Name: finding a cable force from two points
Uploaded: 2025-05-22T09:06:01.000Z
Duration: 1 min 26 s
Channel: JoVE

考虑一个三维静态示例，例如一根杆件通过缆索固定在地面上。此时，张力沿缆索方向作用。
现在，考虑一个以任意原点建立的笛卡尔坐标系，用以将力表示为笛卡尔矢量。
PA 和 PB 表示缆绳两端的位置矢量。力矢量沿位置矢量 PAB 的方向，从 A 点指向 B 点。单位矢量用于表示该方向。
根据矢量加法的三角形法则，可通过从 PB 减去 PA 得到沿 A 点和 B 点的位置矢量。
其次，其大小可通过各分量平方和的平方根来求得。
现在，将位置矢量除以其模长，可得到沿缆绳 AB 方向的单位矢量。
最后，力矢量的大小与单位矢量的乘积以笛卡尔形式表示该力矢量。

<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 59 页</li>
</ol>

沿线的力向量

在三维静力学问题中，往往通过穿过两点的直线来指定其力的方向。思考一个三维的静态杆，同时并将一根电缆固定在地面上。 <img alt="Equation 1" src="/files/ftp_upload/14237/14237_PT_Fig_1v2.svg" style="text-align: center; height: 350px;"> 思考一个笛卡尔坐标系，其原点在支杆的底部，电缆的端点可以表示为A和B。PA和PB分别代表了电缆两端的位置向量。使用向量相加的三角形法则可以得到沿A点和B点的位置向量。为此，需要从PB中减去PA。 <img alt="Equation 1" src="/files/ftp_upload/14237/14237_PT_Equation_1v2.svg" style="text-align: center; height: 15px;"> 位置向量的大小可以通过对其分量进行平方和的平方根的计算而得出。 <img alt="Equation 2" src="/files/ftp_upload/14237/14237_PT_Equation_2v2.svg" style="text-align: center; height: 26px;"> 电缆上的张力沿着点A到点B的方向，该张力遵循了与位置向量PAB相同的方向。沿着电缆的单位向量指向了力的方向。它通过将位置向量与其大小进行相除而得到。 <img alt="Equation 3" src="/files/ftp_upload/14237/14237_PT_Equation_3v2.svg" style="text-align: center; height: 39px;"> 最后，力向量可以用力向量的大小乘以单位向量的笛卡尔形式来进行表示。可以通过该乘积得出一个三维向量，以此来表示作用在杆上的力。 <img alt="Equation 4" src="/files/ftp_upload/14237/14237_PT_Equation_4v2.svg" style="text-align: center; height: 11px;">

在三维静力学问题中，力的方向通常由其作用线通过的两点来确定。考虑一个……

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Force Vectors

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Force Vector along a Line

finding a cable force from two points

A cable force can be found from the two points that define its line of action. In three-dimensional statics, this setup often appears with a pole and a cable anchored to the ground. The cable ends are labeled A and B, and their position vectors are written as P_A and P_B.

The position vector from A to B is found using the triangle law of vector addition. First, subtract P_A from P_B to get P_AB. Then find the magnitude of P_AB by taking the square root of the sum of the squares of its components. This gives the length of the vector in Cartesian form.

The tension force in the cable points from A toward B. It has the same direction as the position vector P_AB. To show that direction, divide P_AB by its magnitude to get the unit vector along the cable. A unit vector is a vector with length 1 that gives direction only.

The force vector is written in Cartesian form by multiplying the force magnitude by the unit vector. This product gives a three-dimensional vector for the force acting on the pole. The method connects the geometry of points A and B to the final force vector.

Finding a Cable Force from Two Points

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Learn how to use two position vectors, a unit vector, and force magnitude to express a cable’s tension as a 3D Cartesian force vector.

Finding a Cable Force from Two Points | JoVE K12

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Finding a Cable Force from Two Points

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