Mechanics | 力学 - aleveler.com

📚 Mechanics | 力学

Mechanics forms the backbone of classical physics, covering the motion of objects and the forces that cause this motion. Mastering mechanics is essential for success in IB Physics, as it provides the conceptual tools to analyze everything from a skydiver’s fall to planetary orbits.

力学构成了经典物理学的核心，涵盖物体的运动及其成因——力。掌握力学对于在 IB 物理中取得成功至关重要，它提供了分析跳伞者下落到行星轨道等各种现象的概念工具。

1. Kinematics – Describing Motion | 运动学 – 运动的描述

Kinematics deals with the description of motion without considering its causes. Displacement (s) is a vector quantity representing the change in position, while distance is the scalar length of the path traveled. Velocity (v) is the rate of change of displacement, and acceleration (a) is the rate of change of velocity. These quantities are linked by fundamental definitions: average velocity v = Δs / Δt, and average acceleration a = Δv / Δt. Direction matters for vectors, and sign conventions must be consistent in one-dimensional motion.

运动学描述物体的运动，而不涉及引起运动的原因。位移是表示位置变化的矢量，而路程是标量。速度是位移的变化率，加速度是速度的变化率。这些量通过基本定义相关联：平均速度 v = Δs / Δt，平均加速度 a = Δv / Δt。对于矢量，方向非常重要，在直线运动中必须保持符号统一。

v = Δs / Δt a = Δv / Δt

Instantaneous velocity and acceleration are found by taking the gradient of a displacement–time or velocity–time graph at a specific point. The area under a velocity–time graph gives displacement. These graphical techniques are powerful tools for analyzing motion, often appearing in IB data‑analysis questions.

瞬时速度和加速度可以通过位移‑时间图或速度‑时间图在某点的斜率求得。速度‑时间图下的面积表示位移。这些图像分析方法是 IB 数据分析题中常见的强大工具。

2. Equations of Uniformly Accelerated Motion | 匀加速运动方程

When acceleration is constant, four kinematic equations describe the relationship between initial velocity (u), final velocity (v), acceleration (a), displacement (s), and time (t). They are derived from the definitions of velocity and acceleration and from simple geometry of a velocity–time graph. These equations only apply when the acceleration is uniform in both magnitude and direction.

当加速度恒定时，四个运动学方程描述了初速度 u、末速度 v、加速度 a、位移 s 和时间 t 之间的关系。它们来源于速度和加速度的定义以及速度‑时间图的简单几何关系。这些方程仅在加速度的大小和方向都保持不变时才适用。

v = u + a t — 无位移
s = u t + ½ a t² — 无末速度
v² = u² + 2 a s — 无时间
s = ½ (u + v) t — 无加速度

For example, calculating the stopping distance of a car with initial speed u and deceleration a uses v² = u² + 2as with v = 0, giving s = u² / (2|a|). Always check that signs are consistent: take one direction as positive and treat deceleration as negative acceleration.

例如，计算一辆初速度为 u、减速为 a 的汽车的制动距离，使用 v² = u² + 2as 并令 v = 0，得 s = u² / (2|a|)。务必检查符号的一致性：选定一个方向为正，并将减速视为负加速度。

3. Projectile Motion | 抛体运动

Projectile motion is the combination of uniform horizontal motion and uniformly accelerated vertical motion under gravity, assuming negligible air resistance. The horizontal component of velocity remains constant (v_x = u cosθ), while the vertical component is affected by gravitational acceleration (a_y = –9.81 m s⁻²). The two components are independent, and time is the same for both.

抛体运动是水平方向的匀速运动和竖直方向在重力作用下的匀加速运动的合成，通常忽略空气阻力。速度的水平分量保持不变（v_x = u cosθ），而竖直方向受重力加速度影响（a_y = –9.81 m s⁻²）。两个分量相互独立，时间对两者相同。

Key quantities include time of flight, maximum height, and range. The maximum height is reached when the vertical velocity becomes zero. The range R = (u² sin 2θ) / g for a projectile launched and landing on the same horizontal level. This expression shows that the maximum range is achieved at a launch angle of 45°.

关键量包括飞行时间、最大高度和射程。当竖直速度为零时达到最大高度。对于起点与终点在同一水平面的抛体，射程 R = (u² sin 2θ) / g。该表达式表明，当抛射角为 45° 时射程最大。

R = (u² sin 2θ) / g t_total = (2 u sinθ) / g h_max = (u² sin²θ) / (2g)

IB questions often ask you to resolve initial velocity into components and apply the equations separately. Remember that at the highest point the vertical speed is zero, not the acceleration—gravity never switches off.

IB 题目常要求将初速度分解为分量并分别应用方程。记住，在最高点竖直速度为零，但加速度不为零——重力始终存在。

4. Newton’s Laws of Motion | 牛顿运动定律

Newton’s three laws form the foundation of dynamics. The first law states that an object remains at rest or in uniform motion in a straight line unless acted upon by an external unbalanced force. This introduces the concept of inertia and the inertial frame of reference. The second law quantifies the relationship: F_net = m a, where the net force is the vector sum of all forces. The third law says that if body A exerts a force on body B, then B exerts an equal and opposite force on A.

牛顿三定律构成了动力学的基础。第一定律指出，除非受到外部的非平衡力作用，物体会保持静止或沿直线匀速运动。这引入了惯性的概念和惯性参考系。第二定律定量描述：F_net = m a，其中合外力是所有力的矢量和。第三定律说明，若物体 A 对物体 B 施加力，则 B 同时对 A 施加大小相等方向相反的力。

When applying F = m a, ensure that a free‑body diagram is drawn and all forces are resolved in chosen axis directions. Mass is measured in kilograms, and force in newtons (1 N = 1 kg m s⁻²). The second law can also be expressed in terms of momentum change, which leads to the impulse–momentum theorem.

应用 F = m a 时，务必先画出受力分析图，并将所有力沿选定坐标轴分解。质量单位是千克，力的单位是牛顿（1 N = 1 kg m s⁻²）。第二定律也可用动量的变化来表示，由此导出冲量‑动量定理。

5. Free-Body Diagrams and Force Analysis | 受力分析与力图

A free‑body diagram (FBD) is a simplified sketch showing all the forces acting on a single object. Each force is represented by an arrow whose length indicates approximate magnitude and whose direction shows the force’s line of action. Common forces include weight (mg), normal reaction (N), tension (T), friction (f), and applied forces.

受力分析图是显示作用在单个物体上所有力的简化示意图。每个力用一个带箭头的矢量表示，其长度大致表示大小，方向表示力的作用线。常见的力有重力 (mg)、支持力 (N)、张力 (T)、摩擦力 (f) 和其他外加力。

In equilibrium problems, the vector sum of forces is zero: ΣF = 0. On inclined planes, resolve weight into components parallel to and perpendicular to the surface. Friction opposes relative motion and can be static or kinetic, with the magnitude often limited by f_max = μ_s N for static friction and f_k = μ_k N for kinetic friction.

在平衡问题中，合外力为零：ΣF = 0。在斜面上，需把重力分解为平行于和垂直于斜面的分量。摩擦力阻碍相对运动，可为静摩擦力或动摩擦力，静摩擦力的最大值常为 f_max = μ_s N，动摩擦力为 f_k = μ_k N。

Never include forces that the object exerts on other objects in its own FBD; only forces acting on the object of interest are drawn. Practice drawing clear FBDs is essential for complex dynamics problems, including connected bodies and pulleys.

切勿在受力分析图中画出该物体对其他物体施加的力；只画出作用在该研究对象上的力。绘制清晰的力图对于处理连接体、滑轮等复杂动力学问题至关重要。

6. Momentum and Impulse | 动量与冲量

Momentum p is defined as the product of mass and velocity: p = m v. It is a vector, and its unit is kg m s⁻¹. Impulse J is the effect of a force over a time interval: J = F Δt, which equals the change in momentum Δp. This is the impulse–momentum theorem.

动量 p 定义为质量与速度的乘积：p = m v。它是矢量，单位为 kg m s⁻¹。冲量 J 是力在时间间隔内的累积效应：J = F Δt，它等于动量的变化 Δp。这就是冲量‑动量定理。

F Δt = Δp = m v – m u

The law of conservation of momentum states that in an isolated system (no external forces), the total momentum remains constant. This principle is crucial for analyzing collisions and explosions. In a collision, if the total kinetic energy is conserved, it is elastic; otherwise it is inelastic. Note that momentum is conserved even in inelastic collisions.

动量守恒定律指出，在孤立系统（无外力）中，总动量保持不变。这一原理对于分析碰撞和爆炸至关重要。在碰撞中，若总动能守恒，则为弹性碰撞；否则为非弹性碰撞。注意，即使在非弹性碰撞中，动量依然守恒。

In two‑dimensional collisions, momentum must be conserved separately along orthogonal axes. Always define a coordinate system and apply conservation component‑wise: Σp_x before = Σp_x after, and similarly for the y‑direction.

在二维碰撞中，动量必须沿正交坐标轴各自守恒。务必定义坐标系并分别应用分量守恒：碰撞前 total p_x = 碰撞后 total p_x，y 方向同理。

7. Work, Energy, and Power | 功、能量与功率

Work is done when a force moves an object. It is defined as W = F d cosθ, where θ is the angle between the force and displacement vectors. Work transfers energy, and its unit is the joule (J). Kinetic energy (KE) is the energy due to motion: KE = ½ m v². Gravitational potential energy (GPE) near Earth’s surface is GPE = m g h.

当力使物体移动时便做了功。功的定义为 W = F d cosθ，其中θ是力与位移之间的夹角。功传递能量，单位是焦耳 (J)。动能是物体由于运动而具有的能量：KE = ½ m v²。地球表面附近的重力势能为 GPE = m g h。

W = F d cosθ KE = ½ m v² GPE = m g h

Power is the rate of doing work: P = W / t, measured in watts (W). An alternative form, useful when a constant force moves an object at constant velocity, is P = F v. This is often applied to engines and moving vehicles.

功率是做功的速率：P = W / t，单位是瓦特 (W)。当恒力使物体以恒定速度运动时，另一种常用形式为 P = F v，常用于引擎和行驶车辆问题。

8. Conservation of Energy | 能量守恒

The principle of conservation of energy states that energy cannot be created or destroyed, only transferred or transformed. In a closed system with no non‑conservative forces, the total mechanical energy E = KE + GPE remains constant. Dropping an object from a height illustrates the conversion: GPE decreases, KE increases, and their sum stays constant.

能量守恒原理指出，能量既不能产生也不能消灭，只能转移或转化。在没有非保守力的封闭系统中，总机械能 E = KE + GPE 保持不变。从高处释放物体就是转换的例证：GPE 减小，KE 增加，而总和恒定。

When external forces like friction or air resistance do work on a system, mechanical energy is no longer conserved, but the total energy (including thermal energy) still is. The work done by non‑conservative forces equals the change in mechanical energy: W_nc = ΔKE + ΔGPE.

当诸如摩擦力或空气阻力等外力对系统做功时，机械能不再守恒，但包括热能在内的总能量仍然守恒。非保守力所做的功等于机械能的变化：W_nc = ΔKE + ΔGPE。

Energy analysis often provides an easier path than resolving forces in complex problems. IB exam questions frequently require applying conservation concepts to roller coasters, pendulums, or spring systems.

能量分析常常比在复杂问题中分解力更容易。IB 考题经常要求在过山车、摆或弹簧系统中应用守恒概念。

9. Uniform Circular Motion | 匀速圆周运动

An object moving in a circle at constant speed is continuously changing direction, hence it has a centripetal acceleration directed towards the centre. The magnitude of this acceleration is a = v² / r, where r is the radius. Using angular speed ω (rad s⁻¹), we also have v = ω r and a = ω² r.

物体以恒定速率作圆周运动时，其速度方向不断改变，因此具有指向圆心的向心加速度。加速度的大小为 a = v² / r，其中 r 为半径。利用角速度 ω（单位为 rad s⁻¹），还可得到 v = ω r 和 a = ω² r。

a = v² / r = ω² r v = ω r

The net force required to produce this acceleration is the centripetal force F_c = m v² / r = m ω² r. It is not a new type of force; it is the resultant of real forces such as tension, friction, gravity, or the normal reaction. In a car rounding a bend, static friction provides the centripetal force. For a planet orbiting the Sun, gravity provides it.

产生该加速度所需的合力即为向心力 F_c = m v² / r = m ω² r。它并不是一种新型的力，而是真实力（如张力、摩擦力、重力或支持力）的合力。在汽车转弯时，静摩擦力提供向心力；对于绕太阳运行的行星，则由引力提供。

Always identify the physical force(s) pointing towards the centre and equate the sum to m v² / r. If the centripetal force is removed, the object moves tangentially in a straight line due to inertia.

务必找出指向圆心的真实力，并将其总和与 m v² / r 等起来。如果向心力消失，物体会因惯性沿切线方向匀速直线运动。

10. Gravitational Fields and Circular Orbits | 引力场与圆轨道

Newton’s law of universal gravitation states that every point mass attracts every other point mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them: F = G M m / r², where G = 6.67 × 10⁻¹¹ N m² kg⁻². This force provides the centripetal force that keeps satellites and planets in orbit.

万有引力定律指出，任何两个质点之间都存在吸引力，力的大小与两质量乘积成正比，与它们距离的平方成反比：F = G M m / r²，其中 G = 6.67 × 10⁻¹¹ N m² kg⁻²。这个力恰好提供使卫星和行星保持轨道的向心力。

For a circular orbit, equate gravitational force to centripetal force: G M m / r² = m v² / r, leading to orbital speed v = √(G M / r). This shows that the speed depends only on the mass of the central body and the orbital radius, not on the satellite’s mass. The orbital period T can then be found from v = 2π r / T, giving T² = (4π² / G M) r³, which is Kepler’s third law.

对于圆轨道，令引力等于向心力：G M m / r² = m v² / r，得到轨道速度 v = √(G M / r)。这表明轨道速度只取决于中心天体的质量和轨道半径，与卫星质量无关。然后可由 v = 2π r / T 求出轨道周期 T，得到 T² = (4π² / G M) r³，正是开普勒第三定律。

v = √(G M / r) T² = (4π² / G M) r³

Gravitational field strength g at a point is the force per unit mass: g = F / m = G M / r². On Earth’s surface, g ≈ 9.81 N kg⁻¹. At satellite altitudes, g is lower but still acts as the centripetal acceleration. Understanding these relationships is key to solving IB problems on orbital motion and apparent weightlessness.

某点的引力场强度 g 定义为每单位质量的引力：g = F / m = G M / r²。在地球表面，g ≈ 9.81 N kg⁻¹。在卫星高度上，g 变小但依然充当向心加速度。理解这些关系是解决 IB 中轨道运动和视重问题的基础。

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Mechanics | 力学