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

Рассмотрим столб, помещенный в трехмерную систему с подключенным кабелем. Когда к кабелю приложено напряжение 15 кН, определите момент вокруг оси z, проходящей через основание.
Момент можно определить двумя методами.
В первом методе рассчитайте проекцию силы вдоль единичного вектора и умножьте ее на величину силы, чтобы получить вектор силы.
Момент начала координат является перекрестным произведением вектора положения и силы. Момент вдоль оси z может быть получен путем скалярного произведения момента вокруг начала координат и единичного вектора вдоль оси z.
В качестве альтернативы можно разложить вектор силы на составляющие. Компоненты, расположенные вдоль осей y и z, не проявляют момента, когда они проходят через ось z и параллельны ей, соответственно.
Напряжение в направлении x может быть получено путем умножения напряжения на косинус направления относительно оси x.
Вспомните уравнение момента силы, и, подставив члены, можно определить момент вокруг оси z.

<ol>
	<li>Хиббелер, Р.К. (2016). Инженерная механика \u2012 Статика и динамика. Хобокен, Нью-Джерси: Пирсон Прентис Холл. С. 133-138</li>
	<li>Бир, Ф..; Джонстон, Э.Р.; Мазурек, Д.Ф; Кромвель,.Д. и Селф, Б.. (2019). Векторная механика для инженеров \u2012 Статика и динамика. Нью-Йорк: Макгроу-Хилл. С. 96 -99</li>
	<li>Мериам, Дж.Л.; Крайге, Л.Г. и Болтон, Д.Н. (2020). Инженерная механика \u2012 Статика. Хобокен, Нью-Джерси: Джон Уайли. С. 78 и 79</li>
</ol>

Принцип моментов: решение задач

Принцип моментов - это основное понятие в физике и инженерии. Он относится к уравновешиванию сил и моментов вокруг точки или оси, также известной как ось вращения. Этот принцип используется во многих реальных ситуациях, включая строительство, спорт и повседневные действия, такие как открывание дверей и толкание предметов.

Одна из таких ситуаций связана с столбом, размещенным в трехмерной системе с кабелем. Когда кабелю приложено натяжение, требуется определить момент относительно оси z, проходящей через основание. Есть два способа решить эту задачу.

Первый способ заключается в вычислении проекции силы вдоль единичного вектора и умножении его на величину силы для получения вектора силы. Момент относительно начала координат можно вычислить, взяв векторное произведение вектора положения и вектора силы. После определения вектора момента, момент вдоль оси z можно вычислить, взяв скалярное произведение момента относительно начала координат и единичного вектора вдоль оси z.

Альтернативно, разложите вектор силы на его компоненты. Компоненты вдоль оси y и оси z не создают момента, так как они проходят через ось z или параллельны ей, соответственно. Натяжение в направлении оси x можно рассчитать, умножив натяжение на косинус направления относительно оси x. Затем используйте уравнение момента силы и подставьте соответствующие значения, чтобы определить момент относительно оси z.

Важно отметить, что принцип моментов критичен для понимания поведения конструкций и машин. Инженеры и дизайнеры широко используют его, чтобы обеспечить стабильность, безопасность и эффективность своих творений. Путем уравновешивания сил и моментов вокруг оси вращения они могут рассчитать напряжения и деформации, которым подвергается конструкция или машина, и выявить потенциальные точки отказа.

В целом, принцип моментов - это мощный инструмент, который помогает решать проблемы, связанные с силами, моментами и движением. Независимо от того, речь идет о строительстве мостов, анализе спортивных техник или простом открытии двери, можно положиться на этот принцип, чтобы убедиться, что все делается правильно и эффективно.

Принцип мгновений является фундаментальным понятием в физике и технике. Это относится к уравновешиванию сил и моментов вокруг точки или оси, ...

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Mechanical Engineering

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Unit 1

Раздел 1

Force System Resultants

Результирующие силы системы

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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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Torque on a Flywheel: Direction and Size

Torque, or the moment of a force, describes how a force can make an object turn about an axis. It is a vector quantity, so it has both size and direction. This idea is important in physics, mechanics, and engineering. A simple flywheel example shows how torque works. The flywheel rotates about point O when a force is applied to it. The moment arm is the perpendicular distance from point O to the line of action of the force. The moment, in newton-meter (N-m), is found by multiplying the force...

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Net Moment Sign Convention in 2D Forces

Net moment in two-dimensional force systems shows how multiple forces can cause rotation about a fixed point. The scalar formulation helps analyze equilibrium in structures that are acted on by several forces. It focuses on the x-y plane and uses a simple sign rule to describe the direction of rotation. To find the resultant moment, the moments from all forces in the system are considered about a chosen point A. Each moment is assigned a sign based on its sense of rotation. Positive moments...

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cartesian form for vector formulation

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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...

resultant moment: vector formulation

Calculating Torque from Multiple Forces

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).

principle of moments

Resolving Forces in Moment Calculations

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...

moment of a force about an axis: scalar

Bicycle Torque and Moment About an Axis

The moment of a force about an axis describes how a force can make an object turn around a fixed line. In scalar analysis, the moment is found from the perpendicular distance between the axis of rotation and the line of action of the force. That distance is called the moment arm. A bicycle gives a clear example of this idea. When the cyclist pushes on the pedal, the chain applies a force to the wheel’s axle and makes it rotate. The axis of rotation passes through the axle, and the moment arm...

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Torque on a Bicycle Wheel: Vector Method

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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Couple Moment in Force Systems

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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Couple Moments and the Right-Hand Rule

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Equivalent Couples in Mechanical Systems

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Torque from a Couple: Solving Forces

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Equivalent Force and Moment Systems

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...

simplification of a force and couple system i

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Resultant Forces in Force and Couple Systems

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simplification of a force and couple system: ii

simplification-of-a-force-and-couple-system-ii

Resultant Force and Wrench Motion

Resultant force and resultant couple moment describe how several forces and moments act together on a rigid body in three dimensions. When many forces act on an object, they can be replaced by one equivalent force. The moments from those forces can also be combined into one equivalent couple moment. In some cases, the resultant force and the resultant couple moment are not mutually perpendicular. That means the angle between them is not 90 degrees. To study this situation, the couple moment is...

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Load Distribution on Shelves and Dams

Load distribution helps engineers describe how force spreads across a surface or structure. A distributed load is a continuous force per unit area, rather than a single point force. It is a common idea in many practical situations. A bookshelf full of stacked books is one simple example. The weight of the books is spread along the shelf, so the pressure changes across the surface. This pressure is measured in Newtons per square meter, or Pascals. A hydroelectric dam gives another clear...

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Structural engineering often requires finding the resultant load on a beam with changing force patterns. Here, a beam of length L carries a varying load made from parabolic and trapezoidal load distributions along the x-axis. The goal is to find the total load, where it acts, and the centroid of the combined area so the beam’s response can be predicted. The parabolic load is handled first. A small force element, dR, acts over a short distance, dx, and these elements are integrated across the...

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Equivalent Resultant Loads on Beams

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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

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Torque About an Axis in 3D Systems

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