📚 AS AQA Engineering: Formula & Theorem Quick Reference Handbook | AS AQA 工程：公式定理速查手册
This quick reference handbook collates the essential formulas, theorems and key relationships required for the AS AQA Engineering syllabus. Each section presents critical equations, concise explanations and paired bilingual commentary to support rapid revision and deeper understanding. Use this guide to reinforce your grasp of mechanics, materials, electrical principles, thermodynamics and more.
本速查手册汇集了 AS AQA 工程课程所需的基本公式、定理和关键关系。每一节都提供关键方程、简明解释以及中英双语的对应说明，帮助快速复习并加深理解。借助本指南，巩固您对力学、材料、电学原理、热力学等领域知识的掌握。
1. Fundamental Mechanics & Forces | 基础力学与力
Newton’s Second Law defines the relationship between resultant force, mass and acceleration. When multiple forces act, the net force determines the motion of the object.
牛顿第二定律定义了合力、质量与加速度之间的关系。当多个力同时作用时，净力决定物体的运动状态。
Fₙₑₜ = m × a
The weight of any object near the Earth’s surface is the gravitational force acting on its mass.
地球表面附近任何物体的重量是作用在其质量上的引力。
W = m × g
Friction force between two solid surfaces is modelled using the coefficient of friction μ and the normal reaction R. The kinetic friction applies during sliding.
两固体表面之间的摩擦力用摩擦系数 μ 和法向反力 R 来建模。滑动时适用动摩擦。
F = μ × R
On an inclined plane, the component of weight parallel to the slope is mg sin θ, while the perpendicular component is mg cos θ. These components are crucial for resolving forces and determining acceleration.
在斜面上，重力的平行分量为 mg sin θ，垂直分量为 mg cos θ。这两个分量对于分解力和确定加速度至关重要。
2. Moments and Equilibrium | 力矩与平衡
The turning effect of a force about a pivot is called the moment. It depends on both the magnitude of the force and the perpendicular distance from the pivot to the line of action.
力绕支点的转动效应称为力矩。它取决于力的大小以及支点到力作用线的垂直距离。
M = F × d
For an object to be in static equilibrium, two conditions must be satisfied: the vector sum of all forces must be zero, and the sum of clockwise moments about any point must equal the sum of anticlockwise moments.
物体处于静力平衡必须满足两个条件：所有力的矢量和为零，且绕任意点的顺时针力矩之和等于逆时针力矩之和。
∑ F = 0 and ∑ Mˡˢ = ∑ Mˢᵒ
Reaction forces at supports and pivot points are calculated by applying these equilibrium principles. A simply supported beam with a central load W experiences vertical reactions R₁ = R₂ = W/2 when symmetric.
支座和支点的反力通过应用这些平衡原理来计算。一根中心加载 W 的简支梁在对称时承受的垂直反力为 R₁ = R₂ = W/2。
3. Stress, Strain and Elasticity | 应力、应变与弹性
Engineering stress is the internal force per unit cross-sectional area. It is measured in pascals (Pa) or N/m².
工程应力是单位横截面积上的内力，单位为帕斯卡 (Pa) 或 N/m²。
σ = F / A
Strain is the dimensionless measure of deformation, given by the change in length relative to the original length.
应变是变形的无量纲量度，由长度变化量与原始长度之比给出。
ε = ΔL / L₀
Young’s modulus E quantifies a material’s stiffness within the elastic limit. It is the gradient of the stress-strain curve in the linear region.
杨氏模量 E 量化材料在弹性限度内的刚度。它是应力-应变曲线在线性区域的斜率。
E = σ / ε
Hooke’s law for a spring relates extension to applied force, where k is the spring constant. This linear behaviour mirrors elastic deformation in solids.
弹簧的胡克定律将伸长量与所施加的力联系起来，其中 k 为弹簧常数。这种线性行为与固体中的弹性变形相似。
F = k × x
4. Kinematics and Motion | 运动学与运动
For uniform acceleration in a straight line, the three equations of motion (SUVAT) relate displacement s, initial velocity u, final velocity v, acceleration a and time t.
对于匀加速直线运动，三个运动学方程 (SUVAT) 将位移 s、初速度 u、末速度 v、加速度 a 和时间 t 联系起来。
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
The average velocity during uniform acceleration can be found as (u + v)/2, and it equals the total displacement divided by time when acceleration is constant.
匀加速过程中的平均速度可求为 (u + v)/2，当加速度恒定时，它等于总位移除以时间。
Projectile motion can be analysed by treating horizontal and vertical components independently. Horizontally, velocity is constant; vertically, acceleration is g downwards.
抛体运动可通过独立分析水平和垂直分量来处理。水平方向速度恒定，垂直方向加速度为向下的 g。
5. Work, Energy and Power | 功、能与功率
Work done by a constant force is the product of the force and the displacement in the direction of the force.
恒力所做的功是力与沿力方向的位移的乘积。
W = F × d × cos θ
Kinetic energy is the energy an object possesses due to its motion, and gravitational potential energy depends on height.
动能是物体由于运动而拥有的能量，重力势能则取决于高度。
Eₖ = ½ m v² Eₚ = m g h
Power is the rate of doing work or transferring energy. For a moving vehicle, power can also be expressed as the product of tractive force and velocity.
功率是做功或传递能量的速率。对于行驶的车辆，功率也可表示为牵引力与速度的乘积。
P = W / t = F × v
Efficiency of any energy conversion is given by the ratio of useful output power to total input power, often expressed as a percentage.
任何能量转换的效率由有用输出功率与总输入功率之比给出，通常以百分比表示。
η = (Pₒᵤₜ / Pᵢₙ) × 100%
6. Fluid Mechanics Principles | 流体力学原理
Pressure exerted by a fluid at rest increases with depth and depends on the fluid’s density and gravitational field strength.
静止流体施加的压强随深度增加，并取决于流体的密度和重力场强度。
p = ρ g h
For an incompressible fluid flowing in a closed pipe, the mass flow rate is conserved. This leads to the continuity equation linking cross-sectional areas and velocities.
对于在封闭管道中流动的不可压缩流体，质量流量守恒。由此得出将横截面积与速度联系起来的连续性方程。
A₁ v₁ = A₂ v₂
Bernoulli’s principle for ideal fluids states that the sum of pressure energy, kinetic energy per unit volume and potential energy per unit volume remains constant along a streamline.
伯努利原理适用于理想流体，指出单位体积的压力能、动能和势能之和沿流线守恒。
p + ½ ρ v² + ρ g h = constant
Archimedes’ principle gives the upthrust on an object submerged in a fluid as equal to the weight of the displaced fluid.
阿基米德原理指出，浸没在流体中的物体所受的浮力等于被排开流体的重量。
Fᵤₚ = ρ g V
7. Thermal Physics and Heat Transfer | 热物理与热传递
Linear thermal expansion describes how the length of a solid material changes with temperature. The coefficient of linear expansion α is a material property.
线性热膨胀描述固体材料的长度如何随温度变化。线性膨胀系数 α 是材料的属性。
ΔL = α L₀ ΔT
The heat energy required to change the temperature of a substance depends on its mass, specific heat capacity and the temperature change.
改变物质的温度所需的热量取决于其质量、比热容和温度变化。
Q = m c ΔT
For an ideal gas, the pressure, volume and temperature are related by the ideal gas equation. R is the universal gas constant (8.31 J mol⁻¹ K⁻¹).
对于理想气体，压强、体积和温度通过理想气体方程相关联。R 是普适气体常数 (8.31 J mol⁻¹ K⁻¹)。
p V = n R T
In heat conduction through a uniform slab, the rate of heat transfer depends on the temperature gradient, area and thermal conductivity k.
在通过均匀平板的热传导中，传热速率取决于温度梯度、面积和导热系数 k。
Q / t = k A (ΔT) / L
8. Electrical Fundamentals and Circuits | 电学基础与电路
Ohm’s law provides the foundational link between voltage, current and resistance for many conductors under constant temperature.
欧姆定律给出了在恒温条件下许多导体的电压、电流和电阻之间的基本关系。
V = I × R
The resistance of a wire or conductor depends on its resistivity, length and cross-sectional area. Resistivity ρ is a property of the material.
导线或导体的电阻由其电阻率、长度和横截面积决定。电阻率 ρ 是材料的属性。
R = ρ L / A
Electrical power dissipated in a component can be expressed in three equivalent forms using Ohm’s law substitution.
元器件中耗散的电功率可以利用欧姆定律代入而表示为三种等效形式。
P = I V = I² R = V² / R
Kirchhoff’s current law (KCL) states that the total current entering a junction equals the total current leaving; Kirchhoff’s voltage law (KVL) states that the sum of all voltages around a closed loop is zero.
基尔霍夫电流定律表明，流入节点的总电流等于流出的总电流；基尔霍夫电压定律表明，闭合回路中所有电压的代数和为零。
∑ Iᵢₙ = ∑ Iₒᵤₜ ∑ V = 0
9. Engineering Mathematics Toolkit | 工程数学工具
Engineers frequently use trigonometric ratios for right-angled triangles to resolve vectors and determine lengths and angles.
工程师经常使用直角三角形的三角比来分解矢量并确定长度和角度。
sin θ = opposite / hypotenuse cos θ = adjacent / hypotenuse tan θ = opposite / adjacent
The sine rule and cosine rule are essential for solving non-right-angled triangles that appear in force diagrams and structural analysis.
正弦定理和余弦定理对于求解力图和结构分析中出现的非直角三角形至关重要。
a / sin A = b / sin B = c / sin C
a² = b² + c² – 2 b c cos A
Areas and volumes of common shapes are needed for centre of mass calculations and mass property estimations.
常见形状的面积和体积对于质心计算和质量属性估算是必需的。
A (circle) = π r² V (cylinder) = π r² h V (sphere) = ⁴/₃ π r³
Percentage error and tolerance calculations are used to assess accuracy in measurement and manufacturing.
百分误差和公差计算用于评估测量和制造中的精度。
% error = (|measured − true| / true) × 100%
10. Material Properties and Testing | 材料性能与测试
Tensile testing provides a stress-strain curve from which key properties are determined: yield strength, ultimate tensile strength (UTS), and fracture point. The UTS is the maximum engineering stress reached.
拉伸测试给出应力-应变曲线，从中可确定关键性能：屈服强度、极限抗拉强度 (UTS) 和断裂点。UTS 是达到的最大工程应力。
Ductility is quantified by the percentage elongation after fracture, comparing the final gauge length L₁ to the original gauge length L₀.
延展性通过断裂后的伸长率来量化，将最终标距 L₁ 与原始标距 L₀ 进行比较。
Elongation % = ((L₁ – L₀) / L₀) × 100%
Reduction in cross-sectional area at fracture also indicates ductility. A larger reduction implies more plastic deformation before failure.
断裂处的横截面积收缩率也能反映延展性。收缩率越大，意味着破坏前发生了更多的塑性变形。
RA% = ((A₀ – A₁) / A₀) × 100%
Hardness is a measure of a material’s resistance to indentation or scratching. While no single formula applies, common scales like Brinell, Vickers and Rockwell are used alongside look-up tables in materials selection.
硬度衡量材料抵抗压入或刮擦的能力。虽然没有单一公式，但在材料选择中常使用布氏、维氏和洛氏等标度并结合查表。
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