Name: calculating torque from multiple forces
Uploaded: 2023-09-22T11:49:21.000Z
Duration: 1 min 30 s
Channel: JoVE

力矩是指当对物体施加力时，物体绕某一点或轴旋转的趋势。
不通过物体重心的力会产生力矩；作用在物体上所有这些力矩的矢量和称为合力矩。
考虑两辆拖拉机拉动电线杆使其倒下。在此情况下，有两个力作用于电线杆上，分别为在点 A 处的力 F1 和在点 B 处的力 F2。这些力以笛卡尔矢量形式表示。
这些力作用在距离为 r 的位置A 和 rB 从参考点 O.
这些力的力矩可通过力与从参考点到力作用线的垂直距离的叉积来计算。
为了求得合力矩，将力矩 M1 和 M2 相加，并进一步表示为笛卡尔向量形式。
最后，通过向量叉乘展开行列式，可将作用在杆上的合力矩确定为笛卡尔向量形式。

<ol>
	<li>罗素 C. 希伯勒。（2016）. 工程力学:静力学与动力学。皮尔森。第 122-131 页。</li>
</ol>

结果力矩: 向量公式

当一个力作用在一个物体上时，物体绕一个点发生旋转的倾向被称为其力矩。如果有多个力作用在一个物体上时，则所有作用在物体上的力矩之和可以表示为系统的结果力矩。结果力矩可以被视为一个可以像其他向量一样可以进行相加和相减的矢量。

系统力的结果力矩可以通过向量公式来进行计算。例如，如果我们假设两个力F1和F2分别作用在杆上的点A和点B上，这些力的结果力矩可以表示为

Mr = (rA x F1) + (rB x F2)。

其中rA和rB分别是从固定点到点A和点B的位置向量。乘积r x F被称为力矩，并且可以用行列式的形式来进行表示。借助位置向量与力向量的叉乘可以得出力的方向和引起旋转运动的点。 这种结果力矩的向量公式有几个实际的应用。例如，工程师使用它来计算机械零件（如齿轮、滑轮和曲柄）上的扭矩。它也能够广泛的应用在运动科学中，并以此来计算在不同类型的体育活动中关节的扭矩。

当对物体施加力时，物体绕某一点旋转的趋势称为力矩。如果有多个力作用于一个……

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Force System Resultants

力系合力

KEY TERMS AND CONCEPTS

关键术语与概念

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). Here, rA and rB are position vectors from a fixed point to A and B.

The cross product r x F gives the moment of a force. It can also be written in determinant form. This vector form includes both the direction of the force and the point about which rotation occurs.

This method has practical uses in engineering and sports science. Engineers use it to find torque on machine parts such as gears, pulleys, and cranks. It is also used to calculate torques on joints during physical activity.

Calculating Torque from Multiple Forces

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resultant moment: vector formulation

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

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principle of moments: problem solving

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couple

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Equivalent Force and Moment Systems

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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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key terms and concepts

Learn how resultant moment uses vector math to combine multiple forces, with examples from poles, machine parts, and sports joints.

Calculating Torque from Multiple Forces | 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 Torque from Multiple Forces

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