📚 IGCSE Cambridge Engineering: Quick Reference Formula & Theorem Handbook | IGCSE 剑桥工程：公式定理速查手册
This bilingual quick reference handbook covers the essential formulas, theorems and key relationships required for the Cambridge IGCSE Engineering syllabus. Each entry pairs a concise English statement with its Chinese translation, helping you to memorise and apply engineering principles confidently. Use it as a rapid revision tool alongside regular practice.
这本中英双语速查手册涵盖了剑桥 IGCSE 工程课程所需的核心公式、定理和重要关系。每个条目都以简明的英文陈述与中文译文配对，帮助您熟练掌握并自信地运用工程原理。请将其用作与日常练习相结合的快速复习工具。
1. Forces and Newton's Second Law | 力与牛顿第二定律
A resultant force acting on an object causes it to accelerate. The relationship is given by F = ma, where F is the net force in newtons (N), m is mass in kilograms (kg), and a is acceleration in metres per second squared (m/s²).
作用在物体上的合力会导致其加速。关系式为 F = ma，其中 F 为净力（牛顿 N），m 为质量（千克 kg），a 为加速度（米每二次方秒 m/s²）。
The weight of an object is the force due to gravity: W = mg. Here g is the gravitational field strength, approximately 9.8 N/kg on Earth.
物体的重量是由重力产生的力：W = mg。其中 g 为重力场强度，地球表面约为 9.8 N/kg。
For an object in equilibrium, the sum of all forces in any direction is zero. This principle is used to analyse static structures and mechanical systems.
对于处于平衡状态的物体，任意方向上所有力的总和为零。这一原理用于分析静定结构和机械系统。
2. Moments and Equilibrium | 力矩与平衡
The moment of a force about a pivot is calculated as M = F × d, where M is the moment in newton-metres (N m), F is the force (N), and d is the perpendicular distance from the pivot to the line of action of the force (m).
关于支点的力矩计算公式为 M = F × d，其中 M 为力矩（牛顿米 N m），F 为力（N），d 为支点到力作用线的垂直距离（m）。
For a lever in rotational equilibrium, the total clockwise moment equals the total anticlockwise moment: ΣM꜀ₗₒ꜀ₖ = ΣMₐₙₜᵢ. This principle underpins the operation of levers, gears and many machines.
对于处于旋转平衡的杠杆，总顺时针力矩等于总逆时针力矩：ΣM꜀ₗₒ꜀ₖ = ΣMₐₙₜᵢ。这一原理是杠杆、齿轮和许多机械运作的基础。
A couple consists of two equal and opposite parallel forces that create a turning effect. Its moment is M = F × d, where d is the perpendicular distance between the forces.
力偶由两个大小相等、方向相反的平行力组成，产生转动效果。其力矩为 M = F × d，其中 d 为两力之间的垂直距离。
3. Pressure and Hydraulics | 压强与液压原理
Pressure is defined as force per unit area: P = F / A, where P is pressure in pascals (Pa), F is force (N), and A is area (m²). A smaller area produces a larger pressure for the same force.
压强定义为每单位面积所受的力：P = F / A，其中 P 为压强（帕斯卡 Pa），F 为力（N），A 为面积（m²）。相同力下面积越小压强越大。
In a hydraulic system, pressure is transmitted equally throughout an enclosed fluid. This gives the relationship F₁/A₁ = F₂/A₂, enabling a small input force to lift a larger load.
在液压系统中，压强在封闭液体内等值传递。由此得出关系式 F₁/A₁ = F₂/A₂，使小的输入力能抬起较大的负载。
The hydraulic mechanical advantage can be expressed as MA = A₂/A₁ = F₂/F₁, where A₂ is the larger piston area.
液压机械优势可表示为 MA = A₂/A₁ = F₂/F₁，其中 A₂ 为较da活塞的面积。
4. Work and Energy | 功与能
Work is done when a force moves an object through a distance: W = Fd, where W is work in joules (J), F is the force (N) in the direction of motion, and d is distance (m).
力使物体移动一段距离即做功：W = Fd，其中 W 为功（焦耳 J），F 为运动方向上的力（N），d 为距离（m）。
Gravitational potential energy gained by an object lifted to a height h is Eₚ = mgh. Kinetic energy of a moving object is Eₖ = ½ mv².
物体被举升至高度 h 所获得的重力势能为 Eₚ = mgh。运动物体的动能为 Eₖ = ½ mv²。
The principle of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. In a mechanical system, total energy input equals work output plus energy losses.
能量守恒原理表明，能量不能被创造或消灭，只能从一种形式转化为另一种形式。在机械系统中，总输入能量等于输出功加上能量损耗。
5. Power | 功率
Power is the rate of doing work or transferring energy: P = W/t or P = ΔE/t, where P is power in watts (W), W is work (J), t is time (s), and ΔE is energy transferred.
功率是做功或传递能量的速率：P = W/t 或 P = ΔE/t，其中 P 为功率（瓦特 W），W 为功（J），t 为时间（s），ΔE 为传递的能量。
For an object moving at constant velocity against a resistive force F, the power is also given by P = Fv. This is useful when calculating the power of engines and motors.
对于物体以恒定速度克服阻力 F 运动时，功率也可表示为 P = Fv。这在计算发动机和电动机的功率时很有用。
joule per second. Larger machines are often rated in kilowatts (kW), where 1 kW = 1000 W.
一瓦特等于每秒一焦耳。较大的机器通常以千瓦（kW）为单位，1 kW = 1000 W。
6. Efficiency | 效率
Efficiency measures how well a system converts input energy into useful output energy: η = (useful output energy / total input energy) × 100% or η = (useful power output / total power input) × 100%.
效率衡量系统将输入能量转化为有用输出能量的能力：η = （有用输出能量 / 总输入能量）× 100% 或 η = （有用输出功率 / 总输入功率）× 100%。
No real machine is 100% efficient because some energy is always lost as heat, sound or due to friction. Efficiency is always less than 1 (or less than 100%).
没有一台真实机器的效率能达到 100%，因为总有一些能量以热量、声音或摩擦的形式损耗。效率始终小于 1（或小于 100%）。
In simple terms, η = work output / work input for a mechanical system. This ratio guides engineers in designing machines that minimise energy waste.
简而言之，对于机械系统 η = 输出功 / 输入功。这一比值指导工程师设计出尽量减少能量浪费的机器。
7. Ohm's Law and Electrical Power | 欧姆定律与电功率
Ohm's Law states that the current through a conductor is directly proportional to the voltage across it, provided temperature remains constant: V = IR, where V is voltage (V), I is current (A), and R is resistance (Ω).
欧姆定律指出，在温度保持不变的条件下，通过导体的电流与其两端的电压成正比：V = IR，其中 V 为电压（伏特 V），I 为电流（安培 A），R 为电阻（欧姆 Ω）。
Electrical power can be calculated using three equivalent forms: P = IV, P = I²R, and P = V²/R. These are used depending on which quantities are known.
电功率可以通过三种等价形式计算：P = IV、P = I²R 和 P = V²/R。根据已知量选择使用相应的公式。
In electric circuits, energy consumed over time is E = Pt = VIt, measured in joules. In practice, domestic energy is often measured in kilowatt-hours (kWh).
在电路中，一段时间内消耗的能量为 E = Pt = VIt，单位为焦耳。实际生活中，家庭用电常用千瓦时（kWh）计量。
8. Resistors in Series and Parallel | 电阻的串联与并联
For resistors connected in series, the total resistance is the sum of individual resistances: Rₜ = R₁ + R₂ + R₃ + .... The current is the same through each resistor.
对于串联连接的电阻，总电阻等于各电阻之和：Rₜ = R₁ + R₂ + R₃ + …。通过每个电阻的电流相同。
For resistors connected in parallel, the reciprocal of the total resistance equals the sum of the reciprocals of individual resistances: 1/Rₜ = 1/R₁ + 1/R₂ + .... The total resistance is always less than the smallest individual resistor.
对于并联连接的电阻，总电阻的倒数等于各电阻倒数之和：1/Rₜ = 1/R₁ + 1/R₂ + …。总电阻始终小于最小的单个电阻。
In a parallel circuit, the voltage across each branch is the same. The total current is the sum of the currents in each branch: Iₜ = I₁ + I₂ + ....
在并联电路中，各支路两端的电压相同。总电流等于各支路电流之和：Iₜ = I₁ + I₂ + …。
9. Stress, Strain and Young's Modulus | 应力、应变与杨氏模量
Tensile stress is the force applied per unit cross-sectional area: σ = F / A, where σ is stress in pascals (Pa), F is force (N), and A is original cross-sectional area (m²).
拉应力是施加在每单位横截面积上的力：σ = F / A，其中 σ 为应力（帕斯卡 Pa），F 为力（N），A 为原始横截面积（m²）。
Tensile strain is the extension per unit original length: ε = ΔL / L₀, where ε has no units, ΔL is change in length, and L₀ is the original length.
拉应变是每单位原始长度的伸长量：ε = ΔL / L₀，其中 ε 无量纲，ΔL 为长度变化，L₀ 为原始长度。
Young's modulus describes the stiffness of a material within its elastic limit: E = σ / ε. It is measured in pascals (Pa) and is constant for a given material.
杨氏模量描述材料在弹性限度内的刚度：E = σ / ε。其单位为帕斯卡（Pa），对于给定材料为常数。
10. Density and Hooke's Law | 密度与胡克定律
Density is mass per unit volume: ρ = m / V, where ρ is density (kg/m³), m is mass (kg), and V is volume (m³). Pure water has a density of 1000 kg/m³ at 4 °C.
密度是每单位体积的质量：ρ = m / V，其中 ρ 为密度（kg/m³），m 为质量（kg），V 为体积（m³）。纯水在 4 °C 时的密度为 1000 kg/m³。
Hooke's Law states that the extension of a spring is directly proportional to the applied force, up to its elastic limit: F
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