📚 IGCSE CCEA Engineering: Quick Revision Handbook of Formulas and Theorems | IGCSE CCEA 工程：公式定理速查手册
This quick revision handbook covers the essential formulas and theorems required for the IGCSE CCEA Engineering syllabus. Each section provides key equations with paired English and Chinese explanations, allowing you to reinforce understanding and memorize facts efficiently. From mechanics to electronics, this guide serves as a last‑minute refresher and a handy reference throughout your course.
本速查手册涵盖 IGCSE CCEA 工程大纲必需的核心公式与定理。每个小节均提供关键方程，并配有中英文解释，帮助你巩固理解、高效记忆。从力学到电子学，无论是考前冲刺还是日常复习，这份指南都可作为便捷的参考工具。
1. Mechanics and Kinematics | 力学与运动学
v = s / t
Average speed equals total distance travelled divided by total time taken.
平均速度等于总位移除以总时间。
a = (v – u) / t
Acceleration is the rate of change of velocity; here u is initial velocity, v is final velocity and t is time.
加速度是速度的变化率；式中 u 为初速度，v 为末速度，t 为时间。
v = u + a t
First equation of motion for uniform acceleration in a straight line.
匀加速直线运动的第一个运动方程。
s = u t + ½ a t²
Displacement s after time t under uniform acceleration.
匀加速运动下经过时间 t 的位移 s。
v² = u² + 2 a s
Relates final velocity, initial velocity, acceleration and displacement without time.
不含时间变量，联系末速度、初速度、加速度和位移。
s = ½ (u + v) t
Displacement equals average velocity multiplied by time.
位移等于平均速度乘以时间。
2. Forces and Moments | 力与力矩
F = m × a
Newton’s second law: the net force acting on a body equals its mass times its acceleration.
牛顿第二定律：作用在物体上的合外力等于质量乘以加速度。
W = m × g
Weight is the gravitational force on a mass; g is the gravitational field strength (≈ 9.8 N/kg on Earth).
重量是质量受到的重力；g 为重力场强度（地球表面约 9.8 N/kg）。
Moment = F × d
The moment of a force about a pivot is the force multiplied by the perpendicular distance from the pivot.
力对支点的力矩等于力乘以力臂（支点到力作用线的垂直距离）。
Σ clockwise moments = Σ anticlockwise moments
Principle of moments: for a system in rotational equilibrium, total clockwise moments equal total anticlockwise moments.
力矩原理：系统转动平衡时，顺时针力矩之和等于逆时针力矩之和。
F₁ d₁ = F₂ d₂
Lever principle derived from moment balance; commonly used for simple machines.
由力矩平衡导出的杠杆原理，常用于简单机械。
Action = – Reaction
Newton’s third law: forces between two interacting bodies are equal in magnitude and opposite in direction.
牛顿第三定律：相互作用的两个物体之间的作用力和反作用力大小相等、方向相反。
3. Material Properties | 材料力学
σ = F / A
Stress (σ) is the applied force per unit cross‑sectional area. Units: N/m² or Pa.
应力（σ）是单位截面积上所受的力，单位为 N/m² 或 Pa。
ε = ΔL / L₀
Strain (ε) is the extension per unit original length; dimensionless.
应变（ε）为单位原长上的伸长量，无量纲。
E = σ / ε
Young’s modulus (E) measures stiffness of a material; valid within the elastic limit.
杨氏模量（E）衡量材料的刚度，仅在弹性限度内成立。
F = k × x
Hooke’s law: the extension (or compression) x of a spring is directly proportional to the applied force F, within the elastic limit.
胡克定律：在弹性限度内，弹簧的伸长（或压缩）量 x 与所受外力 F 成正比。
k_total = k₁ + k₂ + … (parallel)
Equivalent spring constant for springs in parallel: sum of individual constants.
并联弹簧的等效劲度系数等于各弹簧劲度系数之和。
1/k_total = 1/k₁ + 1/k₂ + … (series)
For springs connected end‑to‑end, the reciprocal of the total spring constant equals the sum of the reciprocals.
串联弹簧等效劲度系数的倒数等于各弹簧劲度系数倒数之和。
4. Work, Energy and Power | 功、能与功率
W = F × d
Work done by a constant force equals force times distance moved in the direction of the force.
恒力做功等于力乘以沿力方向移动的距离。
Eₖ = ½ m v²
Kinetic energy of a body of mass m moving at speed v.
质量为 m、速度为 v 的物体的动能。
Eₚ = m g h
Gravitational potential energy gained by raising a mass m through height h.
质量为 m 的物体提升高度 h 所增加的重力势能。
P = W / t = F × v
Power is the rate of doing work; also equals force times velocity for constant motion.
功率是做功的速率；恒力匀速运动时也等于力乘以速度。
η = (useful output / total input) × 100%
Efficiency is the ratio of useful energy or power output to total input, expressed as a percentage.
效率是有用输出能量或功率与总输入之比，以百分数表示。
5. Electrical Fundamentals | 电学基础
V = I × R
Ohm’s law: voltage across a resistor equals current times resistance.
欧姆定律：电阻两端的电压等于电流乘以电阻。
P = I × V = I² R = V² / R
Electrical power dissipated or converted; three equivalent forms.
电功率的三种等效表达式。
R_total = R₁ + R₂ + R₃ + … (series)
Resistors in series: total resistance is the sum of all individual resistances.
串联电阻的总电阻等于各电阻之和。
1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + … (parallel)
For resistors in parallel, the reciprocal of the total resistance equals the sum of the reciprocals of each resistor.
并联电阻总电阻的倒数等于各支路电阻倒数之和。
Q = I × t
Electric charge transferred when a constant current flows for time t.
恒定电流 I 在时间 t 内传递的电荷量。
E = P × t = I × V × t
Electrical energy consumed or transferred; measured in joules or kilowatt‑hours.
消耗或传递的电能，单位为焦耳或千瓦时。
6. Electronics and Voltage Dividers | 电子学与分压器
V_out = V_in × [R₂ / (R₁ + R₂)]
Voltage divider rule for two resistors in series: output voltage across R₂.
两个电阻串联的分压公式：R₂ 两端的输出电压。
Gain = V_out / V_in
Voltage gain of an amplifier or divider stage.
放大器或分压级的电压增益。
I_base, I_collector, I_emitter (transistor)
For a bipolar junction transistor in active mode: I_emitter ≈ I_collector + I_base, and current gain β = I_c / I_b.
双极型晶体管在放大区：发射极电流约等于集电极电流与基极电流之和，电流增益 β = I_c / I_b。
Note: Logic gate symbols and truth tables are also essential for digital electronics sections, but the core formula here is the voltage divider.
注：数字电子学部分还需掌握逻辑门符号与真值表，但此处核心公式为分压器。
7. Thermodynamics | 热力学
ΔL = α L₀ ΔT
Linear thermal expansion: change in length of a solid due to temperature change ΔT; α is the coefficient of linear expansion.
线热膨胀：固体因温度变化 ΔT 而产生的长度变化，α 为线膨胀系数。
Q / t = k A ΔT / d
Rate of conductive heat transfer through a material of thickness d, cross‑sectional area A and thermal conductivity k.
热传导速率：厚度 d、截面积 A、导热系数 k 的材料，单位时间传递的热量。
Q = m c Δθ
Heat energy required to change the temperature of a mass m by Δθ; c is specific heat capacity.
质量为 m 的物体温度变化 Δθ 所需的热量，c 为比热容。
η_thermal = W_out / Q_in
Thermal efficiency of a heat engine is the ratio of net work output to heat input.
热机的热效率等于输出净功与输入热量之比。
8. Fluid Mechanics | 流体力学
p = F / A
Pressure is force per unit area acting perpendicular to a surface.
压强是垂直作用在单位面积上的力。
p = ρ g h
Hydrostatic pressure at a depth h in a fluid of density ρ, under gravity g; assumes fluid is incompressible.
液体内部深度 h 处的静压强，ρ 为液体密度，适用于不可压缩流体。
F_b = ρ_fluid × g × V_displaced
Archimedes’ principle: buoyant force equals the weight of fluid displaced by the immersed object.
阿基米德原理：浮力等于物体排开液体的重量。
A₁ v₁ = A₂ v₂ (continuity)
For incompressible flow, the volume flow rate is constant; cross‑sectional area A times velocity v remains unchanged along a streamline.
对于不可压缩流体，连续性方程：截面积 A 与流速 v 的乘积沿流线保持不变。
9. Engineering Mathematics | 工程数学
c = √(a² + b²) (Pythagoras)
Right‑angled triangle: the square of the hypotenuse equals the sum of squares of the other two sides.
直角三角形中，斜边的平方等于两直角边的平方和。
sin θ = opposite / hypotenuse
Trigonometric ratio for sine of an angle in a right‑angled triangle.
直角三角形中角 θ 的正弦值等于对边比斜边。
cos θ = adjacent / hypotenuse
Cosine ratio: adjacent side divided by hypotenuse.
余弦值等于邻边比斜边。
tan θ = opposite / adjacent
Tangent ratio: opposite side divided by adjacent side.
正切值等于对边比邻边。
A_rectangle = l × w, A_triangle = ½ b h, A_circle = π r²
Common area formulas needed for engineering calculations.
工程计算中常用的面积公式。
V_prism = A_base × h, V_cylinder = π r² h
Volumes of prismatic and cylindrical bodies.
棱柱和圆柱的体积公式。
10. Structural Analysis | 结构分析
Σ F_horizontal = 0, Σ F_vertical = 0
Conditions for static equilibrium: the sum of all horizontal forces and the sum of all vertical forces must each be zero.
静力平衡条件：所有水平方向的力之和为零，所有竖直方向的力之和为零。
Σ M_any point = 0
For a body in rotational equilibrium, the sum of clockwise moments about any point equals the sum of anticlockwise moments; commonly written as Σ M = 0.
转动平衡条件：对任一点的合力矩为零，即顺时针力矩之和等于逆时针力矩之和。
Reaction forces from equilibrium equations
In simply supported beams, support reactions are found by applying Σ F_v = 0 and Σ M = 0.
简支梁的支座反力可通过 Σ F_v = 0 和 Σ M = 0 求解。
Stress in simple bending: σ = M y / I
Bending stress formula used for beam analysis, where M is bending moment, y distance from neutral axis, and I second moment of area.
弯曲应力公式，用于梁分析：M 为弯矩，y 为
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