📚 Formula & Theorems Quick Reference Handbook | 公式定理速查手册
This handbook gathers the essential formulas, laws and theorems required for Year 12 WJEC Engineering. It is designed as a rapid revision aid for the Engineering Principles unit, covering core topics from mechanics and materials to electricity, fluids and thermal physics. Use it to reinforce your understanding and to practise applying each relationship accurately.
本手册汇集了 Year 12 WJEC 工程课程必备的公式、定律和定理，专为《工程原理》单元快速复习而编写，涵盖力学、材料、电学、流体和热物理等核心主题。通过反复练习，帮助你巩固理解并准确应用每个关系式。
1. Kinematics | 运动学
These equations describe motion with constant acceleration along a straight line.
下列方程描述匀加速直线运动。
v = u + at
Relates final velocity v, initial velocity u, constant acceleration a and time t. All quantities are in SI units (m/s, m/s², s).
该式联系末速度 v、初速度 u、恒定加速度 a 和时间 t。各量均使用国际单位（m/s, m/s², s）。
s = ut + ½at²
Calculates displacement s from initial velocity, acceleration and time. The term ½at² accounts for the distance covered due to acceleration.
利用初速度、加速度和时间计算位移 s。½at² 项表示因加速而经过的距离。
v² = u² + 2as
Links final velocity, initial velocity, acceleration and displacement without requiring time. Very useful when time is unknown.
联系末速度、初速度、加速度和位移，不含时间变量。当时间未知时非常实用。
s = ½(u + v)t
Gives displacement as the product of average velocity and time. Assumes uniform acceleration.
位移等于平均速度乘以时间。适用于匀加速运动。
Standard gravitational acceleration on Earth is taken as g = 9.81 m/s².
地球表面标准重力加速度取 g = 9.81 m/s²。
2. Dynamics & Newton's Laws | 动力学与牛顿定律
Newton’s second law is the foundation for analysing forces and motion. Weight is a special case of force.
牛顿第二定律是分析力与运动的基础。重力是一种特殊的力。
F = ma
Resultant force F (newtons, N) equals mass m (kg) multiplied by acceleration a (m/s²). Direction of acceleration matches the resultant force.
合力 F（牛顿，N）等于质量 m（kg）乘以加速度 a（m/s²）。加速度方向与合力方向一致。
W = mg
Weight W is the gravitational force on a mass m. Use g = 9.81 m/s² unless stated otherwise.
重力 W 是质量为 m 的物体所受的引力。除非另有说明，均使用 g = 9.81 m/s²。
p = mv
Momentum p (kg·m/s) is the product of mass and velocity. It is a vector quantity.
动量 p（kg·m/s）是质量与速度的乘积，为矢量。
FΔt = Δp
Impulse equals change in momentum. A force applied over a time interval Δt alters the momentum of an object.
冲量等于动量的变化量。力在时间间隔 Δt 内的作用会改变物体的动量。
3. Moments & Equilibrium | 力矩与平衡
A moment causes rotation. For a body in static equilibrium, the sum of clockwise moments must balance the sum of anticlockwise moments about any pivot.
力矩产生转动效应。静力平衡时，绕任意支点的顺时针力矩之和必须等于逆时针力矩之和。
M = F × d
Moment M (N·m) is the product of force F and perpendicular distance d from the pivot to the line of action. The distance must be perpendicular.
力矩 M（N·m）等于力 F 与支点到力作用线的垂直距离 d 的乘积。距离必须为垂直距离。
Σ clockwise moments = Σ anticlockwise moments
For rotational equilibrium, the total turning effect in one direction equals the total in the opposite direction.
旋转平衡时，一个方向的总转动效应等于反方向的总转动效应。
A couple consists of two equal, parallel but opposite forces. Its moment is force × perpendicular distance between the forces.
力偶由两个大小相等、平行且方向相反的力组成，其力矩等于力乘以两力之间的垂直距离。
4. Work, Energy & Power | 功、能量与功率
Work is done when a force moves its point of application. Energy is the capacity to do work, and power is the rate of doing work.
力使其作用点移动时便做了功。能量是做功的本领，功率是做功的速率。
W = F s cosθ
Work done W (J) by a constant force F moving through displacement s. θ is the angle between force and displacement. If force is parallel to motion, cosθ = 1.
恒定力 F 经过位移 s 所做的功 W（J）。θ 为力与位移的夹角。当力与运动方向平行时，cosθ = 1。
KE = ½mv²
Kinetic energy (J) of a mass m moving at speed v.
质量为 m、以速度 v 运动的物体的动能（J）。
PE = mgh
Gravitational potential energy gained when a mass m is raised through height h. The reference level can be chosen arbitrarily.
质量为 m 的物体被提升高度 h 后获得的重力势能。参考平面可任意选取。
P = W / t = F v
Power P (watt, W) is work done per unit time. For a constant force acting on an object moving at constant speed v, power is also F v (force × velocity).
功率 P（瓦特，W）是单位时间内所做的功。当恒定力作用于以恒定速度 v 运动的物体时，功率也等于 F v。
Efficiency = Useful output energy (or power) / Total input energy (or power)
Efficiency is often expressed as a percentage. It can never exceed 100% due to losses such as friction and heat.
效率常用百分数表示。由于摩擦和热损失，效率不可能超过 100%。
5. Stress & Strain | 应力与应变
When materials are loaded, they experience stress and undergo strain. These quantities describe the internal intensity of force and the resulting deformation.
材料受载时产生应力并发生应变。这两个量分别描述内力的密集程度和由此产生的变形。
σ = F / A
Tensile or compressive stress σ (Pa or N/m²) is the force F applied perpendicularly divided by the original cross‑sectional area A. 1 MPa = 10⁶ Pa.
拉伸或压缩应力 σ（Pa 或 N/m²）等于垂直作用力 F 除以原始横截面积 A。1 MPa = 10⁶ Pa。
ε = ΔL / L₀
Strain ε is the fractional change in length: extension (or compression) ΔL divided by original length L₀. Strain has no units; it is often expressed as a percentage.
应变 ε 是长度的相对变化量：伸长量（或压缩量）ΔL 除以原始长度 L₀。应变无量纲，常用百分数表示。
Percentage elongation = (ΔL / L₀) × 100%
百分比伸长 = (ΔL / L₀) × 100%
6. Young's Modulus & Material Properties | 杨氏模量与材料性质
Young's modulus measures the stiffness of a material within its elastic limit. It is a constant for a given material, independent of shape.
杨氏模量衡量材料在弹性限度内的刚度，它是给定材料的一种特性常数，与形状无关。
E = σ / ε
Young's modulus E (Pa or GPa) is the ratio of tensile stress to tensile strain as long as Hooke’s law is obeyed. Gradient of the linear portion of a stress‑strain curve.
杨氏模量 E（Pa 或 GPa）是在胡克定律适用范围内，拉伸应力与拉伸应变之比，即应力-应变曲线线性段的斜率。
Hooke’s law: F = k x, where k is the spring constant (N/m) and x is extension. For a wire, k = E A / L₀.
胡克定律：F = k x，其中 k 为弹簧常数（N/m），x 为伸长量。对于线材，k = E A / L₀。
Typical values: steel – 200 GPa, aluminium – 70 GPa. Units: 1 GPa = 10⁹ Pa.
典型值：钢 – 200 GPa，铝 – 70 GPa。单位：1 GPa = 10⁹ Pa。
7. Factor of Safety | 安全系数
Engineers apply a factor of safety to ensure that components perform reliably under loads below the material’s failure point.
工程师采用安全系数，确保构件在低于材料失效点的载荷下可靠工作。
Factor of Safety = Ultimate stress / Working stress
Also defined as Failure load / Safe working load. The factor is always greater than 1, typically ranging from 1.5 to 10 depending on application and consequences of failure.
也定义为 失效载荷 / 许用工作载荷。安全系数始终大于 1，通常根据应用场合和失效后果在 1.5 至 10 之间选取。
A higher factor of safety increases reliability but may result in a heavier and more expensive design.
较高的安全系数提高了可靠性，但可能导致更重、更昂贵的设计。
8. Electrical Circuits (Ohm's Law & Kirchhoff's) | 电路（欧姆定律与基尔霍夫定律）
Ohm’s law and Kirchhoff’s rules are the cornerstones of circuit analysis. They describe how current, voltage and resistance are related in direct‑current circuits.
欧姆定律和基尔霍夫定律是电路分析的基石，阐述直流电路中电流、电压和电阻之间的关系。
V = I R
The potential difference V (volts) across a resistor equals the current I (amperes) multiplied by the resistance R (ohms, Ω). Applies to ohmic conductors at constant temperature.
电阻两端的电势差 V（伏特）等于电流 I（安培）乘以电阻 R（欧姆，Ω），适用于恒温下的欧姆导体。
Resistance, R = ρ L / A
Resistance depends on material resistivity ρ (Ω·m), length L and cross‑sectional area A.
电阻取决于材料电阻率 ρ（Ω·m）、长度 L 和横截面积 A。
Kirchhoff’s Current Law: Σ Iin = Σ Iout
The sum of currents entering a junction equals the sum of currents leaving. Charge is conserved.
基尔霍夫电流定律：流入节点的电流之和等于流出节点的电流之和，电荷守恒。
Kirchhoff’s Voltage Law: Σ V = 0
The algebraic sum of potential differences around any closed loop is zero. Energy supplied by sources equals energy dissipated by components.
基尔霍夫电压定律：沿任意闭合回路，电势差的代数和为零。电源提供的能量等于元件消耗的能量。
9. Resistors in Series & Parallel | 电阻的串联与并联
Combining resistors in different ways changes the total resistance and the distribution of voltage and current.
电阻的不同组合方式会改变总电阻以及电压和电流的分配。
Series: Rtotal = R₁ + R₂ + R₃ + …
In series, the same current flows through each resistor. Total resistance is the sum of individual resistances.
串联时，每个电阻流过的电流相同，总电阻等于各电阻之和。
Parallel: 1/Rtotal = 1/R₁ + 1/R₂ + 1/R₃ + …
In parallel, the voltage across each branch is the same. Total resistance is always less than the smallest individual resistance.
并联时，各支路电压相等，总电阻总是小于最小的单个电阻。
For two parallel resistors
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