📚 GCSE CAIE Engineering: Formula & Theorem Quick Reference Guide | GCSE CAIE 工程：公式定理速查手册
This revision booklet gathers the essential formulae, laws, and theorems required for the CAIE GCSE Engineering syllabus. It is designed for rapid lookup and last‑minute review before examinations. Each section presents the core relationships in mechanics, materials, and electrical engineering, followed by clear explanatory notes that link the mathematics to practical engineering situations.
这本复习手册汇集了 CAIE GCSE 工程课程所需的核心公式、定律和定理，旨在帮助考生快速查阅和考前冲刺。每一节都以力学、材料学和电气工程中的基本关系为主线，并提供清晰的解释，将数学公式与实际工程情境联系起来。
1. Basic Motion and Kinematics | 基本运动与运动学
Kinematics describes how objects move without reference to the forces causing the motion. The four SUVAT equations link initial velocity u, final velocity v, acceleration a, time t, and displacement s. They are the starting point for analysing uniformly accelerated linear motion in mechanisms and vehicle systems.
运动学描述物体如何运动，而不涉及引起运动的力。四个 SUVAT 方程将初速度 u、末速度 v、加速度 a、时间 t 和位移 s 联系起来，是分析匀加速直线运动的基础，广泛应用于机构和车辆系统中。
v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t
Always ensure that the signs for direction are consistent – usually one direction is taken as positive. In engineering problems, these equations help to calculate braking distances, piston stroke durations, and conveyor speeds.
使用这些公式时必须确保方向符号一致，通常取一个方向为正。在工程问题中，它们可用于计算制动距离、活塞行程时间和输送带速度。
- v = final velocity (m/s) | 末速度（米/秒）
- u = initial velocity (m/s) | 初速度（米/秒）
- a = acceleration (m/s²) | 加速度（米/秒²）
- t = time (s) | 时间（秒）
- s = displacement (m) | 位移（米）
2. Forces and Newton’s Laws | 力与牛顿定律
Newton’s three laws of motion form the foundation of dynamics. The second law, F = ma, quantifies the resultant force needed to produce a given acceleration on a mass m. In engineering design, this law governs the sizing of actuators, the analysis of vehicle dynamics, and the determination of structural loads.
牛顿运动三定律构成了动力学的基础。第二定律 F = ma 量化了产生给定加速度所需的合力，质量 m 是惯性的量度。在工程设计中，该定律决定了执行器的选型、车辆动力学分析和结构载荷的确定。
F = ma
The resultant force is the vector sum of all forces acting on an object. Always draw a free‑body diagram to identify individual forces: weight (mg), normal reaction, friction, tension, and applied forces. In equilibrium, the resultant force is zero – this is a special case of Newton’s second law where a = 0.
合力是作用在物体上所有力的矢量和。务必画出受力分析图以标示各个力：重力 (mg)、支持力、摩擦力、张力和外加力。平衡状态下合力为零，这是牛顿第二定律在 a = 0 时的特殊情况。
| Force Type | 力类型 | Typical Expression |
|---|---|---|
| Weight | 重力 | W = mg |
| Friction (limiting) | 极限摩擦力 | F = μR |
| Spring force | 弹簧力 | F = kx |
Here μ is the coefficient of friction, R is the normal reaction, k is the spring constant, and x is the extension or compression. The direction of friction always opposes relative motion or the tendency to move.
其中 μ 为摩擦系数，R 为法向反力，k 为弹簧劲度系数，x 为伸长或压缩量。摩擦力的方向总是与相对运动或运动趋势相反。
3. Work, Energy and Power | 功、能量与功率
Work is done when a force moves its point of application. The work done is the product of the force component in the direction of motion and the distance moved. Energy is the capacity to do work; it appears in mechanical forms as kinetic energy and gravitational potential energy.
当力使其作用点移动时即做功。功等于力在运动方向的分量与移动距离的乘积。能量是做功的本领，在力学中以动能和重力势能的形式出现。
W = F s cosθ
Eₖ = ½mv²
Eₚ = mgh
Power is the rate of doing work. In mechanical systems, it is often calculated as the product of force and velocity, or as the torque multiplied by the angular velocity for rotating shafts.
功率是做功的速率。在机械系统中，常表示为力与速度的乘积，或对于旋转轴表示为转矩与角速度的乘积。
P = W/t = Fv
P = τ × ω
In any real machine, some energy is dissipated as heat due to friction. The principle of conservation of energy states that total energy is constant, but it may be transformed from one form to another. Engineers use this principle to conduct energy audits and improve efficiency.
在任何真实机器中，因摩擦部分能量以热的形式耗散。能量守恒定律指出总能量保持不变，但可从一种形式转化为另一种形式。工程师利用这一原则进行能量核算并提高效率。
4. Moments and Equilibrium | 力矩与平衡
A moment is the turning effect of a force about a pivot. The magnitude of a moment depends on both the force and the perpendicular distance from the pivot to the line of action of the force. The principle of moments is fundamental to the design of levers, beams, and rotating components.
力矩是力对支点的转动效应，取决于力的大小和支点到力作用线的垂直距离。力矩原理是杠杆、梁和旋转部件设计的基础。
Moment = F × d
For an object in static equilibrium, two conditions must be satisfied: the vector sum of all forces must be zero, and the sum of clockwise moments about any point must equal the sum of anticlockwise moments about that same point.
物体处于静力平衡时，必须满足两个条件：所有力的矢量和为零；且对任意点，顺时针力矩之和等于逆时针力矩之和。
ΣF = 0 and ΣM = 0
When analysing a beam, the reaction forces at supports can be found by taking moments about one support to eliminate the unknown reaction at that point. This is a standard technique in structural and mechanical design.
分析梁时，可对一支座取矩，以消除该处未知反力，从而求出支座反力。这是结构设计与机械设计中的标准方法。
5. Stress, Strain and Young’s Modulus | 应力、应变与杨氏模量
When a material is subjected to a load, it experiences stress and strain. Direct stress (tensile or compressive) is defined as the force per unit cross‑sectional area, while strain is the ratio of extension to original length. These quantities describe the intensity of internal forces and the resulting deformation.
当材料承受载荷时会产生应力和应变。正应力（拉伸或压缩）定义为单位横截面积上的力，而应变是伸长量与原长的比值。这两个物理量描述了内力的强度以及由此产生的变形。
σ = F / A
ε = ΔL / L₀
For many engineering materials, stress is proportional to strain within the elastic limit – this is Hooke’s Law for continuous materials. Young’s modulus E quantifies the stiffness.
对于许多工程材料，在弹性限度内应力与应变成正比，即胡克定律在连续材料中的表达。杨氏模量 E 用于量化材料的刚度。
σ = E ε
E = stress / strain (Pa or N/m²)
The stress‑strain curve obtained from a tensile test reveals key properties: yield strength, ultimate tensile strength, and ductility. Design engineers must keep working stresses well below the yield point, incorporating a factor of safety.
拉伸试验获得的应力‑应变曲线揭示了屈服强度、抗拉强度和延展性等关键性质。设计工程师必须使工作应力远低于屈服点，并引入安全系数。
6. Properties of Materials | 材料性质
Selecting the right material for a component requires consideration of mechanical properties such as hardness, toughness, ductility, and malleability. Hardness indicates resistance to indentation or scratching; toughness is the ability to absorb energy before fracture; ductility allows plastic deformation without breaking.
为零件选择合适的材料需要考虑硬度、韧性、延展性和可锻性等力学性能。硬度表示抵抗压痕或划痕的能力；韧性是在断裂前吸收能量的能力；延展性允许材料在断裂前发生塑性变形。
Density is a fundamental property that directly affects the weight of a structure. For a given volume V and mass m:
密度是直接影响结构重量的基本属性。对于给定体积 V 和质量 m：
ρ = m / V
Thermal expansion must be accounted for when components are expected to operate over a wide temperature range. The linear expansion ΔL is given by:
若零件在宽温域内工作，必须考虑热膨胀。线性膨胀量 ΔL 由下式给出：
ΔL = α L₀ ΔT
where α is the linear expansivity of the material, L₀ is the original length, and ΔT is the temperature change. In assemblies with dissimilar materials, mismatched expansion can cause thermal stress and failure.
式中 α 是材料的线膨胀系数，L₀ 为原长，ΔT 为温度变化量。在由异种材料构成的组件中，膨胀不匹配可能导致热应力与失效。
7. Electrical Fundamentals (Ohm’s Law) | 电学基础（欧姆定律）
Ohm’s law is the cornerstone of circuit analysis. For many conductors at constant temperature, the current I flowing through the conductor is directly proportional to the potential difference V across it. The constant of proportionality is the resistance R.
欧姆定律是电路分析的基石。对于恒温下的许多导体，流过导体的电流 I 与两端电压 V 成正比，比例常数即电阻 R。
V = I R
Resistance depends on the material’s resistivity ρ, its length L, and cross‑sectional area A. This relation is crucial when selecting cables and designing heating elements.
电阻取决于材料的电阻率 ρ、长度 L 和横截面积 A。这一关系在电缆选型和加热元件设计中至关重要。
R = ρ L / A
Series and parallel resistor networks are encountered in almost every electronic and electrical system. Equivalent resistances are calculated as follows:
串联和并联电阻网络几乎出现在所有电子与电气系统中。等效电阻的计算如下：
Series: R_total = R₁ + R₂ + R₃ + …
Parallel: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …
Current division and voltage division rules are derived from these basics. In engineering, understanding series‑parallel combinations enables the design of reliable control circuits and power distribution networks.
分流和分压规则均可从这些基础关系推导得出。掌握串并联组合有助于设计可靠的控制电路和配电网络。
8. Electrical Power and Energy | 电功率与电能
Electrical power is the rate at which electrical energy is transferred. The fundamental power equation combines voltage and current; using Ohm’s law, two alternative forms can be written, which are useful when voltage or current is unknown.
电功率是电能传递的速率。基本功率公式将电压和电流联系起来；利用欧姆定律可写出两种变体，在电压或电流未知时非常实用。
P = V I
P = I² R
P = V² / R
Energy consumed by an electrical device is the product of power and time. In engineering specifications, energy is often stated in joules (J) or in kilowatt‑hours (kWh) for mains‑powered equipment.
电器消耗的能量是功率与时间的乘积。在工程规范中，能量通常用焦耳 (J) 表示，对于市电设备也常用千瓦时 (kWh) 计量。
E = P t
When selecting power supplies or batteries, engineers must match both the voltage rating and the current capacity to the load, while considering the continuous power dissipation. Excessive power loss in a component leads to overheating and reduced reliability.
选择电源或电池时，工程师必须使电压等级和电流容量与负载匹配，同时考虑持续功耗。元件功耗过大将导致过热和可靠性降低。
9. Mechanical Advantage and Efficiency | 机械优势与效率
Simple machines such as levers, pulleys, gears, and hydraulic systems allow a small effort to overcome a large load. The ideal mechanical advantage (IMA) is determined purely by geometry, while the actual mechanical advantage (AMA) accounts for friction and other losses.
杠杆、滑轮、齿轮和液压系统等简单机械可以用较小的动力克服较大的负载。理想机械利益 (IMA) 仅取决于几何结构，而实际机械利益 (AMA) 则计入摩擦等损失。
MA = Load / Effort
Efficiency = (Work output / Work input) × 100%
In rotary systems, gear trains and belt drives transmit power. The velocity ratio (VR) of two meshing gears is inversely proportional to their numbers of teeth. For a belt drive with pulley diameters d₁ and d₂:
在回转系统中，齿轮系和带传动传递动力。两啮合齿轮的速比 (VR) 与齿数成反比。对于带轮直径为 d₁ 和 d₂ 的带传动：
VR = Output speed / Input speed = N_input / N_output (gears) = d₂ / d₁ (pulleys)
The overall efficiency of a machine is the product of the efficiencies of its individual stages. Engineers strive to maximise efficiency to reduce energy consumption and operating costs.
机器的总效率是各部件效率的乘积。工程师力求将效率最大化，以降低能耗和运行成本。
10. Theorems in Structures | 结构定理
Structures such as trusses and frameworks are analysed using the method of joints and the method of sections. Two key theorems often used are Lami’s theorem (for three co‑planar concurrent forces in equilibrium) and the Triangle of Forces.
桁架和框架结构可通过节点法和截面法进行分析。其中两个常用定理为拉密定理（用于三个共面汇交力的平衡）和力的三角形法则。
Lami’s Theorem: F₁/sin(α) = F₂/sin(β) = F₃/sin(γ)
These geometric approaches allow the determination of unknown forces without solving simultaneous equations. When a structure is statically determinate, the internal forces can be found using equilibrium alone.
这些几何方法使得无需解联立方程即可确定未知力。当结构为静定时，仅凭平衡条件即可求得内力。
For uniformly distributed loads (UDL) on a simply supported beam, the maximum bending moment occurs at the centre and is given by wL²/8, where w is load per unit length and L is the span. This relationship is vital when sizing beams to avoid bending failure.
对于简支梁上的均布载荷 (UDL)，最大弯矩出现在跨中，计算公式为 wL²/8，其中 w 为单位长度载荷，L 为跨度。该关系在梁的尺寸设计以避开弯曲破坏时至关重要。
M_max = wL² / 8
11. Conversion of Units and Constants | 单位换算与常数
Correct handling of units is essential in all engineering calculations. The SI base units are metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), and candela (cd). Derived units such as newton (N), joule (J), watt (W), and pascal (Pa) are routinely used.
正确的单位处理是所有工程计算的基础。国际单位制基本单位为米 (m)、千克 (kg)、秒 (s)、安培 (A)、开尔文 (K) 和坎德拉 (cd)。常用的导出单位有牛顿 (N)、焦耳 (J)、瓦特 (W) 和帕斯卡 (Pa)。
| Quantity | SI Unit | Equivalent |
|---|---|---|
| Force | N | kg·m/s² |
| Pressure / Stress | Pa | N/m² |
| Energy, Work | J | N·m |
| Power | W | J/s |
Common conversions: 1 tonne = 1000 kg, 1 litre = 0.001 m³, 1 km/h = 0.2778 m/s. The acceleration due to gravity, g, is normally taken as 9.8 m/s², though 10 m/s² may be used for approximate calculations unless specified otherwise.
常用换算：1 吨 = 1000 千克，1 升 = 0.001 立方米，1 千米/时 = 0.2778 米/秒。重力加速度 g 通常取 9.8 m/s²，除非另有说明，近似计算中允许使用 10 m/s²。
12. Summary of Key Equations | 关键方程汇总
The table below brings together the most frequently used equations in GCSE CAIE Engineering. Familiarity with these relationships will enable you to solve a wide range of numerical problems under timed examination conditions.
下表汇总了 GCSE CAIE 工程中最常用的方程式。熟练掌握这些关系式将帮助你在限时的考试条件下解决各类计算问题。
| Topic | Formula |
|---|---|
| Acceleration | a = (v − u) / t |
| Resultant Force | F = ma |
| Kinetic Energy | Eₖ = ½mv² |
| Potential Energy | Eₚ = mgh |
| Work Done | W = Fs cosθ |
| Power (mechanical) | P = W / t = Fv |
| Moment | M = Fd |
| Tensile Stress | σ = F / A |
| Tensile Strain | ε = ΔL / L₀ |
| Young’s Modulus | E = σ / ε |
| Ohm’s Law | V = IR |
| Electrical Power | P = VI = I²R = V²/R |
| Resistance (in wire) | R = ρL / A |
| Density | ρ = m / V |
| Efficiency | η = (Output / Input) × 100% |
Keep this table with you when revising and practise applying each formula to context‑based problems. Remember that correct unit conversion and sensible rounding will significantly improve the accuracy of your final answers.
复习时请随身携带本表，并结合情境化问题练习应用每一个公式。请记住，正确的单位换算和合理的舍入将显著提高最终答案的准确性。
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