📚 A-Level CIE Engineering: Formula & Theorem Quick-Reference Handbook | A-Level CIE 工程：公式定理速查手册
This handbook brings together the essential formulae and theorems required for the CIE A-Level Engineering syllabus. It is designed as a one‑stop revision resource, covering mechanics, materials, fluids, thermodynamics, and electrical principles. Every equation is accompanied by a concise explanation in both English and Chinese, ensuring you grasp both the language and the underlying physics.
本手册汇总了 CIE A-Level 工程教学大纲所需的核心公式与定理，是涵盖力学、材料、流体、热力学与电学原理的一站式复习资源。每个方程均附有简明扼要的中英文解释，帮助你同时掌握术语与物理本质。
1. Mechanics: Linear Motion Equations | 力学：直线运动方程
Linear motion with constant acceleration can be described using four fundamental kinematic equations. These relate displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). They are derived from the definitions of velocity and acceleration and are valid only when acceleration remains constant throughout the motion.
匀加速直线运动可用四个基本运动学方程描述，联系位移（s）、初速度（u）、末速度（v）、加速度（a）和时间（t）。这些方程由速度与加速度的定义导出，仅在加速度恒定时适用。
| Formula | Description (English | 中文) |
|---|---|
| v = u + a t | Velocity-time relation | 速度‑时间关系 |
| s = u t + ½ a t² | Displacement with initial velocity | 具有初速度的位移 |
| v² = u² + 2 a s | Velocity‑displacement relation | 速度‑位移关系 |
| s = ½ (u + v) t | Displacement using average velocity | 平均速度求位移 |
Remember that direction matters; assign a positive direction and ensure u, v, a, and s are signed accordingly. These equations form the basis of projectile motion analysis when split into horizontal and vertical components.
注意方向性：先规定正方向，然后为 u、v、a、s 赋予相应符号。这些方程是分解斜抛运动为水平与竖直分量时的基础。
2. Newton’s Laws and Forces | 牛顿定律与力
Newton’s three laws of motion govern how forces affect the state of motion of a body. The first law introduces inertia, the second quantifies the relationship between net force, mass, and acceleration, and the third describes action‑reaction pairs. In engineering, we constantly apply these to static and dynamic systems.
牛顿运动三定律支配着力如何改变物体的运动状态。第一定律引入了惯性；第二定律量化了合力、质量与加速度的关系；第三定律描述作用力与反作用力对。工程中我们经常将其用于静力学与动力学系统。
| Formula | Description (English | 中文) |
|---|---|
| F = m a | Newton’s second law: net force = mass × acceleration | 牛顿第二定律：合力 = 质量 × 加速度 |
| W = m g | Weight = mass × gravitational field strength (g ≈ 9.81 m s⁻²) | 重力 = 质量 × 重力场强 |
| Ff ≤ μ R | Frictional force ≤ coefficient of friction × normal reaction; equality holds at limiting equilibrium | 摩擦力 ≤ 摩擦系数 × 法向反力；极限平衡时取等号 |
When drawing free-body diagrams, include all forces—weight, normal reaction, friction, tension, and applied forces—and resolve them along convenient axes. Equilibrium occurs when the vector sum of forces is zero.
绘制受力图时，应画全所有力——重力、法向反力、摩擦力、张力和外加力——并沿适当轴向分解。当力的矢量和为零时，系统处于平衡。
3. Work, Energy and Power | 功、能与功率
Work is done when a force moves its point of application. Energy is the capacity to do work, and power is the rate at which work is performed. The principle of conservation of energy is fundamental to all engineering systems. Kinetic and potential energies are frequently exchanged.
当力使其作用点发生位移时就做了功。能量是做功的能力，功率则是做功的快慢。能量守恒原理是所有工程系统的基础。动能与势能经常相互转换。
| Formula | Description (English | 中文) |
|---|---|
| W = F s cos θ | Work = force × displacement × cosine of angle between them | 功 = 力 × 位移 × 两者夹角的余弦 |
| KE = ½ m v² | Kinetic energy | 动能 |
| PE = m g h | Gravitational potential energy (near Earth’s surface) | 重力势能（近地面） |
| P = W / t = F v | Average power = work/time; instantaneous power = force × velocity | 平均功率 = 功/时间；瞬时功率 = 力 × 速度 |
| η = useful output / total input | Efficiency (often expressed as a percentage) | 效率（通常以百分数表示） |
In any real machine, some energy is always dissipated as heat due to friction or electrical resistance; thus efficiency is always less than 100%. Engineers aim to minimise these losses.
在任何实际机器中，总有一部分能量因摩擦或电阻耗散为热，因此效率始终低于100%。工程师的目标是最小化这些损失。
4. Stress, Strain and Hooke’s Law | 应力、应变与胡克定律
When a material is subjected to external forces, it experiences stress (internal resistive force per unit area) and strain (proportional deformation). For many materials, stress is proportional to strain within the elastic limit, as stated by Hooke’s Law. This linear relationship is the gateway to understanding structural integrity.
当材料受外力作用时，会产生应力（单位面积上的内部抵抗力）和应变（比例形变）。对许多材料而言，在弹性极限内应力与应变成正比，此即胡克定律。这种线性关系是理解结构完整性的入口。
| Formula | Description (English | 中文) |
|---|---|
| σ = F / A | Direct stress (tensile or compressive) | 正应力（拉伸或压缩） |
| ε = ΔL / L₀ | Tensile strain = extension / original length | 拉伸应变 = 伸长量 / 原长 |
| F = k x | Hooke’s Law for a spring: force = spring constant × extension | 弹簧胡克定律：力 = 劲度系数 × 伸长量 |
The elastic limit is the point beyond which the material no longer returns to its original shape. Beyond the yield point, plastic deformation occurs, and the simple linear relation ceases.
弹性极限是材料不再恢复原状的临界点。超过屈服点后发生塑性变形，简单的线性关系便不再成立。
5. Young’s Modulus and Material Properties | 杨氏模量与材料性质
Young’s modulus (E) is a measure of the stiffness of a material, defined as the ratio of tensile stress to tensile strain within the proportional limit. It is a material constant, independent of the dimensions of a particular sample. A high Young’s modulus indicates a stiff material that resists deformation.
杨氏模量（E）是衡量材料刚性的指标，定义为在比例极限内拉伸应力与拉伸应变的比值。它是一个材料常数，与具体试样的尺寸无关。高杨氏模量意味着材料刚性大，抵抗变形能力强。
| Formula | Description (English | 中文) |
|---|---|
| E = σ / ε | Young’s modulus = tensile stress / tensile strain | 杨氏模量 = 拉伸应力 / 拉伸应变 |
| E = (F L₀) / (A ΔL) | Practical form: F = force, L₀ = original length, A = cross‑sectional area, ΔL = extension | 实用形式：F为力，L₀为原长，A为横截面积，ΔL为伸长量 |
Engineers use Young’s modulus to calculate deflection in beams and to select appropriate materials for load‑bearing applications. The stress‑strain graph also yields other properties such as yield strength, ultimate tensile strength, and ductility.
工程师利用杨氏模量计算梁的挠度，并为承重应用选择合适的材料。应力‑应变曲线还提供其他性质，如屈服强度、极限抗拉强度和延展性。
6. Moments and Equilibrium | 力矩与平衡
A moment is the turning effect of a force about a pivot, equal to the product of the force and the perpendicular distance from the pivot to the line of action. For a body to be in static equilibrium, both the resultant force and the resultant moment must be zero. This principle underpins the analysis of beams, levers, and frameworks.
力矩是力绕支点产生的转动效应，等于力与支点到力作用线垂直距离的乘积。物体处于静力平衡时，合力与合力矩都必须为零。这一原理是梁、杠杆和桁架分析的基础。
| Formula | Description (English | 中文) |
|---|---|
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