📚 Year 13 WJEC Engineering: Formula & Theorem Quick Reference Handbook | Year 13 WJEC工程：公式定理速查手册
This quick reference handbook compiles essential formulas, laws, and theorems required for Year 13 WJEC Engineering. Covering mechanics, materials, thermofluids, electronics, and control, it serves as a rapid revision aid to reinforce understanding and problem-solving skills before examinations. Each formula is accompanied by definitions of symbols and contextual application.
本手册汇集了 Year 13 WJEC 工程学科的核心公式、定律与定理，涵盖力学、材料、热流体、电子学和控制等领域，可作为考前快速复习的工具。每个公式均附有符号解释和工程语境下的应用说明。
1. Mechanics: Newton's Laws & Equations of Motion | 力学：牛顿定律与运动方程
Newton's second law states that the net force acting on a body is equal to the product of its mass and acceleration: F = m a, where F is net force (N), m is mass (kg), and a is acceleration (m s⁻²). This principle governs dynamic equilibrium in structures, vehicles, and mechanisms.
牛顿第二定律指出，作用在物体上的净力等于其质量与加速度的乘积：F = m a，其中 F 为净力（N），m 为质量（kg），a 为加速度（m s⁻²）。该定律主导了结构、车辆和机构中的动态平衡分析。
For motion with uniform acceleration, the SUVAT equations relate displacement s, initial velocity u, final velocity v, acceleration a, and time t: v = u + a t; s = u t + ½ a t²; v² = u² + 2 a s; s = ½(u + v) t. These are used extensively in projectile and linear motion problems.
对于匀加速运动，SUVAT 方程组关联位移 s、初速度 u、末速度 v、加速度 a 和时间 t：v = u + a t；s = u t + ½ a t²；v² = u² + 2 a s；s = ½(u + v) t。这些方程在抛体运动和直线运动问题中广泛使用。
2. Moments, Equilibrium and Centre of Gravity | 力矩、平衡与重心
The moment of a force about a pivot is the product of the force and the perpendicular distance from the pivot: M = F × d (Nm). For a body in static equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any point: Σ M_clockwise = Σ M_anticlockwise.
力对支点的力矩等于力与力臂（垂直距离）的乘积：M = F × d（单位 Nm）。物体处于静力平衡时，对任意点的顺时针力矩之和等于逆时针力矩之和：Σ M_clockwise = Σ M_anticlockwise。
A system is also in translational equilibrium when the resultant force in any direction is zero: Σ Fx = 0 and Σ Fy = 0. The centre of gravity (CoG) is the point where the entire weight of an object can be considered to act; for symmetrical homogeneous objects, it coincides with the geometric centre.
当任意方向上的合力为零时，系统亦处于平动平衡：Σ Fx = 0 且 Σ Fy = 0。重心（CoG）是可认为物体全部重量集中的点；对于对称均质物体，重心与几何中心重合。
3. Stress, Strain and Young's Modulus | 应力、应变与杨氏模量
Tensile or compressive stress is force per unit cross-sectional area: σ = F / A (Pa). Strain is the ratio of extension to original length: ε = ΔL / L₀ (dimensionless). Young's modulus describes the stiffness of a material in the linear elastic region: E = σ / ε (Pa).
拉伸或压缩应力是单位横截面积上的力：σ = F / A（Pa）。应变是伸长量与原长之比：ε = ΔL / L₀（无量纲）。杨氏模量描述材料在线弹性范围内的刚度：E = σ / ε（Pa）。
Hooke's law applies up to the limit of proportionality: σ = E ε. Engineering stress-strain curves yield key points: yield strength, ultimate tensile strength (UTS), and fracture point. Ductile materials exhibit significant plastic deformation, while brittle materials fail with little warning.
胡克定律在比例极限内成立：σ = E ε。工程应力-应变曲线给出关键数据点：屈服强度、抗拉强度（UTS）和断裂点。延性材料表现出显著的塑性变形，而脆性材料则在几乎无预兆的情况下失效。
4. Factor of Safety and Poisson's Ratio | 安全系数与泊松比
The factor of safety (FoS) ensures a structure performs safely under design loads: FoS = Ultimate stress / Working stress. A FoS greater than 1 accounts for uncertainties in material properties, loading, and manufacturing.
安全系数（FoS）确保结构在设计载荷下安全工作：安全系数 = 极限应力 / 工作应力。大于 1 的安全系数考虑了材料性能、载荷和制造中的不确定性。
Poisson's ratio, ν, quantifies the lateral contraction relative to axial extension: ν = -ε_lateral / ε_axial. It typically ranges from 0.2 to 0.4 for metals and is used in three-dimensional stress analysis and pressure vessel design.
泊松比 ν 表示横向收缩与轴向伸长的比值：ν = -ε_lateral / ε_axial。金属的泊松比通常在 0.2 至 0.4 之间，用于三维应力分析和压力容器设计。
5. Thermal Expansion and Heat Transfer | 热膨胀与传热
Linear thermal expansion: ΔL = α L₀ Δθ, where α is the coefficient of linear expansion (°C⁻¹ or K⁻¹), L₀ is original length, and Δθ is temperature change. Bimetallic strips exploit differing α values for thermostat applications.
线性热膨胀：ΔL = α L₀ Δθ，其中 α 为线膨胀系数（°C⁻¹ 或 K⁻¹），L₀ 为原长，Δθ 为温度变化。双金属片利用不同的 α 值实现恒温控制。
Heat energy required to change temperature: Q = m c Δθ, where c is specific heat capacity (J kg⁻¹ K⁻¹). Conduction through a solid of cross-sectional area A and thickness d: P = Q/t = k A (Δθ)/d, where k is thermal conductivity (W m⁻¹ K⁻¹).
改变温度所需的热能：Q = m c Δθ，其中 c 为比热容（J kg⁻¹ K⁻¹）。通过截面积为 A、厚度为 d 的固体的热传导功率：P = Q/t = k A (Δθ)/d，k 为导热系数（W m⁻¹ K⁻¹）。
6. Fluid Mechanics: Continuity and Bernoulli | 流体力学：连续性方程与伯努利方程
For an incompressible fluid in steady flow, the mass flow rate is conserved: A₁ v₁ = A₂ v₂ or ρ A v = constant. This continuity equation explains why fluid speeds up through narrow sections.
对于不可压缩流体的稳定流动，质量流率守恒：A₁ v₁ = A₂ v₂ 或 ρ A v = 常数。该连续性方程解释了流体在狭窄截面处加速的原因。
Bernoulli's equation for ideal, steady, frictionless flow along a streamline: p + ½ ρ v² + ρ g h = constant. It relates pressure energy, kinetic energy, and gravitational potential energy per unit volume, and is fundamental to aerofoil lift and Venturi meter calculations.
理想、稳定、无摩擦流线流动的伯努利方程：p + ½ ρ v² + ρ g h = 常数。它关联了单位体积的压力能、动能和重力势能，是机翼升力和文丘里流量计计算的基础。
7. Electrical Principles: Ohm's Law, Power and Energy | 电学原理：欧姆定律、功率与能量
Ohm's law for a conductor at constant temperature: V = I R, where V is voltage (V), I is current (A), and R is resistance (Ω). Electrical power dissipated is P = I V = I² R = V² / R (W), and energy consumed over time t is W = P t (J).
恒温下导体的欧姆定律：V = I R，其中 V 为电压（V），I 为电流（A），R 为电阻（Ω）。电功率为 P = I V = I² R = V² / R（W），时间内消耗的电能为 W = P t（J）。
For resistors in series, total resistance Rtotal = R₁ + R₂ + .... For resistors in parallel, 1/Rtotal = 1/R₁ + 1/R₂ + .... Voltage divider rule: Vout = Vin × (R₂/(R₁+R₂)) for two series resistors.
串联电阻的总电阻：Rtotal = R₁ + R₂ + …。并联电阻：1/Rtotal = 1/R₁ + 1/R₂ + …。分压公式（两串联电阻）：Vout = Vin × (R₂/(R₁+R₂))。
8. Kirchhoff's Laws and Network Analysis | 基尔霍夫定律与网络分析
Kirchhoff's current law (KCL): The algebraic sum of currents entering a node is zero: Σ I = 0. Equivalently, Iin = Iout. This law reflects conservation of charge.
基尔霍夫电流定律（KCL）：流入节点的电流代数和为零：Σ I = 0。亦常写作 Iin = Iout，反映电荷守恒。
Kirchhoff's voltage law (KVL): The sum of the electromotive forces and potential differences around any closed loop is zero: Σ V = 0. This is based on energy conservation and is used to set up mesh equations in circuits with multiple sources.
基尔霍夫电压定律（KVL）：任一闭合回路中电动势与电位差的代数和为零：Σ V = 0。该定律基于能量守恒，用于多源电路的网孔方程建立。
9. Operational Amplifiers and Feedback | 运算放大器与反馈
An ideal op-amp has infinite input impedance, zero output impedance, and infinite open-loop gain. The closed-loop gain for an inverting amplifier is Av = -Rf / Rin, where Rf is feedback resistor and Rin is input resistor. The negative sign indicates phase inversion.
理想运放具有无限大输入阻抗、零输出阻抗和无限大开环增益。反相放大器的闭环增益为 Av = -Rf / Rin，其中 Rf 为反馈电阻，Rin 为输入电阻，负号表示相位反转。
For a non-inverting amplifier, the gain is Av = 1 + Rf / R₁. The summing amplifier produces an output proportional to the weighted sum of inputs: Vout = -Rf (V₁/R₁ + V₂/R₂ + ...). Comparators use op-amps without feedback to switch outputs based on input threshold.
同相放大器的增益为 Av = 1 + Rf / R₁。加法放大器输出与加权输入和成正比：Vout = -Rf (V₁/R₁ + V₂/R₂ + …)。比较器利用无反馈的运放按输入阈值切换输出状态。
10. Digital Logic: Boolean Algebra and Simplification | 数字逻辑：布尔代数与化简
Fundamental Boolean identities: A + 0 = A, A + 1 = 1, A + A = A, A · 0 = 0, A · 1 = A, A · A = A, and A + A' = 1. De Morgan's theorems are (A·B)' = A' + B' and (A+B)' = A' · B'.
基本布尔恒等式：A + 0 = A，A + 1 = 1，A + A = A，A · 0 = 0，A · 1 = A
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