📚 GCSE Edexcel Maths: Moments and Equilibrium | GCSE Edexcel 数学：力矩与平衡 考点精讲
In GCSE Edexcel Mathematics, the topic of moments and equilibrium bridges the gap between algebraic manipulation and real-world physics. It introduces the concept of a turning effect caused by a force, and how we can use the principle of moments to solve problems involving balanced systems such as beams, levers, and see-saws. Mastering this topic requires a clear understanding of perpendicular distance, the condition for equilibrium, and the ability to set up and solve linear equations from given diagrams.
在 GCSE Edexcel 数学中，力矩与平衡这一主题连接了代数运算与现实世界中的物理知识。它介绍了力的转动效应这一概念，以及如何运用力矩原理解决平衡系统问题，例如横梁、杠杆和跷跷板。掌握这一主题需要清晰地理解垂直距离、平衡条件，并能够根据示意图建立并求解线性方程。
1. What is a Moment? | 什么是力矩？
A moment is the turning effect of a force about a fixed point, called the pivot or fulcrum. 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 moment increases if either the force or the distance is increased. This turning effect is what causes objects to rotate, and it is measured in newton-metres (Nm) when force is in newtons (N) and distance in metres (m).
力矩是力绕一个固定点（称为支点或转轴）产生的转动效应。力矩的大小取决于两个因素：力的大小和从支点到力的作用线的垂直距离。当力或距离增大时，力矩也会增大。这种转动效应使物体发生旋转，当力的单位是牛顿 (N)、距离的单位是米 (m) 时，力矩的单位是牛顿·米 (Nm)。
The formula for calculating a moment is:
计算力矩的公式是：
Moment = Force × Perpendicular distance from pivot
力矩 = 力 × 支点到力作用线的垂直距离
It is essential that the distance used is always measured at 90° to the direction of the force. If the force is applied at an angle, you must either resolve the force to find the component perpendicular to the beam, or find the perpendicular distance from the pivot to the line of action of the force. In GCSE Edexcel questions, forces are typically perpendicular to the beam, so you can directly use the distance along the beam.
必须注意的是，所用的距离始终是沿着与力的方向成 90° 的方向测量的。如果力是倾斜施加的，你必须将力分解以找到垂直于横梁的分量，或者找到从支点到力作用线的垂直距离。在 GCSE Edexcel 的考题中，力通常垂直于横梁，因此你可以直接使用沿横梁的距离。
2. Understanding Perpendicular Distance | 理解垂直距离
The perpendicular distance is the shortest distance from the pivot to the line of action of the force. Imagine a straight line extending infinitely in both directions along which the force acts; the perpendicular distance is the length of the segment that meets this line at a right angle from the pivot. If a force acts vertically downwards at the end of a horizontal beam, the perpendicular distance is simply the horizontal distance from the pivot to that point.
垂直距离是支点到力作用线的最短距离。想象一条沿着力的方向向两边无限延伸的直线，垂直距离就是从支点到这条直线的垂直线段的长度。如果一个力竖直向下作用在水平横梁的末端，那么垂直距离就是从支点到该点的水平距离。
Mistakes often occur when students use the slant distance along a tilted beam rather than the true perpendicular distance. Always check that the distance you are using is measured at 90° to the force. In diagrams, a right-angle symbol is often shown to indicate the perpendicular direction.
学生常犯的错误是使用倾斜横梁的斜向距离，而不是真实的垂直距离。始终要检查你使用的距离是否与力的方向成 90°。在示意图中，常用直角符号来标明垂直方向。
3. The Principle of Moments | 力矩原理
The principle of moments states that for a body to be in rotational equilibrium, the sum of the clockwise moments about any pivot must equal the sum of the anticlockwise moments about that same pivot. In an equation form:
力矩原理指出，一个物体要保持转动平衡，绕任意支点的顺时针力矩之和必须等于绕该支点的逆时针力矩之和。用等式表示为：
Total clockwise moments = Total anticlockwise moments
顺时针力矩总和 = 逆时针力矩总和
This principle is a powerful tool for solving problems where forces act at different distances from a pivot. When a system is balanced, there is no net turning effect, so the beam remains horizontal and stationary. You can take moments about any point, but choosing the pivot as the point where an unknown force acts can simplify the equation by eliminating that force's moment (since its distance is zero).
这一原理是解决力作用在支点不同距离处的问题的有力工具。当系统平衡时，净转动效应为零，因此横梁保持水平静止。你可以对任意点取矩，但选择某个未知力作用点作为支点可以使该力的力矩为零（因为其距离为零），从而简化方程。
4. Conditions for Complete Equilibrium | 完全平衡的条件
For a rigid body to be in complete equilibrium, two separate conditions must be satisfied. The first is that there is no resultant force, meaning the upward forces equal the downward forces. The second is that there is no resultant moment, meaning the clockwise moments equal the anticlockwise moments. Both conditions must hold true simultaneously.
刚体要达到完全平衡，必须同时满足两个独立的条件。第一，合力为零，即向上的力等于向下的力。第二，合力矩为零，即顺时针力矩等于逆时针力矩。这两个条件必须同时成立。
In many Edexcel GCSE problems, you will use the principle of moments to find an unknown force or distance, and then use the vertical force balance to find a reaction force at the pivot. For example, if a beam is supported at its centre, the upward reaction force at the pivot must equal the total weight of the beam plus any additional loads.
在许多 Edexcel GCSE 问题中，你会先用力矩原理求出未知的力或距离，再利用竖直方向的力平衡求出支点处的反作用力。例如，如果一根横梁在中心处被支撑，那么支点向上的反作用力必定等于横梁自身的重量加上所有额外负载的总重量。
5. Worked Example: A Simple See-Saw | 示例解析：简单的跷跷板
Consider a uniform see-saw of length 4 m pivoted at its centre. A child of weight 300 N sits at one end. Where must a second child of weight 400 N sit to balance the see-saw? Let the distance from the pivot to the second child be d metres.
考虑一根长 4 m 的均匀跷跷板，在其中点处支撑。一个重 300 N 的孩子坐在一端。另一名重 400 N 的孩子应坐在何处才能使跷跷板平衡？设第二个孩子到支点的距离为 d 米。
400 N force creates a clockwise moment of 400 × d. According to the principle of moments:
对支点取矩：300 N 的力产生逆时针力矩 300 × 2（因为端点距中心 2 m）。400 N 的力产生顺时针力矩 400 × d。根据力矩原理：
300 × 2 = 400 × d
600 = 400d → d = 600/400 = 1.5 m
tal downward force = 300 + 400 = 700 N, so the upward reaction at the pivot must be 700 N.
第二个孩子应该坐在支点另一侧 1.5 m 处。然后你可以验证竖直方向平衡：向下的总力 = 300 + 400 = 700 N，因此支点处的向上反力必定是 700 N。
6. Uniform and Non-Uniform Rods | 均匀杆和非均匀杆
If a rod or beam is described as uniform, its weight acts at its geometric centre. This simplifies moment calculations because you know exactly where the weight force is applied. For a non-uniform rod, the centre of mass is not at the geometric centre, and its position may be an unknown that you need to find using the principle of moments.
如果一根杆或横梁被描述为均匀的，其重量作用在它的几何中心。这简化了力矩计算，因为你确切地知道重力的作用点。对于非均匀杆，质心不在几何中心，其位置可能是你需要利用力矩原理来求解的未知量。
A typical exam question might provide two different balancing setups with known pivots and forces, allowing you to set up simultaneous equations to determine both the weight of the rod and the location of its centre of mass. Always draw a clear diagram marking all forces and distances before setting up equations.
典型的考题可能会给出两种不同支点和力的平衡设置，从而让你建立方程组，同时求出杆的重量和其质心的位置。在建立方程之前，务必画一个清晰的示意图，标出所有的力和距离。
7. Multiple Forces and Distributed Loads | 多个力和分布载荷
Some problems involve three or more forces acting on a beam. The principle of moments still applies: you simply sum all the clockwise moments and equate them to the sum of all the anticlockwise moments. It is helpful to label each force with its turning direction relative to the chosen pivot. Be systematic and list moments in a table to avoid sign errors.
有些问题涉及作用在横梁上的三个或更多力。力矩原理仍然适用：你只需要将所有顺时针力矩相加，并令其等于所有逆时针力矩之和。为每个力标记绕选定支点的转动方向会很有帮助。系统地列出力矩表格，可以避免符号错误。
A distributed load, such as the weight of the beam itself, can be treated as a single force acting at the centre of mass. If the beam is uniform, this is the midpoint. If a load is spread over a length, you may be given its total weight and its centre of action, which you use exactly like a single point force.
分布载荷，例如横梁自身的重量，可以看作是作用在质心处的单个力。如果横梁是均匀的，这个点就是中点。如果一个载荷分布在一定长度上，题目通常会给出其总重量和作用中心，你完全可以将其当作一个集中力来使用。
8. Moments in Levers and Real-Life Contexts | 杠杆中的力矩与现实情境
Levers are simple machines that use the principle of moments to amplify a force. A small effort applied at a large distance from the pivot can balance a larger load placed closer to the pivot. This is the principle behind crowbars, wheelbarrows, and spanners. In exam questions, you might be asked to calculate the effort needed to lift a load, or the mechanical advantage gained.
杠杆是利用力矩原理来放大力的简单机械。在远离支点处施加一个较小的力，可以平衡放置在离支点较近处的较大负载。这就是撬棍、手推车和扳手背后的原理。在考试题目中，你可能会被要求计算抬起负载所需的力，或者获得的机械效益。
When dealing with levers, identify the pivot clearly. In a wheelbarrow, the pivot is the wheel axle; the load is in the tray, and the effort is applied at the handles. Taking moments about the wheel axle allows you to relate the effort to the load.
处理杠杆问题时，要清楚地确定支点。在手推车中，支点是轮轴；负载在车斗中，力施加在把手上。对轮轴取矩，就可以建立起力与负载的关系式。
9. Common Exam Pitfalls and How to Avoid Them | 常见考试陷阱及应对策略
One frequent mistake is using the wrong distance — forgetting that the distance must be perpendicular. If a diagram shows a force acting at an angle, do not use the length of the beam as the distance unless the force is perpendicular. Another trap is ignoring the weight of the beam itself. If the beam is uniform, its weight acts at the centre; if you omit it, your moment equation will be incomplete.
一个常见错误是使用了错误的距离——忘记了距离必须垂直。如果图中显示力是倾斜作用的，除非这个力垂直于横梁，否则就不能把横梁长度当作距离。另一个陷阱是忽略了横梁自身的重量。如果横梁是均匀的，其重量作用在中心；如果你忽略它，你的力矩方程就会不完整。
Students also sometimes mix up clockwise and anticlockwise directions. Decide on a sign convention or carefully write 'CW' and 'ACW' next to each moment. Finally, when solving for an unknown distance, ensure the answer is physically plausible — it should be less than or equal to the length of the beam.
学生们有时还会混淆顺时针和逆时针方向。可以约定一个正负号规则，或者在每个力矩旁仔细地标注“CW”（顺时针）和“ACW”（逆时针）。最后，当求解未知距离时，要确保答案在物理上是合理的——它应该小于或等于横梁的长度。
10. Summary and Key Formulas | 总结与核心公式
The key formula for a moment: M = F × d, where d is the perpendicular distance. For equilibrium: Sum of CW moments = Sum of ACW moments. And for complete equilibrium: Upward forces = Downward forces. Practice drawing clear diagrams and systematically listing moments before forming equations. Check your units: force in newtons, distance in metres, moments in newton-metres.
力矩的核心公式：M = F × d，其中 d 为垂直距离。平衡条件为：顺时针力矩总和 = 逆时针力矩总和。而完全平衡还需满足：向上的力 = 向下的力。练习绘制清晰的示意图，并在列方程之前系统化地列出各个力矩。检查单位：力用牛顿，距离用米，力矩用牛顿·米。
By approaching each problem methodically, and always verifying both force and moment equilibrium where required, you will be well prepared for the Edexcel GCSE moments questions. Remember that even complex-looking beam problems can be broken down into simple, solvable linear equations.
通过有条不紊地处理每个问题，并在需要时始终验证力和力矩的平衡，你将为 Edexcel GCSE 力矩类题目做好充分准备。记住，即使看似复杂的横梁问题，也可以拆解成简单、可解的线性方程。
Published by TutorHao | Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图