Moments and Equilibrium in IB Mathematics | IB 数学：力矩与平衡 考点精讲

📚 Moments and Equilibrium in IB Mathematics | IB 数学：力矩与平衡 考点精讲

Moments and equilibrium are fundamental concepts in the Mechanics portion of IB Mathematics, particularly within the vectors and calculus applications topics. Understanding how to calculate the turning effect of forces and apply equilibrium conditions equips students with powerful problem-solving tools for rigid-body statics. This article provides a concise yet comprehensive review of the key points you need for the IB exam.

力矩与平衡是IB数学力学部分的基础知识，尤其出现在向量和微积分应用相关题目中。掌握如何计算力的转动效应以及运用平衡条件，能让学生具备解决刚体静力学问题的强大工具。本文将精炼而全面地梳理你在IB考试中需要掌握的核心考点。

1. Introduction to Moments | 力矩导论

A moment (also called torque) quantifies the ability of a force to cause rotation about a specific point or axis. In IB Mathematics, this concept appears when analyzing systems in static equilibrium or when using vector cross products. Every force that does not pass through the reference point produces a turning effect, and the sum of these effects determines whether the body will rotate.

力矩（也称扭矩）量化了力使物体绕某一点或轴产生转动的能力。在IB数学中，分析静态平衡系统或运用向量叉积时都会遇到这个概念。任何不通过参考点的力都会产生转动效应，这些效应的总和决定了物体是否会发生转动。

2. Definition and Formula of Moment | 力矩的定义与公式

The magnitude of the moment of a force F about a point O is the product of the force and the perpendicular distance from O to the line of action of the force. Using scalar notation, the moment M is given by:

力 F 关于点 O 的力矩大小，等于力的大小乘以 O 点到力作用线的垂直距离。用标量记号表示，力矩 M 为：

M = F × d

If the force is applied at an angle, the perpendicular distance is d sinθ, so M = F d sinθ, where θ is the angle between the force vector and the line segment from the pivot to the point of application. In vector terms, the moment is defined by the cross product M = r × F, where r is the position vector from the pivot to the point where the force acts. This formulation captures both magnitude and direction and is essential for three-dimensional problems.

如果力以某个角度施加，垂直距离变成 d sinθ，因此 M = F d sinθ，其中 θ 是力向量与从支点到作用点连线之间的夹角。在向量形式中，力矩定义为叉积 M = r × F，其中 r 是从支点到力作用点的位置向量。这种形式同时包含了力矩的大小和方向，对处理三维问题尤为重要。

3. Moment in Two Dimensions: Sign Convention | 二维力矩：正负符号约定

In a 2D plane, moments can cause either clockwise or counterclockwise rotation. It is standard to adopt a sign convention early in the solution: for instance, take counterclockwise moments as positive and clockwise moments as negative. The opposite convention works equally well as long as it is applied consistently throughout the problem.

在二维平面中，力矩会导致顺时针或逆时针转动。通常的做法是在解题之初就确定符号约定：例如，规定逆时针力矩为正，顺时针为负。相反的约定同样可行，但必须在整个解题过程中前后一致。

When summing moments about a point

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