IGCSE CAIE Engineering: Formula & Theorem Quick Reference | IGCSE CAIE 工程：公式定理速查手册

📚 IGCSE CAIE Engineering: Formula & Theorem Quick Reference | IGCSE CAIE 工程：公式定理速查手册

This quick-reference handbook consolidates the essential formulas, theorems, and engineering principles required for the IGCSE CAIE Engineering syllabus. It is designed to help you revise efficiently, spot connections between topics, and apply the right equation to the right problem under exam pressure.

本速查手册汇总了 IGCSE CAIE 工程课程所需的核心公式、定理与工程原理，旨在帮助你在考试压力下高效复习、把握知识点之间的联系，并在解题时快速找到正确的公式。

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1. Fundamental Units & Dimensions | 基本单位与量纲

Every calculation in engineering begins with the correct unit. The International System (SI) provides the foundation for all measurements you will encounter in the exam.

工程计算的起点是正确使用单位。国际单位制（SI）是考试中所有测量的基础。

• Base quantities: mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol).

• 基本量：质量（kg）、长度（m）、时间（s）、电流（A）、温度（K）、物质的量（mol）。

• Derived quantities include force (N = kg·m/s²), pressure (Pa = N/m²), energy (J = N·m), and power (W = J/s).

• 导出量包括力（N = kg·m/s²）、压强（Pa = N/m²）、能量（J = N·m）和功率（W = J/s）。

Converting prefixes: 1 kN = 10³ N, 1 MPa = 10⁶ Pa, 1 mm = 10⁻³ m

单位前缀换算：1 kN = 10³ N，1 MPa = 10⁶ Pa，1 mm = 10⁻³ m

Always convert all values to base SI units before substituting into a formula; failing to do so is one of the most common causes of lost marks.

代入公式前务必将所有数值转换为 SI 基本单位；这是最常见的失分原因之一。

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2. Forces, Motion & Newton’s Laws | 力、运动与牛顿定律

Force is a vector quantity that causes a change in motion. Three Newton’s laws govern the relationship between forces and motion in engineering contexts.

力是引起运动变化的矢量。牛顿三定律描述了工程中力与运动的基本关系。

F = m × a (Newton’s Second Law) | F = m × a（牛顿第二定律）

Weight: W = m × g (g ≈ 9.81 m/s²) | 重力：W = m × g（g ≈ 9.81 m/s²）

• Newton’s First Law: an object remains at rest or in uniform motion unless acted upon by a resultant force.

• 牛顿第一定律：物体在不受合力作用时，保持静止或匀速直线运动状态。

• Newton’s Third Law: for every action, there is an equal and opposite reaction.

• 牛顿第三定律：每一个作用力都有一个大小相等、方向相反的反作用力。

Momentum: p = m × v | 动量：p = m × v

Impulse: I = F × t = Δp | 冲量：I = F × t = Δp

In engineering, Newton’s Second Law is the foundation for calculating vehicle acceleration, lifting loads, and designing structural supports that withstand dynamic forces.

在工程中，牛顿第二定律是计算车辆加速度、起吊载荷以及设计承受动载荷结构支承的基础。

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3. Moments & Equilibrium | 力矩与平衡

A moment is the turning effect of a force about a pivot. Equilibrium conditions are essential for the design of levers, cranes, and support structures.

力矩是力绕支点产生的转动效应。平衡条件是杠杆、起重机和支承结构设计的基础。

Moment = Force × Perpendicular distance | 力矩 = 力 × 垂直距离

M = F × d (unit: N·m) | M = F × d（单位：N·m）

For an object to be in equilibrium, two conditions must be satisfied simultaneously:

物体处于平衡状态需同时满足两个条件：

• Sum of clockwise moments = Sum of anticlockwise moments (about any point).

• 顺时针力矩之和 = 逆时针力矩之和（对任意取矩点）。

• Resultant force in any direction = 0 (ΣFₓ = 0, ΣFᵧ = 0).

• 任意方向的合力为零（ΣFₓ = 0，ΣFᵧ = 0）。

Beam reactions are solved by taking moments about one support, then using vertical equilibrium to find the other reaction. Always state the direction of reactions in your answer.

求解梁的支座反力时，先对某一支座取矩，再利用竖直方向平衡方程求另一个反力。作答时务必注明反力的方向。

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4. Stress, Strain & Young’s Modulus | 应力、应变与杨氏模量

Materials in engineering structures must withstand loads without excessive deformation or failure. Stress and strain quantify the internal response of a material to external forces.

工程结构中的材料必须能够承受载荷而不发生过大变形或失效。应力与应变量化了材料对外部载荷的内部响应。

Stress: σ = F / A (unit: Pa or N/m²) | 应力：σ = F / A（单位：Pa 或 N/m²）

Strain: ε = ΔL / L (no unit) | 应变：ε = ΔL / L（无量纲）

Young’s Modulus: E = σ / ε = (F·L) / (A·ΔL) | 杨氏模量：E = σ / ε = (F·L) / (A·ΔL)

The elastic limit is the maximum stress a material can withstand and still return to its original shape. Beyond this point, permanent deformation occurs (plastic behaviour).

弹性极限是材料能够完全恢复原状的最大应力。超过该点后将发生永久变形（塑性行为）。

For a tensile test, the gradient of the straight-line portion of the stress-strain graph gives Young’s modulus. The area under the curve represents the energy absorbed per unit volume before fracture.

在拉伸试验中，应力-应变曲线直线段的斜率即为杨氏模量。曲线下的面积表示断裂前单位体积材料吸收的能量。

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5. Energy, Work & Power | 能量、功与功率

Work is done when a force moves an object through a distance. Energy is the capacity to do work, and power is the rate at which work is done.

力使物体移动一段距离时即做功。能量是做功的能力，功率是做功的快慢。

Work: W = F × d (unit: J) | 功：W = F × d（单位：J）

Power: P = W / t = F × v (unit: W) | 功率：P = W / t = F × v（单位：W）

Kinetic Energy: KE = ½ × m × v² | 动能：KE = ½ × m × v²

Potential Energy: PE = m × g × h | 势能：PE = m × g × h

Efficiency is the ratio of useful output energy (or power) to total input energy (or power), usually expressed as a percentage.

效率是有用输出能量（或功率）与总输入能量（或功率）之比，通常以百分比表示。

Efficiency = (Useful output / Total input) × 100% | 效率 =（有用输出 / 总输入）× 100%

In mechanical systems such as gear trains, efficiency loss is primarily due to friction between meshing teeth and bearing resistance.

在齿轮传动等机械系统中，效率损失主要来自轮齿啮合和轴承摩擦。

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6. Simple Machines & Gear Trains | 简单机械与齿轮传动

Simple machines enable a small effort to overcome a large load. Gear trains transmit power and change speed and torque between rotating shafts.

简单机械使较小的施力能够克服较大的负载。齿轮传动在转轴之间传递功率并改变转速与转矩。

Mechanical Advantage (MA) = Load / Effort | 机械优势（MA）= 负载 / 施力

Velocity Ratio (VR) = Distance moved by effort / Distance moved by load | 速度比（VR）= 施力移动距离 / 负载移动距离

Efficiency = MA / VR × 100% | 效率 = MA / VR × 100%

For a gear train, the velocity ratio is determined by the number of teeth on each gear:

对于齿轮传动系统，速度比由各齿轮的齿数决定：

Gear Ratio = Teeth on driven gear / Teeth on driver gear | 齿轮比 = 从动轮齿数 / 主动轮齿数

Speed relation: N₁ × T₁ = N₂ × T₂ | 转速关系：N₁ × T₁ = N₂ × T₂

where N represents rotational speed (rpm) and T represents the number of teeth. A smaller driven gear produces a higher output speed but lower torque.

其中 N 表示转速（rpm），T 表示齿数。较小的从动轮产生更高的输出转速，但转矩较小。

In pulleys and belt drives, the same relationship applies using pulley diameters instead of teeth: D₁ × N₁ = D₂ × N₂.

在带轮传动中，上述关系同样适用，只需用带轮直径代替齿数：D₁ × N₁ = D₂ × N₂。

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7. Thermal Properties & Heat Transfer | 热学性质与热传递

Engineering materials expand when heated, and systems transfer heat through conduction, convection, and radiation. Understanding these principles is vital for designing engines, heat exchangers, and structures exposed to temperature changes.

工程材料受热时膨胀，系统通过传导、对流和辐射传递热量。理解这些原理对于设计发动机、换热器以及暴露在温度变化环境中的结构至关重要。

Linear expansion: ΔL = α × L₀ × ΔT | 线膨胀：ΔL = α × L₀ × ΔT

Heat energy: Q = m × c × ΔT | 热能：Q = m × c × ΔT

Heat capacity: C = m × c | 热容：C = m × c

where α is the coefficient of linear expansion (unit: K⁻¹), c is specific heat capacity (J/(kg·K)), and ΔT is the temperature change.

其中 α 是线膨胀系数（单位：K⁻¹），c 是比热容（单位：J/(kg·K)），ΔT 是温度变化量。

In conduction, the rate of heat transfer through a material is proportional to its thermal conductivity, cross-sectional area, and temperature gradient, and inversely proportional to its thickness. This is described by Fourier’s Law of Conduction.

在传导中，通过材料的热传递速率与其导热系数、横截面积和温度梯度成正比，与其厚度成反比。这由傅里叶导热定律描述：

Q/t = k × A × ΔT / L | Q/t = k × A × ΔT / L

where k is the thermal conductivity (W/(m·K)), A is the cross-sectional area, and L is the thickness of the material.

其中 k 是导热系数（单位：W/(m·K)），A 是横截面积，L 是材料厚度。

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8. Electrical Principles | 电工学基本原理

Electrical engineering principles underpin the design of circuits, motors, and control systems. Ohm’s Law and electrical power equations are central to this topic.

电工学原理是电路、电动机和控制系统设计的基础。欧姆定律和电功率方程是该主题的核心。

Ohm’s Law: V = I × R | 欧姆定律：V = I × R

Electrical Power: P = V × I = I² × R = V² / R | 电功率：P = V × I = I² × R = V² / R

Electrical Energy: E = P × t = V × I × t (unit: J) | 电能：E = P × t = V × I × t（单位：J）

For resistors connected in series:

串联电阻：

Rₜₒₜₐₗ = R₁ + R₂ + R₃ + … | 总电阻 Rₜₒₜₐₗ = R₁ + R₂ + R₃ + …

For resistors connected in parallel:

并联电阻：

1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + 1/R₃ + … | 1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + 1/R₃ + …

In series circuits, current is constant and voltage divides. In parallel circuits, voltage is constant and current divides inversely with resistance.

串联电路中电流处处相等，电压分配；并联电路中电压相等，电流按电阻的反比分配。

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9. Fluid Pressure & Forces | 流体压强与力

Fluid mechanics applies to hydraulic systems, water supply engineering, and submerged structures in civil engineering. Pressure in a static fluid increases with depth.

流体力学应用于液压系统、供水工程及土木工程的水下结构。静流体中的压强随深度增加而增大。

Pressure: P = F / A (unit: Pa) | 压强：P = F / A（单位：Pa）

Pressure at depth: P = ρ × g × h | 深度压强：P = ρ × g × h

Force on submerged surface: F = P × A = ρ × g × h × A | 水下表面受力：F = P × A = ρ × g × h × A

where ρ is fluid density (kg/m³), g is gravitational acceleration (9.81 m/s²), and h is the depth below the free surface.

其中 ρ 为流体密度（kg/m³），g 为重力加速度（9.81 m/s²），h 为距自由液面的深度。

In hydraulic systems, Pascal’s Principle states that pressure applied to an enclosed fluid is transmitted equally to every point in the fluid. This allows a small force on a small area to produce a large force on a larger area:

在液压系统中，帕斯卡原理指出：施加在密闭流体上的压强会等值传递到流体的每一个点。这使得小面积上的小力可产生大面积上的大力：

F₁/A₁ = F₂/A₂ | F₁/A₁ = F₂/A₂

Hydraulic brakes and hydraulic jacks are common IGCSE applications of this principle.

液压制动器和液压千斤顶是 IGCSE 中该原理的常见应用实例。

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10. Key Material Properties & Selection | 关键材料性能与选材

Engineers must compare material properties to select the right material for each application. The key mechanical properties tested in IGCSE include strength, hardness, toughness, ductility, and malleability.

工程师必须比较材料性能，才能为特定用途选择合适的材料。IGCSE 中考查的关键力学性能包括强度、硬度、韧性、延展性和可锻性。

• Strength — ability to withstand force without failure. Categories include tensile, compressive, and shear strength.

• 强度 — 承受外力而不失效的能力。分为抗拉、抗压和抗剪强度。

• Hardness — resistance to surface indentation, scratching, or wear.

• 硬度 — 抵抗表面压痕、划伤或磨损的能力。

• Toughness — ability to absorb energy before fracturing (impact resistance).

• 韧性 — 断裂前吸收能量的能力（抗冲击性）。

• Ductility — ability to deform under tension without fracturing (e.g., steel wire drawing).

• 延展性 — 在拉伸下变形而不断裂的能力（如钢丝拉拔）。

• Malleability — ability to deform under compression without cracking (e.g., hammering sheet metal).

• 可锻性 — 在压缩下变形而不开裂的能力（如锤打金属薄板）。

Material | 材料	Key Property | 关键特性	Typical Use | 典型用途
Carbon steel | 碳钢	High strength, tough | 高强度、韧性好	Structural frames, shafts 结构框架、轴
Aluminium | 铝	Light, corrosion-resistant | 轻质、耐腐蚀	Aircraft bodies, cans 飞机机身、易拉罐
Cast iron | 铸铁	High compressive strength, brittle | 抗压强度高、较脆	Engine blocks, machine bases 发动机缸体、机床底座
Copper | 铜	Excellent conductor, ductile | 优异导电性、延展性好	Electrical wiring, pipes 电线、管道

In short, assessing the required strength, weight, cost, thermal properties, and conductivity of a component will guide material selection.

简言之，综合评估构件所需的强度、重量、成本、热性能和导电性，即可指导材料选择。

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11. Exam Strategies: Choosing the Correct Formula | 应试策略：如何选择正确的公式

In the IGCSE CAIE Engineering examination, selecting the wrong formula is a common mistake. The following decision framework will help you identify the correct equation efficiently.

在 IGCSE CAIE 工程考试中，选错公式是常见错误。以下决策框架可帮助你高效找出正确的方程。

• Step 1 — Read the question and identify the unknown quantity you need to calculate.

• 第 1 步 — 通读题目，确定要求解的未知量。

• Step 2 — Write down every known value with its unit, converting all measurements to SI base units first.

• 第 2 步 — 列出所有已知量并注明单位，先将所有数值转换为 SI 基本单位。

• Step 3 — Determine the topic area: mechanics, materials, thermal, electrical, or fluid.

• 第 3 步 — 判断考点领域：力学、材料、热学、电学或流体。

• Step 4 — Select the formula that contains your unknown and the known quantities. Rearrange if necessary.

• 第 4 步 — 选择同时包含未知量和已知量的公式。必要时进行变换。

• Step 5 — Substitute values, calculate, and include the correct unit in your final answer. Check the reasonableness of magnitude.

• 第 5 步 — 代入数值计算，最终答案注明正确单位，并检查数量级是否合理。

Look for keywords: “moment” suggests a pivot; “stress” and “strain” indicate material properties; “current” or “voltage” indicates electrical principles; “depth” suggests fluid pressure.

注意关键词：”力矩”提示存在支点；”应力”和”应变”指向材料性能；”电流”或”电压”指向电工原理；”深度”提示流体压强。

Always draw a free-body diagram where forces are involved. Visualising the system reduces sign errors and clarifies the direction of forces, moments, and reactions.

涉及力时务必画出受力分析图。将系统图像化可以减少正负号错误，并明确力、力矩和反力的方向。

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12. Common Pitfalls & Markscheme Language | 常见失分点与评分标准用语

Understanding how marks are awarded is just as important as knowing the physics. Marks are often assigned separately for the formula (1 mark), substitution (1 mark), and final answer with unit (1 mark).

理解如何得分与掌握物理知识同样重要。得分点通常按公式（1 分）、代入数据（1 分）和带单位的最终答案（1 分）分别分配。

• Pitfall 1: Mixing up radius and diameter. Check the question carefully.

• 易错点 1：混淆半径与直径。请仔细审题。

• Pitfall 2: Forgetting to convert mm² to m² or cm to m before calculating stress.

• 易错点 2：计算应力前忘记将 mm² 转换为 m²，或将 cm 转换为 m。

• Pitfall 3: Using weight (N) instead of mass (kg) in F = ma calculations.

• 易错点 3：在 F = ma 计算中使用重力值（N）而不是质量（kg）。

• Pitfall 4: Choosing the wrong moment arm — the perpendicular distance, not the distance along a slanted member.

• 易错点 4：取错力臂 — 必须是垂直距离，而不是沿倾斜构件的距离。

Scenario | 情景	Correct Formula | 正确公式
Lifting a load vertically | 竖直提升重物	PE = m × g × h
Braking a moving vehicle | 制动行驶车辆	F = m × a; KE = ½ × m × v²
Suspension bridge cable tension | 悬索桥缆索张力	σ = F / A
Heat exchanger design | 换热器设计	Q = m × c × ΔT
Hydraulic press force | 液压机出力	F₁/A₁ = F₂/A₂
Motor power rating | 电机额定功率	P = V × I

Practice rewriting formulas in different forms until you can rearrange mechanically. In the exam, show every line of working—unstructured numerical answers rarely receive full marks.

练习将公式变换为不同形式，直到你能够机械性地完成变形。考试中要写出每一步演算过程 — 仅仅给出数值答案通常得不到满分。

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