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

力矩定义为位置矢量与力矢量的叉乘积。
力矩的叉积可以用行列式形式表示，使用单位矢量以及位置矢量和力矢量的笛卡尔形式。
通过展开行列式，确定力矩的笛卡尔矢量表达式。
假设一个大小为 10 N 的力作用在旋转门上，作用点的位置矢量以笛卡尔分量形式给出。此处需要确定力矩的矢量表达式。
在这种情况下，力矢量垂直于 y-z 平面，导致力矢量的 y 和 z 分量为零。通过位置矢量和力矢量构建一个行列式。
最后，通过展开行列式，可确定旋转门绕 z 轴的力矩。
此处，力矩方向为正的 z 方向，表示门沿逆时针方向旋转。

<ol>
	<li>希伯勒，RC（2016 年）。工程力学 \u2012 静力学和动力学。 新泽西州霍博肯:Pearson Prentice Hall。第 129 页。</li>
</ol>

用于矢量形式的笛卡尔形式

笛卡尔形式的矢量公式是利用位置矢量和力矢量来计算力矩的过程。力矩被定义为这些矢量的叉积，使其成为向量。位置和力矢量的笛卡尔形式涉及到单位矢量，并且可以通过行列式的形式来表示叉积。 <img alt="Equation 1" src="/files/ftp_upload/14254/14254_ Equation_1v2.svg" style="text-align: center; height: 60px;"> 可以将行列式进行展开，并以此来确定力矩的笛卡尔矢量形式。 <img alt="Equation 2" src="/files/ftp_upload/14254/14254_ Equation_2v2.svg" style="text-align: center; height: 15px;">

例如，假设一个10N的力作用在一个绕x方向旋转的门上，给定的位置矢量r = 0i - 4j + 0k的笛卡尔分量形式。在这种情况下，力矢量垂直于门所位于的y-z平面，这意味着力矢量中的y和z分量都为零。为了计算处力矩，使用位置矢量和力矢量F = 10i + 0j + 0k来构造行列式。

<img alt="Equation 3" src="/files/ftp_upload/14254/14254_ Equation_3v2.svg" style="text-align: center; height: 60px;">

最后，通过将行列式进行展开，我们可以确定旋转门的力矩等于40 N.m关于z轴。正的力矩则意味着门是逆时针旋转的。 用于矢量形式的笛卡尔形式使我们能够直接的使用位置和力矢量来计算力矩。

矢量公式中的笛卡尔形式是一种利用位置矢量和力矢量计算力矩的方法。力矩是……

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Force System Resultants

力系合力

KEY TERMS AND CONCEPTS

关键术语与概念

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.

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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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