IGCSE OCR Engineering: Quick Reference Formula & Theorem Handbook | IGCSE OCR 工程：公式定理速查手册

📚 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

re 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

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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

ple 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

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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; E_k = ½ m v²

动能 = ½ m v²；E_k = ½ m v²

Gravitational potential energy = m g h; E_p = m g h

重力势能 = m g h；E_p = 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%

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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

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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: R_total = R₁ + R₂ + R₃ + …

串联电阻：R_总 = R₁ + R₂ + R₃ + …

Resistors in parallel: 1/R_total = 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

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6. Thermodynamics and Heat | 热力学与热量

eat 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 (L_f for fusion, L_v for vaporisation)

比潜热：Q = m L（L_f 熔化潜热，L_v 汽化潜热）

r’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。

ed 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。

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7. Fluid Mechanics | 流体力学

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。

uid 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

ntinuity 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）。

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8. Engineering Mathematics | 工程数学

Trigonometry in right‑ 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

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9. Manufacturing Processes | 制造工艺

ing 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: F_r = f N, where f is feed per revolution.

进给速率：F_r = 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。

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10. Systems and Control | 系统与控制

Open‑loop gain: G = output / input.

开环增益：G = 输出 / 输入。

s: 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。

eeth 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。

tems: 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。

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11. Statics and Structures | 静力学与结构

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。

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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。

elocity 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%。

R = 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。

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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