📚 Moments and Equilibrium: IGCSE CIE Mathematics Key Points | IGCSE CIE 数学：力矩与平衡 考点精讲
Moments are a fundamental concept in mechanics, essential for understanding how forces can cause rotation. In IGCSE CIE Mathematics, you will need to calculate moments, apply the principle of moments, and solve problems involving equilibrium. This article provides a comprehensive revision of key points, formulas, and problem-solving strategies for the topic of moments and equilibrium.
力矩是力学中的基本概念，对于理解力如何导致转动至关重要。在 IGCSE CIE 数学考试中，你需要计算力矩、运用力矩原理，并解决涉及平衡的问题。本文全面梳理力矩与平衡的考点、公式和解题策略。
1. What is a Moment? | 什么是力矩？
A moment is the turning effect of a force about a pivot. The size of the moment 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 distance, the greater the moment.
力矩是力对支点的转动效应。力矩的大小取决于两个因素：力的大小以及支点到力的作用线的垂直距离。力越大或距离越大，力矩就越大。
The formula for moment is:
力矩的公式为：
Moment = Force × Perpendicular distance
M = F × d⊥
Here M is measured in newton metres (N m), F in newtons (N), and d⊥ in metres (m). The perpendicular distance d⊥ is the shortest distance from the pivot to the line of action of the force, not necessarily the distance to the point where the force is applied. A moment can be clockwise or anticlockwise; by convention, we often take anticlockwise as positive, but in the principle of moments we simply equate the sums of clockwise and anticlockwise moments.
其中 M 的单位是牛顿·米 (N m)，F 的单位是牛顿 (N)，d⊥ 的单位是米 (m)。垂直距离 d⊥ 是从支点到力的作用线的最短距离，不一定是力的作用点到支点的距离。力矩可以是顺时针或逆时针；通常我们规定逆时针为正，但在力矩原理中只需令顺时针与逆时针力矩总和相等即可。
2. Calculating Moments | 力矩的计算
When a force is perpendicular to a rigid body, the moment is simply F × d, where d is the distance from the pivot to the point of application. For example, a spanner of length 0.25 m with a 20 N force applied at the end perpendicular to the handle produces a moment of 20 × 0.25 = 5 N m.
当力垂直于刚体时，力矩就是 F × d，其中 d 是支点到力作用点的距离。例如，一个长 0.25 m 的扳手，在末端施加 20 N 的力且与手柄垂直，产生的力矩为 20 × 0.25 = 5 N m。
If the force is not perpendicular to the object, you must use the perpendicular distance d⊥. This can be found by d⊥ = d sin θ, where θ is the angle between the force and the object. Alternatively, you can resolve the force into a component perpendicular to the object: F⊥ = F sin θ. The moment is then M = F d sin θ = F⊥ d.
如果力不垂直于物体，必须使用垂直距离 d⊥。可通过 d⊥ = d sin θ 求得，其中 θ 是力与物体之间的夹角。或者，你也可以将力分解为垂直物体的分量：F⊥ = F sin θ。此时力矩 M = F d sin θ = F⊥ d。
Example: A rod is pivoted at one end. A force of 10 N is applied at the other end, at an angle of 30° to the rod. The rod’s length is 0.5 m. The moment about the pivot is 10 × 0.5 × sin 30° = 10 × 0.5 × 0.5 = 2.5 N m.
例子：一根杆在一端有支点。在另一端施加 10 N 的力，与杆成 30° 角。杆长 0.5 m。对支点的力矩为 10 × 0.5 × sin 30° = 10 × 0.5 × 0.5 = 2.5 N m。
3. The Principle of Moments | 力矩原理
When a body is in rotational equilibrium (i.e. it is not turning), the total clockwise moment about any point is equal to the total anticlockwise moment about that same point. This is called the principle of moments.
当一个物体处于转动平衡（即不发生转动）时，关于任意一点的顺时针力矩总和等于逆时针力矩总和。这就是力矩原理。
Σ Mclockwise = Σ Manticlockwise
Consider a seesaw balanced at its centre. A child of weight 300 N sits 1.5 m to the left of the pivot, and a child of weight 500 N sits 0.9 m to the right. Anticlockwise moment = 300 × 1.5 = 450 N m; clockwise moment = 500 × 0.9 = 450 N m. The moments are equal, so the seesaw balances.
以一个在中心支点平衡的跷跷板为例。一个重 300 N 的孩子坐在支点左侧 1.5 m 处，另一个重 500 N 的孩子坐在右侧 0.9 m 处。逆时针力矩 = 300 × 1.5 = 450 N m；顺时针力矩 = 500 × 0.9 = 450 N m。两者相等，因此跷跷板平衡。
4. Conditions for Equilibrium | 平衡条件
For a rigid body to be in complete equilibrium (both translationally and rotationally), two conditions must be satisfied:
刚体要处于完全平衡状态（既无平动也无转动），必须满足两个条件：
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The resultant force in any direction is zero (ΣF = 0). This ensures no linear acceleration.
任意方向的合力为零 (ΣF = 0)，确保没有平动加速度。
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The resultant moment about any point is zero (ΣM = 0). This ensures no angular acceleration.
关于任意点的合力矩为零 (ΣM = 0)，确保没有角加速度。
In typical IGCSE problems, you will apply these conditions to a beam or rod that is stationary. Often you choose a pivot point where unknown forces (like a support reaction) act, so that their moments become zero in your equation, simplifying the calculation.
在典型的 IGCSE 题目中，你会把这些条件用于静止的横梁或杆。通常会选择一个支点，使某个未知力（如支撑反力）的力矩为零，从而简化方程。
5. Centre of Gravity | 重心
The centre of gravity (CoG) of an object is the point through which its entire weight appears to act. For a uniform, symmetrical object, the centre of gravity is at its geometrical centre. For example, the CoG of a uniform metre rule is at the 50 cm mark. In moment calculations, the weight of a uniform beam is drawn as a single downward force acting at its midpoint.
物体的重心是全部重量看似集中作用的那一点。对于均匀、对称的物体，重心位于其几何中心。例如，一把均匀的米尺的重心在 50 cm 刻度处。在力矩计算中，均匀杆的重量画为一个作用于中点的向下的力。
If an object is non-uniform, its centre of gravity will not be at the centre. You can find the position of the centre of gravity experimentally by suspending the object from different points, or mathematically using equilibrium conditions and the principle of moments (see Section 9).
如果物体不均匀，重心就不在几何中心。你可以通过将物体从不同点悬挂来实验确定重心，或利用平衡条件和力矩原理进行计算（见第 9 节）。
6. Types of Equilibrium | 平衡的种类
There are three types of equilibrium, depending on what happens when a body is slightly displaced:
根据物体微微偏离平衡位置后发生的情况，平衡可分为三种：
| Type of Equilibrium 平衡类型 |
Behaviour after small displacement 被轻微移动后的行为 |
Moment produced 产生的力矩 |
Example 例子 |
|---|---|---|---|
| Stable 稳定平衡 |
Returns to original position
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