📚 IGCSE CCEA Engineering: Formula and Theorem Quick Reference Handbook | IGCSE CCEA 工程：公式定理速查手册
This quick reference handbook brings together the essential formulas, theorems and unit conversions required for the IGCSE CCEA Engineering course. It is designed for revision, problem solving and practical applications, covering mechanics, materials, electricity, energy, fluid power and manufacturing calculations.
本速查手册汇集 IGCSE CCEA 工程课程所必需的重要公式、定理与单位换算。手册适用于复习、解题和工程应用，涵盖力学、材料、电学、能量、流体动力与制造计算。
Every equation below is written in the form used by the CCEA specification, with the standard symbols, units and rearrangement hints you need in the examination. Work through each section, check that the units are consistent, and always state the formula before substituting numbers.
下文每个方程均采用 CCEA 考纲所使用的形式，并给出标准符号、单位与变形提示，方便考试作答。请逐节复习，确认单位一致，并牢记先写公式、再代入数字的答题顺序。
1. SI Units and Prefixes | 国际单位制与词头
Engineering calculations always start with standard SI base units: length in metres (m), mass in kilograms (kg), time in seconds (s), electric current in amperes (A) and temperature in kelvin (K). Derived units include newton (N) for force, pascal (Pa) for pressure, joule (J) for energy and watt (W) for power.
工程计算始终从标准国际单位制基本单位开始：长度用米（m）、质量用千克（kg）、时间用秒（s）、电流用安培（A）、温度用开尔文（K）。导出单位包括力用牛顿（N）、压力用帕斯卡（Pa）、能量用焦耳（J）、功率用瓦特（W）。
Derived units can always be broken back into base units, which is the quickest way to check whether an answer is dimensionally correct.
导出单位总能拆回基本单位，这是检验答案量纲是否正确的最快方法。
1 N = 1 kg·m/s² 1 Pa = 1 N/m² 1 J = 1 N·m 1 W = 1 J/s
Prefixes allow very large and very small values to be written clearly. Use the table below to convert before substituting into formulas.
词头可用于清晰表示极大或极小的数值。代入公式前，应使用下表完成单位换算。
| Prefix | Symbol | Factor | 中文 |
|---|---|---|---|
| giga | G | 10⁹ | 吉 |
| mega | M | 10⁶ | 兆 |
| kilo | k | 10³ | 千 |
| centi | c | 10⁻² | 厘 |
| milli | m | 10⁻³ | 毫 |
| micro | μ | 10⁻⁶ | 微 |
| nano | n | 10⁻⁹ | 纳 |
Apply these three rules every time: convert millimetres to metres by dividing by 1000; convert grams to kilograms by dividing by 1000; convert minutes to seconds by multiplying by 60. Most lost marks in CCEA Engineering come from unit errors rather than algebra.
每次解题请遵守三条规则：毫米换算为米除以 1000；克换算为千克除以 1000；分钟换算为秒乘以 60。CCEA 工程考试中失去的分数大多来自单位错误，而非代数错误。
2. Mechanics: Force, Newton's Laws and Moments | 力学：力、牛顿定律与力矩
Newton's three laws underpin every mechanical calculation. The first law states that a body stays at rest or moves at constant velocity unless acted on by a resultant force. The second law defines the relationship between force, mass and acceleration. The third law states that every action has an equal and opposite reaction.
牛顿三大定律支撑着所有力学计算。第一定律指出，若无合外力作用，物体保持静止或匀速直线运动。第二定律给出力、质量与加速度之间的关系。第三定律指出，每一个作用力都有一个大小相等、方向相反的反作用力。
F = m × a W = m × g （g = 9.81 m/s²）
Weight is a force and is measured in newtons; mass is measured in kilograms. Never mix the two in the same equation.
重量是一种力，单位为牛顿；质量单位为千克。切勿在同一方程中把两者混用。
A moment is the turning effect of a force about a pivot, and the principle of moments states that for a body in equilibrium the total clockwise moment equals the total anticlockwise moment.
力矩是力绕支点的转动效果；力矩平衡原理指出，处于平衡状态的物体，顺时针力矩总和等于逆时针力矩总和。
M = F × d（d 为垂直距离） ΣM clockwise = ΣM anticlockwise
For linear motion, use the standard equations of motion; for circular or rotary systems, convert linear speed to angular speed with the relationships below.
直线运动使用标准运动学方程；圆周或旋转系统则用下列关系把线速度换算为角速度。
v = s ÷ t a = (v − u) ÷ t v² = u² + 2as s = ut + ½at² ω = 2πN ÷ 60
- Balanced forces: resultant force = 0, velocity constant | 平衡力：合力为 0，速度不变
- Unbalanced forces: resultant force ≠ 0, acceleration occurs | 非平衡力：合力不为 0，产生加速度
- Beam problems: take moments about one support to find the reaction at the other | 梁的问题：对一侧支座取矩，求出另一支座的支反力
3. Stress, Strain and Young's Modulus | 应力、应变与杨氏模量
When a material is loaded, engineers describe the internal effect using stress (force per unit area) and strain (extension per unit original length). Both are essential for selecting materials in CCEA Engineering design questions.
材料受载时，工程师用应力（单位面积上的力）和应变（单位原长上的伸长量）描述其内部效应。两者都是 CCEA 工程设计中选材题目的必备知识。
σ = F ÷ A ε = ΔL ÷ L₀ E = σ ÷ ε
s in pascals (Pa) or newtons per square millimetre (N/mm²); strain is dimensionless; Young's modulus in GPa or N/mm². 1 N/mm² = 1 MPa.
单位：应力用帕斯卡（Pa）或牛顿每平方毫米（N/mm²）；应变无单位；杨氏模量用 GPa 或 N/mm²。1 N/mm² = 1 MPa。
On a stress-strain graph, the straight-line section is the elastic region, the gradient equals Young's modulus, and the point where the line stops being straight is the limit of proportionality.
在应力-应变图上，直线段为弹性区，其斜率等于杨氏模量，直线结束的那一点即为比例极限。
- Elastic deformation is reversible | 弹性变形是可逆的
- Plastic deformation is permanent | 塑性变形是永久性的
- Brittle materials (cast iron, glass) fracture with little strain | 脆性材料（铸铁、玻璃）应变很小时即断裂
- Ductile materials (mild steel, copper) stretch before fracture | 延性材料（低碳钢、铜）断裂前会明显伸长
4. Energy, Work, Power and Efficiency | 能量、功、功率与效率
Work is done when a force moves an object through a distance. Energy is the capacity to do work and is conserved: it may change form but is never created or destroyed.
力使物体移动一段距离时便做了功。能量是做功的能力，并且守恒：能量可以转化形式，但不会凭空产生或消失。
W = F × s Eₖ = ½mv² Eₚ = mgh
Power is the rate of doing work, and efficiency compares useful output with total input.
功率是做功的速率，效率则比较有用输出与总输入。
P = W ÷ t P = F × v Efficiency = (useful output ÷ total input) × 100%
Machines such as pulleys, gears and levers trade force for distance. The velocity ratio and mechanical advantage let you predict the output of a system and estimate its efficiency before it is built.
滑轮、齿轮和杠杆等机械以距离换取力。速度比与机械利益可用来预测系统的输出，并在制造之前估算效率。
VR = distance moved by effort ÷ distance moved by load MA = load ÷ effort
- 1 kWh = 3.6 × 10⁶ J = 3.6 MJ | 1 千瓦时 = 3.6 × 10⁶ 焦耳
- Energy cost = power (kW) × time (h) × unit price | 电费 = 功率(kW) × 时间(h) × 单价
- No machine is 100% efficient: friction and heat losses always occur | 没有机械效率为 100%：摩擦与热损失始终存在
5. Electricity: Ohm's Law and Circuits | 电学：欧姆定律与电路
Ohm's law links voltage, current and resistance in a linear conductor. Voltage is the energy transferred per unit charge, current is the rate of flow of charge, and resistance opposes that flow.
欧姆定律把线性导体中的电压、电流和电阻联系起来。电压是单位电荷所转移的能量，电流是电荷流动的速率，电阻则阻碍这种流动。
V = I × R Q = I × t E = V × I × t
Electrical power can be calculated in three equivalent ways depending on which quantities are known.
根据已知量不同，电功率有三种等价的计算方式。
P = V × I = I²R = V² ÷ R
Resistors combine differently in series and parallel circuits, and these two rules must be memorised exactly.
电阻在串联与并联电路中的组合方式不同，这两条规则必须准确记忆。
Series: Rtotal = R₁ + R₂ + R₃ Parallel: 1 ÷ Rtotal = 1 ÷ R₁ + 1 ÷ R₂ + 1 ÷ R₃
- Series circuit: current is the same everywhere, voltages add | 串联电路：各处电流相同，电压相加
- Parallel circuit: voltage is the same across each branch, currents add | 并联电路：各支路电压相同，电流相加
- Total parallel resistance is always less than the smallest individual resistor | 并联总电阻总是小于其中最小的单个电阻
- Fuse rating should be slightly above the normal operating current | 保险丝额定电流应略高于正常工作电流
6. Fluid Power: Pressure and Hydraulics | 流体动力：压力与液压
Pressure in a fluid acts equally in all directions, and this is the basis of hydraulic and pneumatic systems. Pressure is defined as force per unit area and is measured in pascals, bar or N/mm².
流体中的压力向各个方向等量传递，这正是液压与气动系统的基础。压力定义为单位面积上的力，单位可用帕斯卡、巴或 N/mm²。
p = F ÷ A 1 bar = 100 kPa = 10⁵ Pa 1 MPa = 10⁶ Pa
Pascal's principle states that pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid. A small force on a small piston therefore produces a large force on a large piston, giving mechanical advantage.
帕斯卡原理指出，施加在密闭流体上的压力会大小不变地传递到流体各处。因此小活塞上的小力可在大活塞上产生大力，从而获得机械利益。
F₁ ÷ A₁ = F₂ ÷ A₂ A₁ × d₁ = A₂ × d₂ （体积守恒）
Because liquids are almost incompressible, the volume pushed by the input piston must equal the volume displaced at the output piston, so a large output force moves through a smaller distance.
由于液体几乎不可压缩，输入活塞推入的体积必然等于输出活塞排出的体积，因此输出力大时其移动距离就小。
- Hydraulic systems use oil and operate at high pressure | 液压系统使用液压油，工作压力高
- Pneumatic systems use compressed air and are cleaner but less precise | 气动系统使用压缩空气，更洁净但精度较低
- Flow rate: Q = A × v, with Q in m³/s, A in m², v in m/s | 流量：Q = A × v，Q 单位 m³/s，A 单位 m²，v 单位 m/s
7. Materials: Density, Thermal Properties and Expansion | 材料：密度、热学性质与热膨胀
Density tells you how much mass is packed into a given volume, and it is the first property to check when comparing candidate materials for a lightweight component.
密度表示单位体积内所含质量的多少，在为轻量化零件筛选材料时，它是首要考察的性能。
ρ = m ÷ V m = ρ × V V = m ÷ ρ
When a material is heated it expands; the change in length depends on the original length, the temperature rise and the coefficient of linear expansion of the material.
材料受热会膨胀；长度变化量取决于原长、温升以及材料的线膨胀系数。
ΔL = α × L₀ × Δθ θ(K) = θ(°C) + 273.15
The energy needed to raise the temperature of a substance depends on its mass, its specific heat capacity and the temperature change.
使物质升温所需能量取决于其质量、比热容以及温度变化量。
E = m × c × Δθ
- Thermal conductivity: high in metals, low in polymers and ceramics | 热导率：金属高，聚合物与陶瓷低
- Thermal expansion joints are used in bridges and rails | 桥梁与铁轨需设置热膨胀缝
- Bimetallic strips use different expansion rates to bend and switch circuits | 双金属片利用不同膨胀率弯曲，从而通断电路
8. Manufacturing Calculations: Speeds, Feeds and Tolerances | 制造计算：转速、进给与公差
Manufacturing questions in CCEA Engineering test whether you can set a machine correctly. Cutting speed is the surface speed of the tool relative to the workpiece, and it determines the spindle speed for a given diameter.
CCEA 工程的制造类题目考查你能否正确设定机床。切削速度是刀具相对于工件的表面速度，它决定了给定直径下的主轴转速。
v = π × D × N ÷ 1000 → N = 1000 × v ÷ (π × D)
Here v is cutting speed in metres per minute, D is diameter in millimetres and N is spindle speed in revolutions per minute. The 1000 converts millimetres to metres.
其中 v 为切削速度（m/min），D 为直径（mm），N 为主轴转速（r/min）。除以 1000 是把毫米换算为米。
Feed rate is how fast the tool advances along the workpiece, calculated from the feed per tooth, the number of teeth on the cutter and the spindle speed.
进给速度是刀具沿工件推进的快慢，由每齿进给量、刀具齿数和主轴转速算出。
Feed rate = fz × z × N Material removal rate = width × depth × feed rate
Tolerance is the permitted variation in a dimension, and it always equals the difference between the upper and lower limits. Interchangeable manufacture depends on tolerances being held.
公差是尺寸允许的变动量，总等于上极限尺寸与下极限尺寸之差。互换性生产依赖于公差的保证。
Tolerance = upper limit − lower limit Nominal size = (upper limit + lower limit) ÷ 2
- Clearance fit: shaft is always smaller than the hole | 间隙配合：轴始终小于孔
- Interference fit: shaft is always larger than the hole | 过盈配合：轴始终大于孔
- Transition fit: may be either, depending on the actual sizes produced | 过渡配合：视实际尺寸，可能为间隙或过盈
- Quality control uses go / no-go gauges to check limits quickly | 质量控制用通规/止规快速检验极限尺寸
9. Geometry and Trigonometry for Engineering Drawing | 工程制图的几何与三角
Engineering drawings rely on accurate geometric and trigonometric calculation. Right-angled triangles are solved with Pythagoras' theorem and the three trigonometric ratios.
工程制图依赖精确的几何与三角计算。直角三角形用勾股定理和三个三角函数求解。
a² + b² = c² sin θ = opposite ÷ hypotenuse cos θ = adjacent ÷ hypotenuse tan θ = opposite ÷ adjacent
Circle and cylinder formulas are needed for stock calculation, volume and surface area, while area formulas are used to work out material quantities and costs.
圆的公式与圆柱公式用于算料、体积与表面积；面积公式则用于计算材料用量与成本。
Circumference = π × D Area of circle = π × r² Volume of cylinder = π × r² × h
- Rectangle area = length × width | 矩形面积 = 长 × 宽
- Triangle area = ½ × base × height | 三角形面积 = ½ × 底 × 高
- Volume of a prism = cross-sectional area × length | 棱柱体积 = 截面积 × 长度
- Angles in a triangle add to 180°; in a quadrilateral, 360° | 三角形内角和 180°；四边形内角和 360°
- Scale drawing: use the stated ratio, e.g. 1:10 means 1 mm on paper = 10 mm actual | 比例图：按标注比例，如 1:10 表示图上 1 mm 等于实际 10 mm
10. Unit Conversions and Derived Quantities | 单位换算与导出量
Examination questions frequently include data in non-SI units so that you must convert before calculating. Learn the conversions in the table below and apply them at the start of a solution.
考题常给出非国际单位制数据，要求你在计算前先换算。请掌握下表中的换算关系，并在解题开头就完成换算。
| Quantity | Conversion | 中文 |
|---|---|---|
| Length | 1 m = 1000 mm; 1 inch = 25.4 mm | 1 米 = 1000 毫米；1 英寸 = 25.4 毫米 |
| Area | 1 m² = 10⁶ mm²; 1 cm² = 100 mm² | 1 平方米 = 10⁶ 平方毫米 |
| Volume | 1 m³ = 10⁹ mm³; 1 litre = 1000 cm³ | 1 立方米 = 10⁹ 立方毫米；1 升 = 1000 立方厘米 |
| Pressure | 1 bar = 10⁵ Pa; 1 N/mm² = 1 MPa | 1 巴 = 10⁵ 帕；1 N/mm² = 1 兆帕 |
| Energy | 1 kWh = 3.6 MJ; 1 kJ = 1000 J | 1 千瓦时 = 3.6 兆焦 |
| Power | 1 kW = 1000 W; 1 hp ≈ 746 W | 1 千瓦 = 1000 瓦；1 马力 ≈ 746 瓦 |
| Speed | 1 m/s = 3.6 km/h; 1 m/min = 1000 mm/min | 1 米/秒 = 3.6 千米/时 |
f the linear factor: 1 m² = 1000² mm² = 10⁶ mm². Cubed units use the cube: 1 m³ = 10⁹ mm³.
请记住：面积换算用长度系数的平方，1 m² = 1000² mm² = 10⁶ mm²；体积换算用立方，1 m³ = 10⁹ mm³。
