📚 Year 12 AQA Engineering: Formula & Theorem Quick Reference Handbook | Year 12 AQA 工程：公式定理速查手册
This handbook consolidates the essential formulas, laws, and theorems from the Year 12 AQA Engineering curriculum. It is designed for rapid revision and as a ready reference for problem-solving in mechanics, materials, fluids, thermodynamics, electronics, and more. Every entry is paired with a concise explanation to reinforce understanding and application.
本手册汇总了 Year 12 AQA 工程课程的核心公式、定律和定理。专为快速复习而设计，也可作为解决力学、材料学、流体力学、热力学、电子学等问题的即时参考。每个条目都配有简明解释，以加深理解和应用。
1. Mathematical Toolkit | 数学工具
Pythagoras' theorem relates the sides of a right-angled triangle: c² = a² + b², where c is the hypotenuse.
勾股定理关联直角三角形的边：c² = a² + b²，其中 c 为斜边。
Trigonometric ratios: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent.
三角比：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。
Sine and cosine rules are used for non‑right‑angled triangles. Sine rule: a / sin A = b / sin B = c / sin C. Cosine rule: a² = b² + c² − 2bc cos A.
正弦定理和余弦定理适用于非直角三角形。正弦定理：a / sin A = b / sin B = c / sin C。余弦定理：a² = b² + c² − 2bc cos A。
Vectors can be resolved into perpendicular components: if a vector F makes an angle θ with the horizontal, its components are F cos θ (horizontal) and F sin θ (vertical).
矢量可以分解为垂直分量：若矢量 F 与水平方向夹角为 θ，则其分量为 F cos θ（水平）和 F sin θ（垂直）。
2. Linear Motion and Kinematics | 直线运动与运动学
Average speed: v = s / t, where s is distance travelled in time t. Velocity is a vector; speed is scalar.
平均速率：v = s / t，其中 s 为时间 t 内通过的距离。速度是矢量，速率是标量。
Acceleration: a = (v − u) / t, or change in velocity over time. Constant acceleration leads to the SUVAT equations.
加速度：a = (v − u) / t，即速度随时间的变化率。匀加速运动可使用 SUVAT 方程组。
SUVAT equations (valid only when acceleration is constant):
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
s = ½ (u + v) t
SUVAT 方程（仅适用于加速度恒定）：
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
s = ½ (u + v) t
avity, a = g = 9.81 m s⁻² (downwards), and motion is often analysed with sign conventions.
自由落体运动中，a = g = 9.81 m s⁻²（向下），分析时常需要规定正方向。
3. Forces and Newton's Laws | 力与牛顿定律
Newton's first law: An object remains at rest or in uniform motion unless acted upon by a resultant external force.
牛顿第一定律：除非受到合外力的作用，物体将保持静止或匀速直线运动状态。
Newton's second law: F = m a, where F is the resultant force, m is mass, a is acceleration.
牛顿第二定律：F = m a，F 为合力，m 为质量，a 为加速度。
Weight is the gravitational force: W = m g. The normal reaction force often balances weight on horizontal surfaces.
重量是重力作用：W = m g。在水平面上，法向反作用力通常与重力平衡。
Newton's third law: For every action, there is an equal and opposite reaction. These forces act on different bodies.
牛顿第三定律：每一个作用力都有一个大小相等、方向相反的反作用力，作用在不同的物体上。
Friction: The maximum static friction is fₛ ≤ μₛ R, and kinetic friction is fₖ = μₖ R, where R is the normal reaction, μ is the coefficient of friction.
摩擦力：最大静摩擦力 fₛ ≤ μₛ R，动摩擦力 fₖ = μₖ R，R 为法向反力，μ 为摩擦系数。
4. Momentum and Impulse | 动量与冲量
Linear momentum: p = m v. It is a vector quantity, unit kg m s⁻¹.
线动量：p = m v。它是一个矢量，单位为 kg m s⁻¹。
Conservation of momentum: In a closed system, total momentum before interaction equals total momentum after interaction: Σ pbefore = Σ pafter.
动量守恒：在封闭系统中，相互作用前的总动量等于相互作用后的总动量：Σ pbefore = Σ pafter。
Impulse is the change in momentum: Impulse = F t = Δ p = m v − m u. Force–time graphs give impulse as the area under the curve.
冲量等于动量的变化：冲量 = F t = Δ p = m v − m u。力-时间图线下面积表示冲量。
5. Work, Energy and Power | 功、能与功率
Work done by a constant force: W = F s cos θ, where θ is the angle between the force and displacement direction.
恒力做功：W = F s cos θ，θ 为力与位移方向间的夹角。
Kinetic energy: Eₖ = ½ m v². Gravitational potential energy: Eₚ = m g h.
动能：Eₖ = ½ m v²。重力势能：Eₚ = m g h。
Principle of conservation of energy: energy cannot be created or destroyed, only transferred. In the absence of non‑conservative forces, total mechanical energy is constant.
能量守恒原理：能量既不能被创造也不能被消灭，只能转化。在没有非保守力的情况下，总机械能守恒。
Power is the rate of doing work: P = W / t. For constant force moving at velocity v parallel to the force, P = F v.
功率是做功的速率：P = W / t。当恒力与速度 v 方向一致时，P = F v。
Efficiency: η = (useful energy output / total energy input) × 100%.
效率：η = (有用能量输出 / 总能量输入) × 100%。
6. Mechanics of Materials | 材料力学
Stress: σ = F / A, force per unit cross‑sectional area, unit Pa (N m⁻²).
应力：σ = F / A，单位横截面积上的力，单位为 Pa (N m⁻²)。
Strain: ε = Δ L / L₀, dimensionless ratio of extension to original length.
应变：ε = Δ L / L₀，伸长量与原始长度的比值，无量纲。
Young's modulus: E = σ / ε, provided the material obeys Hooke's law and is within the proportional limit.
杨氏模量：E = σ / ε，前提是材料遵循胡克定律且处于比例极限内。
Hooke's law for springs: F = k x, where k is the spring constant and x is extension. For a material, the linear portion of the stress‑strain curve gives σ = E ε.
弹簧的胡克定律：F = k x，k 为弹簧常数，x 为伸长量。对于材料，应力-应变曲线的线性部分满足 σ = E ε。
Poisson's ratio: ν = − (lateral strain / axial strain).
泊松比：ν = − (横向应变 / 轴向应变)。
Ultimate tensile strength (UTS) and yield stress are key values from a stress‑strain diagram. Brittle materials fail with little plastic deformation; ductile materials show substantial plastic deformation and necking.
极限抗拉强度和屈服应力是应力-应变图上的关键值。脆性材料在很少的塑性变形后即断裂；韧性材料则表现出显著的塑性变形和颈缩。
7. Fluid Mechanics | 流体力学
Density: ρ = m / V (kg m⁻³). Pressure: p = F / A (Pa). In a stationary fluid, pressure increases with depth: p = ρ g h.
密度：ρ = m / V (kg m⁻³)。压强：p = F / A (Pa)。在静止流体中，压强随深度增加：p = ρ g h。
Pascal's principle: pressure applied to an enclosed fluid is transmitted equally in all directions. In hydraulic systems, p₁ = p₂, hence F₁ / A₁ = F₂ / A₂.
帕斯卡原理：施加在封闭流体上的压强会大小不变地向各个方向传递。液压系统中，p₁ = p₂，因此 F₁ / A₁ = F₂ / A₂。
Archimedes' principle: upthrust on an object equals the weight of fluid displaced. U = ρ_fluid V_displaced g.
阿基米德原理：物体在流体中受到的浮力等于排开流体的重量。U = ρ_fluid V_displaced g。
Continuity equation for incompressible flow: A₁ v₁ = A₂ v₂, volumetric 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 for steady, inviscid, incompressible flow along a streamline: p + ½ ρ v² + ρ g h = constant.
理想流体沿流线稳态流动的伯努利原理：p + ½ ρ v² + ρ g h = 常数。
8. Thermodynamics | 热力学
Temperature is a measure of average kinetic energy of particles. The Celsius and Kelvin scales are related by T(K) = θ(°C) + 273.15.
温度是粒子平均动能的量度。摄氏温标与开尔文温标的关系为：T(K) = θ(°C) + 273.15。
Linear thermal expansion: Δ L = α L₀ Δ T, where α is the coefficient of linear expansion.
线性热膨胀：Δ L = α L₀ Δ T，α 为线膨胀系数。
Ideal gas equation: p V = n R T, where n is number of moles, R = 8.31 J mol⁻¹ K⁻¹. Can also be written as p V = N k T, k being Boltzmann's constant.
理想气体状态方程：p V = n R T，n 为摩尔数，R = 8.31 J mol⁻¹ K⁻¹。也可写为 p V = N k T，k 为玻尔兹曼常数。
Boyle's law (T constant): p₁ V₁ = p₂ V₂. Charles's law (p constant): V₁ / T₁ = V₂ / T₂. Pressure law (V constant): p₁ / T₁ = p₂ / T₂.
波义耳定律（恒温）：p₁ V₁ = p₂ V₂。查理定律（恒压）：V₁ / T₁ = V₂ / T₂。压强定律（恒容）：p₁ / T₁ = p₂ / T₂。
First law of thermodynamics: Δ U = Q − W, where Δ U is change in internal energy, Q is heat added to the system, W is work done by the system.
热力学第一定律：Δ U = Q − W，Δ U 为内能变化，Q 为系统吸收的热量，W 为系统对外做的功。
Specific heat capacity: Q = m c Δ T. Specific latent heat: Q = m L (L for fusion or vaporisation).
比热容：Q = m c Δ T。比潜热：Q = m L（L 为熔化潜热或汽化潜热）。
9. Electrical Principles | 电学原理
Electric charge: Q = I t, where I is current (A) and t is time (s). Charge is measured in coulombs (C).
电荷：Q = I t，I 为电流 (A)，t 为时间 (s)。电荷单位为库仑 (C)。
Ohm's law: V = I R, valid for ohmic conductors at constant temperature.
欧姆定律：V = I R，适用于恒温下的欧姆导体。
Resistance depends on material and geometry: R = ρ L / A, where ρ is resistivity (Ω m).
电阻取决于材料和几何尺寸：R = ρ L / A，ρ 为电阻率 (Ω m)。
Electrical power: P = V I = I² R = V² / R. Energy transferred: E = P t = V I t.
电功率：P = V I = I² R = V² / R。电能耗：E = P t = V I t。
Kirchhoff's current law (KCL): sum of currents entering a junction equals sum leaving. Kirchhoff's voltage law (KVL): sum of e.m.f.s in a closed loop equals sum of potential drops.
基尔霍夫电流定律 (KCL)：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 (KVL)：闭合回路中电动势之和等于各元件上电压降之和。
Voltage dividers: for two resistors in series, Vout = Vin (R₂ / (R₁ + R₂)).
分压电路：对于两个串联电阻，Vout = Vin (R₂ / (R₁ + R₂))。
10. Digital Electronics and Logic | 数字电子学与逻辑
Boolean algebra defines logical operations on binary variables (0 and 1). Basic operations are AND (conjunction), OR (disjunction), and NOT (negation).
布尔代数定义了二进制变量（0和1）的逻辑运算。基本运算包括 AND（与）、OR（或）和 NOT（非）。
AND gate output: Q = A · B; OR gate output: Q = A + B; NOT gate output: Q = Ā.
与门输出：Q = A · B；或门输出：Q = A + B；非门输出：Q = Ā。
Combined gates: NAND is NOT (A AND B); NOR is NOT (A OR B); XOR gives 1 when inputs are different.
复合门：NAND 是 NOT (A AND B)；NOR 是 NOT (A OR B)；XOR 在输入不同时输出 1。
Truth tables list all possible input combinations and corresponding outputs. Boolean identities such as A + A = A, A · A = A, A + 0 = A, A · 1 = A, and De Morgan's laws are used for simplification.
真值表列出所有可能的输入组合及对应输出。布尔恒等式如 A + A = A, A · A = A, A + 0 = A, A · 1 = A，以及德·摩根定律，可用于化简。
Analog-to-digital conversion: resolution of an ADC = Vref / (2ⁿ − 1) for an n-bit converter.
模数转换：n位转换器的分辨率 = Vref / (2ⁿ − 1)。
11. Statics and Moments | 静力学与力矩
For a body in static equilibrium, both the resultant force and resultant moment must be zero. Σ F = 0 and Σ M = 0.
刚体处于静力平衡时，合力与合力矩均为零：Σ F = 0，Σ M = 0。
Moment of a force about a point: M = F d, where d is the perpendicular distance from the pivot to the line of action of the force. Unit: N m.
力对一点的力矩：M = F d，其中 d 为转轴到力作用线的垂直距离。单位：N m。
Principle of moments: for equilibrium, sum of clockwise moments equals sum of anticlockwise moments about any pivot.
力矩原理：平衡状态下，对任意转轴，顺时针力矩之和等于逆时针力矩之和。
A couple consists of two equal, parallel but opposite forces. The moment of a couple is M = F × (perpendicular distance between lines of action).
力偶由一对大小相等、平行但方向相反的力组成。力偶矩为 M = F × (力作用线间的垂直距离)。
Centres of mass and gravity are used to determine stability. The centre of mass of a uniform lamina can be found by symmetry or calculation.
质心与重心的概念用于判断稳定性。均匀薄片的质心可通过对称性或计算求得。
12. Rotational Dynamics | 转动动力学
Angular displacement θ is measured in radians. Angular velocity: ω = Δ θ / Δ t (rad s⁻¹). Angular acceleration: α = Δ ω / Δ t (rad s⁻²).
角位移 θ 以弧度衡量。角速度：ω = Δ θ / Δ t (rad s⁻¹)。角加速度：α = Δ ω / Δ t (rad s⁻²)。
For uniform angular acceleration, angular SUVAT equations apply: ω₂ = ω₁ + α t, θ = ω₁ t + ½ α t², ω₂² = ω₁² + 2 α θ.
对于匀角加速度，角运动学方程组适用：ω₂ = ω₁ + α t，θ = ω₁ t + ½ α t²，ω₂² = ω₁² + 2 α θ。
Torque turns a body. T = F r sin θ; for a force perpendicular to the radius, T = F r. In rotational form, Newton's second law: T = I α, where I is the moment of inertia (kg m²).
转矩使物体转动。T = F r sin θ；当力垂直于半径时，T = F r。转动的牛顿第二定律：T = I α，I 为转动惯量 (kg m²)。
Rotational kinetic energy: Eₖ_rot = ½ I ω². Work done by a constant torque: W = T θ.
转动动能：Eₖ_rot = ½ I ω²。恒转矩做功：W = T θ。
Power in rotation: P = T ω. These analogies link linear and rotational mechanics.
转动功率：P = T ω。这些类比将线性力学与转动联系起来。
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