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

Рассмотрим трехмерный статический пример, например столб с тросом, прикрепленным к земле. Здесь вдоль кабеля действует сила растяжения.
Теперь рассмотрим декартову систему координат с произвольным началом координат, чтобы представить силу в виде декартова вектора.
PA иP B представляют векторы положения для двух концов кабеля. Вектор силы направлен из точки А в точку В в том же направлении, что и вектор положения PAB. Единичный вектор определяет это направление.
Применяя закон сложения векторов треугольника, вектор положения по точкам А и В может быть получен путем вычитания PA из PB.
Во-вторых, его величина может быть найдена с помощью квадратного корня из суммы квадратов его компонентов.
Теперь деление вектора положения на его величину дает единичный вектор вдоль кабеля AB.
Наконец, произведение величины вектора силы на единичный вектор выражает вектор силы в декартовой форме.

<ol>
	<li>Хиббелер, Р.К. (2016). Инженерная механика \u2012 Статика и динамика. Хобокен, Нью-Джерси: Пирсон Прентис Холл. С. 59</li>
</ol>

Сила вектора вдоль линии

Довольно часто в задачах трёхмерной статики направление силы определяется двумя точками, через которые проходит её линия действия. Рассмотрим трёхмерный статический столб с кабелем, закреплённым в земле.

<img alt="Equation 1" src="/files/ftp_upload/14237/14237_PT_Fig_1v2.svg" style="text-align: center; height: 350px;" />

Учитывая декартову систему координат с началом в основании столба, конечные точки кабеля можно обозначить как A и B. PA и PB представляют собой векторы положения для двух концов кабеля. Для получения вектора положения для точек A и B применяется правило сложения векторов по треугольнику. В этих целях вектор положения PA вычитают из вектора положения PB.

<img alt="Equation 1" src="/files/ftp_upload/14237/14237_PT_Equation_1v2.svg" style="text-align: center; height: 15px;" />

Величину вектора положения можно получить из квадратного корня из суммы квадратов его компонентов.

<img alt="Equation 2" src="/files/ftp_upload/14237/14237_PT_Equation_2v2.svg" style="text-align: center; height: 26px;" />

Сила натяжения, действующая на кабель, направлена от точки A к точке B в том же направлении, что и вектор положения PAB. Единичный вектор вдоль кабеля указывает направление силы. Его определяют путём деления вектора положения на его величину.

<img alt="Equation 3" src="/files/ftp_upload/14237/14237_PT_Equation_3v2.svg" style="text-align: center; height: 39px;" />

Наконец, вектор силы можно выразить в декартовой форме, умножив величину вектора силы на единичный вектор. Это произведение даёт трёхмерный вектор, представляющий силу, действующую на столб.

<img alt="Equation 4" src="/files/ftp_upload/14237/14237_PT_Equation_4v2.svg" style="text-align: center; height: 11px;" />

Довольно часто в трехмерных статических задачах направление силы определяется двумя точками, через которые проходит линия ее действия. Рассмотрим ...

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Mechanical Engineering

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Unit 1

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Force Vectors

Векторы силы

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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

scalar and vectors

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force vector along a line

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...

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key terms and concepts

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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