📚 Formula & Theorems Quick Reference Handbook | 工程公式定理速查手册
This handbook compiles the essential formulas, theorems, and principles required for the Year 11 AQA Engineering course. It is designed as a quick revision tool, covering mechanics, electricity, materials, fluid mechanics, thermodynamics, and mathematical foundations. Each entry is presented with standard notation and SI units to support problem-solving and exam preparation.
本手册汇编了 AQA 工程 Year 11 课程所需的核心公式、定理和原理。作为快速复习工具，它涵盖力学、电学、材料、流体力学、热力学和数学基础。每个条目均使用标准符号和国际单位制，便于解题和备考。
1. Forces and Equilibrium | 力与平衡
Newton's Second Law states that the resultant force on an object equals its mass multiplied by its acceleration. In equilibrium, the vector sum of all forces acting on a body is zero, meaning no net force and no acceleration.
牛顿第二定律指出，物体所受合力等于其质量乘以加速度。在平衡状态下，作用在物体上的所有力的矢量和为零，即无净力，无加速度。
- Resultant force: F = m × a (N, kg, m/s²)
- Equilibrium condition: ΣF = 0 (translational equilibrium)
- Weight: W = m × g, where g ≈ 9.81 m/s²
- Friction (limiting): Ff ≤ μ × R, where μ is coefficient of friction, R is normal reaction
F = m × a
ΣFₓ = 0, ΣFᵧ = 0
In static equilibrium problems, resolve forces into components and apply the equilibrium equations separately to each direction.
在静力平衡问题中，将力分解为分量，并对每个方向分别应用平衡方程。
2. Moments and Torque | 力矩与扭矩
A moment is the turning effect of a force about a pivot. It is calculated as the product of the force and the perpendicular distance from the pivot to the line of action of the force. The principle of moments states that for a system in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any point.
力矩是力绕支点的转动效应，等于力与支点到力作用线的垂直距离的乘积。力矩原理指出，对于转动平衡的系统，任一点处顺时针力矩之和等于逆时针力矩之和。
M = F × d
ion: torque T = F × d for a couple with two equal opposite forces separated by distance d.
其中 M 为力矩（牛·米），F 为力（牛），d 为垂直距离（米）。力偶产生纯转动：力偶的扭矩 T = F × d，其中两个等值反向力的间距为 d。
3. Stress and Strain | 应力与应变
Stress measures the internal force per unit area within a material, while strain is the proportional deformation. They are fundamental to understanding material strength and stiffness. Hooke's Law applies within the elastic limit, where stress is proportional to strain.
应力度量材料内部单位面积上的内力，而应变是比例变形量。它们是理解材料强度和刚度的基础。在弹性极限内，胡克定律成立，即应力与应变成正比。
Stress σ = F / A
Strain ε = ΔL / L₀
Young's Modulus E = σ / ε
original length, ΔL is change in length. For a wire, the spring constant k = (E × A) / L₀.
单位：应力为帕斯卡（Pa）或牛/米²；应变无量纲。L₀ 为原始长度，ΔL 为长度变化。对于线材，弹簧常数 k = (E × A) / L₀。
4. Kinematics Equations (SUVAT) | 运动学方程
For uniform acceleration in a straight line, the five variables — displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t) — are linked by four equations. These are essential for analysing motion in mechanical systems.
对于匀加速直线运动，位移（s）、初速度（u）、末速度（v）、加速度（a）和时间（t）这五个变量由四个方程联系。它们是分析机械系统运动的基础。
| v = u + a t | 无位移 s |
| s = u t + ½ a t² | 无末速度 v |
| s = ½ (u + v) t | 无加速度 a |
| v² = u² + 2 a s | 无时间 t |
Acceleration due to gravity g = 9.81 m/s² (downwards) is often used in free-fall problems.
选择包含三个已知变量和一个未知量的方程。在自由落体问题中，常使用重力加速度 g = 9.81 m/s²（向下）。
5. 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 potential difference across it, provided temperature remains constant. Electrical power can be expressed in three forms depending on known quantities.
欧姆定律表明，在温度不变的条件下，通过导体的电流与其两端的电势差成正比。电功率可根据已知量表示为三种形式。
V = I × R
P = I × V = I² × R = V² / R
Where V is voltage (volts, V), I is current (amperes, A), R is resistance (ohms, Ω), and P is power (watts, W). These relations are critical for circuit analysis and component selection.
其中 V 为电压（伏特），I 为电流（安培），R 为电阻（欧姆），P 为功率（瓦特）。这些关系对于电路分析和元件选型至关重要。
6. Series and Parallel Circuits | 串联与并联电路
In series circuits, the current is the same through all components, and the total resistance is the sum of individual resistances. In parallel circuits, the voltage is the same across each branch, and the reciprocal of total resistance equals the sum of reciprocals of branch resistances.
串联电路中，通过所有元件的电流相同，总电阻等于各电阻之和。并联电路中，各支路电压相同，总电阻的倒数等于各支路电阻的倒数之和。
Series: Rₜₒₜₐₗ = R₁ + R₂ + R₃ + …
Parallel: 1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + 1/R₃ + …
e Law (KVL): ΣV around any closed loop = 0. These laws are used for more complex networks.
基尔霍夫电流定律：流入节点的电流之和等于流出电流之和。基尔霍夫电压定律：任一闭合回路中电压的代数和为零。这些定律用于更复杂的网络分析。
7. Energy, Work and Power | 能量、功与功率
Work is done when a force moves an object in the direction of the force. Energy is the capacity to do work and is conserved. Mechanical power is the rate of doing work or transferring energy.
当力使物体沿力的方向移动时，做功。能量是做功的能力，且能量守恒。机械功率是做功或传递能量的速率。
Work done W = F × d × cosθ
Kinetic energy Eₖ = ½ m v²
Gravitational potential energy Eₚ = m g h
Power P = W / t = F × v (for constant speed)
Efficiency = (useful output energy / total input energy) × 100%. In real machines, some energy is always dissipated as heat due to friction.
效率 = (有用输出能量 / 总输入能量) × 100%。在真实机械中，由于摩擦，总有一部分能量以热的形式耗散。
8. Fluid Mechanics Basics | 流体力学基础
For incompressible fluids in a closed system, the volume flow rate is constant. This continuity equation relates cross-sectional area and velocity. Bernoulli’s equation describes conservation of energy in a flowing fluid.
对于封闭系统中的不可压缩流体，体积流量恒定。连续性方程关联横截面积与流速。伯努利方程描述流动流体中的能量守恒。
Volume flow rate Q = A × v (constant)
Bernoulli: p + ½ ρ v² + ρ g h = constant
incompressible flow. The simplified form p₁ + ½ρv₁² = p₂ + ½ρv₂² is used when height changes are negligible.
其中 p 为静压，ρ 为流体密度，v 为流速，h 为高度。该方程沿流线适用于定常、无粘、不可压缩流动。当高度变化可忽略时，使用简化式 p₁ + ½ρv₁² = p₂ + ½ρv₂²。
9. Trigonometry and Vectors | 三角学与矢量
Trigonometric functions are used to resolve forces, velocities, and other vector quantities into perpendicular components. The sine and cosine rules handle non-right-angled triangles, which frequently appear in force diagrams.
三角函数用于将力、速度等矢量分解为垂向分量。正弦定理和余弦定理处理非直角三角形，在受力分析中常见。
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
For any triangle with sides a, b, c and opposite angles A, B, C: a / sin A = b / sin B = c / sin C (sine rule), and a² = b² + c² − 2bc cos A (cosine rule). Vector addition is performed tip-to-tail or by adding components Rₓ = Aₓ + Bₓ, Rᵧ = Aᵧ + Bᵧ.
对于边长为 a、b、c，对角为 A、B、C 的任意三角形：正弦定理 a/sin A = b/sin B = c/sin C；余弦定理 a² = b² + c² − 2bc cos A。矢量加法采用首尾相接法或分量相加：Rₓ = Aₓ + Bₓ, Rᵧ = Aᵧ + Bᵧ。
10. Units and Dimensional Analysis | 单位与量纲分析
Correct use of SI base units (metre, kilogram, second, ampere, kelvin, mole, candela) and derived units (newton, joule, watt, pascal, volt, ohm) is fundamental to engineering calculations. Dimensional homogeneity ensures every term in an equation has the same fundamental dimensions.
正确使用国际基本单位（米、千克、秒、安培、开尔文、摩尔、坎德拉）和导出单位（牛顿、焦耳、瓦特、帕斯卡、伏特、欧姆）是工程计算的基础。量纲一致性确保方程中每一项目具有相同的基本量纲。
| Quantity | Unit (Symbol) | Dimensions |
| Force (F) | newton (N) | MLT⁻² |
| Pressure (p) | pascal (Pa) | ML⁻¹T⁻² |
| Energy/Work | joule (J) | ML²T⁻² |
| Power | watt (W) | ML²T⁻³ |
10⁻³), micro (μ = 10⁻⁶). Converting units correctly avoids errors in formula application.
词头：千 (k=10³)、兆 (M=10⁶)、厘 (c=10⁻²)、毫 (m=10⁻³)、微 (μ=10⁻⁶)。正确换算单位可避免公式应用错误。
11. Material Properties and Density | 材料性能与密度
Density measures mass per unit volume; it is intrinsic to a material. Other key properties include ultimate tensile strength, toughness, hardness, and stiffness (Young's modulus). The engineering stress-strain curve shows elastic, yielding, and plastic regions.
密度为单位体积的质量，是材料的固有属性。其他重要性能包括极限抗拉强度、韧性、硬度和刚度（杨氏模量）。工程应力-应变曲线显示弹性、屈服和塑性区域。
Density ρ = m / V
Stiffness k = F / ΔL (for a spring)
s. Brittle materials fracture with little plastic strain; ductile materials undergo significant strain before failure.
弹性极限是材料在不发生永久变形的前提下所能承受的最大应力。安全系数 = 极限强度 / 工作应力。脆性材料断裂前塑性应变很小；延性材料在失效前经历显著应变。
12. Heat Transfer and Thermodynamics | 热传递与热力学
Heat transfer occurs via conduction, convection, and radiation. The First Law of Thermodynamics states that energy cannot be created or destroyed, only transferred or converted. For engineering systems, the steady-flow energy equation is frequently applied.
热量通过传导、对流和辐射传递。热力学第一定律指出，能量不能被创造或毁灭，只能传递或转换。在工程系统中，稳态流动能量方程常被应用。
Heat gained/lost: Q = m c ΔT
Conduction rate: P = (k A ΔT) / L
ference. For thermal expansion: ΔL = α L₀ ΔT, where α is the linear expansion coefficient.
其中 c 为比热容（焦/千克·开），k 为导热系数（瓦/米·开），A 为横截面积，L 为厚度，ΔT 为温差。热膨胀公式：ΔL = α L₀ ΔT，α 为线膨胀系数。
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