Equilibrium in Mechanics: Conditions and Determination | 力学中平衡状态的条件与判定

📚 Equilibrium in Mechanics: Conditions and Determination | 力学中平衡状态的条件与判定

Equilibrium is one of the most fundamental concepts in mechanics, serving as the foundation for solving problems in statics, dynamics, and engineering analysis. A body is in equilibrium when the resultant of all external forces acting upon it is zero and, for a rigid body, when the resultant moment about any point is also zero. Under these conditions, the body either remains at rest or moves with constant velocity, with its acceleration exactly equal to zero.

平衡是力学中最基本的概念之一，是解决静力学、动力学和工程分析问题的基石。当一个物体所受全部外力的合力为零，并且对于刚体而言，绕任意一点的合力矩也为零时，该物体就处于平衡状态。在这些条件下，物体要么保持静止，要么以恒定速度运动，其加速度恰好为零。

Mastering the conditions of equilibrium is essential for A-level mathematics students preparing for mechanics examinations. Whether we are analysing a particle suspended by strings, a ladder resting against a wall, or a block on an inclined plane, the same core principles apply: write down all forces, resolve them appropriately, and apply the equilibrium conditions to solve for unknowns.

掌握平衡条件对准备力学考试的 A-Level 数学学生来说至关重要。无论是分析由绳子悬挂的质点、靠在墙壁上的梯子，还是斜面上的物块，核心原则都是相同的：列出所有力、进行适当分解，并应用平衡条件来求解未知量。

1. The Fundamental Conditions of Equilibrium | 平衡的基本条件

For a rigid body to be in complete equilibrium, two independent conditions must be satisfied simultaneously. The first is translational equilibrium: the vector sum of all external forces acting on the body must be zero. The second is rotational equilibrium: the sum of all moments of the external forces about any chosen point must be zero.

刚体若要处于完全平衡状态，必须同时满足两个相互独立的条件。第一个是平动平衡：物体所受全部外力的矢量和必须为零。第二个是转动平衡：所有外力对任意选定点的力矩之和必须为零。

ΣF = 0 且 ΣM = 0

For a single particle, only the first condition applies, because a particle has no physical dimensions and therefore cannot experience rotation. However, for extended bodies such as beams, ladders, and bridges, both conditions must hold; neglecting either one leads to incorrect solutions in practical problems.

对于单个质点，只需满足第一个条件，因为质点没有实际尺寸，因而不会发生转动。但对于横梁、梯子和桥梁等具有一定几何尺寸的物体，两个条件必须同时成立；忽略任何一个都会导致实际问题的错误解。

It is also important to recognise that equilibrium does not imply rest. According to Newton’s first law, a body moving in a straight line with constant velocity also has zero acceleration and is therefore in equilibrium. This distinction is crucial in dynamics problems where objects move at steady speeds.

同样重要的是，平衡并不等同于静止。根据牛顿第一定律，沿直线以恒定速度运动的物体加速度也为零，因此同样处于平衡状态。在物体匀速运动的动力学问题中，这种区分至关重要。

2. Translational Equilibrium: Resultant Force Equal to Zero | 平动平衡：合力为零

Translational equilibrium requires that the resultant force acting on a body is exactly zero. In a two-dimensional plane, this vector condition can be expressed as two scalar equations: the sum of all horizontal components must be zero, and the sum of all vertical components must be zero.

平动平衡要求作用在物体上的合力恰好为零。在二维平面内，这个矢量条件可以表示为两个标量方程：所有水平分量之和必须为零，所有竖直分量之和必须为零。

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