📚 IGCSE CAIE Engineering: Formula and Theorem Quick Reference Handbook | IGCSE CAIE 工程：公式定理速查手册
Mastering IGCSE CAIE Engineering requires a firm command of essential formulas, constants, and fundamental theorems. This quick reference handbook consolidates the key mathematical relationships that you must recall accurately under examination conditions. Use this resource for structured revision, ensuring each equation is understood in context rather than simply memorised in isolation. The sections are organised by topic to mirror the CAIE syllabus, covering mechanics, materials, thermodynamics, electrical principles, and applied mathematics relevant to engineering problem-solving.
掌握 IGCSE CAIE 工程学需要牢固掌握基本公式、常数和基础定理。本速查手册整理了你在考试条件下必须准确回忆的关键数学关系。请将本资源用于结构化复习，确保每个方程都在具体情境中加以理解，而非孤立记忆。各节按主题组织，与 CAIE 考纲相呼应，涵盖力学、材料、热力学、电学原理以及工程问题求解所需的应用数学。
1. Fundamental Units and Dimensions | 基本单位与量纲
All engineering quantities derive from a set of base SI units. Length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity form the foundations. Derived units such as the newton (N) for force and the pascal (Pa) for pressure are expressed in terms of base units. Dimensional analysis serves as a powerful verification tool; if the dimensions on both sides of an equation do not match, the equation is physically invalid. In the IGCSE examination, you may be asked to check the homogeneity of a given relationship or to deduce the units of a constant.
所有工程量都来源于一组基本国际单位。长度、质量、时间、电流、热力学温度、物质的量和发光强度构成基础。导出单位如力的牛顿 (N) 和压强的帕斯卡 (Pa) 均以基本单位表示。量纲分析是一种强大的验证工具；若方程两侧量纲不匹配，则该方程在物理上不成立。在 IGCSE 考试中，你可能需要检验给定关系的齐次性，或推导某个常数的单位。
Force: F = m × a (Unit: kg·m·s⁻² = N)
力：F = m × a (单位：kg·m·s⁻² = N)
Pressure: P = F / A (Unit: N·m⁻² = Pa)
压强：P = F / A (单位：N·m⁻² = Pa)
2. Mechanics: Kinematics and Linear Motion | 力学：运动学与直线运动
The equations of uniformly accelerated motion, often referred to as SUVAT equations, are central to solving linear kinematics problems. These relationships assume constant acceleration and motion along a straight line. The variables are displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). Selecting the correct equation depends on identifying which quantities are known and which variable is required without involving an unwanted unknown.
匀加速运动方程（常称 SUVAT 方程）是解决直线运动学问题的核心。这些关系假设加速度恒定且运动沿直线进行。变量包括位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t)。选择正确方程取决于识别已知量和所需变量，同时避免引入不需要的未知量。
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
s = ½ (u + v) t
When an object falls freely under gravity near the Earth’s surface, the acceleration a is replaced by g = 9.81 m·s⁻², directed downward. Air resistance is neglected in most IGCSE calculations unless explicitly included in a modelling scenario as a resistive force proportional to speed or speed squared.
当物体在地球表面附近自由下落时，加速度 a 替换为 g = 9.81 m·s⁻²，方向向下。除非在建模情境中明确包含空气阻力作为与速度或速度平方成正比的阻力，否则大多数 IGCSE 计算均忽略空气阻力。
3. Forces, Moments, and Equilibrium | 力、力矩与平衡
Newton’s three laws of motion provide the conceptual framework for all force analysis. The first law defines inertia and equilibrium; the second law quantifies the relationship between resultant force, mass, and acceleration; the third law states that forces occur in equal and opposite pairs acting on different bodies. In statics problems, the conditions for equilibrium require both the vector sum of forces and the sum of moments about any point to be zero.
牛顿运动三定律为所有受力分析提供了概念框架。第一定律定义了惯性和平衡；第二定律量化了合力、质量与加速度之间的关系；第三定律指出力以大小相等、方向相反的成对形式作用于不同物体上。在静力学问题中，平衡条件要求力的矢量和以及对任意点的力矩之和均为零。
Moment of a force: M = F × d (perpendicular distance from pivot)
力矩：M = F × d (到支点的垂直距离)
Principle of moments: Σ clockwise moments = Σ anticlockwise moments
力矩原理：顺时针力矩总和 = 逆时针力矩总和
The concept of centre of gravity is essential for stability analysis. An object will topple if the line of action of its weight falls outside its base area. In engineering structures, understanding moment distribution prevents catastrophic failure, as seen in beam design and cantilever applications.
重心的概念对于稳定性分析至关重要。若物体重力的作用线落在其基底面积之外，物体将倾覆。在工程结构中，理解力矩分布可防止灾难性失效，这在梁设计和悬臂应用中尤为关键。
4. Stress, Strain, and Material Properties | 应力、应变与材料性质
The mechanical behaviour of materials under load is described by stress-strain relationships. Stress quantifies internal resistive force per unit area, while strain measures the resulting deformation relative to the original dimension. Young’s modulus characterises the stiffness of a material within its elastic limit, where the relationship between stress and strain is linear and reversible, following Hooke’s law.
材料在载荷作用下的力学行为通过应力-应变关系加以描述。应力量化了单位面积上的内部阻力，而应变则衡量相对于原始尺寸的变形量。杨氏模量表征了材料在其弹性限度内的刚度，在此限度内，应力与应变呈线性且可逆关系，遵循胡克定律。
Stress: σ = F / A (Unit: N·m⁻² or Pa)
应力：σ = F / A (单位：N·m⁻² 或 Pa)
Strain: ε = ΔL / L₀ (dimensionless)
应变：ε = ΔL / L₀ (无量纲)
Young’s modulus: E = σ / ε (Unit: Pa)
杨氏模量：E = σ / ε (单位：Pa)
The load-extension graph for a ductile material reveals key points: the limit of proportionality, the elastic limit, the yield point, and the ultimate tensile strength before necking and fracture. IGCSE candidates must interpret such graphs and calculate stiffness from the linear region as k = F / Δx for a spring, or E from the slope of a stress-strain curve.
韧性材料的载荷-伸长曲线揭示了关键点：比例极限、弹性极限、屈服点以及颈缩和断裂前的抗拉强度。IGCSE 考生必须解读此类曲线图，并从线性区域计算弹簧刚度 k = F / Δx，或根据应力-应变曲线斜率计算 E。
5. Thermal Physics and Heat Transfer | 热物理与传热
Thermal expansion is a critical consideration in engineering design, as temperature changes cause dimensional variations in structures. Linear expansion, area expansion, and volume expansion each have corresponding coefficients. Bridges incorporate expansion joints, and railway tracks require gaps to accommodate thermal movement. The heat equation relates temperature change to energy transfer when a substance undergoes heating or cooling without a change of state.
热膨胀是工程设计中一项关键考量，因为温度变化会导致结构尺寸变化。线膨胀、面膨胀和体膨胀各有相应的系数。桥梁设有伸缩缝，铁轨需要间隙以适应热运动。当物质经历加热或冷却而不发生物态变化时，热方程将温度变化与能量传递联系起来。
Linear expansion: ΔL = α L₀ ΔT
线膨胀：ΔL = α L₀ ΔT
Heat energy: Q = m c Δθ (specific heat capacity method)
热能：Q = m c Δθ (比热容法)
Latent heat: Q = m L (during phase change at constant temperature)
潜热：Q = m L (恒温相变过程中)
Heat transfer occurs through conduction, convection, and radiation. Conduction is governed by Fourier’s law in its simplest form, where the rate of heat transfer depends on thermal conductivity, cross-sectional area, and temperature gradient. Engineers select materials with appropriate thermal properties for insulation, heat exchangers, and thermal management systems.
热传递通过传导、对流和辐射进行。传导受傅里叶定律在其最简形式下的支配，热传递速率取决于导热系数、横截面积和温度梯度。工程师为隔热、热交换器和热管理系统选择具有适当热性能的材料。
6. Electrical Principles and Circuit Analysis | 电学原理与电路分析
Ohm’s law establishes the proportionality between potential difference across a conductor and the current flowing through it, provided temperature remains constant. Resistance is a material and geometric property; for a uniform wire, resistance depends on resistivity, length, and cross-sectional area. IGCSE problems frequently require combining resistors in series and parallel to find equivalent resistance.
欧姆定律确立了导体两端电位差与流过电流之间的正比关系，前提是温度保持恒定。电阻是材料和几何特性；对于均匀导线，电阻取决于电阻率、长度和横截面积。IGCSE 题目经常要求结合串联和并联电阻以求等效电阻。
V = I × R (Ohm’s law)
V = I × R (欧姆定律)
Resistance: R = ρ L / A
电阻：R = ρ L / A
Series: R_total = R₁ + R₂ + R₃ + …
串联：R_total = R₁ + R₂ + R₃ + …
Parallel: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …
并联：1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …
Electrical power is the product of potential difference and current. Using substitution with Ohm’s law, power can also be expressed as I²R or V²/R. These forms are particularly useful when analysing heating effects in resistive components and determining appropriate power ratings for engineering components.
电功率是电位差与电流的乘积。利用欧姆定律代入，功率也可表示为 I²R 或 V²/R。这些形式在分析电阻元件中的热效应以及确定工程元件的适当额定功率时尤为有用。
P = I × V = I² R = V² / R
功率：P = I × V = I² R = V² / R
7. Energy, Work, and Power in Mechanical Systems | 机械系统中的能量、功与功率
Energy exists in various forms, and the principle of conservation of energy states that energy cannot be created or destroyed, only transferred or transformed. In mechanical engineering, the interplay between gravitational potential energy, kinetic energy, and work done against resistive forces governs the behaviour of moving systems. Efficiency calculations quantify the proportion of useful energy output relative to total energy input.
能量以多种形式存在，能量守恒原理指出能量不能创生或消灭，只能转移或转化。在机械工程中，重力势能、动能与克服阻力所做的功之间的相互作用支配着运动系统的行为。效率计算量化了有用能量输出相对于总能量输入的比例。
Gravitational potential energy: Eₚ = m g h
重力势能：Eₚ = m g h
Kinetic energy: Eₖ = ½ m v²
动能：Eₖ = ½ m v²
Work done: W = F d cos θ
做功：W = F d cos θ
Power: P = W / t = F v (for constant force and velocity aligned)
功率：P = W / t = F v (当恒力与速度同向时)
Efficiency is expressed as a percentage or decimal fraction. In any practical machine, some input energy is dissipated as heat due to friction, air resistance, or electrical resistance, ensuring that efficiency is always less than 100%. The IGCSE syllabus expects candidates to perform energy audits and suggest design improvements to reduce losses.
效率以百分比或小数表示。在任何实际机器中，由于摩擦、空气阻力或电阻，部分输入能量会以热的形式耗散，这确保效率始终低于 100%。IGCSE 考纲要求考生进行能量核算，并提出减少损耗的设计改进建议。
8. Fluid Mechanics: Pressure and Buoyancy | 流体力学：压强与浮力
Fluid pressure at a given depth depends on the density of the fluid, gravitational field strength, and the vertical height of the fluid column above the point. This hydrostatic pressure acts equally in all directions at a given depth. Pascal’s principle states that pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid, forming the basis of hydraulic systems used in brakes, lifts, and presses.
流体中某给定深度处的压强取决于流体密度、重力场强度以及该点上方液柱的垂直高度。这种静水压强在给定深度处向各个方向等大作用。帕斯卡原理指出，施加于封闭流体上的压强会不减地传递至整个流体，这构成了制动器、升降机和压力机中液压系统的基础。
Hydrostatic pressure: P = ρ g h
静水压强：P = ρ g h
Hydraulic advantage: F₁ / A₁ = F₂ / A₂
液压增益：F₁ / A₁ = F₂ / A₂
Archimedes’ principle describes the buoyant force on an object immersed in a fluid: the upthrust equals the weight of fluid displaced. An object floats when its average density is less than that of the fluid; it sinks when its density exceeds that of the fluid. Ship design and submarine ballast systems apply these principles directly.
阿基米德原理描述了浸没在流体中的物体所受的浮力：向上推力等于排开流体的重量。当物体的平均密度小于流体密度时，它上浮；当密度大于流体密度时，它下沉。船舶设计和潜艇压载系统直接应用了这些原理。
Upthrust force: F_upthrust = ρ_fluid × V_displaced × g
上推力：F_upthrust = ρ_fluid × V_displaced × g
9. Engineering Mathematics: Trigonometry and Vector Resolution | 工程数学：三角学与矢量分解
Trigonometric ratios are indispensable for resolving forces, determining component dimensions, and analysing inclined planes. The sine, cosine, and tangent functions relate angles to side ratios in right-angled triangles. The sine rule and cosine rule extend these relationships to non-right-angled triangles, which appear frequently in truss analysis and navigation problems within engineering contexts.
三角比对于分解力、确定构件尺寸以及分析斜面不可或缺。正弦、余弦和正切函数在直角三角形中将角度与边长比关联起来。正弦定理和余弦定理将这些关系扩展到非直角三角形，这在工程背景下的桁架分析和导航问题中频繁出现。
sin θ = opposite / hypotenuse; cos θ = adjacent / hypotenuse; tan θ = opposite / adjacent
sin θ = 对边 / 斜边；cos θ = 邻边 / 斜边；tan θ = 对边 / 邻边
Sine rule: a / sin A = b / sin B = c / sin C
正弦定理：a / sin A = b / sin B = c / sin C
Cosine rule: a² = b² + c² – 2 b c cos A
余弦定理：a² = b² + c² – 2 b c cos A
A vector quantity possesses both magnitude and direction. Resolving a vector into perpendicular components simplifies the addition of multiple forces or velocities. The resultant of two perpendicular vectors is found using Pythagoras’ theorem, and the direction is given by the inverse tangent function. Engineers use vector diagrams and resolution techniques to design structures that withstand combined loading from multiple directions.
矢量具有大小和方向。将矢量分解为正交分量可简化多个力或速度的相加。两个相互垂直矢量的合力通过勾股定理求得，方向由反正切函数给出。工程师使用矢量图和分解技术来设计能够承受多方向组合载荷的结构。
10. Circuits, Logic Gates, and Digital Principles | 电路、逻辑门与数字原理
Digital electronics relies on Boolean logic and the behaviour of logic gates to process binary signals. The fundamental gates — AND, OR, NOT, NAND, NOR, and XOR — each follow defined truth tables. In engineering systems, combinations of these gates form control circuits for automation, safety interlocks, and signal processing. Truth table construction and Boolean expression simplification are tested skills in the IGCSE Engineering specification.
数字电子学依赖于布尔逻辑和逻辑门的行为来处理二进制信号。基本逻辑门——与门、或门、非门、与非门、或非门和异或门——各自遵循确定的真值表。在工程系统中，这些门的组合构成了自动化、安全联锁和信号处理的控制电路。真值表构建和布尔表达式简化是 IGCSE 工程规范中考查的技能。
AND: Output is 1 only if all inputs are 1
与门：仅当所有输入均为 1 时输出才为 1
OR: Output is 1 if at least one input is 1
或门：若至少一个输入为 1，则输出为 1
Potential dividers are a core analogue circuit concept, producing a fraction of the input voltage that depends on the ratio of two resistances. Sensors such as thermistors and light-dependent resistors (LDRs) are used in potential divider configurations to convert environmental changes into voltage signals for processing. The output voltage formula is a direct application of Ohm’s law and series resistance principles.
分压器是一个核心模拟电路概念，产生的输入电压比例取决于两个电阻的比值。热敏电阻和光敏电阻等传感器用于分压器配置中，将环境变化转换为电压信号以供处理。输出电压公式是欧姆定律和串联电阻原理的直接应用。
Potential divider: V_out = V_in × (R₂ / (R₁ + R₂))
分压器：V_out = V_in × (R₂ / (R₁ + R₂))
11. Materials and Manufacturing Calculations | 材料与制造计算
Engineering manufacture requires accurate calculation of material quantities, cutting speeds, feed rates, and material removal rates. Density links mass and volume, enabling cost estimation and weight prediction. In machining, the cutting speed is related to spindle rotation rate and workpiece diameter, while the material removal rate determines productivity. These formulas bridge theoretical design and practical workshop execution.
工程制造需要精确计算材料数量、切削速度、进给率和材料去除率。密度将质量与体积联系起来，使得成本估算和重量预测成为可能。在机械加工中，切削速度与主轴转速和工件直径相关，而材料去除率则决定生产效率。这些公式连接了理论设计与实际车间操作。
Density: ρ = m / V
密度：ρ = m / V
Cutting speed: v = π D N / 1000 (for metric units, m/min)
切削速度：v = π D N / 1000 (公制单位，米/分钟)
Wastage and tolerance calculations ensure that components fit together within acceptable limits. Bilateral and unilateral tolerances specify the allowable variation from a nominal dimension. Engineers must interpret engineering drawings, calculate clearance or interference fits, and ensure that assembly specifications are met without compromising function or safety.
损耗与公差计算确保组件在可接受范围内装配吻合。双边和单边公差规定了相对于标称尺寸的允许变动量。工程师必须解读工程图纸，计算间隙或过盈配合，并确保装配规格在不影响功能或安全性的前提下得到满足。
12. Key Constants and Conversion Factors | 关键常数与换算系数
Examination success often depends on correctly recalling and applying physical constants. The acceleration due to gravity at Earth’s surface, the speed of light in a vacuum, specific heat capacities, and standard atmospheric pressure are frequently required. In addition, you must be adept at converting between non-SI units commonly used in engineering contexts, such as revolutions per minute to radians per second, or litres to cubic metres.
考试成功往往取决于正确回忆和应用物理常数。地球表面的重力加速度、真空中的光速、比热容和标准大气压是常需使用的。此外，你必须熟练地在工程情境中常用的非 SI 单位之间进行换算，例如将每分钟转数换算为弧度每秒，或将升换算为立方米。
| Quantity / 物理量 | Symbol / 符号 | Value / 数值 |
| Acceleration of free fall / 重力加速度 | g | 9.81 m·s⁻² |
| Speed of light in vacuum / 真空中光速 | c | 3.00 × 10⁸ m·s⁻¹ |
| Specific heat capacity of water / 水的比热容 | c_water | 4200 J·kg⁻¹·K⁻¹ |
| Standard atmospheric pressure / 标准大气压 | P_atm | 1.01 × 10⁵ Pa |
| Density of water / 水的密度 | ρ_water | 1000 kg·m⁻³ |
Systematic revision using this handbook, coupled with repeated practice of past paper questions, will build the fluency required for the IGCSE CAIE Engineering examination. Always show full working, state the formula before substituting values, and check dimensional consistency in your final answer.
使用本手册进行系统复习，并结合反复练习历年真题，将培养出 IGCSE CAIE 工程学考试所需的熟练度。务必展示完整的解题步骤，先写出公式再代入数值，并检验最终答案的量纲一致性。
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