Newton’s Laws for IGCSE Edexcel Maths | IGCSE Edexcel 数学：牛顿定律考点精讲

📚 Newton’s Laws for IGCSE Edexcel Maths | IGCSE Edexcel 数学：牛顿定律考点精讲

Newton’s Laws of Motion form a fundamental bridge between force and motion in mechanics, and they appear regularly in IGCSE Edexcel Mathematics questions involving connected particles, pulleys, lifts, and inclined planes. Understanding these laws allows you to model real-world situations mathematically and solve for acceleration, tension, normal reaction, and driving forces.

牛顿运动定律是力学中联系力与运动的基本桥梁，在 IGCSE Edexcel 数学考试中频繁出现，涉及连接体、滑轮、电梯和斜面等问题。掌握这些定律，你就能用数学方法模拟现实情境，求解加速度、张力、法向反作用力和驱动力等物理量。

1. Newton’s First Law: The Law of Inertia | 牛顿第一定律：惯性定律

Newton’s First Law states that an object will remain at rest or move with constant velocity unless acted upon by a resultant external force. In IGCSE maths problems, this means that when the net force is zero, acceleration is zero, and the object is in equilibrium.

牛顿第一定律指出，除非受到合外力作用，否则物体将保持静止或匀速直线运动状态。在 IGCSE 数学问题中，这意味着当合外力为零时，加速度为零，物体处于平衡状态。

For example, if a car is moving at a steady speed on a straight road, the driving force exactly balances the resistive forces. Mathematically, we use the equilibrium condition:

例如，如果一辆汽车在笔直的公路上匀速行驶，驱动力恰好与阻力平衡。数学上，我们使用平衡条件：

ΣF = 0

This is crucial for setting up equations for objects at rest or moving with constant velocity on horizontal surfaces, lifts moving at constant speed, or bodies just about to move (limiting equilibrium).

这对于建立物体静止或匀速直线运动的方程至关重要，例如水平面上的物体、匀速运动的电梯、或即将运动的物体（极限平衡）。

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

The Second Law tells us that the net force acting on a particle is equal to the product of its mass and acceleration. The direction of the acceleration is the same as that of the resultant force. This is the most widely used law in IGCSE mechanics questions.

第二定律告诉我们，作用在质点上的合力等于质点的质量与加速度的乘积。加速度的方向与合外力方向相同。这是 IGCSE 力学问题中使用最广泛的定律。

Resultant force = mass × acceleration → F = ma

Units: Force in newtons (N), mass in kilograms (kg), acceleration in metres per second squared (m/s²). Always resolve forces and apply F = ma along the direction of motion or parallel to the plane.

单位：力的单位是牛顿 (N)，质量的单位是千克 (kg)，加速度的单位是米每二次方秒 (m/s²)。始终沿运动方向或平行于平面的方向分解力，并应用 F = ma。

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

Newton’s Third Law states that if body A exerts a force on body B, then body B exerts an equal and opposite force on body A. These forces are of the same type, act on different bodies, and are equal in magnitude but opposite in direction.

牛顿第三定律指出，如果物体 A 对物体 B 施加一个力，那么物体 B 同时对物体 A 施加一个大小相等、方向相反的力。这两个力属于同种类型，作用在不同物体上，大小相等，方向相反。

In connected particle problems, this law is the reason the tension in a light inextensible string is the same throughout, and why the force exerted by one block on another in contact is equal and opposite. It helps us isolate systems and draw correct force diagrams.

在连接体问题中，该定律解释了为什么轻绳不可伸长时张力处处相等，也解释了为什么互相接触的物块之间的作用力是等大反向的。它帮助我们隔离系统并绘制正确的受力图。

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4. Drawing Free-Body Diagrams | 画受力分析图

A clear free-body diagram is essential for applying Newton’s laws. Start by isolating the particle or body, then draw all forces acting on it: weight (mg downwards), normal reaction (perpendicular to the surface), tension (along the string, away from the body), driving/resistive forces, friction (opposing motion), and any externally applied forces.

清晰的受力分析图是应用牛顿定律的关键。首先隔离质点或物体，然后画出作用于它的所有力：重力 (mg 竖直向下)、法向反作用力 (垂直于接触面)、张力 (沿着绳子方向，远离物体)、驱动力/阻力、摩擦力 (与运动方向相反) 以及任何外加力。

Never include forces exerted by the body on other objects; only show those acting on the body. For inclined planes, resolve weight into components parallel and perpendicular to the plane.

切勿包含该物体施加给其他物体的力；只画作用在该物体上的力。对于斜面问题，将重力分解为平行于斜面和垂直于斜面的分量。

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5. Connected Particles: Horizontal Surface and Pulley Systems | 连接体：水平面与滑轮系统

When two or more particles are connected by a light inextensible string, they have the same acceleration and the same tension throughout the string. You treat the whole system as one to find acceleration, then isolate individual particles to find tension or contact forces.

当两个或多个质点通过轻绳连接且绳子不可伸长时，它们的加速度相同，且绳中的张力处处相等。你可以将整个系统视为一个整体求加速度，然后隔离单个质点求张力或接触力。

Common scenarios: one mass on a smooth horizontal table, connected via a pulley to a hanging mass. For the system: driving force = weight of hanging mass; total mass = sum of both masses. Then a = (m₂g) / (m₁ + m₂). To find tension, apply F = ma to the mass on the table: T = m₁a.

常见场景：一个物体放在光滑水平桌面上，通过滑轮与悬挂物体相连。对系统整体：驱动力 = 悬挂物体的重量；总质量 = 两物体质量之和。于是 a = (m₂g) / (m₁ + m₂)。为求张力，对桌面上的物体应用 F = ma：T = m₁a。

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6. Inclined Planes with Connected Particles | 斜面上的连接体问题

When one or both particles lie on smooth inclined planes, resolve the weight component along the plane. For a mass on a smooth incline, the force causing acceleration (or opposing motion) is mg sin θ. Use the same system approach: write an equation of motion for the whole system parallel to the direction of motion, then solve for a.

当连接体系统中有一个或两个质点位于光滑斜面上时，需要分解重力沿斜面的分量。对于光滑斜面上的物体，引起加速度（或阻碍运动）的力是 mg sin θ。采用相同的整体法：沿运动方向对整个系统列出运动方程，然后求解加速度 a。

If the system contains different inclinations, the net driving force becomes the sum of the down-plane components of the hanging masses, with careful treatment of sign depending on which direction the system moves.

如果系统中包含不同倾角的斜面，净驱动力变为悬挂物体沿斜面向下分量的代数和，需根据系统运动方向谨慎处理正负号。

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7. Lifts and Vertical Motion | 电梯与竖直运动

A lift moving with acceleration changes the apparent weight of objects inside it. Consider a person standing on scales in a lift. The reading on the scales is the normal reaction R. Applying F = ma: R − mg = ma (if accelerating upwards), or mg − R = ma (if accelerating downwards).

加速运动的电梯会改变内部物体的表观重量。考虑一个人站在电梯内的秤上，秤的读数就是法向反作用力 R。应用 F = ma：向上加速时 R − mg = ma；向下加速时 mg − R = ma。

R = mg + ma (upward acceleration)
R = mg − ma (downward acceleration)

If the lift moves at constant velocity or is at rest, a = 0 and R = mg. These problems test your ability to correctly state the direction of acceleration and set up the net force equation.

如果电梯匀速运动或静止，a = 0，则 R = mg。这类问题考察你是否能正确指出加速度方向并建立合力方程。

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8. Friction and Limiting Equilibrium | 摩擦力与极限平衡

In IGCSE Edexcel Maths, friction F is often modelled by F ≤ μR, where μ is the coefficient of friction and R is the normal reaction. When a body is on the point of moving, friction reaches its maximum: F = μR. This is called limiting equilibrium.

在 IGCSE Edexcel 数学中，摩擦力 F 通常用 F ≤ μR 来建模，其中 μ 是摩擦系数，R 是法向反作用力。当物体处于将动未动的临界状态时，摩擦力达到最大值：F = μR。这称为极限平衡。

To solve, resolve perpendicular to the surface to find R, then use F = μR for limiting friction. Apply equilibrium conditions or F = ma parallel to the surface. Remember that friction always opposes motion or the tendency to move.

解题时，先垂直表面方向分解力求 R，然后用 F = μR 表示极限摩擦力。沿表面方向应用平衡条件或 F = ma。记住摩擦力总是阻碍运动或运动趋势。

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9. Applying F = ma in Vector Form | 向量形式下的 F = ma

When forces are given in i, j notation, Newton’s Second Law can be applied directly in vector form. If a particle of mass m is acted upon by several forces, the resultant vector force equals mass times the acceleration vector.

当力以 i, j 向量形式给出时，牛顿第二定律可直接以向量形式应用。若质量为 m 的质点受到若干个力的作用，合力向量等于质量乘以加速度向量。

ΣF = m a (all quantities as vectors)

Find the resultant force by adding all force vectors, then set it equal to m(ai + bj). This gives two scalar equations to solve for a and b, the components of acceleration.

通过将所有力向量相加求出合力，然后令其等于 m(ai + bj)。这将给出两个标量方程，用来求解加速度分量 a 和 b。

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10. Momentum, Impulse and Newton’s Laws | 动量、冲量与牛顿定律

Newton’s Second Law is often expressed in terms of momentum: Force = rate of change of momentum. For constant mass, this reduces to F = ma. Impulse is the change in momentum: Impulse = F × t = mv − mu. This appears in IGCSE questions involving collisions or sudden forces.

牛顿第二定律常用动量表述：力 = 动量变化率。对于质量不变的情况，此式简化为 F = ma。冲量是动量的变化量：冲量 = F × t = mv − mu。这在涉及碰撞或突然施加的力的 IGCSE 题目中出现。

Remember that momentum is a vector: p = mv. In onedimensional situations, assign a positive direction. Impulse can be positive or negative depending on whether it increases or decreases the momentum in that direction.

记住动量是向量：p = mv。在一维情形下，先规定正方向。冲量可以为正或负，取决于它是使该方向的动量增加还是减少。

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11. Common Exam Pitfalls and Tips | 常见考试陷阱与技巧

Many students forget to convert mass from grams to kilograms or to use consistent units. Always check that mass is in kg, acceleration in m/s², and force in N. Also, be careful with the direction of forces: when a particle is decelerating, the acceleration is negative relative to the chosen positive direction.

许多学生会忘记将质量从克换算为千克，或未使用一致的单位。务必检查质量的单位是 kg，加速度是 m/s²，力的单位是 N。此外，注意力的方向：当质点减速时，相对于选定的正方向，加速度为负。

When treating systems, don’t doublecount internal forces like tension. Use the system as a whole to find acceleration, then isolate. Always draw a force diagram. For lift problems, draw the normal reaction acting upward on the person, not downward. Think: ‘What is the force that gives me the acceleration?’

处理系统问题时，不要重复计算如张力这样的内力。先用整体法求加速度，再隔离分析。始终要画受力图。对于电梯问题，画出作用在人身上的法向反作用力方向向上，而非向下。思考：“什么力使我获得加速度？”

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