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

ケーブルが取り付けられた3次元システムにポールが配置されたとします。15 kNの張力がケーブルに印加されたら、Z軸がベースを通過するモーメントを決定します。
モーメントは、2つの方法を使用して決定できます。
最初の方法では、単位ベクトルに沿った力の投影を計算し、それに力の大きさを掛けて力ベクトルを求めます。
原点に関するモーメントは、位置ベクトルと力の外積です。z軸に沿ったモーメントは、原点を中心としたモーメントとz軸に沿った単位ベクトルのドット積によって取得できます。
または、力ベクトルをそのコンポーネントに解決します。y軸とz軸に沿ったコンポーネントは、それぞれz軸を通過して平行になるときにモーメントを及ぼしません。
x方向の張力は、張力とx軸に対するcosine方向を掛けることによって得ることができます。
力のモーメント方程式を思い出し、項を代入することにより、z軸の周りのモーメントを決定できます。

<ol>
	<li>ヒベラー、RC(2016)。エンジニアリングメカニクス \u2012 静力学と力学。 ニュージャージー州ホーボーケン:ピアソンプレンティスホール。133-138ページ</li>
	<li>ビール、FP;ジョンストン、ER;マズレック、D.F;クロムウェル、PJおよびセルフ、BP(2019)。エンジニアのためのベクトル力学 \u2012 静力学と動力学。ニューヨーク:マグロウヒル。96-99ページ</li>
	<li>メリアム、J.L.;Kraige、LGおよびBolton、JN(2020)。エンジニアリングメカニクス \u2012 静力学。ニュージャージー州ホーボーケン:ジョン・ワイリー。78 および 79</li>
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モーメントの原理：問題解決

モーメントの原理は、物理学や工学における基本的な概念です。これは、ポイントまたは軸、つまりピボット周りでの力とモーメントのバランスを指します。この原理は、建築、スポーツ、ドアの開閉や物体の押し込みといった日常の活動を含む、多くの現実のシナリオで使用されています。

そのようなシナリオの一つに、ケーブルが取り付けられた三次元システムに設置されたポールがあります。ケーブルに張力がかかった場合、基盤を通るz軸周りのモーメントを求める必要があります。この問題には2つの方法があります。

第一の方法は、単位ベクトルに沿った力の射影を計算し、それに力の大きさを掛けることで力ベクトルを求める方法です。原点周りのモーメントは、位置ベクトルと力ベクトルの外積を取ることで計算できます。モーメントベクトルが求められたら、原点周りのモーメントとz軸に沿った単位ベクトルの内積を取ることで、z軸に沿ったモーメントを評価できます。

または、力ベクトルをその成分に分解することもできます。y軸およびz軸に沿った成分は、z軸を通過し、それぞれ平行ですので、モーメントは発生しません。x方向の張力は、x軸に対する方向余弦と張力を掛けて計算できます。それから、力のモーメントの式を使用し、適切な項目を代入することで、z軸周りのモーメントを求めることができます。

重要なことは、モーメントの原理が構造や機械の振る舞いを理解する上で重要であるということです。エンジニアやデザイナーは、構造物や機械が安定、安全、効果的であることを確認するために、これを広く使用しています。ピボットポイントを中心に力とモーメントを平衡化させることで、構造物や機械が受ける応力と歪みを計算し、潜在的な故障点を特定することができます。

全体的に、モーメントの原理は、力、トルク、および運動に関連する問題を解決するのに役立つ強力なツールです。橋の建設、スポーツ技術の分析、あるいは単純なドアの開閉に関することでも、この原理に頼ることで正確かつ効率的な解決策を確保することができます。

モーメントの原理は、物理学と工学の基本的な概念です。これは、点または軸の周りの力とモーメントのバランスを指します...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Force System Resultants

力の系の合力

KEY TERMS AND CONCEPTS

主要用語と概念

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

moment of a force: scalar formulation

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...

moment of a force: problem solving

Calculating Force Moments in Problem Solving

Calculating force moments in problem solving helps students analyze rotation in mechanical systems. The scalar form of the moment of a force is used to work through these problems step by step. Start by drawing a free-body diagram, or FBD. An FBD is a sketch that shows every force acting on the system. Each force must be identified and included. Next, identify the axis or point about which the moment is calculated. In some problems, that axis or point is given. In other cases, you must choose...

resultant moment: scalar formulation

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...

moment of a force: vector formulation

Moment of Force and Direction Rules

The moment of force describes the twisting effect a force can have on an object. It shows why a force may cause rotation instead of straight-line motion. The moment arm is the perpendicular distance from the force’s line of action to the axis of rotation. The moment of force is a vector quantity, not a scalar. Its vector form comes from the cross product of the position vector and the force vector. A cross product gives a vector that is perpendicular to both input vectors. The moment about a...

cartesian form for vector formulation

Calculating Moment with Cartesian Vectors

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...

moment of a force about an axis: vector

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...

couple

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...

couples: scalar and vector formulation

Couple Moments and the Right-Hand Rule

Couple moments describe the turning effect of two parallel forces that are equal in magnitude and opposite in sense. This force pair creates rotation instead of straight-line motion. A common example is a ship’s steering wheel, where a small turn can guide the ship through the ocean. A couple moment can rotate clockwise or anticlockwise. The right-hand rule helps identify its direction. To use it, curl the fingers of your right hand in the direction of rotation. Your thumb points in the...

equivalent couples

Equivalent Couples in Mechanical Systems

Equivalent couples in mechanical systems describe pairs of forces that create the same rotational effect on a rigid body. Two couples are equivalent when they have the same moment magnitude and act in the same direction. They then cause the same angular displacement or angular acceleration in the body. A couple is made of equal and opposite forces separated by a perpendicular distance. The moment of the couple depends on both the force size and the spacing between the forces. For example, one...

moment of a couple: problem solving

Torque from a Couple: Solving Forces

The moment of a couple is a key physics and engineering idea for finding torque, or rotational force. A couple is made of two equal and opposite forces that act on an object. When those forces work together, they create a twisting effect that can rotate the object. The moment of a couple is calculated by multiplying the size of one force by the perpendicular distance between the two lines of action. That distance is the shortest straight-line distance between the force directions. This gives...

system of forces and couples

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

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

Resultant Forces in Force and Couple Systems

Force and couple systems can be reduced to a simpler equivalent system. This makes rigid body analysis easier because the external effect of the forces is captured with fewer elements. When the resultant force and resultant couple moment have perpendicular lines of action, the system may be reduced further to a single resultant force along a new line of action. Three common force systems can be simplified this way: concurrent, coplanar, and parallel. A concurrent force system has lines of...

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...

distributed loads

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...

load along a single axis

Finding Resultant Loads on a Beam

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...

distributed loads: problem solving

Equivalent Resultant Loads on Beams

Equivalent resultant loads on beams can be found by breaking a distributed load into smaller regions. This makes the beam easier to analyze. Each region can then be handled on its own before the results are combined. The magnitude of the equivalent resultant load for each region comes from the area of that region. In this setting, area represents the force applied by the distributed load over that part of the beam. After finding the force for each region, the position of each resultant load...

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 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

mechanical engineering

Torque About an Axis in 3D Systems

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