Moments and Equilibrium | 力矩与平衡

📚 Moments and Equilibrium | 力矩与平衡

In mechanics, a moment is the turning effect produced by a force acting on a body about a pivot or a fixed point. It is a fundamental concept used to analyse levers, beams, see-saws and many everyday structures. Understanding moments and the conditions for equilibrium allows you to solve problems where forces cause rotation, not just translation. This article covers the essential theory, calculations and problem-solving techniques required for the AQA Level 2 Certificate in Further Mathematics, focusing on the topic of moments and equilibrium.

在力学中，力矩是指力作用在物体上绕一个支点或固定点产生的转动效应。它是分析杠杆、横梁、跷跷板以及许多日常结构的基本概念。理解力矩及平衡条件可以帮助你解决力不仅引起移动、还会引起转动的问题。本文涵盖 AQA 二级进阶数学证书所需的力矩与平衡基本理论、计算方法和解题技巧。

1. Introduction to Moments | 力矩简介

A moment measures how effectively a force can rotate an object around a pivot. It depends on two factors: the magnitude of the force and the perpendicular distance from the pivot to the line of action of the force. The larger the force or the greater the distance, the bigger the turning effect. In diagrams, the pivot is often labelled P or O, and the force is applied at some point away from the pivot.

力矩衡量一个力绕支点转动物体的有效程度。它取决于两个因素：力的大小以及从支点到力作用线的垂直距离。力越大或距离越大，转动效应就越强。在示意图中，支点常标为 P 或 O，力作用在离支点一定距离的某点。

2. Calculating a Moment | 计算力矩

The moment M of a force F about a pivot is given by the formula:

力矩 M 通过以下公式计算：

M = F × d

where F is the force in newtons (N) and d is the perpendicular distance in metres (m) from the pivot to the line of action of the force. The distance must be measured at right angles to the direction of the force. If the force is not perpendicular to the distance, you need to resolve the force or find the perpendicular component.

其中 F 是力，单位为牛顿 (N)；d 是从支点到力作用线的垂直距离，单位为米 (m)。该距离必须与力的方向成直角。如果力不垂直于这段距离，你需要将力分解或找出垂直分量。

3. Units of Moment | 力矩的单位

Since moment is force multiplied by distance, its unit is the newton metre (N m). Note that this is not the same as the joule, even though a joule can be expressed as N m. In moments, the distance is measured perpendicularly, not along the direction of the force; the physical meaning is turning effect, not energy. In calculations, always write the unit as N m.

由于力矩是力乘以距离，其单位是牛顿米 (N m)。注意这与焦耳不同，尽管焦耳也可表示为 N m。在力矩中，距离按垂直方向测量，而非沿着力的方向；其物理意义是转动效应，而不是能量。计算时单位务必写成 N m。

4. Clockwise and Anticlockwise Moments | 顺时针与逆时针力矩

Moments can act in two rotational directions: clockwise (CW) and anticlockwise (ACW). When multiple forces act on a body, you must assign a positive sign to one direction, for example, taking anticlockwise as positive. Then, the net moment is the algebraic sum of all individual moments. If the sum is zero, the object has no resultant turning effect and is in rotational equilibrium.

力矩可以沿两个旋转方向作用：顺时针 (CW) 和逆时针 (ACW)。当多个力作用在物体上时，你必须给其中一个方向赋予正号，例如取逆时针为正。那么合力矩就是所有单个力矩的代数和。如果代数和为零，则该物体没有净转动效应，处于转动平衡状态。

5. The Principle of Moments | 力矩原理

The principle of moments states that for a body in rotational equilibrium, the sum of the clockwise moments about any pivot equals the sum of the anticlockwise moments about that same pivot. This is the fundamental tool for solving equilibrium problems. Mathematically:

力矩原理指出，对于处于转动平衡的物体，绕任意支点的顺时针力矩之和等于逆时针力矩之和。这是解决平衡问题的基本工具。数学表达为：

Σ clockwise moments = Σ anticlockwise moments

This equation can be used to find unknown forces or distances when a beam or lever is balanced.

当横梁或杠杆平衡时，可以用这个方程求未知的力或距离。

6. Equilibrium Conditions | 平衡条件

An object is in complete equilibrium when two conditions are satisfied: (1) The resultant force in any direction is zero (translational equilibrium), and (2) the resultant moment about any point is zero (rotational equilibrium). In many GCSE Further Maths problems, you will only need to consider moments when the beam is simply supported and there are no horizontal forces, but remember to check vertical forces balance as well.

当满足两个条件时，物体处于完全平衡：(1) 任意方向上的合力为零（平动平衡）；(2) 绕任意点的合力矩为零（转动平衡）。在许多 GCSE 进阶数学问题中，当横梁简单支撑且无水平力时，你只需考虑力矩，但也要记住检查竖直方向的力是否平衡。

7. Centre of Mass and Uniform Rods | 质心与均匀杆

The weight of a uniform rod or beam acts at its centre of mass, which is at the midpoint of the rod. This is crucial because the weight is a force that produces a moment about any support. For a non-uniform rod, the centre of mass may not be at the geometric centre, and its position is often given or needs to be found. When drawing force diagrams, mark the weight acting vertically downwards at the centre of mass.

均匀杆或横梁的重量作用在其质心处，质心位于杆的中点。这一点很关键，因为重量是对任意支撑点产生力矩的一个力。对于非均匀杆，质心可能不在其几何中心，其位置通常会给出或需要求解。画受力图时，要在质心处标出竖直向下的重量。

8. Solving Problems: Uniform Beam with Supports | 解题：带支座的均匀横梁

Consider a uniform beam of weight W and length L, resting on two supports at its ends. A load is placed at some point. To find the reaction forces, you can take moments about one support. This eliminates that reaction from the equation. Write the moment equation: clockwise moments (due to weight and load) = anticlockwise moments (due to the other reaction). Then, use the vertical force balance to find the remaining reaction.

考虑一根重量为 W、长度为 L 的均匀横梁，两端放在支座上。在某点施加一个负载。为求支座反力，你可以绕其中一个支座取矩。这样就把该反力从方程中消去了。写出力矩方程：顺时针力矩（由重量和负载产生）= 逆时针力矩（由另一个反力产生）。然后利用竖直方向力的平衡求出剩下的那个反力。

R₁ × L = W × (L/2) + load × distance from pivot

9. Non-uniform Beams and Unknown Masses | 非均匀梁与未知质量

If the beam is non-uniform, its centre of mass is not at the midpoint. You might be asked to find the position of the centre of mass or an unknown mass placed on the beam. The method is the same: take moments about a chosen pivot. The weight of the beam acts at an unknown distance x from one end. Set up the equation with clockwise and anticlockwise moments, and solve for x. Always draw a clear diagram with all distances marked from the pivot.

如果横梁是非均匀的，其质心就不在中点。你可能要计算质心的位置或梁上某个未知质量。方法是一样的：绕选定的支点取矩。梁的重量作用在距一端未知距离 x 处。建立顺时针和逆时针力矩的方程，然后解出 x。一定要画清晰的示意图，标明所有到支点的距离。

10. Multiple Forces and Taking Moments about a Point | 多个力与绕某点取矩

When several forces act on a rigid body, you can take moments about any point, but strategic choice simplifies the algebra. Select a point where an unknown force acts – that way, its moment is zero and it disappears from the equation. For systems with inclined forces, only the perpendicular component of the force contributes to the moment. Use trigonometry to resolve the force if needed.

当多个力作用在刚体上时，你可以绕任意点取矩，但选择恰当的点可以简化计算。选择一个未知力作用点，这样该力的力矩为零，就从方程中消失了。对于含有倾斜力的系统，只有力的垂直分量才产生力矩。如果需要，可用三角函数对力进行分解。

11. Common Pitfalls and Tips | 常见错误与提示

A common mistake is using the slanted distance instead of the perpendicular distance. Always check that d is measured at right angles to the force. Another error is forgetting to include the weight of the beam itself. Even if the beam is ‘light’ (mass negligible), the question will state it; otherwise, assume it has weight acting at its centre. Be consistent with the sign convention for clockwise and anticlockwise moments. Finally, ensure all units are in newtons and metres.

一个常见错误是使用了斜边距离而非垂直距离。务必检查 d 是否与力成直角测量。另一个错误是忘记计入横梁自身的重量。即使横梁是“轻质”的（质量可忽略），题目也会说明；否则应假设它有重量并作用在其中心。顺时针和逆时针力矩的符号规定要保持一致。最后，确保所有单位均为牛顿和米。

12. Conclusion and Exam Advice | 总结与考试建议

Moments and equilibrium problems test your ability to model physical situations with algebra. Practice drawing clear free-body diagrams, labelling all forces and distances. Identify a pivot, write a moment equation using the principle of moments, then solve. These skills are highly transferable to A-level mechanics. In the exam, show each step clearly and double-check that your distances are perpendicular and that you have included all forces producing a moment.

力矩与平衡问题考查你将物理情景转化为代数建模的能力。练习画清晰的受力图，标注所有力和距离。选取一个支点，利用力矩原理列出方程，然后求解。这些技巧对 A-level 力学非常有用。考试时，每一步都要写清楚，并复查距离是否为垂直距离，以及是否包含了所有产生力矩的力。

Published by TutorHao | Further Mathematics Revision Series | aleveler.com

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