📚 A-Level WJEC Engineering: Formula and Theorem Quick Reference Guide | A-Level WJEC 工程：公式定理速查手册
This quick reference guide brings together the most essential formulas and theorems needed for the WJEC A-Level Engineering course. Use it to reinforce your understanding, check key relationships during problem-solving, and ensure you have the mathematical tools at your fingertips. Each section presents the formula, a brief explanation, and direct applications in engineering contexts.
本速查手册汇总了 WJEC A-Level 工程课程中最核心的公式和定理。你可以用它来巩固理解、在解题时核实关键关系，并确保手中握有所需的数学工具。每一节都给出公式、简要说明以及在工程中的直接应用。
1. Foundation Mathematics and Units | 基础数学与单位
Right-angled trigonometry is fundamental in resolving forces and calculating components. For a triangle with angle θ: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
直角三角形三角学是分解力和计算分量的基础。对于角为 θ 的三角形：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
When dealing with non-right triangles, the sine rule and cosine rule are used: a/sin A = b/sin B = c/sin C, and a² = b² + c² – 2bc cos A.
处理非直角三角形时，使用正弦定理和余弦定理：a/sin A = b/sin B = c/sin C，以及 a² = b² + c² – 2bc cos A。
All engineering calculations rely on a consistent set of SI base units: length (metre, m), mass (kilogram, kg), time (second, s), electric current (ampere, A), temperature (kelvin, K), amount of substance (mole, mol), and luminous intensity (candela, cd). Derived units such as newton (N = kg·m/s²) and pascal (Pa = N/m²) follow from these.
所有工程计算都依赖一套一致的 SI 基本单位：长度（米，m），质量（千克，kg），时间（秒，s），电流（安培，A），温度（开尔文，K），物质的量（摩尔，mol），发光强度（坎德拉，cd）。导出单位如牛顿（N = kg·m/s²）和帕斯卡（Pa = N/m²）由此而来。
Area of circle: A = πr²; Volume of cylinder: V = πr²h
2. Statics and Equilibrium | 静力学与平衡
For a body to be in static equilibrium, both the resultant force and the resultant moment must be zero: ΣF = 0 and ΣM = 0. This is the foundation for analysing trusses, beams, and frames.
物体处于静力平衡时，合力和合力矩都必须为零：ΣF = 0 且 ΣM = 0。这是分析桁架、梁和框架的基础。
The moment of a force about a point is given by M = F × d, where d is the perpendicular distance from the point to the line of action of the force. The principle of moments states that for equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any point.
力对某点的力矩为 M = F × d，其中 d 是点到力作用线的垂直距离。力矩原理指出，平衡时对任意点顺时针力矩之和等于逆时针力矩之和。
Friction between two dry surfaces is modelled by F ≤ μR, where μ is the coefficient of friction and R is the normal reaction. For limiting equilibrium, F = μR.
两干表面之间的摩擦用 F ≤ μR 建模，μ 为摩擦系数，R 为法向反力。极限平衡时 F = μR。
Resultant of two forces: R = √(F₁² + F₂² + 2F₁F₂ cos θ)
3. Dynamics and Motion | 动力学与运动
The equations of uniformly accelerated motion (SUVAT) are used extensively in engineering mechanics: v = u + at; s = ut + ½at²; v² = u² + 2as; s = ½(u + v)t.
匀加速运动方程（SUVAT）在工程力学中广泛使用：v = u + at；s = ut + ½at²；v² = u² + 2as；s = ½(u + v)t。
Newton’s second law links force, mass, and acceleration: F = ma. This vector equation can be applied in each independent direction.
牛顿第二定律将力、质量和加速度联系起来：F = ma。这个矢量方程可沿各个独立方向分别应用。
Momentum is defined as p = mv. The impulse-momentum principle states that impulse = change in momentum: Ft = mv – mu.
动量定义为 p = mv。冲量-动量原理指出冲量等于动量的变化：Ft = mv – mu。
For circular motion at constant speed, the centripetal acceleration is a = v²/r = ω²r, and the centripetal force is F = mv²/r = mω²r.
匀速圆周运动中，向心加速度为 a = v²/r = ω²r，向心力为 F = mv²/r = mω²r。
4. Work, Energy and Power | 功、能量与功率
Work done by a constant force is W = Fd cos θ, where θ is the angle between the force and displacement. When the force is parallel to displacement, W = Fd.
恒力做功为 W = Fd cos θ，其中 θ 是力与位移之间的夹角。当力与位移平行时，W = Fd。
Kinetic energy (KE) and gravitational potential energy (GPE) are the two main mechanical energy forms: KE = ½mv²; GPE = mgh.
动能（KE）和重力势能（GPE）是两种主要的机械能形式：KE = ½mv²；GPE = mgh。
The principle of conservation of energy states that total energy in an isolated system remains constant. In the presence of non-conservative forces, work done against friction is converted into heat.
能量守恒原理指出孤立系统的总能量保持不变。存在非保守力时，克服摩擦做的功转化为热量。
Power is the rate of doing work: P = W/t. For a constant force moving at velocity v, P = Fv.
功率是做功的速率：P = W/t。对于以速度 v 移动的恒力，P = Fv。
Efficiency η = (useful energy output / total energy input) × 100%
5. Materials and Stress Analysis | 材料与应力分析
Direct stress is defined as force per unit area: σ = F/A. Strain is the ratio of change in length to original length: ε = ΔL/L₀. Both are essential in describing material behaviour under load.
正应力定义为单位面积上的力：σ = F/A。应变是长度变化与原始长度的比值：ε = ΔL/L₀。两者对描述载荷下的材料行为至关重要。
Young’s modulus (E) measures stiffness in the elastic region: E = σ/ε. For many ductile materials, the stress-strain graph shows a linear region obeying Hooke’s law up to the limit of proportionality.
杨氏模量（E）衡量弹性区的刚度：E = σ/ε。对于许多韧性材料，应力-应变图显示在线弹性区内遵循胡克定律，直到比例极限。
Factor of safety = ultimate stress / allowable working stress. It accounts for uncertainties in loading, material properties, and manufacturing.
安全系数 = 极限应力 / 许用工作应力。它考虑了载荷、材料特性和制造中的不确定性。
Poisson’s ratio ν relates lateral strain to axial strain: ν = – (lateral strain / axial strain). It typically ranges from 0 to 0.5 for engineering materials.
泊松比 ν 将横向应变与轴向应变联系起来：ν = – (横向应变 / 轴向应变)。工程材料通常取值范围为 0 到 0.5。
Shear stress τ = F/A (where force is parallel to area)
6. Thermodynamics and Heat Transfer | 热力学与热传递
The ideal gas law connects pressure, volume, and temperature for a given amount of gas: pV = nRT, where n is the number of moles and R = 8.31 J/(mol·K).
理想气体定律将定量气体的压力、体积和温度联系起来：pV = nRT，其中 n 为摩尔数，R = 8.31 J/(mol·K)。
For conduction through a solid slab, Fourier’s law gives the rate of heat transfer: Q = kA(ΔT/L)t, where k is thermal conductivity, A the cross-sectional area, ΔT the temperature difference, and L the thickness.
对于固体平板导热，傅里叶定律给出传热速率：Q = kA(ΔT/L)t，其中 k 为导热系数，A 为横截面积，ΔT 为温差，L 为厚度。
Specific heat capacity c relates the energy needed to raise the temperature of a mass m: Q = mcΔθ. For a change of state, latent heat L is used: Q = mL.
比热容 c 将升高质量 m 的温度所需能量关联起来：Q = mcΔθ。发生状态变化时，使用潜热 L：Q = mL。
The first law of thermodynamics states that the change in internal energy ΔU equals the heat added Q minus the work done by the system W: ΔU = Q – W.
热力学第一定律指出内能的变化 ΔU 等于加入的热量 Q 减去系统对外所做的功 W：ΔU = Q – W。
7. Fluid Mechanics | 流体力学
Density is mass per unit volume: ρ = m/V. Pressure at a depth h in a static fluid is given by P = ρgh, assuming uniform density and incompressible fluid.
密度是单位体积的质量：ρ = m/V。静止流体中深度 h 处的压力为 P = ρgh，假设密度均匀且流体不可压缩。
Pascal’s principle states that pressure applied to an enclosed fluid is transmitted undiminished to all parts of the fluid. This is the basis of hydraulic systems: F₁/A₁ = F₂/A₂.
帕斯卡原理指出，施加给密闭流体的压力会不变地传递到流体的各个部分。这是液压系统的基础：F₁/A₁ = F₂/A₂。
For steady, incompressible flow, the continuity equation applies: A₁v₁ = A₂v₂, where A is cross-sectional area and v is flow velocity.
对于稳定、不可压缩流动，适用连续性方程：A₁v₁ = A₂v₂，其中 A 为横截面积，v 为流速。
Bernoulli’s equation expresses conservation of energy along a streamline: P + ½ρv² + ρgh = constant. It is used to relate pressure, velocity, and elevation in a flowing fluid.
伯努利方程表达了沿流线的能量守恒：P + ½ρv² + ρgh = 常数。它用于关联流动流体中的压力、速度和高度。
Volume flow rate Q = Av
8. Electrical Principles | 电学原理
Ohm’s law states that the current through a resistor is directly proportional to the voltage across it: V = IR. Resistance depends on material and geometry: R = ρL/A, where ρ is resistivity.
欧姆定律指出通过电阻的电流与其两端的电压成正比：V = IR。电阻取决于材料和几何形状：R = ρL/A，其中 ρ 为电阻率。
Electrical power is the product of voltage and current: P = IV. Using Ohm’s law, this can also be expressed as P = I²R or P = V²/R.
电功率是电压与电流的乘积：P = IV。利用欧姆定律，它也可表示为 P = I²R 或 P = V²/R。
Kirchhoff’s laws govern circuit analysis: Kirchhoff’s current law (KCL) states that the sum of currents entering a junction equals the sum leaving; Kirchhoff’s voltage law (KVL) states that the sum of electromotive forces round any closed loop equals the sum of voltage drops.
基尔霍夫定律支配电路分析：基尔霍夫电流定律（KCL）指出进入节点的电流之和等于离开的电流之和；基尔霍夫电压定律（KVL）指出任意闭合回路中电动势之和等于电压降之和。
For resistors in series: R_total = R₁ + R₂ + … For resistors in parallel: 1/R_total = 1/R₁ + 1/R₂ + …
电阻串联：R_total = R₁ + R₂ + … 电阻并联：1/R_total = 1/R₁ + 1/R₂ + …
Capacitance C = Q/V; time constant τ = RC
9. Structural Analysis | 结构分析
Simple beam theory relates bending moment M, the second moment of area I, stress σ, and the distance from the neutral axis y: M/I = σ/y = E/R. This formula is fundamental for calculating bending stresses in beams.
简单梁理论将弯矩 M、截面二次矩 I、应力 σ 以及到中性轴的距离 y 联系起来：M/I = σ/y = E/R。该公式是计算梁弯曲应力的基础。
For pin-jointed trusses, the method of joints and method of sections are used to determine internal forces. Each joint must satisfy equilibrium conditions ΣF_x = 0 and ΣF_y = 0.
对于铰接桁架，使用节点法和截面法确定内力。每个节点都必须满足平衡条件 ΣF_x = 0 和 ΣF_y = 0。
The deflection of a cantilever beam under a point load at the free end is δ = FL³/(3EI), where F is the load, L the length, E the Young’s modulus, and I the second moment of area.
自由端受集中载荷的悬臂梁挠度为 δ = FL³/(3EI)，其中 F 为载荷，L 为长度，E 为杨氏模量，I 为截面二次矩。
Buckling of slender columns is predicted by Euler’s formula: critical load P_cr = π²EI/L_eff², where L_eff is the effective length depending on end conditions.
细长柱的屈曲由欧拉公式预测：临界载荷 P_cr = π²EI/L_eff²，其中 L_eff 是取决于端部条件的有效长度。
10. Engineering Constants and Conversions | 工程常数与换算
Familiarity with key constants and multipliers speeds up problem-solving. Standard acceleration due to gravity: g = 9.81 m/s². Atmospheric pressure: 101 kPa. Speed of light: c ≈ 3.00 × 10⁸ m/s.
熟悉关键常数和倍数可以加快解题速度。标准重力加速度：g = 9.81 m/s²。大气压力：101 kPa。光速：c ≈ 3.00 × 10⁸ m/s。
Common SI prefixes: kilo (k = 10³), mega (M = 10⁶), giga (G = 10⁹); milli (m = 10⁻³), micro (μ = 10⁻⁶), nano (n = 10⁻⁹). Conversion of units often uses these multipliers.
常用 SI 前缀：千（k = 10³）、兆（M = 10⁶）、吉（G = 10⁹）；毫（m = 10⁻³）、微（μ = 10⁻⁶）、纳（n = 10⁻⁹）。单位换算经常使用这些乘数。
Temperature conversions: T(K) = θ(°C) + 273.15; θ(°F) = 9/5 θ(°C) + 32. For thermodynamic calculations, always use kelvin.
温度换算：T(K) = θ(°C) + 273.15；θ(°F) = 9/5 θ(°C) + 32。热力学计算中务必使用开尔文。
Density of water: 1000 kg/m³; 1 litre = 10⁻³ m³
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