📚 IGCSE OCR Engineering: Quick Reference Formula & Theorem Handbook | IGCSE OCR 工程：公式定理速查手册
This handbook consolidates all the essential formulas and theorems required for the IGCSE OCR Engineering syllabus. It is designed as a rapid revision tool, covering mechanics, materials, electricity, thermodynamics, fluid mechanics, manufacturing, and systems. Each formula is presented in a standard engineering notation with concise English and Chinese explanations, enabling you to recall and apply key principles efficiently during coursework and examinations.
本手册汇总了 IGCSE OCR 工程课程中的所有核心公式与定理，是一份快速复习工具，涵盖力学、材料、电学、热力学、流体力学、制造与系统控制。每个公式均采用标准工程符号，配以简洁的中英文说明，帮助你在课程作业和考试中高效回忆并准确应用关键原理。
1. Mechanics of Motion | 运动力学
The SUVAT equations are used for uniform acceleration in a straight line. Where u = initial velocity, v = final velocity, a = acceleration, s = displacement, t = time.
SUVAT 方程用于直线匀加速运动。其中 u = 初速度，v = 末速度，a = 加速度，s = 位移，t = 时间。
v = u + a t
v = u + a t（末速度 = 初速度 + 加速度 × 时间）
s = u t + ½ a t²
s = u t + ½ a t²（位移 = 初速度 × 时间 + ½ × 加速度 × 时间²）
v² = u² + 2 a s
v² = u² + 2 a s（末速度² = 初速度² + 2 × 加速度 × 位移）
s = ½ (u + v) t
s = ½ (u + v) t（位移 = ½ × (初速度 + 末速度) × 时间）
Average speed = total distance ÷ total time
平均速率 = 总距离 ÷ 总时间
2. Forces and Newton’s Laws | 力与牛顿定律
Newton’s Second Law: resultant force = mass × acceleration; F = m a
牛顿第二定律：合力 = 质量 × 加速度；F = m a
Weight = mass × gravitational field strength; W = m g
重量 = 质量 × 重力场强度；W = m g（g ≈ 9.8 m/s² on Earth）
Moment of a force = force × perpendicular distance from pivot; M = F d
力矩 = 力 × 到支点的垂直距离；M = F d
Principle of moments: sum of clockwise moments = sum of anticlockwise moments (for equilibrium).
力矩原理：顺时针力矩之和 = 逆时针力矩之和（平衡条件）。
Pressure = force ÷ area; p = F ÷ A
压强 = 力 ÷ 面积；p = F ÷ A
3. Energy, Work and Power | 能量、功和功率
Work done = force × distance moved in the direction of the force; W = F d
做功 = 力 × 沿力方向移动的距离；W = F d
Kinetic energy = ½ m v²; Ek = ½ m v²
动能 = ½ m v²；Ek = ½ m v²
Gravitational potential energy = m g h; Ep = m g h
重力势能 = m g h；Ep = m g h（h 为高度变化）
Power = work done ÷ time; P = W ÷ t, also P = F v for constant velocity motion.
功率 = 做功 ÷ 时间；P = W ÷ t。匀速运动时也可写为 P = F v。
Efficiency = (useful energy output ÷ total energy input) × 100%
效率 = (有用能量输出 ÷ 总能量输入) × 100%
4. Materials and Mechanical Properties | 材料与机械性能
Stress (tensile or compressive) = force ÷ cross‑sectional area; σ = F ÷ A
应力（拉伸或压缩）= 力 ÷ 横截面积；σ = F ÷ A
Strain = change in length ÷ original length; ε = ΔL ÷ L0
应变 = 长度变化 ÷ 原始长度；ε = ΔL ÷ L0
Young modulus = stress ÷ strain (within the limit of proportionality); E = σ ÷ ε
杨氏模量 = 应力 ÷ 应变（在比例极限范围内）；E = σ ÷ ε
Stiffness k = force ÷ extension for a spring; k = F ÷ x (Hooke’s Law: F = k x)
弹簧刚度 k = 力 ÷ 伸长量；k = F ÷ x（胡克定律：F = k x）
Spring energy stored = ½ k x²
弹簧储存的弹性势能 = ½ k x²
Factor of safety = ultimate stress ÷ working stress
安全系数 = 极限应力 ÷ 许用应力
5. Electricity and Circuits | 电学与电路
Ohm’s Law: voltage = current × resistance; V = I R
欧姆定律：电压 = 电流 × 电阻；V = I R
Electrical power: P = I V, also P = I² R = V² ÷ R
电功率：P = I V，也等于 I² R 或 V² ÷ R
Energy transferred = power × time; E = P t, often in kilowatt‑hours: kWh = kW × h
电能 = 功率 × 时间；E = P t，常用单位千瓦时：kWh = kW × h
Resistors in series: Rtotal = R₁ + R₂ + R₃ + …
串联电阻：R总 = R₁ + R₂ + R₃ + …
Resistors in parallel: 1/Rtotal = 1/R₁ + 1/R₂ + 1/R₃ + …
并联电阻：1/R总 = 1/R₁ + 1/R₂ + 1/R₃ + …
Charge = current × time; Q = I t
电荷量 = 电流 × 时间；Q = I t
6. Thermodynamics and Heat | 热力学与热量
Specific heat capacity: heat energy = mass × specific heat capacity × temperature change; Q = m c Δθ
比热容：热量 = 质量 × 比热容 × 温度变化；Q = m c Δθ
Specific latent heat: Q = m L (Lf for fusion, Lv for vaporisation)
比潜热：Q = m L（Lf 熔化潜热，Lv 汽化潜热）
Thermal conductivity (Fourier’s law in one dimension): Q/t = k A (Δθ / d), where k is thermal conductivity, A is cross‑sectional area, d is thickness.
热传导（一维傅里叶定律）：Q/t = k A (Δθ / d)，k 为导热系数，A 为截面积，d 为厚度。
Linear expansion: ΔL = α L0 Δθ, where α is coefficient of linear expansion.
线膨胀：ΔL = α L0 Δθ，α 为线膨胀系数。
Ideal gas law (often used in pneumatics): p V = n R T, but simplified as p V = constant for isothermal processes.
理想气体状态方程（气动系统中常用）：p V = n R T，等温过程简化为 p V = 常数。
7. Fluid Mechanics | 流体力学
Pressure in a fluid: p = ρ g h, where ρ is density, g is gravitational field strength, h is depth.
流体压强：p = ρ g h，ρ 为密度，g 为重力场强度，h 为深度。
Pascal’s principle: pressure applied to an enclosed fluid is transmitted undiminished in all directions. In hydraulic systems: F₁ / A₁ = F₂ / A₂
帕斯卡原理：施加于封闭流体的压强会大小不变地向各个方向传递。液压系统中：F₁ / A₁ = F₂ / A₂
Continuity equation for incompressible fluids: A₁ v₁ = A₂ v₂ (volume flow rate is constant).
不可压缩流体的连续性方程：A₁ v₁ = A₂ v₂（体积流量守恒）。
Bernoulli’s principle (simplified): p + ½ ρ v² + ρ g h = constant (for streamline flow).
伯努利原理（简化形式）：p + ½ ρ v² + ρ g h = 常数（适用于流线型流动）。
8. Engineering Mathematics | 工程数学
Trigonometry in right‑angled triangles: sin θ = opposite / hypotenuse; cos θ = adjacent / hypotenuse; tan θ = opposite / adjacent.
直角三角形中的三角函数：sin θ = 对边 / 斜边；cos θ = 邻边 / 斜边；tan θ = 对边 / 邻边。
Sine rule: a / sin A = b / sin B = c / sin C
正弦定理：a / sin A = b / sin B = c / sin C
Cosine rule: a² = b² + c² − 2 b c cos A
余弦定理：a² = b² + c² − 2 b c cos A
Area of a triangle = ½ a b sin C
三角形面积 = ½ a b sin C
Circle: circumference = 2 π r; area = π r²
圆：周长 = 2 π r；面积 = π r²
Volumes: cylinder V = π r² h; sphere V = ⁴⁄₃ π r³; cone V = ⅓ π r² h
体积：圆柱 V = π r² h；球 V = ⁴⁄₃ π r³；圆锥 V = ⅓ π r² h
9. Manufacturing Processes | 制造工艺
Cutting speed (lathe): V = π D N, where D is workpiece diameter, N is spindle speed (rev/min).
切削速度（车床）：V = π D N，D 为工件直径，N 为主轴转速（转/分钟）。
Feed rate: Fr = f N, where f is feed per revolution.
进给速率：Fr = f N，f 为每转进给量。
Material removal rate (turning) = V × f × d, where d is depth of cut.
材料切除率（车削）= V × f × d，d 为切削深度。
Drill speed: N = (1000 V) / (π D), where V is cutting speed in m/min, D in mm.
钻孔主轴转速：N = (1000 V) / (π D)，V 单位为 m/min，D 单位为 mm。
Welding heat input (arc welding) = (V × I × 60) / (travel speed), often expressed in J/mm.
焊接热输入（电弧焊）= (V × I × 60) / (焊接速度)，常用 J/mm 表示。
10. Systems and Control | 系统与控制
Open‑loop gain: G = output / input.
开环增益：G = 输出 / 输入。
Feedback systems: for a negative feedback loop, overall gain = G / (1 + G H), where H is feedback factor.
反馈系统：负反馈回路的总增益 = G / (1 + G H)，H 为反馈因子。
Gear ratio = number of teeth on driven gear / number of teeth on driver gear. Speed ratio = driver speed / driven speed = reciprocal of gear ratio.
齿轮比 = 从动齿轮齿数 / 主动齿轮齿数。速比 = 主动轮转速 / 从动轮转速 = 齿轮比的倒数。
Pulley systems: velocity ratio = number of rope sections supporting the load (for block and tackle).
滑轮系统：速比 = 支持负载的绳索股数（滑轮组）。
Belt drive: speed ratio = driver pulley diameter / driven pulley diameter.
带传动：速比 = 主动轮直径 / 从动轮直径。
11. Statics and Structures | 静力学与结构
Equilibrium conditions: sum of horizontal forces = 0, sum of vertical forces = 0, sum of moments about any point = 0.
平衡条件：水平方向合力 = 0，竖直方向合力 = 0，对任一点的合力矩 = 0。
Newton’s third law: action and reaction are equal in magnitude and opposite in direction.
牛顿第三定律：作用力与反作用力大小相等、方向相反。
Centre of gravity: the point where the entire weight of an object appears to act.
重心：物体整个重量看似集中的作用点。
Stability: a body is stable if its centre of gravity is low and its base is wide; tipping occurs when the line of action of weight falls outside the base.
稳定性：重心较低、基底较宽的物体更稳定；当重力作用线超出基底时物体会倾倒。
Stress‑strain graphs: yield stress, ultimate tensile stress, breaking stress are key values taken from the curve.
应力‑应变图：屈服应力、极限抗拉应力、断裂应力是曲线上关键取值。
12. Mechanical Advantage and Efficiency | 机械效益与效率
Mechanical advantage (MA) = load / effort.
机械效益 (MA) = 负载 / 动力。
Velocity ratio (VR) = distance moved by effort / distance moved by load (in the same time).
速比 (VR) = 动力移动的距离 / 负载移动的距离（在同一时间内）。
Efficiency = (MA / VR) × 100%, also = (useful work output / work input) × 100%.
效率 = (MA / VR) × 100%，也等于 (有用功输出 / 功输入) × 100%。
Inclined plane: VR = length of slope / height. Screw jack: VR = 2 π R / pitch, where R is lever arm length.
斜面：VR = 斜面长 / 高度。螺旋千斤顶：VR = 2 π R / 螺距，R 为力臂长度。
Gear trains: overall VR = product of teeth on driven gears / product of teeth on driver gears (for compound gears).
齿轮系：总 VR = 从动齿轮齿数乘积 / 主动齿轮齿数乘积（复合轮系）。
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