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

Момент силы определяется как перекрестное произведение векторов положения и силы.
Перекрестное произведение на данный момент может быть выражено в определяющей форме с использованием единичных векторов и декартовой формы векторов положения и силы.
Определитель расширяется для определения формулировки декартова вектора момента силы.
Предположим, что сила в 10 Н приложена к вращающейся двери в точке, заданной вектором положения в виде декартовых компонент. Здесь необходимо определить момент силы в векторной формулировке.
В этом случае вектор силы перпендикулярен плоскости y-z , в результате чего компоненты y и z вектора силы равны нулю. Определитель строится с использованием вектора положения и вектора силы.
Наконец, путем разложения определителя определяется момент силы вращающейся двери вокруг оси z.
Здесь момент находится в положительном направлении z, что означает, что дверь вращается против часовой стрелки.

<ol>
	<li>Хиббелер, Р.К. (2016). Инженерная механика \u2012 Статика и динамика. Хобокен, Нью-Джерси: Пирсон Прентис Холл. С. 129.</li>
</ol>

Картезианская форма для векторной формулировки

Картезианская форма для векторной формулировки - это процесс расчета момента силы с использованием векторов положения и силы. Момент силы определяется как векторное произведение этих векторов, что делает его векторной величиной. Картезианская форма векторов положения и силы включает в себя базисные векторы, которые могут быть использованы для выражения векторного произведения в детерминантной форме.

<img alt="Equation 1" src="/files/ftp_upload/14254/14254_ Equation_1v2.svg" style="text-align: center; height: 60px;" />

Детерминант можно раскрыть, чтобы определить картезианскую векторную формулировку момента силы.

<img alt="Equation 2" src="/files/ftp_upload/14254/14254_ Equation_2v2.svg" style="text-align: center; height: 15px;" />

Например, предположим, что к вращающейся двери применяется сила 10 Н вдоль направления x в точке, заданной вектором положения r = 0i - 4j + 0k в картезианской компонентной форме. В этом случае вектор силы перпендикулярен плоскости y-z, на которой стоит дверь, что означает, что компоненты y и z вектора силы равны нулю. Чтобы рассчитать момент силы, конструируется детерминант, используя вектор положения и вектор силы, F = 10i + 0j + 0k.

<img alt="Equation 3" src="/files/ftp_upload/14254/14254_ Equation_3v2.svg" style="text-align: center; height: 60px;" />

Наконец, раскрыв детерминант, мы можем определить, что момент силы вращающейся двери равен 40 Н.м относительно оси z. Положительный момент означает, что дверь вращается против часовой стрелки.

Картезианская форма для векторной формулировки позволяет нам расчитать момент силы, используя векторы положения и силы простым способом.

Декартова форма векторной формулировки — это процесс вычисления момента силы с использованием положения и векторов силы. Момент силы – это...

Education

JoVE Core

Mechanical Engineering

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Unit 1

Раздел 1

Force System Resultants

Результирующие силы системы

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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

calculating-moment-with-cartesian-vectors

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cartesian form for vector formulation

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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Equivalent Force and Moment Systems

Structural systems often experience several forces and couple moments at the same time. Engineers simplify these loading cases by replacing them with an equivalent force and moment system at a chosen point O. This makes the analysis easier while keeping the same external effects on the member. A key idea in this process is the principle of transmissibility. A force is a sliding vector, which means it can be moved anywhere along its line of action without changing how the body behaves...

simplification of a force and couple system i

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simplification of a force and couple system: ii

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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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