Name: torque about an axis in 3d systems
Uploaded: 2023-09-22T11:49:21.000Z
Duration: 1 min 30 s
Channel: JoVE

Stellen Sie sich einen Mast vor, der in einem 3-dimensionalen System platziert ist und an dem ein Kabel befestigt ist. Wenn eine Spannung von 15 kN auf das Kabel ausgeübt wird, bestimmen Sie das Moment, das um die Z-Achse verläuft, die durch den Sockel verläuft.
Der Zeitpunkt kann mit zwei Methoden bestimmt werden.
Berechnen Sie bei der ersten Methode die Projektion der Kraft entlang des Einheitsvektors und multiplizieren Sie sie mit der Größe der Kraft, um den Kraftvektor zu erhalten.
Das Moment um den Ursprung ist das Kreuzprodukt aus dem Ortsvektor und der Kraft. Das Moment entlang der z-Achse kann durch das Punktprodukt des Moments um den Ursprung und den Einheitsvektor entlang der z-Achse erhalten werden.
Alternativ können Sie den Kraftvektor in seine Bestandteile auflösen. Die Komponenten entlang der y-Achse und der z-Achse üben beim Durchgang kein Moment aus und sind parallel zur z-Achse.
Die Spannung in x-Richtung erhält man, indem man die Spannung mit dem Richtungskosinus in Bezug auf die x-Achse multipliziert.
Rufen Sie die Kraftmomentgleichung auf, und durch Ersetzen der Terme kann das Moment um die z-Achse bestimmt werden.

<ol>
	<li>Hibbeler, R.C. (2016). Technische Mechanik \u2012 Statik und Dynamik. Hoboken, New Jersey: Pearson Prentice Hall. Seiten 133-138</li>
	<li>Bier, F.P.; Johnston, E.R.; Mazurek, D.F.; Cromwell, P.J. und Self, B.P. (2019). Vektormechanik für Ingenieure \u2012 Statik und Dynamik. New York: McGraw-Hill. Seiten 96 -99</li>
	<li>Meriam, J.L.; Kraige, L.G. und Bolton, J.N. (2020). Technische Mechanik \u2012 Statik. Hoboken, New Jersey: John Wiley. S. 78 und 79</li>
</ol>

prinzip der momente: problemlösung

Das Prinzip der Momente ist ein grundlegendes Konzept in der Physik und Ingenieurwissenschaft. Es bezieht sich auf das Ausbalancieren von Kräften und Momenten um einen Punkt oder eine Achse, auch bekannt als Drehpunkt. Dieses Prinzip wird in vielen realen Szenarien angewendet, einschließlich Bauwesen, Sport und täglichen Aktivitäten wie dem Öffnen von Türen und dem Drücken von Gegenständen. Ein solches Szenario beinhaltet eine Stange, die in ein dreidimensionales System mit einem angeschlossenen Kabel gesteckt wird. Wenn eine Spannung auf das Kabel ausgeübt wird, muss das Moment um die z-Achse, die durch die Basis verläuft, bestimmt werden. Es gibt zwei Methoden, um dieses Problem anzugehen. Die erste Methode beinhaltet die Berechnung der Projektion der Kraft entlang des Einheitsvektors und die Multiplikation mit der Magnitude der Kraft, um den Kraftvektor zu erhalten. Das Moment bezüglich des Ursprungs kann berechnet werden, indem das Kreuzprodukt des Positionsvektors und des Kraftvektors genommen wird. Sobald der Momentenvektor bestimmt ist, kann das Moment entlang der z-Achse bewertet werden, indem das Skalarprodukt des Moments bezüglich des Ursprungs und des Einheitsvektors entlang der z-Achse genommen wird. Alternativ kann der Kraftvektor in seine Komponenten aufgelöst werden. Die Komponenten entlang der y-Achse und der z-Achse üben kein Moment aus, da sie jeweils durch die z-Achse verlaufen bzw. dieser parallel sind. Die Spannung in Richtung x kann berechnet werden, indem die Spannung mit dem Richtungskosinus bezüglich der x-Achse multipliziert wird. Dann verwendet man die Momentengleichung und setzt die entsprechenden Begriffe ein, um das Moment um die z-Achse zu bestimmen. Es ist wichtig anzumerken, dass das Prinzip der Momente entscheidend ist, um das Verhalten von Strukturen und Maschinen zu verstehen. Ingenieure und Designer verwenden es ausgiebig, um sicherzustellen, dass ihre Kreationen stabil, sicher und effektiv sind. Durch das Ausbalancieren von Kräften und Momenten um einen Drehpunkt können sie die Spannungen und Dehnungen berechnen, denen eine Struktur oder Maschine ausgesetzt sein wird, und potenzielle Fehlerstellen identifizieren. Insgesamt ist das Prinzip der Momente ein leistungsstolles Werkzeug, das hilft, Probleme im Zusammenhang mit Kräften, Drehmomenten und Bewegung zu lösen. Ob es darum geht, Brücken zu bauen, Sporttechniken zu analysieren oder einfach eine Tür zu öffnen, man kann sich auf dieses Prinzip verlassen, um sicherzustellen, dass Dinge korrekt und effizient erledigt werden.

Prinzip der Momente: Problemlösung

Das Prinzip der Momente ist ein grundlegendes Konzept in der Physik und im Ingenieurwesen. Es bezieht sich auf das Ausbalancieren von Kräften und Momenten um einen Punkt oder eine Achse, ...

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Principle of Moments: Problem Solving

torque about an axis in 3d systems

The principle of moments helps solve torque problems in physics and engineering. It describes how forces and moments balance around a pivot point or axis. It is used in many real-life cases, including construction, sports, and everyday actions like opening doors and pushing objects.

The problem here uses a pole in a three-dimensional system with a cable attached. When tension acts in the cable, the goal is to find the moment about the z-axis through the base. One method is to project the force along a unit vector to build the force vector. The moment about the origin is then found with the cross product of the position vector and the force vector. After that, the moment along the z-axis is found by taking the dot product of the moment vector with the z-axis unit vector.

A second method is to split the force into components. The y-axis and z-axis components create no moment in this setup. The y-axis component passes through the axis, and the z-axis component is parallel to it. The x-direction tension is found by multiplying the cable tension by the direction cosine with respect to the x-axis. Then the moment of force equation is used to determine the moment about the z-axis.

This kind of analysis is important in structures and machines. Engineers and designers use the principle of moments to check stability, safety, and performance. It helps them estimate the stresses and strains a structure or machine may face and spot possible failure points. The same idea also appears in bridges, sports technique, and simple tasks like opening a door.

Torque About an Axis in 3D Systems

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Resultant moment describes the total turning effect of multiple forces acting on an object. It shows how several forces can combine to make a body rotate about a point. In physics, moment is the tendency of a force to cause rotation. The resultant moment of a force system is a vector quantity. That means it can be added and subtracted like other vectors. For a system with two forces, F1 and F2, acting on a pole at points A and B, the resultant moment is written as MR = (rA x F1) + (rB x F2).

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The principle of moments, also called Varignon's theorem, explains how a force creates rotation about a point. It is a key idea in physics and engineering. It also helps describe when a rigid body stays in equilibrium under external forces. The moment of a force is found by multiplying the force by the perpendicular distance from the point of application to the point being studied. A larger force or a larger distance gives a larger moment. This makes the idea useful for checking how strongly a...

principle of moments: problem solving

The principle of moments helps solve torque problems in physics and engineering. It describes how forces and moments balance around a pivot point or axis. It is used in many real-life cases, including construction, sports, and everyday actions like opening doors and pushing objects. The problem here uses a pole in a three-dimensional system with a cable attached. When tension acts in the cable, the goal is to find the moment about the z-axis through the base. One method is to project the force...

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Torque, or the moment of a force, explains how a force can make an object rotate about an axis. A bicycle is a clear example. When a cyclist presses on the pedal, the crankshaft turns and helps drive the wheel. To analyze this motion, a coordinate system is set up first. The x-axis is tangential to the wheel, the y-axis is perpendicular to the x-axis and points upward, and the z-axis points outward from the wheel. In this setup, the force on the wheel axle lies along the x-axis. The axle...

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A couple is a pair of parallel forces that are equal in magnitude and opposite in direction. The two forces are separated by a perpendicular distance called the couple arm. Together, they create a turning effect on a body instead of a net push. This turning effect is called the couple moment. It rotates the body about an axis that is perpendicular to the plane of the forces. The SI unit for a couple moment is the Newton-meter (N-m). A common example is tightening a bolt with a lug wrench. In...

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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...

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Resultant Force and Wrench Motion

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key terms and concepts

Learn how the principle of moments is used to find torque about the z-axis in a 3D cable-and-pole setup using cross products and force components.

Torque About an Axis in 3D Systems | JoVE K12

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Torque About an Axis in 3D Systems

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