📚 A-Level CAIE Engineering: Formula & Theorem Quick Reference Handbook | A-Level CAIE 工程：公式定理速查手册
This quick reference handbook summarises the essential formulae, principles and theorems required for the Cambridge International AS & A Level Engineering syllabus. It is designed for revision: read a section, close the page, then reproduce the formulae and definitions from memory. Always write the formula before substituting values, and always state the unit of the final answer. Used alongside past paper practice, this handbook turns scattered notes into a single, exam-ready revision spine.
本速查手册总结了剑桥国际 AS 与 A Level 工程课程所需的核心公式、原理和定理。它专为复习设计：读完一节，合上页面，再凭记忆默写公式与定义。务必先写出公式再代入数值，并且每个答案都要标明单位。配合真题练习使用，本手册能把零散的笔记整合成一条完整、可直接应试的复习主线。
1. SI Units, Prefixes and Derived Units | 国际单位制、词头与导出单位
Every calculation should begin with a unit check. Convert all data into consistent SI units first. This avoids systematic errors in equations such as F = ma, which requires mass in kg and acceleration in m s⁻², and in W = Fs, which requires force in N and displacement in m.
每次计算前，应把所有数据转换为一致的 SI 单位。这样可以避免 F = ma 等方程因单位不统一而产生系统性错误，该方程要求质量用 kg、加速度用 m s⁻²；同理 W = Fs 要求力用 N、位移用 m。
The seven SI base units are the metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol) and candela (cd). Everything else in engineering is a derived unit built from these.
SI 的七个基本单位是米 (m)、千克 (kg)、秒 (s)、安培 (A)、开尔文 (K)、摩尔 (mol) 和坎德拉 (cd)。工程中其他所有单位都是由这七个基本单位组合而成的导出单位。
| Prefix | Factor | Example |
|---|---|---|
| giga (G) | 10⁹ | 1 GHz = 10⁹ Hz |
| mega (M) | 10⁶ | 1 MPa = 10⁶ Pa |
| kilo (k) | 10³ | 1 kN = 10³ N |
| centi (c) | 10⁻² | 1 cm = 10⁻² m |
| milli (m) | 10⁻³ | 1 mm = 10⁻³ m |
| micro (μ) | 10⁻⁶ | 1 μm = 10⁻⁶ m |
¹, Pa = N m⁻², Hz = s⁻¹, C = A s, V = J C⁻¹ = W A⁻¹, Ω = V A⁻¹, F = C V⁻¹, and T = Wb m⁻².
必须记住的常见导出单位：N = kg m s⁻²，J = N m = kg m² s⁻²，W = J s⁻¹，Pa = N m⁻²，Hz = s⁻¹，C = A s，V = J C⁻¹ = W A⁻¹，Ω = V A⁻¹，F = C V⁻¹，T = Wb m⁻²。
2. Statics: Forces, Moments and Equilibrium | 静力学：力、力矩与平衡
A body is in equilibrium when the resultant force and the resultant moment are both zero. The two conditions are written as ΣF = 0 and ΣM t point cleverly - usually where an unknown force acts - removes that unknown from the moment equation.
当合外力与合力矩都为零时，物体处于平衡状态，即对任意点满足 ΣF = 0 与 ΣM = 0。巧妙地选择取矩点——通常选在某个未知力的作用点上——就能让该未知量从力矩方程中消去。
M = F × d (N m) ；couple 力偶 M = F × d
The moment (torque) of a force equals the force multiplied by the perpendicular distance from the pivot. A couple is a pair of equal, opposite and parallel forces; its moment is the force times the perpendicular distance between them, and it produces rotation without translation.
力矩等于力乘以力的作用线到转轴的垂直距离。力偶是一对大小相等、方向相反、互相平行的力；其力矩等于力乘以两力之间的垂直距离，只产生转动而不产生平动。
For three coplanar forces in equilibrium, the triangle of forces closes, and Lami's theorem applies: F₁ / sin α = F₂ / sin β = F₃ / sin γ, where α, β, γ are the angles opposite each force. The centre of gravity of a uniform rectangle lies at half its height, and of a uniform triangle at one third of its height from the base.
三个共面力平衡时，力的三角形闭合，且满足拉密定理：F₁ / sin α = F₂ / sin β = F₃ / sin γ，其中 α、β、γ 分别是各力对面的夹角。匀质矩形的重心在其高度的一半处，匀质三角形的重心在距底边三分之一高度处。
3. Kinematics and Dynamics | 运动学与动力学
For motion in a straight line at constant acceleration, use the four equations of motion (the SUVAT equations). They apply only when acceleration is uniform, so check the question wording before using them.
匀加速直线运动使用四个运动学公式（SUVAT 公式）。它们仅在加速度恒定时成立，因此使用前务必确认题干条件。
v = u + at ；s = ut + ½at² ；v² = u² + 2as ；s = ½(u + v)t
Newton's three laws underpin dynamics: a body remains at rest or in uniform motion unless acted on by a resultant force; F = ma; and for every action there is an equal and opposite reaction. Momentum p = mv is conserved in collisions when no external resultant force acts, and impulse Ft = Δp explains why crumple zones reduce injury.
牛顿三大定律构成动力学基础：物体不受合外力时保持静止或匀速直线运动；F = ma；作用力与反作用力大小相等、方向相反。当无外力作用时，碰撞中动量 p = mv 守恒；冲量 Ft = Δp 则解释了为何溃缩吸能区能减轻伤害。
For rotation, angular velocity is ω = 2πf = 2π / T, linear speed is v = ωr, and centripetal acceleration is a = v² / r = ω²r, requiring a centripetal force F = mv² / r directed towards the centre. Torque is τ = Fr.
转动情况下，角速度 ω = 2πf = 2π / T，线速度 v = ωr，向心加速度 a = v² / r = ω²r，所需向心力 F = mv² / r 指向圆心。转矩 τ = Fr。
4. Work, Energy, Power and Efficiency | 功、能、功率与效率
Work is done when a force moves its point of application: W = Fs cos θ, where θ is the angle between the force and the displacement. Work transfers energy, and energy is never created or destroyed, only converted between forms.
当力的作用点发生位移时即做功：W = Fs cos θ，其中 θ 是力与位移之间的夹角。做功传递能量，而能量既不会创生也不会消失，只会在不同形式之间转化。
KE = ½mv² ；PE = mgh ；spring energy = ½kx²
Power is the rate of doing work or transferring energy: P = W / t, and for a force moving a body at constant velocity, P = Fv. Efficiency compares useful output with total input.
功率是做功或传递能量的速率：P = W / t；当力使物体匀速运动时，P = Fv。效率是比较有用输出与总输入的指标。
efficiency = (useful output energy / total input energy) × 100%
tio VR = distance moved by effort / distance moved by load, and efficiency = MA / VR × 100% An ideal machine would have MA = VR, but friction always makes the real efficiency lower.
在机械中，机械效益 MA = 负载 / 施力，速度比 VR = 施力移动距离 / 负载移动距离，效率 = MA / VR × 100%。理想机械的 MA = VR，但摩擦总是使实际效率更低。
5. Stress, Strain and Material Behaviour | 应力、应变与材料性能
Direct stress is the internal force per unit cross-sectional area, and strain is the fractional change in length. Both are essential when selecting materials for load-bearing components.
正应力是单位横截面积上的内力，应变是长度的相对变化量。在为承载构件选材时，这两个量都必不可少。
σ = F / A (Pa) ；ε = ΔL / L₀ (no unit) ；E = σ / ε (Pa)
Hooke's law states that extension is proportional to load up to the limit of proportionality: F = kx, where the stiffness constant is k = AE / L. The elastic strain energy stored is ½FΔx, equal to the area under a force-extension graph.
胡克定律指出：在比例极限以内，伸长量与载荷成正比，即 F = kx，其中刚度系数 k = AE / L。储存的弹性应变能为 ½FΔx，也就是力—伸长量图像下的面积。
- Elastic region: material returns to its original shape when the load is removed. | 弹性区：卸载后材料恢复原状。
- Yield point: stress at which plastic deformation begins. | 屈服点：开始发生塑性变形的应力。
- Ultimate tensile strength (UTS): maximum stress the material can withstand. | 抗拉强度极限：材料能承受的最大应力。
- Fracture point: stress at which the specimen breaks. | 断裂点：试样断裂时的应力。
Poisson' lateral strain / axial strain, typically about 0.3 for metals. Thermal expansion follows ΔL = αL₀Δθ, and if expansion is restrained, a thermal stress develops that can be large enough to buckle rails or crack concrete.
泊松比 ν = 横向应变 / 轴向应变，金属通常约为 0.3。热膨胀遵循 ΔL = αL₀Δθ；若膨胀受到约束，就会产生热应力，足以使钢轨弯曲或混凝土开裂。
6. Fluid Mechanics: Pressure, Buoyancy and Flow | 流体力学：压力、浮力与流动
Density is mass per unit volume, and pressure in a fluid acts equally in all directions at a given depth. In a static fluid, gauge pressure increases linearly with depth.
密度是单位体积的质量；在液体中某一深度处，压强向各个方向大小相等。在静止流体中，表压随深度线性增加。
ρ = m / V ；p = F / A ；p = ρgh
Atmospheric pressure is about 101.3 kPa. Pascal's principle states that pressure applied to an enclosed fluid is transmitted undiminished throughout it, which is the basis of the hydraulic press: F₁ / A₁ = F₂ / A₂. Archimedes' principle states that the upthrust on a body equals the weight of fluid displaced; a body floats when upthrust equals its weight.
大气压约为 101.3 kPa。帕斯卡原理指出：施加于密闭流体的压强会大小不变地传递到流体各处，这是液压机的基础：F₁ / A₁ = F₂ / A₂。阿基米德原理指出物体所受浮力等于其排开流体的重力；当浮力等于物重时物体漂浮。
For steady, incompressible flow, the continuity equation gives A₁v₁ = A₂v₂, so a narrower pipe means a faster flow. Bernoulli's equation shows the trade-off between pressure, speed and height along a streamline.
对于稳定、不可压缩流动，连续性方程给出 A₁v₁ = A₂v₂，因此管道越窄流速越快。伯努利方程则表明沿同一流线压力、流速与高度之间的相互转换关系。
p + ½ρv² + ρgh = constant
The Reynolds number Re = ρvd / μ predicts whether flow is laminar (Re below about 2000) or turbulent, which matters in pipe design and in drag calculations.
雷诺数 Re = ρvd / μ 用于判断流动是层流（Re 约低于 2000）还是湍流，这在管道设计与阻力计算中十分重要。
7. Thermodynamics and Heat Transfer | 热力学与传热
Heat is energy transferred because of a temperature difference. The specific heat capacity equation gives the energy needed to change temperature without a change of state, while latent heat applies to a change of state at constant temperature.
热量是由于温度差而传递的能量。比热容公式给出在不发生物态变化时改变温度所需的能量，而潜热则适用于恒温下的物态变化。
Q = mcΔθ ；Q = mL ；pV = nRT
For a fixed mass of gas, the combined gas law is p₁V₁ / T₁ = p₂V₂ / T lvin. In a heat engine the maximum theoretical efficiency is the Carnot efficiency η = 1 − T_cold / T_hot, which shows why a large temperature difference is desirable.
对一定质量的气体，联合气体定律为 p₁V₁ / T₁ = p₂V₂ / T₂，其中温度必须用开尔文。热机的最大理论效率为卡诺效率 η = 1 − T_cold / T_hot，这说明为何要尽量扩大温差。
Heat is transferred by conduction, convection and radiation. For steady conduction through a uniform bar of length L, the rate of heat flow is P = kAΔθ / L, where k is thermal conductivity. Radiation from a surface obeys the Stefan-Boltzmann law P = εσAT⁴, with T in kelvin and σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴.
传热方式有传导、对流和辐射。对长度为 L 的均匀棒材的稳态传导，热流率为 P = kAΔθ / L，其中 k 为热导率。表面辐射遵循斯特藩—玻尔兹曼定律 P = εσA
