📚 GCSE Cambridge Engineering: Formula & Theorem Quick Reference Handbook | GCSE 剑桥工程：公式定理速查手册
This handbook is a compact revision aid covering the essential formulas, principles, and theorems required for the Cambridge IGCSE Engineering syllabus. It brings together mechanics, materials, electricity, simple machines, thermal concepts, and structural analysis into a single quick-reference list designed to support both coursework and examination preparation.
本手册是一本紧凑的复习工具，涵盖剑桥 IGCSE 工程课程所必需的核心公式、原理和定理。它将力学、材料、电学、简单机械、热学概念和结构分析整合到一份快速参考列表中，旨在为课程作业和考试备考提供支持。
1. Fundamental Mechanics | 基本力学定律
Newton’s second law: The resultant force F equals mass m multiplied by acceleration a: F = m × a. Force is measured in newtons (N), mass in kilograms (kg), and acceleration in metres per second squared (m/s²).
牛顿第二定律：合力 F 等于质量 m 乘以加速度 a：F = m × a。力以牛顿 (N) 为单位，质量以千克 (kg) 为单位，加速度以米每二次方秒 (m/s²) 为单位。
Weight: Weight W is the gravitational force on a mass: W = m × g, where g = 9.81 m/s² on Earth. Weight is a force and is measured in newtons.
重量：重量 W 是作用在质量上的重力：W = m × g，其中地面上 g = 9.81 m/s²。重量是一种力，单位为牛顿。
Equilibrium conditions: For a body to be in static equilibrium, the vector sum of all forces must be zero (ΣF = 0) and the sum of all moments about any point must be zero (ΣM = 0).
平衡条件：物体处于静力平衡时，所有力的矢量和为零 (ΣF = 0)，且对任意点的力矩代数和为零 (ΣM = 0)。
2. Linear Motion | 直线运动
Uniform acceleration equations (SUVAT): These apply when acceleration a is constant. v = u + a t, where u = initial velocity, v = final velocity, t = time. s = u t + ½ a t², where s = displacement.
匀加速运动方程 (SUVAT)：适用于加速度 a 恒定的情况。v = u + a t，其中 u = 初速度，v = 末速度，t = 时间。s = u t + ½ a t²，其中 s = 位移。
v² = u² + 2 a s. s = ½ (u + v) t (average velocity × time). All quantities are in SI units: m/s, m, s, m/s².
v² = u² + 2 a s。s = ½ (u + v) t (平均速度 × 时间)。所有量均采用国际单位制：m/s、m、s、m/s²。
These equations allow engineers to calculate stopping distances, launch velocities, and component motion in machines.
这些方程使工程师能够计算机器的制动距离、发射速度和部件运动。
3. Forces and Moments | 力与力矩
Moment of a force: The turning effect of a force about a pivot. Moment M = F × d, where d is the perpendicular distance from the pivot to the line of action of the force. The SI unit is the newton-metre (N m).
力矩：力对支点的转动效应。力矩 M = F × d，其中 d 为支点到力作用线的垂直距离。国际单位制为牛顿·米 (N m)。
Principle of moments: For a system in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any pivot. This is used to analyse levers, beams, and crane arms.
力矩原理：系统处于转动平衡时，对任意支点的顺时针力矩之和等于逆时针力矩之和。这用于分析杠杆、梁和起重机臂。
Resultant force: The net effect of multiple forces acting on a body. If forces are collinear, they are added algebraically; if at an angle, they can be resolved into perpendicular components using Pythagoras and trigonometry.
合力：作用在物体上的多个力的净效果。若力共线，则代数相加；若成角度，可用勾股定理和三角函数分解为垂直分量。
4. Work, Energy and Power | 功、能与功率
Work done: Work W = F × d, where force F is constant and acts in the direction of displacement d. 1 joule (J) = 1 N m.
功：功 W = F × d，其中力 F 恒定且作用方向与位移 d 一致。1 焦耳 (J) = 1 N m。
Kinetic energy: KE = ½ m v². The energy possessed by a moving body. Measured in joules.
动能：KE = ½ m v²。运动物体所具有的能量，单位为焦耳。
Gravitational potential energy: GPE = m g h. Energy stored due to an object’s height h relative to a reference level.
重力势能：GPE = m g h。物体因高度 h 相对于参考面而储存的能量。
Power: The rate of doing work. P = W / t, where W is work done in time t. Also, for constant speed, P = F × v. The unit of power is the watt (W): 1 W = 1 J/s.
功率：做功的速率。P = W / t，其中 W 是在时间 t 内做的功。对于匀速运动，P = F × v。功率的单位是瓦特 (W)：1 W = 1 J/s。
Efficiency: η = (useful energy output / total energy input) × 100%. No machine can be 100% efficient due to friction and other losses.
效率：η = (有用能量输出 / 总能量输入) × 100%。由于摩擦和其他损失，任何机器的效率都无法达到 100%。
5. Stress and Strain | 应力与应变
Tensile and compressive stress: σ = F / A, where F is the applied force (tensile or compressive) and A is the original cross‑sectional area. Unit: pascal (Pa) or N/m².
拉伸与压缩应力：σ = F / A，其中 F 为施加的力（拉伸或压缩），A 为原始横截面积。单位：帕斯卡 (Pa) 或 N/m²。
Strain: ε = ΔL / L₀, where ΔL is the change in length (extension or contraction) and L₀ is the original length. Strain has no units; it is a ratio.
应变：ε = ΔL / L₀，其中 ΔL 为长度变化量（伸长或缩短），L₀ 为原始长度。应变无单位，是一个比值。
Young’s modulus: E = σ / ε, within the elastic limit. It measures the stiffness of a material. A high Young’s modulus indicates a stiffer material. Unit: Pa.
杨氏模量：E = σ / ε，在弹性限度内有效。它衡量材料的刚度。杨氏模量高表示材料刚度大。单位：Pa。
Ultimate tensile strength: The maximum stress a material can withstand while being stretched or pulled before necking and failure. It is obtained from the stress‑strain curve.
极限抗拉强度：材料在拉伸过程中发生颈缩和断裂之前所能承受的最大应力。可通过应力‑应变曲线获得。
6. Material Properties and Hooke’s Law | 材料性能与胡克定律
Hooke’s law: For an elastic material, the extension x is directly proportional to the applied force F, provided the elastic limit is not exceeded: F = k x. k is the spring constant (N/m).
胡克定律：对于弹性材料，只要不超过弹性极限，伸长量 x 与施加的力 F 成正比：F = k x。k 为弹簧常数 (N/m)。
Spring constant: k = F / x. A stiff spring has a large k; a soft spring has a small k. The gradient of a force‑extension graph gives k.
弹簧常数：k = F / x。刚度大的弹簧 k 值大，柔软的弹簧 k 值小。力‑伸长量图的斜率即为 k。
Elastic potential energy: The energy stored in a stretched or compressed spring: E = ½ k x². This energy is recoverable when the spring returns to its original shape.
弹性势能：拉伸或压缩的弹簧中储存的能量：E = ½ k x²。当弹簧恢复原状时，这部分能量可被释放。
Plastic deformation: Beyond the elastic limit, the material deforms permanently and does not return to its original length. Hooke’s law no longer applies.
塑性变形：超过弹性极限后，材料发生永久变形且不再恢复原长。胡克定律不再适用。
7. Pressure and Fluid Systems | 压力与流体系统
Pressure: p = F / A, where a force F acts perpendicularly over an area A. Unit: pascal (Pa); 1 Pa = 1 N/m². Often used to design pneumatic and hydraulic systems.
压强：p = F / A，其中力 F 垂直作用在面积 A 上。单位：帕斯卡 (Pa)；1 Pa = 1 N/m²。常用于设计气动和液压系统。
Hydraulic principle: For an incompressible fluid in a closed system, pressure is transmitted equally: p = F₁ / A₁ = F₂ / A₂. A small force on a small piston can produce a large force on a large piston, giving a mechanical advantage.
液压原理：对于密闭系统中不可压缩的流体，压强等值传递：p = F₁ / A₁ = F₂ / A₂。小活塞上的小力可在大活塞上产生大力，从而获得机械增益。
Pressure due to a liquid column: p = ρ g h, where ρ is the density of the liquid, g is gravitational field strength, and h is the depth below the surface. This formula is used for designing dams, tanks, and underwater structures.
液柱压强：p = ρ g h，其中 ρ 为液体密度，g 为重力场强度，h 为液面下的深度。该公式用于设计水坝、储罐和水下结构。
8. Simple Machines and Efficiency | 简单机械与效率
Mechanical advantage (MA): MA = load / effort. It tells us how many times the machine multiplies the input force. An ideal machine without friction would have MA = VR.
机械利益 (MA)：MA = 负载 / 动力。表示机器将输入力放大了多少倍。理想无摩擦机器的 MA = VR。
Velocity ratio (VR): VR = distance moved by effort / distance moved by load. It is a purely geometric quantity determined by the design of levers, pulleys, gears, or inclined planes.
速比 (VR)：VR = 动力移动的距离 / 负载移动的距离。它完全由杠杆、滑轮、齿轮或斜面等设计决定的几何量。
Efficiency of a machine: η = (MA / VR) × 100%. Real machines always have an efficiency less than 100% because some input work is converted into heat through friction.
机械效率：η = (MA / VR) × 100%。真实机器的效率总是低于 100%，因为部分输入功会因摩擦而转化为热量。
Lever principle: For a first‑class lever, effort × effort arm = load × load arm. This is a direct consequence of the principle of moments.
杠杆原理：对于一类杠杆，动力 × 动力臂 = 负载 × 负载臂。这是力矩原理的直接结果。
9. Electricity and Electronic Systems | 电学与电子系统
Ohm’s law: The current I through a resistor is directly proportional to the voltage V across it, provided temperature remains constant: V = I R. R is resistance in ohms (Ω).
欧姆定律：在温度恒定的条件下，通过电阻的电流 I 与其两端的电压 V 成正比：V = I R。R 为电阻，单位欧姆 (Ω)。
Electrical power: P = I V = I² R = V² / R. Power is measured in watts (W). This is used to select appropriate fuses and cables in circuits.
电功率：P = I V = I² R = V² / R。功率的单位为瓦特 (W)。这用于选择电路中合适的保险丝和电缆。
Resistors in series: Rtotal = R₁ + R₂ + R₃ + … The same current flows through each resistor, but the voltage divides.
串联电阻：Rtotal = R₁ + R₂ + R₃ + … 通过每个电阻的电流相同，但电压分配。
Resistors in parallel: 1/Rtotal = 1/R₁ + 1/R₂ + 1/R₃ + … The voltage across each branch is the same, but the current divides.
并联电阻：1/Rtotal = 1/R₁ + 1/R₂ + 1/R₃ + … 各支路电压相同，但电流分配。
Potential divider: Vout = Vin × (R₂ / (R₁ + R₂)), used to obtain a desired fraction of the input voltage, often with a variable resistor or sensor.
分压器：Vout = Vin × (R₂ / (R₁ + R₂))，用于获得输入电压的所需比例，常与可变电阻或传感器配合使用。
10. Thermal Principles | 热学原理
Specific heat capacity: The energy required to raise the temperature of 1 kg of a substance by 1 °C. Q = m c ΔT, where Q is thermal energy (J), m is mass (kg), c is specific heat capacity (J/kg °C), and ΔT is temperature change (°C).
比热容：将 1 kg 物质温度升高 1 °C 所需的能量。Q = m c ΔT，其中 Q 为热能 (J)，m 为质量 (kg)，c 为比热容 (J/kg °C)，ΔT 为温度变化 (°C)。
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