Statics of a Particle | 质点静力学

📚 Statics of a Particle | 质点静力学

In A-Level Mechanics, statics of a particle is the study of forces acting on a body that remains at rest. A particle is an idealised model in which all mass is concentrated at a single point, so rotational effects are ignored. The central principle is equilibrium: the vector sum of all forces acting on the particle is zero. Mastering resolution of forces, free-body diagrams, friction and limiting equilibrium is essential for Edexcel Mechanics questions.

在 A-Level 力学中，质点静力学研究作用于静止物体上的力。质点是一种理想化模型，将全部质量集中于一个点，因此忽略转动效应。核心原理是平衡：作用在质点上的所有力的矢量和为零。掌握力的分解、受力分析图、摩擦和极限平衡是应对 Edexcel 力学考题的关键。

1. Modelling Assumptions and Particle Model | 建模假设与质点模型

In statics, a body is treated as a particle when its size and shape do not affect the problem. This removes moments and rotations, so all forces can be regarded as acting at a single point. A particle has mass but no dimensions.

在静力学中，当物体的大小和形状不影响问题时，可将其视为质点。这样就去除了力矩和转动效应，所有力都可看作作用于同一点。质点有质量但没有尺寸。

Common modelling assumptions include: light string means zero mass, inextensible string means no extension, smooth pulley means no friction at the pulley, smooth surface means no frictional force, and rough surface means friction may act.

常见的建模假设包括：轻绳表示质量为零；不可伸长绳表示无伸长；光滑滑轮表示滑轮处无摩擦；光滑表面表示无摩擦力；粗糙表面表示可能存在摩擦。

Model	Meaning	中文含义
Particle	Mass at a point, no rotation	质量集中一点，无转动
Light string	Zero mass, tension constant	零质量，张力恒定
Inextensible string	No change in length	长度不变
Smooth pulley	No friction, changes direction only	无摩擦，仅改变方向
Smooth surface	No frictional force	无摩擦力
Rough surface	Friction can act	可能存在摩擦

2. Forces as Vectors | 力作为矢量

A force is a vector quantity: it has magnitude, direction and point of application. In SI units, force is measured in newtons, written N. Statics questions involve several standard forces that must be identified correctly.

力是矢量：它有大小、方向和作用点。在国际单位制中，力的单位是牛顿，写作 N。静力学题目涉及几种标准力，必须正确识别。

The weight W acts vertically downwards from the centre of mass. The normal reaction R acts perpendicular to the contact surface. Tension T acts along a string, pulling away from the particle. Friction F acts along the contact surface and opposes potential sliding.

重力 W 从质心竖直向下作用。法向反力 R 垂直于接触面作用。张力 T 沿绳子作用，方向远离质点。摩擦力 F 沿接触面作用，阻碍可能的滑动。

W = mg

Here g is the acceleration due to gravity, usually taken as 9.8 m s⁻² in Edexcel mechanics unless otherwise stated.

这里 g 是重力加速度，在 Edexcel 力学中通常取 9.8 m s⁻²，除非题目另有说明。

Force	Symbol	Direction	中文方向
Weight	W	Vertically down	竖直向下
Normal reaction	R	Perpendicular to surface	垂直于表面
Tension	T	Along string, away from particle	沿绳，远离质点
Friction	F	Along surface, opposes sliding	沿表面，阻碍滑动

3. Resolving Forces | 力的分解

Resolving a force means splitting it into perpendicular components. This is usually done along horizontal and vertical directions, or along directions parallel and perpendicular to an inclined plane.

分解力是指将一个力拆分为互相垂直的分力。通常沿水平和竖直方向分解，或沿平行和垂直于斜面的方向分解。

If a force P makes an angle θ with the horizontal, then its horizontal component is P cos θ and its vertical component is P sin θ. If the angle is given to the vertical, the components swap.

如果一个力 P 与水平方向成 θ 角，则水平分力为 P cos θ，竖直分力为 P sin θ。如果角度是与竖直方向的夹角，则两个分量互换。

Pₓ = P cos θ, P_y = P sin θ

Always draw a clear diagram and mark the angle carefully. Many errors come from using sin instead of cos, or from resolving in the wrong pair of directions.

一定要绘制清晰的图并仔细标出角度。很多错误来自把 sin 和 cos 用反，或在错误的方向对上分解。

Resolving is used because equilibrium requires the sum of components in any direction to be zero. Choosing convenient perpendicular axes makes the algebra much simpler.

使用分解是因为平衡要求任意方向上的分力之和为零。选择方便的垂直坐标轴可以大幅简化代数运算。

4. Equilibrium Conditions | 平衡条件

A particle is in equilibrium if it remains at rest or moves with constant velocity. In statics, we focus on rest. By Newton’s first law, the resultant force acting on the particle must be zero.

如果质点保持静止或匀速直线运动，则它处于平衡状态。在静力学中，我们关注静止。根据牛顿第一定律，作用在质点上的合力必须为零。

In two dimensions, this gives two independent equations: the sum of horizontal components is zero, and the sum of vertical components is zero.

在二维情形下，这给出两个独立的方程：水平分力之和为零，竖直分力之和为零。

ΣFₓ = 0, ΣF_y = 0

If three non-parallel forces act on a particle in equilibrium, they can be represented by the sides of a triangle taken in order. This is the triangle of forces. It is a useful alternative to resolving when only three forces are involved.

如果三个不平行力作用于平衡的质点上，它们可以按顺序表示为三角形的三条边。这就是力的三角形法则。当只涉及三个力时，这是分解法之外的一种有用方法。

For three forces in equilibrium, Lami’s theorem states that each force is proportional to the sine of the angle between the other two forces.

对于三个平衡力，拉米定理指出每个力与另外两个力之间夹角的正弦成正比。

F₁ / sin α = F₂ / sin β = F₃ / sin γ

5. Free-Body Diagrams | 受力分析图

A free-body diagram shows a particle isolated from its surroundings with all forces acting on it drawn as arrows from the particle. This is the most important step in any statics problem.

受力分析图将质点从周围环境中隔离出来，所有作用在质点上的力都用从质点出发的箭头表示。这是任何静力学问题中最重要的一步。

To draw a correct free-body diagram, first identify the particle, then draw weight if mass is given, normal reaction at contact surfaces, tension in strings, and friction if the surface is rough. Do not include forces exerted by the particle on other objects.

要画出正确的受力分析图，首先要确定质点，然后画出重力（如果给出质量）、接触面处的法向反力、绳中的张力，以及粗糙表面上的摩擦力。不要包括质点施加给其他物体的力。

After drawing the diagram, choose axes, resolve forces, and write the equilibrium equations. Always check that every force has been included exactly once.

画好图后，选择坐标轴，分解力，并列出平衡方程。始终检查每个力是否恰好被包含一次。

Draw the particle alone, not the whole object.
Draw arrows from the particle, not to the particle.
Label forces with symbols, not just numbers.
Mark angles and chosen axes clearly.

6. Inclined Planes | 斜面问题

On an inclined plane, the weight is usually resolved into components parallel and perpendicular to the plane. If the plane makes angle θ with the horizontal, the component of weight perpendicular to the plane is W cos θ, and the component parallel down the plane is W sin θ.

在斜面上，通常将重力分解为平行和垂直于斜面的分量。如果斜面与水平面成 θ 角，则重力垂直于斜面的分量为 W cos θ，沿斜面向下的分量为 W sin θ。

Perpendicular component: W cos θ, Parallel component: W sin θ

If the particle is in equilibrium perpendicular to the plane and no other perpendicular force acts, the normal reaction is equal to W cos θ. Along the plane, the sum of forces must also be zero.

如果质点在垂直于斜面方向上平衡，且没有其他垂直力作用，则法向反力等于 W cos θ。沿斜面方向，力的总和也必须为零。

For example, if a particle is held at rest on a smooth inclined plane by a string parallel to the plane, then the tension T satisfies T = W sin θ.

例如，如果质点被一根平行于斜面的绳子拉住并静止在光滑斜面上，则张力 T 满足 T = W sin θ。

Always be careful about the direction of friction: if both weight and the applied force try to pull the particle down the slope, friction acts up the slope. If the applied force pulls up the slope, friction may act down the slope when the particle is about to move up.

始终注意摩擦力的方向：如果重力和外力都试图使质点沿斜面下滑，则摩擦力沿斜面向上。如果外力沿斜面向上拉，当质点即将向上运动时，摩擦力可能沿斜面向下。

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

Friction is a contact force that opposes sliding between two rough surfaces. For a particle at rest on a rough surface, friction can take any value needed to keep the particle in equilibrium, up to a maximum value.

摩擦是一种接触力，阻碍两个粗糙表面之间的滑动。对于静止在粗糙表面上的质点，摩擦力可以取维持平衡所需的任何值，直至达到最大值。

The maximum possible static friction is proportional to the normal reaction: F_max = μR, where μ is the coefficient of friction. The general condition for rest is F ≤ μR.

最大可能的静摩擦力与法向反力成正比：F_max = μR，其中 μ 是摩擦系数。静止的一般条件为 F ≤ μR。

F ≤ μR, F_max = μR

Limiting equilibrium occurs when the particle is just about to slide. In this state, friction has reached its maximum value, so F = μR. If the particle is not at limiting equilibrium, friction is less than μR and must be found from equilibrium equations.

极限平衡指质点即将开始滑动的状态。此时摩擦力达到最大值，因此 F = μR。如果质点未处于极限平衡，摩擦力小于 μR，必须由平衡方程求出。

The angle of friction λ is defined by tan λ = μ. It can be interpreted as the steepest angle of a rough plane on which a particle can remain in equilibrium without sliding.

摩擦角 λ 定义为 tan λ = μ。它可以解释为质点能在粗糙斜面上保持静止而不滑动的最大倾角。

tan λ = μ

State	Friction condition	中文条件
Stationary, not limiting	F < μR	F < μR
Limiting equilibrium	F = μR	F = μR
Sliding	F = μR	F = μR

8. Tension, Thrust and Pulleys | 张力、推力与滑轮

Tension is the pulling force transmitted through a string, cable or chain. In a light inextensible string, the tension is the same throughout the string. If the string passes over a smooth pulley, the pulley changes the direction of the tension but does not change its magnitude.

张力是通过绳、缆或链传递的拉力。在轻且不可伸长的绳中，绳中各处的张力相同。如果绳绕过光滑滑轮，滑轮改变张力的方向但不改变其大小。

This means a particle on one side of a smooth pulley and a particle on the other side experience the same tension T in the connecting string, provided the string is light and the pulley is smooth.

这意味着在光滑滑轮两侧的质点，在连接绳中受到的张力 T 相同，前提是绳为轻绳且滑轮光滑。

Thrust is the compressive force in a rod. A rod can push or pull, whereas a string can only pull. In statics, thrust is usually modelled as a force acting away from the particle along the rod when compressed.

推力是杆中的压缩力。杆既可以推也可以拉，而绳只能拉。在静力学中，当杆被压缩时，推力通常沿杆方向远离质点作用。

When drawing forces for a particle connected to a string, always draw tension away from the particle. If connected to a rod in compression, draw thrust away from the particle; if the rod is in tension, draw the pull toward the rod.

当为连接绳的质点画受力图时，始终将张力画为远离质点。如果连接的是受压杆，则将推力画为远离质点；如果杆受拉，则将拉力画为指向杆。

9. Connected Particles in Statics | 连接体静力学

Connected particle problems involve two or more particles linked by strings or rods. To solve them, draw a separate free-body diagram for each particle. Apply equilibrium conditions to each particle.

连接体问题涉及由绳或杆连接的两个或更多质点。求解时，为每个质点分别绘制受力分析图。对每个质点分别应用平衡条件。

Because a light inextensible string has constant tension, the tension in the connecting string appears in the equations for both particles. This links the two sets of equilibrium equations.

由于轻且不可伸长的绳中张力恒定，连接绳中的张力会同时出现在两个质点的方程中。这就把两组平衡方程联系起来了。

For example, suppose a particle of mass m rests on a rough slope and is connected by a light inextensible string over a smooth pulley to a hanging particle of mass M. If the system is in equilibrium, the tension pulling up the slope on the first particle equals the tension supporting the hanging mass, so T = Mg.

例如，质量为 m 的质点静止在粗糙斜面上，并通过绕过光滑滑轮的轻绳与悬挂的质量为 M 的质点连接。如果系统平衡，则拉第一个质点沿斜面向上的张力等于支撑悬挂质量的张力，因此 T = Mg。

Then the slope equations are solved using T = Mg. This illustrates the

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Statics of a Particle | 质点静力学