AS Mathematics: Newton's Laws Key Points | AS 数学：牛顿定律考点精讲

📚 AS Mathematics: Newton’s Laws Key Points | AS 数学：牛顿定律考点精讲

Newton’s laws of motion form the foundation of classical mechanics. In AS Mathematics, you are required to apply these laws to solve problems involving forces, motion, and equilibrium. This revision guide covers all essential concepts, formulas, and exam techniques to help you master Newton’s laws.

牛顿运动定律是经典力学的基础。在 AS 数学中，你需要运用这些定律来解决涉及力、运动和平衡的问题。本复习指南涵盖了所有核心概念、公式和考试技巧，帮助你掌握牛顿定律。

1. Introduction to Newton’s Laws | 牛顿定律简介

Newton’s three laws describe how forces affect the motion of objects. They are empirical laws based on observation and experiment. Understanding these laws is crucial for modelling real-world situations, from a car accelerating to a mass sliding down a slope.

牛顿三大定律描述了力如何影响物体的运动。它们是建立在观察和实验基础上的经验定律。理解这些定律对于模拟从汽车加速到物体沿斜面下滑等现实情况至关重要。

2. Newton’s First Law: Inertia | 牛顿第一定律：惯性

Newton’s first law states that an object at rest stays at rest, and an object in motion stays in uniform motion with the same speed and in the same direction, unless acted upon by a net external force. This property is called inertia. It implies that a zero resultant force produces no change in velocity.

牛顿第一定律指出，任何物体都将保持静止或匀速直线运动状态，直到有外力迫使它改变这种状态。这种性质称为惯性。这意味着合力为零时，速度不变。

Mathematically, if the resultant force is zero, acceleration is zero. This leads to the equilibrium condition:

数学上，如果合力为零，加速度也为零。由此可得到平衡条件：

ΣF = 0 ⇔ a = 0 and v = constant

In exam questions, you may need to use this fact to find unknown forces when the object is stationary or moving at constant velocity.

在考试题目中，当物体静止或匀速运动时，你可能需要利用这一事实来求未知力。

3. Newton’s Second Law: F = ma | 牛顿第二定律：F = ma

The net (resultant) force acting on an object is equal to the product of its mass and acceleration. This law allows us to relate forces to motion quantitatively.

作用在物体上的净（合）力等于物体质量与其加速度的乘积。该定律使得我们可以定量地将力与运动联系起来。

F = m a

Here, F is the resultant force in newtons (N), m is the mass in kilograms (kg), and a is the acceleration in metres per second squared (m·s⁻²). When multiple forces act, you must sum them as vectors, taking direction into account. In one dimension, choose a positive direction; forces in that direction are positive, those opposite are negative.

这里 F 是合力，单位牛顿 (N)；m 是质量，单位千克 (kg)；a 是加速度，单位米每二次方秒 (m·s⁻²)。当多个力作用时，必须按矢量求和并考虑方向。在一维问题中，选定正方向；沿正方向的力为正，相反的力为负。

The resultant force causes acceleration in the direction of the net force. If you know the forces, you can find the acceleration; conversely, if you know the acceleration of a system, you can deduce unknown forces.

合力产生沿其方向的加速度。如果已知力，可求加速度；反之，若已知系统加速度，可推求未知力。

4. Newton’s Third Law: Action-Reaction | 牛顿第三定律：作用力与反作用力

If object A exerts a force on object B, then object B simultaneously exerts an equal and opposite force on object A. These two forces are of the same type (e.g., both gravitational, both contact) and act on different bodies.

如果物体 A 对物体 B 施加一个力，那么物体 B 会同时对物体 A 施加一个大小相等、方向相反的力。这两个力类型相同（例如均为引力或均为接触力），且作用在不同物体上。

It is vital to remember that these paired forces do not cancel each other out in the equation of motion for a single object, because they act on different objects. This is one of the most common pitfalls in mechanics problems.

必须牢记：这对力不会在单个物体的运动方程中相互抵消，因为它们作用在不同物体上。这是力学问题中最常见的陷阱之一。

For example, when a book rests on a table, the book exerts a downward force on the table (its weight transferred via contact), and the table exerts an upward normal reaction on the book. These two forces act on different objects; thus, they do not form a Newton’s third-law pair. The third-law pair to the book’s weight is the gravitational pull exerted by the book on the Earth.

例如，一本书放在桌子上，书对桌子施加向下的力（通过接触传递的重量），桌子对书施加向上的法向反力。这两个力作用在不同物体上，因此不是牛顿第三定律的作用力-反作用力对。与书的重力配对的第三定律力是书对地球的引力。

5. Free-Body Diagrams | 受力分析图

Drawing a clear free-body diagram is the first step in solving any mechanics problem. Isolate the object of interest and draw all forces acting on that object as arrows pointing in the direction of the force. Do not include forces exerted by the object on its surroundings. Label each force with its symbol and direction.

绘制清晰的受力分析图是解决任何力学问题的第一步。隔离所研究的物体，将所有作用在该物体上的力用指向力方向的箭头画出。不要包含该物体施加给周围物体的力。用符号和方向标记每个力。

Common forces to show:

需要标示的常见力有：

Force	Symbol	Direction
Weight	mg	Vertically downwards
Normal reaction	R or N	Perpendicular to the contact surface
Tension	T	Along the string, away from the object
Friction	F or f	Opposite to motion or tendency of motion
Applied force	P or F	Given direction

These conventions form the basis for resolving forces and writing equations. Always double-check that you have considered all forces before applying Newton’s second law.

这些惯例构成了分解力和列方程的基础。在应用牛顿第二定律之前，务必仔细检查是否考虑了所有力。

6. Resolving Forces | 力的分解

Forces are vector quantities. To analyse situations where forces act at different angles, you must resolve forces into two perpendicular components, typically horizontally and vertically, or parallel and perpendicular to an inclined plane.

力是矢量。当力以不同角度作用时，必须将力分解为两个相互垂直的分量，通常为水平和竖直方向，或沿斜面方向和垂直斜面方向。

If a force F makes an angle θ with the horizontal, the components are:

若一个力 F 与水平方向成 θ 角，则其分量为：

Horizontal component = F cos θ, Vertical component = F sin θ

These trigonometric relations assume that θ is measured from the horizontal. If the angle is given relative to the vertical, you must adjust accordingly (e.g., horizontal component becomes F sin θ). Always draw a diagram and confirm the correct side of the triangle.

这些三角关系假定 θ 是从水平方向开始测量的。如果给出的角度是相对于竖直方向，你必须相应调整（例如水平分量变为 F sin θ）。始终画出图示并确认三角形的正确边对应关系。

Resolving is essential for solving statics problems (finding unknown forces when a = 0) and for setting up equations of motion along a chosen axis.

力的分解对于求解静力学问题（当 a = 0 时求未知力）以及沿选定轴建立运动方程至关重要。

7. Connected Particles | 连接体问题

When two or more particles are connected by a light, inextensible string, they share the same magnitude of acceleration, and the tension throughout the string is constant (assuming a smooth light pulley or a straight connection). In such problems, treat each particle separately and write its own equation of motion.

当两个或多个物体由轻质不可伸长的绳子连接时，它们具有大小相等的加速度，整根绳子张力恒定（假设光滑轻滑轮或直线连接）。在这类问题中，应分别考虑每个物体并分别列出其运动方程。

A typical setup involves a mass on a horizontal table connected by a string passing over a pulley to a hanging mass. For the mass on the table (mass m₁), horizontal forces include tension and possibly friction. For the hanging mass (m₂), forces are weight and tension. Choose a consistent positive direction, for example, the direction of motion of the system.

典型装置为一个物体放在水平桌面上，通过绕过滑轮的绳子与另一悬挂物体相连。对于桌面上的物体（质量 m₁），水平方向受力包括张力和可能存在的摩擦力。对于悬挂物体（质量 m₂），受力为重力和张力。选定一致的正方向，例如系统的运动方向。

The equations are:

方程如下：

For m₁: T – friction = m₁ a

For m₂: m₂ g – T = m₂ a

Solve simultaneously to find acceleration a and tension T. Remember that if friction is negligible, the friction term is zero.

联立求解得到加速度 a 和张力 T。切记若忽略摩擦，则摩擦力项为零。

8. Pulley Problems | 滑轮问题

Problems with smooth pulleys assume the pulley is light and frictionless, so the tension is the same on both sides of the string. The two hanging objects may both move vertically, in which case their accelerations have equal magnitude but opposite directions (one up, one down).

涉及光滑滑轮的问题中，假设滑轮轻质且无摩擦，因此绳子两边的张力相等。两个悬挂物体可能都作竖直运动，此时它们的加速度大小相等但方向相反（一个向上，一个向下）。

Define the positive direction as the direction of motion of the heavier mass. Then for the heavier mass (m₁ descending), the equation is m₁ g – T = m₁ a. For the lighter mass (m₂ ascending), the equation is T – m₂ g = m₂ a. Solve for a and T. If the string passes over a pulley and connects a mass on a slope, the principle remains the same: each mass has its own equation, and a is common.

定义较重物体的运动方向为正方向。则对于较重的质量（m₁ 下降），方程为 m₁ g – T = m₁ a。对于较轻的质量（m₂ 上升），方程为 T – m₂ g = m₂ a。联立求解 a 和 T。如果绳子绕过滑轮并连接斜面上的物体，原理相同：每个物体有各自的方程，且加速度 a 相同。

9. Friction | 摩擦力

Friction is a force that opposes relative motion or the tendency to move between two surfaces. For two surfaces in contact, the maximum possible static friction is proportional to the normal reaction R: F_max = μₛ R, where μₛ is the coefficient of static friction. Once motion occurs, kinetic friction takes over, given by F = μₖ R, where μₖ is the coefficient of kinetic friction. In many AS contexts,

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AS Mathematics: Newton’s Laws Key Points | AS 数学：牛顿定律 考点精讲