📚 Pre-U WJEC Engineering: Essential Formulae and Theorems Quick Reference Handbook | Pre-U WJEC 工程：必备公式定理速查手册
This quick reference handbook compiles the fundamental formulae, laws, and theorems required for the WJEC Pre-U Engineering course. Each entry is presented with a concise explanation in both English and Chinese, accompanied by accurate scientific notation. Use this resource to reinforce key concepts and sharpen your problem-solving skills for the examination.
本速查手册汇集了WJEC Pre-U工程课程所需的基本公式、定律和定理。每一条目均配有简洁的中英文解释和规范的科学符号。请利用此资源巩固核心概念，提升解题能力，为考试做好充分准备。
1. Kinematics & Linear Motion | 运动学与直线运动
v = u + at
This equation gives the final velocity v after time t when accelerating uniformly from initial velocity u with acceleration a.
该方程给出了从初速度u以加速度a匀加速运动时间t后的末速度v。
s = ut + ½at²
It calculates the displacement s during uniform acceleration, combining the initial velocity contribution and the additional displacement due to acceleration.
它计算匀加速运动中的位移s，包含了初速度贡献的位移和加速度引起的附加位移。
v² = u² + 2as
This time-independent form links initial and final velocities, acceleration, and displacement, useful when time is not known.
此与时间无关的形式将初速度、末速度、加速度和位移联系起来，在时间未知时非常有用。
s = ½(u + v)t
Displacement is the average velocity multiplied by time, valid only for constant acceleration.
位移等于平均速度乘以时间，仅适用于匀加速运动。
2. Newton’s Laws & Forces | 牛顿定律与力
F = ma
Newton’s second law: the net force on an object is equal to its mass times its acceleration.
牛顿第二定律：物体所受的合力等于其质量乘以加速度。
W = mg
Weight is the gravitational force acting on a mass m, with gravitational field strength g (≈9.81 N/kg on Earth).
重量是作用在质量m上的引力，g为引力场强度（地球表面约为9.81 N/kg）。
f ≤ μR
The maximum static or kinetic friction force f is proportional to the normal reaction force R, with coefficient of friction μ.
最大静摩擦力或动摩擦力f与正压力R成正比，μ为摩擦系数。
p = mv
Linear momentum is the product of mass and velocity. Impulse = Δp = FΔt.
线动量是质量与速度的乘积。冲量 = Δp = FΔt。
3. Equilibrium, Moments & Couples | 平衡、力矩与力偶
ΣF = 0 , ΣM = 0
For a body in static equilibrium, the vector sum of all forces and the sum of moments about any point must both be zero.
处于静态平衡的物体，所有力的矢量和以及对任意一点的力矩总和都必须为零。
M = Fd
The moment of a force about a pivot is the force multiplied by the perpendicular distance from the pivot to the line of action.
力对支点的力矩等于力乘以支点到力作用线的垂直距离。
C = F × s
A couple consists of two equal, opposite, parallel forces separated by a distance; its torque is the product of one force and the perpendicular separation.
力偶由相距一定距离的两个大小相等、方向相反的平行力构成；其转矩等于其中一个力与垂直间距的乘积。
4. Stress, Strain & Elasticity | 应力、应变与弹性
σ = F / A
Engineering stress σ (sigma) is the applied force divided by the original cross-sectional area.
工程应力σ为施加的力除以原始截面积。
ε = ΔL / L₀
Strain ε (epsilon) is the extension per unit original length. L₀ is the original length.
应变ε为单位原始长度的伸长量。L₀为原始长度。
E = σ / ε
Young’s modulus E measures a material’s stiffness, valid within the linear elastic region.
杨氏模量E衡量材料的刚度，适用于线弹性区域。
F = kx
Hooke’s law for an elastic spring or wire: extension x is proportional to load F, where k is the spring constant.
弹性弹簧或金属丝的胡克定律：伸长量x与载荷F成正比，k为劲度系数。
5. Pressure & Buoyancy | 压力与浮力
P = F / A
Pressure is defined as force per unit area; 1 Pa = 1 N/m².
压强定义为单位面积上的力；1 Pa = 1 N/m²。
P = ρgh
Hydrostatic pressure at depth h in a fluid of density ρ under gravity g.
密度为ρ的流体在重力场g下深度h处的静水压强。
Fb = ρVg
Archimedes’ principle: buoyant force equals the weight of the displaced fluid. V is the submerged volume.
阿基米德原理：浮力等于排开流体的重量。V为浸没体积。
6. Work, Energy & Power | 功、能量与功率
W = Fs cosθ
Work done by a constant force is the force component along displacement multiplied by distance; θ is the angle between force and displacement.
恒力所做的功等于力沿位移方向的分量与距离的乘积；θ为力与位移的夹角。
Ek = ½mv²
Kinetic energy of a mass m moving at velocity v.
质量为m、速度为v的物体的动能。
Ep = mgh
Gravitational potential energy change near the Earth’s surface.
地表附近的重力势能变化。
P = W/t = Fv
Power is the rate of doing work; for a constant force and velocity, P = Fv.
功率是做功的速率；当力和速度恒定时，P = Fv。
7. Thermodynamics & Heat Transfer | 热力学与传热
Q = mcΔT
Sensible heat transfer required to change a material’s temperature, where c is specific heat capacity.
改变物体温度所需的热量，c为比热容。
Q = mL
Latent heat for phase change at constant temperature; L is the specific latent heat.
温度不变时相变所需的潜热；L为比潜热。
pV = nRT
Ideal gas law: pressure p, volume V, amount of substance n, gas constant R (≈8.31 J/mol·K), and absolute temperature T.
理想气体状态方程：p为压强，V为体积，n为物质的量，R为气体常数，T为绝对温度。
ΔU = Q − W
First law of thermodynamics: change in internal energy equals heat added to the system minus work done by the system.
热力学第一定律：内能的变化等于系统吸收的热量减去系统对外做的功。
8. Electrical Principles | 电学原理
V = IR
Ohm’s law: the potential difference V across a conductor is proportional to the current I and resistance R.
欧姆定律：导体两端的电势差V与电流I和电阻R成正比。
P = IV = I²R = V²/R
Electrical power dissipated as heat in a resistive element; multiple forms are useful for series/parallel analysis.
电阻元件以热形式耗散的电功率；不同表达形式便于串并联分析。
Rtotal = R₁ + R₂ + ⋯ (series)
Resistors in series add directly.
串联电阻直接相加。
1/Rtotal = 1/R₁ + 1/R₂ + ⋯ (parallel)
For resistors in parallel, the reciprocal of total resistance is the sum of reciprocals.
对于并联电阻，总电阻的倒数等于各电阻倒数之和。
9. Fluid Mechanics | 流体力学
A₁v₁ = A₂v₂
Continuity equation for incompressible flow: the volumetric flow rate remains constant along a streamline.
不可压缩流的连续性方程：沿流线的体积流量保持恒定。
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Bernoulli’s equation: conservation of energy per unit volume for an ideal fluid in steady flow along a streamline.
伯努利方程：理想流体沿流线定常流动时单位体积的能量守恒。
Re = ρvd / μ
Reynolds number characterizes flow regime (laminar or turbulent) using fluid density ρ, velocity v, characteristic length d, and dynamic viscosity μ.
雷诺数用流体密度ρ、流速v、特征长度d和动力粘度μ来表征流动状态（层流或湍流）。
10. Material Failure & Safety Factors | 材料失效与安全系数
n = σfail / σallow
Factor of safety n is the ratio of the material’s failure stress to the allowable working stress, ensuring reliable design.
安全系数n为材料失效应力与允许工作应力之比，用以保证可靠设计。
Kt = σmax / σnom
Stress concentration factor Kt quantifies the local stress increase near geometric discontinuities.
应力集中系数Kt量化了几何不连续处附近的局部应力增加程度。
σ = My / I
Simple bending theory: bending stress σ at distance y from the neutral axis, where M is bending moment and I is second moment of area.
简单弯曲理论：弯曲应力σ与距中性轴的垂直距离y、弯矩M和截面二次矩I的关系。
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