📚 Year 12 AQA Engineering: Formula & Theorem Quick Reference Handbook | AQA 工程 Year 12：公式定理速查手册
This quick reference handbook compiles all essential formulas and theorems for the Year 12 AQA Engineering course. It is designed to help you quickly locate the equations you need for mechanics, materials, thermodynamics, fluid dynamics, electrical circuits, digital electronics, and more. Each formula is accompanied by a brief explanation in both English and Chinese, making revision efficient and exam‑ready.
本速查手册汇集了 Year 12 AQA 工程课程的所有核心公式和定理。它旨在帮助你快速查找到力学、材料、热力学、流体、电路、数字电子等方面的公式。每条公式都配有中英文简要说明，让复习更加高效，为考试做好充分准备。
1. Mechanics and Statics | 力学与静力学
Resolving a force into perpendicular components: the horizontal component is F cos θ and the vertical component is F sin θ, where θ is measured from the horizontal.
将力分解为垂直分量：水平分量为 F cos θ，垂直分量为 F sin θ，θ 为力与水平方向的夹角。
Fₓ = F cos θ, Fᵧ = F sin θ
The moment of a force about a pivot is the product of the force and the perpendicular distance from the pivot to the line of action.
力对支点的力矩等于力与力臂（支点到力作用线的垂直距离）的乘积。
M = F × d
For a body in static equilibrium, the vector sum of all forces and the sum of all moments about any point must be zero.
刚体平衡时，所有力的矢量和以及所有力矩对任一点的代数和均为零。
Σ Fₓ = 0, Σ Fᵧ = 0, Σ M = 0
2. Kinematics | 运动学
The four equations of uniformly accelerated motion relate initial velocity u, final velocity v, acceleration a, displacement s, and time t.
匀加速运动的四个方程描述了初速度 u、末速度 v、加速度 a、位移 s 和时间 t 之间的关系。
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
s = ½ (u + v) t
3. Dynamics and Momentum | 动力学与动量
Newton’s second law states that the net force acting on an object equals the product of its mass and acceleration.
牛顿第二定律指出物体受到的合力等于其质量乘以加速度。
F = m a
Linear momentum is the product of mass and velocity. Impulse is the change in momentum and equals the average force multiplied by the time over which it acts.
线动量等于质量乘以速度。冲量是动量的变化量，也等于平均力乘以作用时间。
p = m v, Impulse = Δp = Favg Δt
In a closed system, total momentum is conserved. For a two‑body collision along a straight line:
在封闭系统中，总动量守恒。对于沿直线的两体碰撞：
m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂
4. Work, Energy and Power | 功、能量与功率
Work done by a constant force is the product of the force, the displacement, and the cosine of the angle between them.
恒力做的功等于力、位移以及二者夹角余弦的乘积。
W = F s cos θ
Kinetic energy and gravitational potential energy near the Earth’s surface are given by:
地面附近的动能和重力势能分别为：
KE = ½ m v², GPE = m g h
Power is the rate of doing work. When a force moves an object at constant velocity, power is force times velocity.
功率是做功的快慢。当力使物体匀速运动时，功率等于力乘以速度。
P = W / t = F v
Efficiency is the ratio of useful output energy (or power) to total input energy (or power).
效率是有用输出能量（或功率）与总输入能量（或功率）之比。
η = Euseful / Einput = Puseful / Pinput
5. Material Properties and Stress–Strain | 材料性能与应力应变
Tensile (or compressive) stress is the force applied per unit cross‑sectional area. Strain is the extension (or contraction) per unit original length.
拉（压）应力是单位横截面积上的力。应变是单位原始长度上的伸长（或缩短）量。
σ = F / A, ε = ΔL / L₀
Young’s modulus describes the stiffness of a material within the elastic limit; it is the ratio of stress to strain.
杨氏模量描述材料在弹性范围内的刚度，是应力与应变的比值。
E = σ / ε
Poisson’s ratio is the negative ratio of lateral strain to axial strain for a uniaxially loaded material.
泊松比是单轴受载材料横向应变与轴向应变之比的负值。
ν = –εlateral / εaxial
The factor of safety is the ratio of ultimate stress to allowable working stress.
安全系数是极限应力与许用工作应力之比。
FoS = σult / σworking
6. Beams and Bending | 梁与弯曲
The simple bending equation relates bending moment M, second moment of area I, bending stress σ at a distance y from the neutral axis, Young’s modulus E, and radius of curvature R.
简单弯曲方程将弯矩 M、截面二次矩 I、距中性轴 y 处的弯曲应力 σ、杨氏模量 E 和曲率半径 R 联系起来。
M / I = σ / y = E / R
Maximum bending stress in a symmetrical cross‑section occurs at the outermost fibre, where y = ymax.
对称截面上的最大弯曲应力出现在最外层纤维处，即 y = ymax。
σmax = M ymax / I
For a rectangular beam of width b and depth d, the second moment of area about the neutral axis is:
对于宽度 b、高度 d 的矩形梁，关于中性轴的截面二次矩为：
I = b d³ / 12
Average shear stress in a beam is V/A; for a rectangular section the maximum shear stress is 1.5 times the average.
梁内的平均剪应力为 V/A；矩形截面梁的最大剪应力是平均值的 1.5 倍。
τavg = V / A, τmax = 1.5 V / A
7. Thermodynamics and Heat Transfer | 热力学与热传递
Linear thermal expansion: a solid of original length L₀ expands by ΔL when the temperature changes by ΔT; α is the linear expansion coefficient.
线热膨胀：原始长度 L₀ 的固体在温度变化 ΔT 时伸长 ΔL；α 为线膨胀系数。
ΔL = α L₀ ΔT
The ideal gas law connects pressure p, volume V, amount of substance n, the gas constant R, and absolute temperature T.
理想气体定律将压强 p、体积 V、物质的量 n、气体常数 R 和绝对温度 T 联系起来。
p V = n R T
Sensible heat transfer: the heat Q required to change the temperature of a mass m by Δθ is proportional to its specific heat capacity c.
显热传递：使质量为 m 的物体温度改变 Δθ 所需的热量 Q 正比于比热容 c。
Q = m c Δθ
Latent heat: the heat Q required to change the phase of a mass m without temperature change is Q = m L, where L is the specific latent heat.
潜热：使质量为 m 的物质在无温度变化的情况下发生相变所需的热量 Q = m L，L 是比潜热。
Q = m L
First law of thermodynamics: the increase in internal energy ΔU equals the heat added to the system Q minus the work done by the system W.
热力学第一定律：内能的增加 ΔU 等于系统吸收的热量 Q 减去系统对外做的功 W。
ΔU = Q – W
8. Fluid Mechanics | 流体力学
Hydrostatic pressure at a depth h in a fluid of density ρ is given by p = ρ g h, plus the surface pressure if applicable.
密度为 ρ 的流体在深度 h 处的静水压强为 p = ρ g h，若有表面压强则需加和。
p = ρ g h
The continuity equation for an incompressible fluid states that volume flow rate is constant: A₁ v₁ = A₂ v₂.
不可压缩流体的连续性方程表明体积流量恒定：A₁ v₁ = A₂ v₂。
A₁ v₁ = A₂ v₂
Bernoulli’s equation for a steady, inviscid, incompressible flow along a streamline:
沿流线的定常、无黏、不可压缩流动的伯努利方程：
p₁ + ½ρ v₁² + ρ g h₁ = p₂ + ½ρ v₂² + ρ g h₂
The Reynolds number indicates whether flow is laminar or turbulent; it is the ratio of inertia forces to viscous forces.
雷诺数用于判断流动是层流还是湍流，它是惯性力与黏性力的比值。
Re = ρ v d / μ
9. Electrical Principles | 电路原理
Ohm’s law relates voltage V, current I, and resistance R.
欧姆定律将电压 V、电流 I 和电阻 R 联系起来。
V = I R
Electrical power can be expressed in several equivalent forms.
电功率可用多种等价形式表达。
P = I V = I² R = V² / R
Resistors in series: total resistance is the sum. Resistors in parallel: reciprocals add.
串联电阻：总电阻为各电阻之和。并联电阻：总电导为各电导之和。
Rseries = R₁ + R₂ + …
1 / Rparallel = 1 / R₁ + 1 / R₂ + …
Kirchhoff’s current law: the sum of currents entering a junction equals the sum leaving it. Kirchhoff’s voltage law: the algebraic sum of voltages around any closed loop is zero.
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。
∑ Iin = ∑ Iout, ∑ Vloop = 0
For a capacitor, charge Q = C V; the energy stored in a charged capacitor is ½ C V². The RC time constant τ = R C determines charge/discharge rates.
电容的电荷量 Q = C V；储存的能量为 ½ C V²。RC 时间常数 τ = R C 决定充放电速率。
Q = C V, E = ½ C V², τ = R C
10. Digital Electronics | 数字电子
Basic Boolean operations: AND is logical multiplication, OR is logical addition, NOT is complement (denoted by an overbar).
基本布尔运算：与为逻辑乘，或为逻辑加，非为补运算（以上划线表示）。
A AND B = A · B, A OR B = A + B, NOT A = Ā
De Morgan’s theorems allow the transformation of logic expressions.
德摩根定理可用于逻辑表达式的变换。
(A · B)̄ = Ā + B̄
(A + B)̄ = Ā · B̄
A truth table lists all possible input combinations and the resulting output for a logic gate or circuit.
真值表列出逻辑门或电路所有可能的输入组合及其对应输出。
Binary, octal, and hexadecimal number systems are fundamental to digital engineering; conversions between bases use weighted place values.
二进制、八进制、十六进制是数字工程的基础；不同进制间的转换使用加权位值。
11. Useful Constants and Conversions | 常用常数与单位换算
Gravitational acceleration near the Earth’s surface: g = 9.81 m/s² (often taken as 9.8). Atmospheric pressure at sea level: 101.3 kPa. Density of water: 1000 kg/m³. Universal gas constant: R = 8.31 J/(mol·K). Charge on an electron: e = 1.60 × 10⁻¹⁹ C. Speed of light in vacuum: c = 3.00 × 10⁸ m/s.
重力加速度（近地面）：g = 9.81 m/s²（常取 9.8）。海平面大气压强：101.3 kPa。水的密度：1000 kg/m³。普适气体常数：R = 8.31 J/(mol·K)。电子电荷：e = 1.60 × 10⁻¹⁹ C。真空光速：c = 3.00 × 10⁸ m/s。
g = 9.81 m/s², p₀ = 1.013 × 10⁵ Pa
Common unit conversions: 1 litre = 0.001 m³; 1 tonne = 1000 kg; 1 bar = 10⁵ Pa; 1 degree = π/180 radians.
常见单位换算：1 升 = 0.001 m³；1 吨 = 1000 kg；1 巴 = 10⁵ Pa；1 度 = π/180 弧度。
1 L = 10⁻³ m³, 1 bar = 10⁵ Pa, 1° = π/180 rad
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