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

Stellen Sie sich ein dreidimensionales statisches Beispiel vor, z. B. einen Mast mit einem Kabel, das im Boden verankert ist. Hierbei wirkt eine Zugkraft entlang des Seils.
Betrachten Sie nun ein kartesisches Koordinatensystem mit einem beliebigen Ursprung, um die Kraft als kartesischen Vektor darzustellen.
PA und PB stellen die Positionsvektoren für die beiden Enden des Kabels dar. Der Kraftvektor wird von Punkt A nach B in der gleichen Richtung wie der Ortsvektor PAB gerichtet. Der Einheitsvektor gibt diese Richtung an.
Unter Anwendung des Dreiecksgesetzes der Vektoraddition kann der Ortsvektor entlang der Punkte A und B durch Subtraktion von PA von PB erhalten werden.
Zweitens kann seine Größe anhand der Quadratwurzel der Summe der Quadrate seiner Komponenten bestimmt werden.
Wenn man nun den Positionsvektor durch seine Größe dividiert, erhält man den Einheitsvektor entlang des Kabels AB.
Schließlich drückt das Produkt aus der Größe des Kraftvektors und dem Einheitsvektor den Kraftvektor in der kartesischen Form aus.

<ol>
	<li>Hibbeler, R.C. (2016). Technische Mechanik \u2012 Statik und Dynamik. Hoboken, New Jersey: Pearson Prentice Hall. Seite 59</li>
</ol>

kraftvektor entlang einer linie

Häufig wird in dreidimensionalen Statikproblemen die Richtung einer Kraft durch zwei Punkte angegeben, durch die ihre Wirkungslinie verläuft. Betrachten Sie einen dreidimensionalen statischen Mast mit einem zum Boden verankerten Seil. <img alt="Equation 1" src="/files/ftp_upload/14237/14237_PT_Fig_1v2.svg" style="text-align: center; height: 350px;"> Unter Betrachtung eines kartesischen Koordinatensystems mit dem Ursprung an der Mastbasis können die Endpunkte des Seils als A und B bezeichnet werden. PA und PB repräsentieren die Positionsvektoren für die beiden Enden des Seils. Das Dreiecksgesetz der Vektoraddition wird verwendet, um den Positionsvektor entlang der Punkte A und B zu erhalten. Zu diesem Zweck wird der Positionsvektor PA von PB subtrahiert. <img alt="Equation 1" src="/files/ftp_upload/14237/14237_PT_Equation_1v2.svg" style="text-align: center; height: 15px;"> Die Größe des Positionsvektors kann aus der Quadratwurzel der Summe der Quadrate seiner Komponenten erhalten werden. <img alt="Equation 2" src="/files/ftp_upload/14237/14237_PT_Equation_2v2.svg" style="text-align: center; height: 26px;"> Die Spannkraft, die auf das Seil wirkt, ist von Punkt A in Richtung Punkt B gerichtet, folgend der gleichen Richtung wie der Positionsvektor PAB. Der Einheitsvektor entlang des Seils gibt die Richtung der Kraft an. Er wird bewertet, indem der Positionsvektor durch seine Größe geteilt wird. <img alt="Equation 3" src="/files/ftp_upload/14237/14237_PT_Equation_3v2.svg" style="text-align: center; height: 39px;"> Zuletzt kann der Kraftvektor in der kartesischen Form ausgedrückt werden, indem die Größe des Kraftvektors und der Einheitsvektor multipliziert werden. Dieses Produkt ergibt einen dreidimensionalen Vektor, der die auf den Mast wirkende Kraft repräsentiert. <img alt="Equation 4" src="/files/ftp_upload/14237/14237_PT_Equation_4v2.svg" style="text-align: center; height: 11px;">

Kraftvektor entlang einer Linie

In dreidimensionalen Statikproblemen wird die Richtung einer Kraft häufig durch zwei Punkte angegeben, durch die ihre Wirkungslinie verläuft. Betrachten Sie eine ...

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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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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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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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