A-Level OCR Engineering: Formula & Theorem Quick Reference | A-Level OCR 工程：公式定理速查手册

📚 A-Level OCR Engineering: Formula & Theorem Quick Reference | A-Level OCR 工程：公式定理速查手册

Engineering at A-Level demands a deep familiarity with core formulae and theorems that span mechanics, materials, thermodynamics, fluids and electrical systems. This quick reference handbook collects the essential equations you will need throughout the OCR Engineering course, from resolving forces to analysing circuits and calculating efficiency. Use it as a revision checklist, a problem-solving companion or a last-minute refresher before exams.

A-Level 工程课程要求学生熟练掌握跨越力学、材料、热力学、流体和电气系统的核心公式与定理。本速查手册整理了 OCR 工程课程中你需要的关键方程，从力的分解到电路分析和效率计算。可以将其用作复习清单、解题参考或考前快速回顾。

1. Mathematical Toolkit & Units | 数学工具与单位

Make sure you are fluent with basic trigonometry and geometry, as they appear in almost every engineering problem.

确保你熟练掌握基础三角学与几何，因为它们几乎出现在每一个工程问题中。

Pythagoras’ theorem: a² + b² = c²

勾股定理：a² + b² = c²

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 – 2bc·cos A

余弦定理：a² = b² + c² – 2bc·cos A

Area of a circle: A = π r²; Circumference: C = 2π r

圆的面积：A = π r²；周长：C = 2π r

Area of a triangle (when height is known): A = ½ b h

三角形面积（已知高度）：A = ½ b h

Volume of a cylinder: V = π r² h; Sphere: V = ⁴/₃ π r³

圆柱体积：V = π r² h；球体积：V = ⁴/₃ π r³

Common unit conversions: 1 N = 1 kg·m/s², 1 Pa = 1 N/m², 1 J = 1 N·m, 1 W = 1 J/s

常用单位换算：1 N = 1 kg·m/s²，1 Pa = 1 N/m²，1 J = 1 N·m，1 W = 1 J/s

sin θ = opposite / hypotenuse cos θ = adjacent / hypotenuse tan θ = opposite / adjacent

sin θ = 对边 / 斜边 cos θ = 邻边 / 斜边 tan θ = 对边 / 邻边

2. Statics: Force, Moments & Equilibrium | 静力学：力、力矩与平衡

When a body is in static equilibrium, the resultant force and resultant moment both must be zero.

当物体处于静力平衡时，合外力与合力矩都必须为零。

Sum of horizontal forces: ΣFₓ = 0

水平方向合力：ΣFₓ = 0

Sum of vertical forces: ΣFᵧ = 0

垂直方向合力：ΣFᵧ = 0

Sum of moments about any point: ΣM = 0

对任一点力矩之和：ΣM = 0

Moment of a force: M = F × d (where d is the perpendicular distance from the pivot to the line of action)

力矩：M = F × d （d 为转动轴到力作用线的垂直距离）

Principle of moments: for equilibrium, total clockwise moments = total anticlockwise moments

力矩原理：平衡时，总顺时针力矩 = 总逆时针力矩

A couple produces pure rotation: moment of a couple = F × d (d = perpendicular distance between the forces)

力偶产生纯转动：力偶矩 = F × d（d = 两力之间的垂直距离）

ΣF = 0 and ΣM = 0

ΣF = 0 且 ΣM = 0

3. Linear Kinematics | 直线运动学

These four SUVAT equations describe uniformly accelerated motion along a straight line. Identify the unknowns and select the right equation.

以下四个 SUVAT 方程描述匀加速直线运动。识别未知量并选择正确的方程。

v = u + a t

s = u t + ½ a t²

v² = u² + 2 a s

s = ½ (u + v) t

Variables: s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time

变量：s = 位移，u = 初速度，v = 末速度，a = 加速度，t = 时间

For projectile motion, separate the motion into horizontal (constant velocity) and vertical (constant acceleration g = 9.81 m/s² downwards) components.

对于抛体运动，将运动分解为水平方向（匀速）和垂直方向（恒定加速度 g = 9.81 m/s² 向下）的分量。

v² = u² + 2 a s

4. Dynamics & Newton’s Laws of Motion | 动力学与牛顿运动定律

Newton’s second law is the foundation for linking force, mass and acceleration. Always draw a free-body diagram to resolve forces.

牛顿第二定律是联系力、质量和加速度的基础。始终画出受力分析图来分解力。

Newton II: F = m a

牛顿第二定律：F = m a

Weight: W = m g

重量：W = m g

Friction (static limiting): F_f ≤ μₛ Fₙ (kinetic): F_f = μₖ Fₙ

摩擦力（最大静摩擦）：F_f ≤ μₛ Fₙ （动摩擦）：F_f = μₖ Fₙ

Momentum: p = m v

动量：p = m v

Impulse = change in momentum: F Δt = Δp = m v – m u

冲量 = 动量的变化：F Δt = Δp = m v – m u

Conservation of momentum (no external force): m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂

动量守恒（无外力）：m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂

F = m a and F Δt = Δp

F = m a 与 F Δt = Δp

5. Work, Energy & Power | 功、能量与功率

Energy conservation helps you analyse systems from first principles. Watch out for elastic potential energy in springs and strained members.

能量守恒帮助你从基本原理分析系统。注意弹簧和受载构件中的弹性势能。

Work done by a constant force: W = F d cos θ

恒力做功：W = F d cos θ

Kinetic energy: Eₖ = ½ m v²

动能：Eₖ = ½ m v²

Gravitational potential energy: Eₚ = m g h

重力势能：Eₚ = m g h

Elastic potential energy (spring): Eₑ = ½ k x²

弹性势能（弹簧）：Eₑ = ½ k x²

Power: P = W / t = F v (for constant force and velocity in the same direction)

功率：P = W / t = F v（当力和速度方向相同且恒定时）

Efficiency: η = (useful energy output / total energy input) × 100%

效率：η =（有用能量输出 / 总能量输入）× 100%

Work–energy principle: net work done = change in kinetic energy

功能原理：合力所做的功 = 动能的变化量

W = F d cos θ Eₖ = ½ m v² P = F v

6. Stress, Strain & Young’s Modulus | 应力、应变与杨氏模量

Understanding material behaviour is crucial for design. These definitions apply to uniaxial tension and compression within the elastic limit.

理解材料行为对设计至关重要。以下定义适用于弹性极限内的单轴拉伸和压缩。

Engineering stress (normal stress): σ = F / A

工程应力（正应力）：σ = F / A

Engineering strain: ε = ΔL / L₀

工程应变：ε = ΔL / L₀

Young’s modulus (stiffness): E = σ / ε (only valid in the linear elastic region)

杨氏模量（刚度）：E = σ / ε （仅在线弹性范围内有效）

Hooke’s law for a spring: F = k x

弹簧胡克定律：F = k x

Poisson’s ratio: ν = – (lateral strain / axial strain) = – ε_lateral / ε_axial

泊松比：ν = –（横向应变 / 轴向应变）= – ε_lateral / ε_axial

Shear stress: τ = F / A (shearing force over area parallel to the force)

剪切应力：τ = F / A（剪切力与平行于力的面积之比）

Factor of safety = ultimate stress / allowable working stress

安全系数 = 极限应力 / 许用工作应力

E = σ / ε σ = F / A ε = ΔL / L₀

7. Fluid Mechanics | 流体力学

Fluid principles appear in hydraulic systems, aerodynamics and pipe flow. The Bernoulli equation is central to energy conservation in a streamline flow.

流体原理出现在液压系统、空气动力学和管道流动中。伯努利方程是流线流动中能量守恒的核心。

Pressure: p = F / A

压强：p = F / A

Hydrostatic pressure: Δp = ρ g h

静水压强：Δp = ρ g h

Pascal’s principle: pressure applied to an enclosed fluid is transmitted undiminished, giving F₁ / A₁ = F₂ / A₂ in a hydraulic press.

帕斯卡原理：施加于密闭流体的压强将大小不变地传递，在液压机中有 F₁ / A₁ = F₂ / A₂。

Continuity equation (incompressible flow): A₁ v₁ = A₂ v₂ → Q = A v = constant

连续性方程（不可压缩流动）：A₁ v₁ = A₂ v₂ → Q = A v = 常数

Bernoulli’s equation: p + ½ ρ v² + ρ g h = constant

伯努利方程：p + ½ ρ v² + ρ g h = 常数

Here p = static pressure, ½ ρ v² = dynamic pressure, ρ g h = hydrostatic pressure.

其中 p 为静压，½ ρ v² 为动压，ρ g h 为静水压强。

p + ½ ρ v² + ρ g h = constant

p + ½ ρ v² + ρ g h = 常数

8. Thermal Physics & Thermodynamics | 热物理与热力学

Thermodynamics links heat, work and internal energy. Engineers use these laws to analyse engines, heat pumps and thermal processes.

热力学将热量、功和内能联系起来。工程师运用这些定律分析发动机、热泵和热过程。

Heat capacity and specific heat capacity: Q = m c Δθ

热容与比热容：Q = m c Δθ

Latent heat: Q = m L (L = specific latent heat of fusion or vaporisation)

潜热：Q = m L（L 为熔化或汽化比潜热）

First law of thermodynamics: ΔU = Q – W (U = internal energy, Q = heat added to the system, W = work done BY the system)

热力学第一定律：ΔU = Q – W （U 为内能，Q 为加入系统的热量，W 为系统对外做的功）

Ideal gas equation: p V = n R T (R = 8.31 J/mol·K)

理想气体状态方程：p V = n R T （R = 8.31 J/mol·K）

Thermal efficiency of a heat engine: η = W / Qₕ = (Qₕ – Q_c) / Qₕ = 1 – (Q_c / Qₕ)

热机热效率：η = W / Qₕ = (Qₕ – Q_c) / Qₕ = 1 – (Q_c / Qₕ)

Maximum possible Carnot efficiency: η_Carnot = 1 – T_c / Tₕ (temperatures in kelvin)

最大卡诺效率：η_Carnot = 1 – T_c / Tₕ （温度用开尔文）

ΔU = Q – W and pV = nRT

ΔU = Q – W 与 pV = nRT

9. Electrical Principles | 电学原理

Electrical circuits obey precise laws that control current, voltage, resistance and power. These appear in control systems, actuators and sensors.

电路遵循精确的定律来控制电流、电压、电阻和功率。这些出现在控制系统、执行器和传感器中。

Ohm’s law: V = I R

欧姆定律：V = I R

Resistance in series: R_total = R₁ + R₂ + R₃ + …

串联电阻：R_total = R₁ + R₂ + R₃ + …

Resistance in parallel: 1/R_total = 1/R₁ + 1/R₂ + …

并联电阻：1/R_total = 1/R₁ + 1/R₂ + …

Electrical power: P = I V = I² R = V² / R

电功率：P = I V = I² R = V² / R

Energy: E = P t = I V t

电能：E = P t = I V t

Kirchhoff’s current law (KCL): Σ I_in = Σ I_out at any junction

基尔霍夫电流定律 (KCL)：对任一节点，Σ I_in = Σ I_out

Kirchhoff’s voltage law (KVL): Σ V = 0 around any closed loop

基尔霍夫电压定律 (KVL)：沿任一闭合回路，Σ V = 0

For a potential divider: V_out = V_in × (R₂ / (R₁ + R₂))

对于电位分压器：V_out = V_in × (R₂ / (R₁ + R₂))

V = I R P = I V R_series = ΣR

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A-Level OCR Engineering: Formula & Theorem Quick Reference | A-Level OCR 工程：公式定理速查手册