Ostatecznie, rozwijając wyznacznik, możemy ustalić, że moment siły obrotowych drzwi wynosi 40 N·m względem osi z. Dodatni moment oznacza, że drzwi obracają się w kierunku przeciwnym do ruchu wskazówek zegara.

Postać kartezjańska dla formułowania wektorów pozwala nam na prosty sposób obliczania momentu siły za pomocą wektorów położenia i siły.

Postać kartezjańska dla formułowania wektorów

Forma kartezjańska do formułowania wektorów to proces obliczania momentu siły przy użyciu wektorów położenia i siły. Momentem siły jest ...

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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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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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