📚 Resultant Moments | 合力矩
In A-level Mechanics, resultant moments tell us whether a rigid body will rotate, stay balanced, or begin to tip. This article covers moment calculation, sign conventions, couples, equilibrium and typical Edexcel exam applications.
在 A-level 力学中，合力矩告诉我们刚体是否会转动、保持平衡或开始翻倒。本文介绍力矩计算、正负号规定、力偶、平衡以及 Edexcel 考试的典型应用。
1. What Is a Moment? | 什么是力矩
The moment of a force about a point is a measure of its turning effect. It is defined by the product of the force and the perpendicular distance from the point to the line of action of the force.
力对某点的力矩是力产生转动效果的度量。它定义为力的大小乘以该点到力的作用线的垂直距离。
M = F × d
Here F is the force and d is the perpendicular distance. The SI unit is the newton metre (N m), not the joule.
其中 F 是力，d 是垂直距离。国际单位制单位是牛顿米（N m），不是焦耳。
The distance d must always be measured at right angles to the line of action of the force. If the force acts at an angle, only the perpendicular component creates a moment.
距离 d 必须始终与力的作用线垂直。如果力以某个角度作用，只有垂直分量才会产生力矩。
2. Sign Convention and Perpendicular Distance | 正负号规定与垂直距离
When several forces act in a plane, choose one direction of rotation as positive. Usually anticlockwise is positive and clockwise is negative, but the choice must be consistent throughout the calculation.
当多个力作用在同一平面内时，应选择一个转动方向为正。通常取逆时针为正、顺时针为负，但整个计算过程中必须保持一致。
| Anticlockwise moment | Positive (+) by default |
| Clockwise moment | Negative (-) by default |
The distance used in M = F × d is the shortest distance from the pivot to the line of action. If the force is not perpendicular to the lever arm, use d = r sin θ, where θ is the angle between the force and the lever arm.
计算 M = F × d 时使用的距离是支点到力作用线的最短距离。如果力不垂直于杠杆臂，要用 d = r sin θ，其中 θ 是力与杠杆臂之间的夹角。
3. Resultant Moment of Coplanar Forces | 共面力的合力矩
The resultant moment about a point is the algebraic sum of all individual moments about that point. If the sum is zero, there is no net turning effect about that point.
关于某点的合力矩是所有单个力关于该点的力矩的代数和。如果代数和为零，则该点没有净转动效果。
For forces F₁, F₂, F₃ with perpendicular distances d₁, d₂, d₃ from the pivot, taking anticlockwise as positive:
对于力 F₁、F₂、F₃，其到支点的垂直距离分别为 d₁、d₂、d₃，以逆时针为正：
Σ M = F₁d₁ – F₂d₂ + F₃d₃
The sign of each term depends on whether it would turn the body anticlockwise or clockwise about the chosen pivot.
每一项的符号取决于它会使刚体绕选定支点逆时针转动还是顺时针转动。
A positive total means a resultant anticlockwise moment; a negative total means a resultant clockwise moment.
总和为正表示合力矩是逆时针方向；总和为负表示合力矩是顺时针方向。
4. Principle of Moments | 力矩原理
If a rigid body is in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any point.
如果刚体处于转动平衡状态，则关于任意点，顺时针力矩之和等于逆时针力矩之和。
This is the Principle of Moments. It is equivalent to saying the resultant moment is zero. The point chosen does not have to be a pivot; any point in the plane can be used.
这就是力矩原理。它等价于说合力矩为零。所选择的点不一定是支点；平面内任意点都可以使用。
Σ clockwise = Σ anticlockwise or Σ M = 0
顺时针力矩总和 = 逆时针力矩总和，或 Σ M = 0
This principle is the key tool for solving static rigid body problems in Edexcel Mechanics, particularly for beams, rods and ladders.
该原理是解决 Edexcel 力学中刚体静力学问题的关键工具，尤其是梁、杆和梯子问题。
5. Equilibrium of a Rigid Body | 刚体的平衡
For complete equilibrium, a rigid body must satisfy both linear and rotational conditions: the resultant force is zero and the resultant moment about any point is zero.
刚体完全平衡必须同时满足平动和转动条件：合力为零，且关于任意点的合力矩为零。
In two dimensions, this gives three independent equations: ΣFₓ = 0, ΣFᵧ = 0 and ΣM = 0 about a convenient point. These equations are used to find unknown reaction forces and distances.
在二维问题中，这给出三个独立方程：ΣFₓ = 0、ΣFᵧ = 0 和关于某方便点的 ΣM = 0。这些方程用来求未知反力和距离。
When taking moments, choose a point through which as many unknown forces as possible pass. Their moments become zero and the equation is simpler.
取矩时，应选择尽可能多未知力作用线经过的点。这样这些力的力矩为零，方程更简单。
6. Couples and Their Moment | 力偶及其力矩
A couple consists of two equal and opposite parallel forces whose lines of action are separated by a perpendicular distance d. A couple produces rotation but no resultant linear force.
力偶由两个大小相等、方向相反的平行力组成，它们的作用线之间的垂直距离为 d。力偶产生转动但不产生合力。
The moment of a couple is constant about any point in the plane and is given by:
力偶的力矩在平面内关于任意点都是常数，公式为：
M = F × d
Here F is the magnitude of one of the forces and d is the perpendicular distance between the lines of action.
其中 F 是其中一个力的大小，d 是两条作用线之间的垂直距离。
Because the resultant force of a couple is zero, a couple cannot be balanced by a single force. It can only be balanced by another couple of equal but opposite moment.
由于力偶的合力为零，单个力无法平衡力偶。力偶只能由另一个大小相等、方向相反的力偶来平衡。
7. Resultant Moment about Different Pivots | 关于不同支点的合力矩
The numerical value of the resultant moment depends on the point chosen. However, if a body is in equilibrium, the resultant moment is zero about every point, so you can choose the point that simplifies the algebra.
合力矩的数值取决于所选取的点。但如果刚体处于平衡状态，关于任意点的合力矩都为零，因此你可以选择简化代数运算的点。
If a body is not in equilibrium, the resultant moment about different points generally differs. A non-zero resultant moment indicates that the body will rotate about its centre of mass unless constrained by a pivot.
如果刚体不处于平衡状态，关于不同点的合力矩通常会不同。非零合力矩表示刚体将绕其质心转动，除非受到支点约束。
This is why many Edexcel questions ask for the resultant moment about a specific hinge, support or point, not just an arbitrary point.
这就是为什么许多 Edexcel 题目要求的是关于特定铰链、支撑或点的合力矩，而不仅仅是任意点。
8. Tilting and Toppling | 倾斜与翻倒
Tilting problems are common in Edexcel Mechanics. A rigid body on supports or a hinge is about to tilt when the reaction at one support becomes zero.
倾斜问题是 Edexcel 力学中的常见题型。当某一支撑处的反力变为零时，刚体即将
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