Static Particles | 静态粒子：力的平衡

📚 Static Particles | 静态粒子：力的平衡

In Edexcel A-Level Mechanics, a static particle is a point mass that remains at rest. This topic builds on Newton’s laws and vector addition to show that rest is not the absence of forces, but a perfect balance of forces. You will learn to model forces, resolve components, use friction, and solve inclined-plane and pulley problems.

在 Edexcel A-Level 力学中，静态粒子（质点）指保持静止的点质量。本主题基于牛顿定律和矢量加法，说明静止并不是没有力，而是力的完美平衡。你将学习如何建立力的模型、分解分量、运用摩擦力，并解决斜面和滑轮问题。

1. The Core Principle of Static Equilibrium | 静态平衡的核心原理

For a particle to remain static, its velocity is constant at zero, so its acceleration is exactly 0. Newton’s second law, F = ma, then gives the fundamental condition for equilibrium: the resultant force on the particle must be zero. A particle can be acted on by many forces, but if the vector sum is zero, it stays at rest.

粒子保持静止时，速度恒为零，因此加速度恰好为 0。根据牛顿第二定律 F = ma，可以得到平衡的基本条件：作用在粒子上的合力必须为零。粒子可以受到多个力的作用，但只要矢量合力为零，它就会保持静止。

∑F = 0, a = 0

If a net force exists, the particle accelerates and cannot be static. Therefore, every static equilibrium problem is really an application of the vector equation ∑F = 0.

如果存在净力，粒子就会加速，不可能保持静止。因此，每一个静态平衡问题实际上都是矢量方程 ∑F = 0 的应用。

2. Force as a Vector | 力作为矢量

A force has both magnitude and direction, so it is a vector. In Edexcel statics, forces are usually marked in newtons (N) on a clear diagram. When adding forces, you cannot simply add magnitudes unless they act in the same line; you must add them by vector methods.

力既有大小又有方向，因此它是矢量。在 Edexcel 静力学中，力通常以牛顿（N）为单位，并在清晰的受力图中标出。合成力时，除非力作用在同一直线上，否则不能简单地把大小相加，而必须采用矢量方法合成。

Resultant force = F₁ + F₂ + F₃ + … (vector sum)

This is why drawing a force diagram is the first step in almost every statics problem. The diagram shows all forces acting on the particle from the same point.

这就是为什么在几乎所有静力学问题中，第一步都是画出受力图。受力图从同一点出发标出作用在粒子上的所有力。

3. Resolving Forces into Components | 力的分解

Resolving a force splits it into perpendicular components, usually horizontal and vertical. For a force F at angle θ to the horizontal, the horizontal component is F cosθ and the vertical component is F sinθ. Equilibrium then requires the sum of horizontal components and the sum of vertical components to each be zero.

分解力就是将一个力拆分为互相垂直的分量，通常沿水平和竖直方向。如果力 F 与水平方向成 θ 角，则水平分量为 F cosθ，竖直分量为 F sinθ。平衡要求水平分量之和与竖直分量之和都分别为零。

Fₓ = F cosθ, F_y = F sinθ

∑Fₓ = 0, ∑F_y = 0

Choosing the most convenient directions for resolution can greatly simplify the calculation. In inclined-plane problems, for example, it is usually best to resolve parallel and perpendicular to the slope.

选择最方便的方向进行分解可以大大简化计算。例如，在斜面问题中，通常最好沿斜面和垂直斜面方向进行分解。

4. The Particle Model and Its Assumptions | 质点模型及其假设

In statics, objects are often modelled as particles: the mass is concentrated at a single point, shape and size are ignored, and rotation is neglected. This allows all forces to be treated as concurrent at one point. The particle model is valid when forces meet at a point and turning effects are not relevant.

在静力学中，物体通常被建模为质点：质量集中在一个点上，形状和大小忽略不计，也不考虑转动。这样所有力都可视为共点力。当力交于一点且无需考虑转动效应时，粒子模型是合适的。

This assumption underpins most Edexcel M1 questions involving blocks, beads, smooth rings, and strings. It means all lines of action can be drawn from the centre of mass, making vector addition and resolution manageable.

这一假设是 Edexcel M1 中涉及滑块、珠子、光滑环和绳子的绝大多数问题的基础。它意味着所有力的作用线都可以从质心画出，从而使矢量合成和分解更加容易处理。

5. Common Forces in Static Problems | 静力学问题中的常见力

You must be able to draw and label the standard forces. Weight, W = mg, always acts vertically downwards. The normal reaction, R, acts perpendicular to the contact surface. Tension, T, is a pulling force along a string or rod, directed away from the particle. Friction, F, opposes the direction of relative motion or the tendency to move.

你必须能够画出并标注标准力。重力 W = mg 总是竖直向下。法向反作用力 R 垂直于接触面。拉力（张力）T 是沿着绳子或杆的拉力，方向远离粒子。摩擦力 F 阻碍相对运动或相对运动趋势的方向。

W = mg

Weight: W = mg downwards（重力：W = mg 竖直向下）
Normal reaction: R perpendicular to surface（法向反力：R 垂直于接触面）
Tension: T along string or rod away from object（张力：T 沿绳或杆方向远离物体）
Friction: F ≤ μR, opposite to sliding tendency（摩擦力：F ≤ μR，方向与滑动趋势相反）

Correct labelling is essential because sign mistakes in these forces are a common source of lost marks.

正确标注至关重要，因为这些力的符号错误是失分的常见原因。

6. Triangle and Polygon of Forces | 力的三角形法则与多边形法则

If three forces acting on a particle are in equilibrium, they can be represented in magnitude and direction by the sides of a closed triangle, taken in order. This is the triangle of forces. You can then use the sine rule or cosine rule to find unknown forces or angles.

如果作用于粒子的三个力平衡，它们可以按大小和方向依次表示为一个闭合三角形的三条边，这就是力的三角形法则。接着可以用正弦定理或余弦定理求未知力或角度。

a/sin A = b/sin B = c/sin C

For more than three forces, the polygon of forces gives a closed polygon when the forces are drawn head-to-tail. This is a direct geometric statement of ∑F = 0.

对于三个以上的力，力的多边形法则指出，当这些力首尾相接时，会构成一个闭合多边形。这正是 ∑F = 0 的几何表述。

7. Friction and Limiting Equilibrium | 摩擦与极限平衡

Static friction can take any value up to a maximum. In limiting equilibrium, the object is on the point of moving, so friction reaches its maximum value F = μR, where μ is the coefficient of friction. Friction always acts to oppose the direction of potential sliding.

静摩擦力可以取零到最大值之间的任意值。在极限平衡状态下，物体处于即将运动的临界点，因此摩擦力达到最大值 F = μR，其中 μ 为摩擦系数。摩擦力总是阻碍可能滑动的方向。

F ≤ μR, limiting: F = μR

If a force keeps the particle stationary without reaching limiting friction, F is determined by the equilibrium equations, not by μR. You should only use F = μR when the particle is about to slide.

如果某个力使粒子保持静止但未达到极限摩擦，则 F 的大小由

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Static Particles | 静态粒子：力的平衡