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

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

This quick reference handbook distills the essential formulas and theorems you must master for the IGCSE CIE Engineering syllabus. Use it to reinforce your understanding of core principles: from mechanics and electricity to materials and thermodynamics. Each entry pairs a succinct explanation with the corresponding formula, enabling rapid revision and deeper learning.

本速查手册提炼了 IGCSE CIE 工程课程必须掌握的核心公式与定理。借助它，你可以巩固对力学、电学、材料、热力学等基本原理的理解。每个条目都配有简明解释与相应公式，帮助你快速复习并深化学习。

1. Mechanics and Motion | 力学与运动

Engineering mechanics describes how objects move under the action of forces. The SUVAT equations relate initial velocity (u), final velocity (v), acceleration (a), displacement (s) and time (t), assuming constant acceleration.

工程力学描述物体在力作用下的运动。SUVAT 方程组将初速度 (u)、末速度 (v)、加速度 (a)、位移 (s) 和时间 (t) 关联起来，前提是加速度恒定。

v = u + at

This linear relationship gives velocity after a period of constant acceleration. It is essential for analysing rectilinear motion, such as a vehicle accelerating along a straight road.

该线性关系给出了匀加速一段时间后的速度。它对分析直线运动至关重要，例如车辆在直路上加速的场景。

s = ut + ½at²

The displacement equation enables engineers to predict how far an object travels when starting with an initial speed and undergoing constant acceleration, which is fundamental in designing transport systems.

位移方程使工程师能够预测物体以初速度开始并经历匀加速时所行进的距离，这是设计运输系统的基础。

v² = u² + 2as

Useful when time is not known, this formula links velocity and displacement directly. It is applied in safety engineering to calculate stopping distances.

当时间未知时，该公式可直接关联速度和位移，用于安全工程计算制动距离。

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2. Forces and Moments | 力与力矩

Newton's laws govern the interaction between force, mass and acceleration. The principle of moments explains equilibrium in structures and levers.

牛顿定律支配着力、质量与加速度的相互作用。力矩原理则解释结构和杠杆中的平衡。

F = ma

Newton's second law: the net force acting on a body equals its mass multiplied by its acceleration. This is the cornerstone of dynamics, used to analyse everything from lifting systems to vehicle crash performance.

牛顿第二定律：作用在物体上的净力等于质量乘以加速度。这是动力学的基石，用于分析从起重系统到车辆碰撞性能的各种问题。

M = F × d

The moment of a force about a pivot is the product of the force and the perpendicular distance from the pivot to its line of action. For a body to be in rotational equilibrium, the sum of clockwise moments must equal the sum of anti-clockwise moments.

力对支点的力矩等于力与支点到力作用线垂直距离的乘积。物体要处于转动平衡，顺时针力矩之和必须等于逆时针力矩之和。

ΣM_cw = ΣM_ccw

This is the principle of moments, used when designing balanced beams, cranes, and levers. It ensures stability and safe load distribution.

这就是力矩原理，用于设计平衡梁、起重机和杠杆，确保稳定性和安全负载分配。

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3. Energy, Work and Power | 能量、功和功率

Energy can neither be created nor destroyed, only converted from one form to another. Work is done when a force moves its point of application, and power is the rate of doing work.

能量既不能创造也不能消灭，只能从一种形式转化为另一种形式。力使其作用点移动时做功，功率则是做功的速率。

W = F × d (when force and displacement are parallel)

Work done equals force multiplied by distance moved in the direction of the force. It is measured in joules (J), and underpins efficiency calculations in all mechanical systems.

功等于力乘以沿力方向移动的距离。其单位为焦耳 (J)，是所有机械系统效率计算的基础。

E_p = mgh

Gravitational potential energy stored in an object raised to a height h is the product of mass, gravitational field strength g (9.8 m/s² on Earth) and height. Engineers use this to design elevators and energy storage systems.

升高到高度 h 的物体所储存的重力势能是质量、重力场强度 g（地球上为 9.8 m/s²）和高度的乘积。工程师用它设计电梯和储能系统。

E_k = ½mv²

Kinetic energy of a moving object depends on its mass and the square of its speed. This formula is crucial in vehicle design for understanding braking energy and crash dynamics.

运动物体的动能取决于质量和速度的平方。这一公式在车辆设计中用于理解制动能量和碰撞动力学。

P = W / t = F × v

Power (watts) is the rate of energy transfer or work done per unit time. For a constant force moving at speed v, power is force multiplied by velocity. It guides motor selection and engine analysis.

功率（瓦特）是能量转换或单位时间做功的速率。对于以速度 v 运动的恒力，功率等于力乘以速度。它为电机选型和发动机分析提供依据。

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4. Materials and Stress-Strain | 材料与应力-应变

Understanding how materials respond to forces is vital for structural design. Stress, strain and Young's modulus characterise stiffness, while yield strength and ultimate tensile strength indicate failure points.

了解材料对力的响应对于结构设计至关重要。应力、应变和杨氏模量表征刚度，而屈服强度和极限抗拉强度则指示失效点。

σ = F / A

Stress (σ) is the internal force per unit area within a material, measured in pascals (Pa) or N/m². Tensile stress stretches, compressive stress shortens, and shear stress causes sliding. Designers must keep stress below allowable limits.

应力 (σ) 是材料内部单位面积所受的力，单位为帕斯卡 (Pa) 或 N/m²。拉应力使材料伸长，压应力使之缩短，剪应力则引起滑移。设计者必须将应力控制在许用限度内。

ε = ΔL / L₀

Strain (ε) is the extension per original length, a dimensionless ratio. It quantifies how much a material deforms under load.

应变 (ε) 是单位原长的伸长量，是一个无量纲比值，用于量化材料在载荷下的变形程度。

E = σ / ε

Young's modulus (E) measures the stiffness of a material within the elastic region. A high E means the material resists deformation. Steel has a value around 200 GPa, while polymers may be less than 5 GPa.

杨氏模量 (E) 衡量材料在弹性区域的刚度。E 值高表示材料抗变形能力强。钢的 E 值约为 200 GPa，而聚合物可能低于 5 GPa。

Factor of Safety = Ultimate Stress / Working Stress

Engineers apply a factor of safety to account for uncertainties in loads, material flaws and environmental conditions. Typical values range from 1.5 to 4 for many structural applications.

工程师采用安全系数来考虑载荷、材料缺陷和环境条件的不确定性。许多结构应用中典型值在 1.5 至 4 之间。

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5. Electrical Circuits | 电路

Ohm's law and the power law underpin the analysis of electrical and electronic circuits encountered in engineering projects, from simple switches to control systems.

欧姆定律和功率定律是分析工程项目中电气与电子电路的基础，涵盖从简单开关到控制系统的各种应用。

V = IR

Ohm's law states that the potential difference (voltage) across a resistor equals the current flowing through it multiplied by its resistance. It allows calculation of current, resistance, or voltage drop in DC circuits.

欧姆定律指出，电阻两端的电位差（电压）等于流过它的电流乘以电阻。它可用于计算直流电路中的电流、电阻或电压降。

P = IV = I²R = V²/R

Electrical power (in watts) is the product of current and voltage. For resistive loads, alternative forms using resistance help size components like heating elements and resistors.

电功率（瓦特）是电流与电压的乘积。对于电阻性负载，利用电阻的等价形式有助于确定加热元件和电阻器等元件的规格。

R_series = R₁ + R₂ + R₃ …

Resistors in series add directly. This principle is used to design voltage dividers and to limit current in circuits.

串联电阻直接相加。这一原理用于设计分压器和限制电路中的电流。

1/R_parallel = 1/R₁ + 1/R₂ + 1/R₃ …

For parallel resistors, the reciprocal of the total resistance equals the sum of reciprocals. Parallel arrangement reduces overall resistance and is common in power distribution to share loads.

对于并联电阻，总电阻的倒数等于各电阻倒数之和。并联连接可降低总电阻，常用于配电系统以分担负载。

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6. Thermodynamics | 热力学

Thermodynamics deals with heat, work, and energy conversion. The relationships between heat energy, temperature change and phase change are central to designing engines, heating systems and cooling devices.

热力学研究热、功和能量转换。热能、温度变化和相变之间的关系是设计发动机、供暖系统和冷却装置的核心。

Q = mcΔθ

The heat energy required to raise the temperature of a mass m by Δθ is proportional to the mass and the specific heat capacity c (J/kg°C). Water has a high specific heat capacity (4184 J/kg°C), making it an excellent coolant.

将质量 m 加热升温 Δθ 所需的热能正比于质量与比热容 c（J/kg°C）。水的比热容高达 4184 J/kg°C，使其成为优异的冷却剂。

Q = mL

During a phase change (melting, boiling), the temperature remains constant while the substance absorbs latent heat L. Specific latent heat of fusion and vaporisation explain the large energy involved in changing state.

在相变（熔化、沸腾）过程中，温度保持不变，物质吸收潜热 L。比熔化潜热和气化潜热解释了状态变化所涉及的大量能量。

η = (W_out / Q_in) × 100%

The thermal efficiency of a heat engine is the ratio of useful work output to the heat input, expressed as a percentage. Even ideal Carnot engines are limited by temperature differences; real engines achieve lower efficiencies due to friction and heat losses.

热机的热效率是有用输出功与输入热量之比，以百分数表示。即便是理想的卡诺热机也受温差限制；实际发动机因摩擦和热损失，效率更低。

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7. Fluid Mechanics | 流体力学

Fluid behaviour is governed by pressure, flow rate and energy conservation. These principles are essential for hydraulic systems, pipeline design and aerodynamic analysis.

流体的行为受压力、流量和能量守恒支配。这些原理对液压系统、管道设计和空气动力学分析至关重要。

p = F / A

Pressure is the force exerted per unit area (Pa). In hydraulic pistons, a small force on a small area creates high pressure that can generate a large force on a larger area, as used in car brakes and industrial presses.

压力是单位面积上作用的力（Pa）。在液压活塞中，作用于小面积上的小力产生的压强可在较大面积上产生巨力，这用于汽车制动和工业压机中。

p = ρgh

The hydrostatic pressure at a depth h in a fluid of density ρ is the weight of the fluid column above. This explains why dam walls are thicker at the base and why submarine hulls must be immensely strong.

密度为 ρ 的流体中深度 h 处的静水压力等于上方流体柱的重量。这解释了为何大坝底部更厚、潜艇壳体必须极其坚固。

Q = Av

The volumetric flow rate Q (m³/s) through a pipe equals the cross‑sectional area A multiplied by the fluid velocity v. For an incompressible fluid, Q is constant along a streamline (continuity), so velocity increases where the pipe narrows.

管道中的体积流量 Q（m³/s）等于横截面积 A 乘以流速 v。对于不可压缩流体，Q 沿流线恒定（连续性），因此管道变窄处流速增大。

p₁ + ½ρv₁² + ρgh₁ = p₂ + ½ρv₂² + ρgh₂

Bernoulli's equation expresses conservation of energy in a flowing fluid: the sum of pressure energy, kinetic energy per unit volume and potential energy per unit volume is constant along a streamline. It underpins aerofoil lift and venturi meters.

伯努利方程表达了流动流体中的能量守恒：单位体积的压力能、动能和势能之和沿流线恒定。它是机翼升力和文丘里流量计的原理基础。

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8. Control Systems and Mechanical Advantage | 控制系统与机械效益

Mechanical advantage and velocity ratio describe how simple machines multiply force or speed. In control systems, block diagrams and transfer functions model feedback loops, which are prevalent in robotic and automated engineering.

机械效益和速比描述简单机械如何增力或增速。在控制系统中，框图与传递函数对反馈回路建模，这在机器人与自动化工程中普遍存在。

MA = Load / Effort

Mechanical advantage (MA) is the ratio of output force (load) to input force (effort). A lever or gear system with MA > 1 amplifies force at the expense of distance moved.

机械效益 (MA) 是输出力（负载）与输入力（施力）之比。MA 大于 1 的杠杆或齿轮系统以牺牲移动距离为代价来放大力。

VR = Distance moved by effort / Distance moved by load

Velocity ratio (VR) is the ratio of the distance the effort moves to the distance the load moves in the same time. It depends solely on the geometry of the machine and does not change with friction.

速比 (VR) 是相同时间内施力移动距离与负载移动距离之比，仅取决于机器的几何结构，与摩擦无关。

Efficiency = (MA / VR) × 100%

Efficiency evaluates how well a machine converts input work into useful output work. Friction always reduces efficiency below 100%; understanding losses helps engineers select lubricants and refine designs.

效率衡量机器将输入功转化为有用输出功的程度。摩擦总会使效率低于 100%；了解损耗有助于工程师选择润滑剂并优化设计。

G(s) = C(s) / R(s)

In control theory, the transfer function G(s) relates the output C(s) to the input R(s) in the Laplace domain. It enables analysis of stability, speed of response and accuracy in systems like motor speed controllers and autopilots.

在控制理论中，传递函数 G(s) 将拉普拉斯域中的输出 C(s) 与输入 R(s) 联系起来，可用于分析电机速度控制器和自动驾驶仪等系统的稳定性、响应速度和精度。

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9. Statics and Structures | 静力学与结构力学

Statically determinate structures can be analysed using equilibrium equations. Trusses, beams and frames transmit loads safely when internal forces are correctly calculated.

静定结构可通过平衡方程进行分析。正确计算内力时，桁架、梁和框架可以安全传载。

ΣFₓ = 0, ΣFᵧ = 0, ΣM = 0

For any body in static equilibrium, the vector sum of horizontal forces, vertical forces, and moments about any point must be zero. These three equations solve reaction forces at supports.

对于任何处于静力平衡的物体，水平力、竖直力以及绕任一点的力矩之和必须为零。这三个方程可求解支座反力。

Shear force and bending moment diagrams

By plotting how shear force and bending moment vary along a beam, engineers locate maximum values and design cross‑sections that resist failure. The area under the shear diagram gives the change in moment.

通过绘制剪力和弯矩沿梁的变化图，工程师可确定最大值并设计抵抗失效的截面。剪力图下的面积给出弯矩的变化量。

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10. Dynamics of Rotation | 旋转动力学

Rotational motion obeys analogues of linear laws. Torque, moment of inertia and angular acceleration are key to designing rotating machinery such as turbines, flywheels and gear trains.

旋转运动遵循与直线运动类似的定律。扭矩、转动惯量和角加速度是设计涡轮机、飞轮和齿轮系等旋转机械的关键。

τ = Iα

he equation parallels F = ma, governing how rotational speed changes under applied torque.

其中 τ 为扭矩 (N·m)，I 为转动惯量 (kg·m²)，α 为角加速度 (rad/s²)。该方程与 F = ma 平行，支配外加扭矩下转速的变化。

ω = ω₀ + αt

is used to calculate how quickly a spinning disc reaches its operational speed, which is critical for motor sizing.

v = u + at 的角量对应式，用于计算旋转盘达到工作转速的快慢，这对电机选型至关重要。

P = τω

Rotational power is the product of torque and angular velocity. This formula permits conversion between shaft power and electrical power in generators and motors.

旋转功率是扭矩与角速度的乘积。该公式可用于发电机和电动机中轴功率与电功率的换算。

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11. Wave and Sound Engineering | 波与声学工程

Wave phenomena, including sound, obey the universal wave equation. Acoustics plays a role in noise control, ultrasonic testing and communication systems.

包括声波在内的波动现象遵循通用波动方程。声学在噪声控制、超声波检测和通信系统中起作用。

v = fλ

The speed of a wave (m/s) equals its frequency (Hz) multiplied by its wavelength (m). This relationship is fundamental for specifying antennas, audio systems and non‑destructive testing equipment.

波速 (m/s) 等于频率 (Hz) 乘以波长 (m)。此关系是确定天线、音频系统和无损检测设备规格的基础。

Sound intensity level (dB) = 10 log₁₀ (I / I₀)

The decibel scale compares sound intensity I to a reference I₀ (10⁻¹² W/m²). Engineers use it to assess hearing protection requirements and to design quiet machinery.

分贝标度将声强 I 与基准 I₀ (10⁻¹² W/m²) 进行比较。工程师用它评估听力保护需求及设计低噪声机械。

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12. Engineering Design and Project Calculations | 工程设计计算

Successful engineering integrates formulas with practical constraints. Tolerance, gear ratios, costing and reliability are quantified through straightforward arithmetic that supports decision‑making.

成功的工程实践将公式与实际约束相结合。公差、齿轮比、成本和可靠性通过简明算术量化，为决策提供支持。

Gear ratio = N₂ / N₁ = ω₁ / ω₂

The ratio of teeth numbers or angular velocities determines speed and torque transformation between meshing gears. It is essential for drivetrain design.

齿数或角速度之比决定了啮合齿轮之间的速度和扭矩变换，是传动系统设计的关键。

Tolerance = Upper limit − Lower limit

Tolerance defines the permissible variation in a component's dimension. Tight tolerances improve fit but raise manufacturing cost; engineers balance function and economy.

公差定义了零件尺寸的允许变动范围。紧公差可改善配合但增加制造成本；工程师需平衡功能与经济性。

Overall efficiency = η₁ × η₂ × η₃ …

When several machine elements (e.g., motor, gearbox, pulley) are combined in series, the total efficiency is the product of individual efficiencies. This multiplicative effect means even small losses can accumulate significantly.

当多个机器元件（如电机、减速箱、皮带轮）串联时，总效率为各效率的乘积。这种乘积效应意味着即使很小的损耗也会显著累积。

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