Name: calculating moment with cartesian vectors
Uploaded: 2025-05-22T09:06:02.000Z
Duration: 1 min 26 s
Channel: JoVE

힘의 모멘트는 위치와 힘 벡터의 외적으로 정의됩니다.
순간의 교차곱은 단위 벡터와 데카르트 형식의 위치 및 힘 벡터를 사용하여 행렬식 형식으로 표현할 수 있습니다.
행렬식은 힘의 순간에 대한 데카르트 벡터 공식을 결정하기 위해 확장됩니다.
10 N의 힘이 데카르트 구성 요소 형식에서 위치 벡터에 의해 주어진 점에서 회전문에 적용된다고 가정합니다. 여기서 벡터 공식화의 힘 모멘트를 결정해야 합니다.
이 경우 힘 벡터는 y-z 평면 에 수직이며, 이로 인해 힘 벡터의 y 및 z 요소가 0이 됩니다. 행렬식은 위치 벡터와 힘 벡터를 사용하여 구성됩니다.
마지막으로, 행렬식을 확장함으로써 z축에 대한 회전문의 힘 모멘트가 결정됩니다.
여기서 모멘트는 양의 z 방향이며, 이는 도어가 시계 반대 방향으로 회전한다는 것을 의미합니다.

<ol>
	<li>히벨러, R.C. (2016). Engineering Mechanics \u2012 정역학 및 동역학. 뉴저지 호보켄: 피어슨 프렌티스 홀. 129쪽.</li>
</ol>

벡터 공식화를 위한 데카르트 형식

벡터 공식화를 위한 데카르트 형식은 위치 벡터와 힘 벡터를 사용하여 힘의 모멘트를 계산하는 과정입니다. 힘의 모멘트는 이 벡터들의 교차곱으로 정의되므로 벡터량이 됩니다. 위치와 힘 벡터의 데카르트 형식은 단위 벡터를 포함하며, 이는 교차곱을 행렬식으로 표현하는 데 사용될 수 있습니다.

<img alt="Equation 1" src="/files/ftp_upload/14254/14254_ Equation_1v2.svg" style="text-align: center; height: 60px;" />

그런 다음 행렬식을 확장하여 힘 모멘트의 데카르트 벡터 공식을 결정할 수 있습니다.

<img alt="Equation 2" src="/files/ftp_upload/14254/14254_ Equation_2v2.svg" style="text-align: center; height: 15px;" />

예를 들어, 위치 벡터 r = 0i - 4j + 0k을 따라 회전문에 10 N의 힘을 가한다고 가정합니다. 이 경우 힘 벡터는 문이 서 있는 y-z 평면에 수직이며, 이는 힘 벡터의 y 및 z 성분이 0임을 의미합니다. 힘의 모멘트를 계산하기 위해 위치 벡터와 힘 벡터인 F = 10i + 0j + 0k를 사용하여 행렬식을 구성합니다.

<img alt="Equation 3" src="/files/ftp_upload/14254/14254_ Equation_3v2.svg" style="text-align: center; height: 60px;" />

마지막으로 행렬식을 확장하면 회전문의 힘 모멘트가 z-축에 대해 40 Nm임을 알 수 있습니다. 양의 모멘트는 문이 반시계 방향으로 회전한다는 것을 의미합니다.

벡터 공식화를 위한 데카르트 형식을 사용하면 위치와 힘 벡터를 사용하여 간단한 방식으로 힘의 모멘트를 계산할 수 있습니다.

벡터 공식화를 위한 데카르트 형식은 위치 벡터와 힘 벡터를 사용하여 힘의 모멘트를 계산하는 프로세스입니다. 힘의 순간은 ...

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Mechanical Engineering

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Unit 1

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Force System Resultants

힘 체계의 합력

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 example shows the process in action. A force of 10 N is applied in the x-direction at the position vector r = 0 i - 4 j + 0 k. The force vector has no y or z components, so it can be written as F = 10 i + 0 j + 0 k.

Using these vectors, the determinant is built and expanded. The result is a moment of 40 N.m about the z-axis. The positive sign means the door rotates counterclockwise.

Calculating Moment with Cartesian Vectors

calculating-moment-with-cartesian-vectors

moment of a force: scalar formulation

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cartesian form for vector formulation

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...

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principle of moments

Resolving Forces in Moment Calculations

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Torque About an Axis in 3D Systems

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Bicycle Torque and Moment About an Axis

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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...

couple

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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...

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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

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Resultant Force and Wrench Motion

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key terms and concepts

Learn how Cartesian vector form is used to calculate moment of force with unit vectors, determinants, and a revolving door example in a simple step-by-step method.

Calculating Moment with Cartesian Vectors | 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

Calculating Moment with Cartesian Vectors

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