Newton's Laws of Motion for IB and CIE Physics | 牛顿运动定律 IB 与 CIE 考点精讲

📚 Newton’s Laws of Motion for IB and CIE Physics | 牛顿运动定律 IB 与 CIE 考点精讲

Newton’s laws of motion are the bedrock of classical mechanics and appear in virtually every IB and CIE Physics exam. This article distills the key concepts, common applications, and typical pitfalls you must master—from free-body diagrams and friction to connected bodies, momentum, and circular motion. Whether you are preparing for Paper 1 multiple-choice or tackling a structured question, a clear understanding of these three laws and their consequences will give you a solid foundation for problem-solving.

牛顿运动定律是经典力学的基石，几乎出现在每一次 IB 与 CIE 物理考试中。本文提炼了必须掌握的核心概念、常见应用和典型易错点——从隔离体受力图、摩擦力到连接体、动量以及圆周运动。无论你是在准备选择题还是结构题，透彻理解这三条定律及其推论都将为你解题打下坚实基础。

1. The First Law: Inertia | 第一定律：惯性

An object at rest stays at rest, and an object in motion continues in uniform motion in a straight line, unless acted upon by a resultant external force. This idea—called inertia—directly links to the concept of translational equilibrium: when the net force is zero, velocity is constant (which may be zero). In IB and CIE questions, you will often need to identify situations where no net force acts and then deduce that either the speed, direction, or both remain unchanged.

静止的物体保持静止，运动的物体沿直线做匀速运动，除非受到合外力的作用。这一概念——惯性——直接联系到平动平衡：当合力为零时，速度恒定（可能为零）。在 IB 与 CIE 考题中，经常需要识别无合力作用的情境，进而推断速度大小、方向或两者均保持不变。

If ΣF = 0, then v = constant (or zero).

若 ΣF = 0，则 v = 恒量（或零）。

Be careful: a body moving at constant speed in a circle is NOT in equilibrium because the direction of velocity changes—there must be a centripetal force. The first law does not apply in such a case.

注意：物体若做匀速圆周运动，速度方向时刻改变，并非平衡状态——必须存在向心力，此时第一定律不适用。

2. The Second Law: F = ma | 第二定律：加速度定律

The rate of change of momentum of a body is directly proportional to the resultant force and takes place in the direction of that force. In its most used form, we write ΣF = ma, where ΣF is the vector sum of all forces, m is the inertial mass, and a is the acceleration. Mass measured this way is independent of location, which exam questions may contrast with weight (W = mg).

物体的动量变化率与所受合外力成正比，并沿合外力的方向。最常见的形式为 ΣF = ma，其中 ΣF 是各力的矢量和，m 是惯性质量，a 是加速度。这样测得的惯性质量与位置无关，考题经常会与重力 (W = mg) 做对比。

ΣF = ma (vector form)

Always resolve forces into components along perpendicular axes—usually parallel and perpendicular to the inclined plane or the direction of motion. Remember that a is the acceleration of the system; you must identify the system, draw a free-body diagram, and sum the forces contributing to ΣF in that direction.

解题时始终要将力沿相互垂直的轴分解——通常平行和垂直于斜面或运动方向。记住 a 是系统的加速度；必须明确系统，画出受力图，并沿该方向合成得到 ΣF。

3. The Third Law: Action-Reaction | 第三定律：作用力与反作用力

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 exist only in pairs. A classic IB/CIE question asks why you can walk: your foot pushes backward on the ground; the ground pushes forward on you. The pair of forces are equal in magnitude and opposite in direction but act on different objects—so they do not cancel.

若物体 A 对物体 B 施加一个力，则物体 B 同时对物体 A 施加一个大小相等、方向相反的力。这对力同种性质、作用在不同物体上，并成对出现。经典的 IB / CIE 问题常问为何人能走路：脚向后推地，地向前推人。这对力大小相等、方向相反，但作用在不同物体，因此不能抵消。

Fᴀʙ = –Fʙᴀ

Misidentifying the reaction force is a common error. For a book on a table, the weight of the book (Earth pulls book) has a reaction: the book pulls Earth upward. The normal force (table pushes book) has a reaction: the book pushes down on the table. These two pairs are separate. Always name the two bodies involved.

错认反作用力是常见错误。例如桌面上的书，重力（地球拉书）的反作用力是书向上拉地球。支持力（桌推书）的反作用力是书向下推桌。这两对力是不同的，务必指明所涉及的两个物体。

4. Free-Body Diagrams | 隔离体受力图

Drawing a clear free-body diagram is the single most important step in solving mechanics problems. Represent the body as a dot or a simple box, and draw every force vector acting ON that body: weight (mg, acting from the centre of mass), normal reaction, tension, friction, applied forces, etc. Do NOT include forces exerted by the body on its surroundings. Label each force and its direction unambiguously.

画出一幅清晰的隔离体受力图是解决力学问题最关键的一步。将物体表示为一点或方框，画出所有作用在该物体上的力：重力 (mg, 作用在质心)、法向力、张力、摩擦力、施加力等。不要画物体施加给外界的力。明确标出每个力及其方向。

IB and CIE mark schemes award points for correct force diagrams even if the subsequent calculation contains an error. Start by defining your coordinate axes, then decompose forces that are not aligned with these axes. The acceleration vector should also be indicated to remind you of the direction of the net force.

IB 与 CIE 评分方案中即便后续计算出错，正确的受力图仍能拿分。先定义坐标轴，再分解不沿轴的力。还应标出加速度矢量，以提醒自己合力的方向。

5. Friction and Inclined Planes | 摩擦力与斜面

Friction is modelled as f = μN, where N is the normal reaction force. Two coefficients are specified: static friction (μₛ) for surfaces not sliding, giving a maximum value fₘₐₓ = μₛN; and kinetic friction (μₖ) for surfaces in relative motion, fₖ = μₖN. For an object on an inclined plane at angle θ to the horizontal, the weight component down the slope is mg sinθ, and the normal reaction is mg cosθ.

摩擦力模型为 f = μN，其中 N 是法向反力。区分两个系数：静摩擦系数 (μₛ) 用于尚未滑动的表面，产生最大静摩擦力 fₘₐₓ = μₛN；动摩擦系数 (μₖ)用于相对滑动的表面，fₖ = μₖN。对于倾角为 θ 的斜面上的物体，重力沿斜面的分量为 mg sinθ，法向反力为 mg cosθ。

fₘₐₓ = μₛ N fₖ = μₖ N

A typical exam question asks whether an object will slide: compare the downhill component mg sinθ with the maximum static friction μₛ mg cosθ. If mg sinθ > μₛ mg cosθ, the object accelerates down the slope.

典型考题会问物体是否会滑动：比较下滑分量 mg sinθ 与最大静摩擦 μₛ mg cosθ。若 mg sinθ > μₛ mg cosθ，物体将沿斜面加速下滑。

6. Connected Bodies and Tension | 连接体与张力

When two or more masses are connected by a light inextensible string over a smooth pulley, the tension is the same throughout the string (massless string) and the magnitudes of acceleration of the masses are equal. Write separate equations of motion for each mass using ΣF = ma, choosing the positive direction along the direction of acceleration. Solve simultaneously for acceleration and tension.

当两个或多个物体通过轻质且不可伸长的绳子跨过光滑滑轮相连时，绳中张力处处相等（轻绳）且各物体的加速度大小相等。对每个物体单独应用 ΣF = ma 列方程，沿加速度方向选为正方向，联立求解加速度和张力。

For a heavier mass M descending and a lighter mass m ascending, with the slight complication that the rope is vertical on both sides: Mg – T = Ma, and T – mg = ma. Adding gives a = (M – m)g/(M + m). Exam questions may tilt one side or add friction on a table; always draw free-body diagrams for each mass.

对于较重的 M 下降、较轻的 m 上升的情况，并假设绳子两侧垂直，可列：Mg – T = Ma，T – mg = ma。相加得 a = (M – m)g/(M + m)。考题可能让一侧倾斜或在桌面上引入摩擦；务必为每个物体单独画受力图。

7. Lift Problems and Apparent Weight | 升降机与视重

In a lift accelerating upward, your apparent weight (the normal force from the floor) is greater than your true weight. The scale reads N = mg + ma. When accelerating downward, the normal force becomes N = mg – ma. In free fall (a = g), the reading becomes zero—apparent weightlessness. If the lift moves at constant velocity, a = 0, and the reading equals mg.

在加速上升的电梯中，视重（地板的支持力）大于真实重量。秤的读数为 N = mg + ma。加速下降时，N = mg – ma。自由落体时 (a = g)，读数为零——视失重。若电梯匀速运动，a = 0，读数等于 mg。

N = m(g ± a)

These principles are frequently tested in the context of astronauts in a space station or during launch. In orbit, astronauts feel weightless not because gravity is zero, but because they and their craft are in free fall around the Earth—the normal force is zero.

这类原理常在空间站或发射情境中考查。在轨道上，宇航员感觉失重并非因为引力为零，而是因为他们与航天器一起绕地球自由落体——支持力为零。

8. Momentum and Impulse: The General Second Law | 动量与冲量：更普遍的牛顿第二定律

Newton originally stated his second law in terms of momentum: the net force equals the rate of change of momentum. F = Δp/Δt. For constant mass, this reduces to F = ma. However, for situations where mass changes (e.g. rocket propulsion) or during collisions, the momentum form is essential. Impulse J = F Δt equals the change in momentum Δp. The area under a force–time graph gives impulse.

牛顿最初用动量表述第二定律：合外力等于动量的变化率。即 F = Δp/Δt。在质量不变时化简为 F = ma。但对于质量变化（如火箭推进）或碰撞过程，动量形式至关重要。冲量 J = F Δt 等于动量的变化 Δp。力–时间图线下的面积即为冲量。

p = mv J = Δp = F Δt

In IB Physics, a classic data-analysis question asks you to use a force sensor and motion sensor to verify the impulse–momentum theorem. CIE often embeds momentum ideas in collision conservation problems, linking Newton’s third law to the conservation of momentum in isolated systems.

IB 物理中，经典的数据分析题会要求用力传感器和运动传感器验证冲量–动量定理。CIE 常将动量概念嵌入碰撞守恒问题，通过牛顿第三定律关联孤立系统动量守恒。

9. Circular Motion and Centripetal Force | 圆周运动与向心力

An object moving in a circle at constant speed is accelerating towards the centre because its velocity direction continuously changes. Newton’s second law demands a resultant centripetal force given by F = mv²/r = mω²r. This force is always perpendicular to velocity—hence it does no work. Common sources are tension (string), gravitational force (orbits), friction (car on a bend), or the normal component on a banked track.

匀速圆周运动的物体始终朝圆心加速，因为速度方向不断改变。根据牛顿第二定律，必须有向心力 F = mv²/r = mω²r。该力始终垂直于速度，因此不做功。常见力源有：张力（绳）、引力（轨道）、摩擦力（弯道汽车）或倾斜轨道上的法向分量。

F_c = m v²/r = m ω² r

Exam problems ask for the maximum speed around a curve without skidding, requiring frictional force to provide the centripetal force: μmg ≥ mv²/r. In vertical circular motion (e.g., a bucket of water), tension varies with position and must be evaluated at the top and bottom using energy conservation plus ΣF = ma.

考题会问弯道不侧滑的最大速度，要求摩擦力提供向心力：μmg ≥ mv²/r。在竖直面内的圆周运动（如水桶问题）中，张力随位置变化，需结合机械能守恒和 ΣF = ma，分别分析最高点与最低点。

10. Non-Inertial Frames (IB Higher) | 非惯性参考系（IB拓展）

In a reference frame that accelerates relative to an inertial frame, Newton’s laws appear not to hold unless fictitious forces (pseudo-forces) are introduced. For a linearly accelerating frame with acceleration a_frame, an observer inside feels a fictitious force –m a_frame. The famous example is the apparent sideways force felt by a passenger when a car turns—actually inertia resisting the change in direction.

在相对惯性系加速运动的参考系中，牛顿定律似乎不成立，除非引入虚拟力（惯性力）。对于以加速度 a_frame 平动的非惯性系，内部观察者感受到虚拟力 –m a_frame。典型例子是汽车转弯时乘客感觉被甩向外侧——实际上是惯性抵抗方向变化。

IB higher-level candidates may be asked to calculate the “g” vector in a lift or on a rotating space station. The effective gravity is g_eff = g – a, where a is the acceleration of the frame. You must be able to transform between inertial and non-inertial descriptions.

IB 高等级考生可能需计算电梯内或旋转空间站中的“表观重力”。有效重力加速度 g_eff = g – a，其中 a 为参考系加速度。考生应能在惯性系与非惯性系描述之间转换。

11. Experiment: Verifying Newton’s Second Law | 实验：验证牛顿第二定律

A standard investigation uses a trolley of mass M on a friction-compensated runway, pulled by a hanging mass m. The tension T pulling the trolley provides the accelerating force for the entire system. By varying m and measuring acceleration (using a motion sensor or light gates and picket fence), you can verify that a ∝ F for constant total mass, and a ∝ 1/(M+m) for constant force. A common systematic error is not compensating friction properly; the track should be tilted until the trolley moves at constant speed.

标准实验使用置于摩擦补偿轨道上质量为 M 的小车，由悬挂质量 m 拉动。绳子对小车拉力 T 提供整个系统加速所需的力。改变 m 并测量加速度（用运动传感器或光电门与挡光片），可以验证总质量不变时 a ∝ F，以及力不变时 a ∝ 1/(M+m)。常见的系统误差是未妥善补偿摩擦：应倾斜轨道直到小车匀速滑下。

a = (m g) / (M + m) (assumed massless string, frictionless pulley)

CIE Paper 3 and IB Internal Assessment often require a full uncertainty analysis and a linearised graph. Plotting a vs m gives a curve; instead, plot a vs mg or a vs F after calculating tension T = M a. Always discuss whether the string remains taut and whether the pulley is truly massless.

CIE 试卷 3 和 IB 内部评估常要求完整的不确定度分析和线性化图像。直接作 a–m 图是曲线；应转换为 a–F 图（F = mg），或计算张力 T = M a 后分析。务必讨论绳子是否始终紧绷、滑轮是否真正无质量。

12. Common Pitfalls and Exam Tips | 常见失分点与应试技巧

1) Confusing mass and weight: mass is scalar (inertia), weight is force (vector). Use W = mg. 2) Forgetting to include all forces in free-body diagrams—especially air resistance, normal force, or tension. 3) Misapplying the third law: always check that the two forces act on different bodies. 4) Mixing up the direction of friction for rolling vs sliding; kinetic friction opposes relative motion. 5) Assuming that a constant velocity implies no forces—it implies no net force. 6) Neglecting vector nature of forces: resolve components BEFORE applying ΣF = ma. 7) Using mg as the net force on an incline when friction or tension is also present.

1) 混淆质量与重量：质量是标量（惯性），重量是矢量（力），使用 W = mg。2) 隔离体图中遗漏某些力——尤其是空气阻力、法向力或张力。3) 误用第三定律：务必检查两个力是否作用在不同物体上。4) 混淆滚动与滑动摩擦的方向；动摩擦与相对运动方向相反。5) 以为匀速运动意味着不受力——应是无净力。6) 忽略力的矢量性：在代入 ΣF = ma 前须分解分量。7) 在斜面问题中有摩擦或张力时仍把重力直接当成合力。

In structured questions, always state the equation you are using before substituting numbers. Show the direction of positive acceleration clearly. If you obtain a negative acceleration, interpret it as opposite to your chosen positive direction. Check limiting cases—for example, if m = 0 in a connected-body problem, does your expression give a = 0 or a = g? This sanity check helps catch algebraic mistakes.

在做结构题时，务必先写出所用公式再代入数值。明确标出正方向。若算出负加速度，应解释为与规定正方向相反。检查极限情况——例如连接体问题中若 m=0，你的表达式是否得出 a=0 或 a=g？这种合理性检查有助于发现代数错误。

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Newton’s Laws of Motion for IB and CIE Physics | 牛顿运动定律 IB 与 CIE 考点精讲