📚 Pre-U CIE Engineering: Quick Reference Formula & Theorem Handbook | Pre-U CIE 工程：公式定理速查手册
Welcome to the ultimate quick-reference handbook for CIE Pre-U Engineering. This resource consolidates essential formulas, theorems, and key principles across mechanics, materials, thermodynamics, fluids, electricity, and applied mathematics. Use it to reinforce your understanding and to efficiently recall critical relationships during revision and exam practice.
欢迎使用 CIE Pre-U 工程终极速查手册。本资源汇集了力学、材料、热力学、流体、电学和应用数学中的基本公式、定理和关键原理。用它来巩固你的理解，并在复习和考试练习中高效回忆关键关系。
1. Linear Motion & Kinematics | 直线运动与运动学
Displacement s (m) is the change in position of an object; velocity v (m/s) is the rate of change of displacement; acceleration a (m/s²) is the rate of change of velocity. For uniform acceleration, the four SUVAT equations apply.
位移 s (m) 是物体位置的改变量；速度 v (m/s) 是位移的变化率；加速度 a (m/s²) 是速度的变化率。对于匀加速运动，以下四个 SUVAT 方程适用。
v = u + at
Final velocity equals initial velocity plus acceleration multiplied by time.
末速度等于初速度加上加速度与时间的乘积。
s = ut + ½at²
Displacement is the product of initial velocity and time plus half the acceleration times the square of time.
位移等于初速度与时间的乘积加上一半加速度乘时间的平方。
v² = u² + 2as
This equation links final velocity, initial velocity, acceleration, and displacement without involving time.
该方程联系了末速度、初速度、加速度和位移，不涉及时间。
s = ½(u + v)t
Displacement can be found from the average velocity multiplied by time.
位移可由平均速度与时间的乘积求得。
For free fall near Earth's surface, acceleration a is replaced by gravitational acceleration g = 9.81 m/s², with downward direction usually taken as positive.
对于地球表面附近的自由落体，加速度 a 被重力加速度 g = 9.81 m/s² 替代，通常取向下为正方向。
2. Forces & Newton's Laws | 力与牛顿定律
Newton's first law: a body remains at rest or in uniform motion unless acted upon by a net external force. Newton's second law quantifies this: F = ma, where net force F (N) equals mass m (kg) times acceleration a (m/s²).
牛顿第一定律：物体若不受净外力作用，则保持静止或匀速直线运动状态。牛顿第二定律将其量化：F = ma，其中净力 F (N) 等于质量 m (kg) 乘以加速度 a (m/s²)。
Weight is the gravitational force on a mass: W = mg. The maximum static friction is f = μₛN and kinetic friction is f = μₖN, where μ is the coefficient of friction and N the normal reaction.
重量是作用在质量上的重力：W = mg。最大静摩擦为 f = μₛN，动摩擦为 f = μₖN，其中 μ 是摩擦系数，N 为法向反力。
Hooke's law for an elastic spring: F = kΔx, where k is the spring constant and Δx the extension from natural length.
弹性弹簧的胡克定律：F = kΔx，其中 k 是劲度系数，Δx 为从自然长度的伸长量。
For circular motion at constant speed, the centripetal force needed is F = mv²/r = mω²r, where v is tangential speed, ω angular velocity, and r radius.
对于匀速圆周运动，所需向心力为 F = mv²/r = mω²r，其中 v 是线速度，ω 是角速度，r 是半径。
3. Energy, Work & Power | 能量、功与功率
Work done by a constant force: W = Fs cosθ, where θ is the angle between force and displacement. Work transfers energy, measured in joules (J).
恒力所做的功：W = Fs cosθ，其中 θ 为力与位移之间的夹角。功传递能量，单位为焦耳 (J)。
Kinetic energy: KE = ½mv². Gravitational potential energy near Earth's surface: PE = mgh, where h is height above a reference level.
动能：KE = ½mv²。地球表面附近的重力势能：PE = mgh，其中 h 为相对参考水平面的高度。
Elastic potential energy stored in a spring: Eₑ = ½k(Δx)². Efficiency is the ratio of useful work output to total energy input: η = Wout / Ein (often expressed as a percentage).
弹簧中储存的弹性势能：Eₑ = ½k(Δx)²。效率是有用功输出与总能量输入之比：η = Wout / Ein（通常以百分数表示）。
Power is the rate of doing work: P = W/t = Fv, where v is the velocity of the point of application of the force. Unit: watt (W).
功率是做功的速率：P = W/t = Fv，其中 v 是力作用点的速度。单位为瓦特 (W)。
4. Momentum & Collisions | 动量与碰撞
Linear momentum: p = mv, a vector quantity with unit kg m/s. Newton's second law can be written as F = Δp/Δt. Impulse is J = FΔt = Δp.
线动量：p = mv，为矢量，单位 kg m/s。牛顿第二定律可写为 F = Δp/Δt。冲量为 J = FΔt = Δp。
The principle of conservation of momentum: in a closed system with no external forces, total momentum before a collision equals total momentum after: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
动量守恒定律：在无外力的封闭系统中，碰撞前的总动量等于碰撞后的总动量：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。
For perfectly elastic collisions, kinetic energy is conserved. For perfectly inelastic collisions, the bodies stick together and move with a common velocity.
完全弹性碰撞中，动能守恒。完全非弹性碰撞中，物体粘在一起并以共同速度运动。
5. Statics & Equilibrium | 静力学与平衡
For a body in static equilibrium, both the resultant force and the resultant moment must be zero: ΣF = 0 and Στ = 0 about any point.
处于静力平衡的物体，合力和合力矩都必须为零：ΣF = 0 且对任意点满足 Στ = 0。
The moment of a force about a point: τ = Fd, where d is the perpendicular distance from the point to the line of action of the force. Moments causing clockwise or anticlockwise rotation are assigned opposite signs.
力对某点的力矩：τ = Fd，其中 d 为该点到力作用线的垂直距离。引起顺时针或逆时针转动的力矩取相反符号。
A couple consists of two equal, opposite, parallel forces and produces a pure moment: C = F × (distance between forces). The centre of mass of a uniform object can be located by symmetry or calculation.
力偶由一对大小相等、方向相反、不共线的平行力组成，产生纯力矩：C = F × (力间距离)。均匀物体的质心可通过对称性或计算确定。
6. Stress, Strain & Material Properties | 应力、应变与材料性质
Tensile or compressive stress: σ = F/A, measured in pascals (Pa) or N/m². Strain is the extension per original length: ε = ΔL/L (dimensionless).
拉伸或压缩应力：σ = F/A，单位为帕斯卡 (Pa) 或 N/m²。应变是伸长量与原长之比：ε = ΔL/L（无量纲）。
Young's modulus, a measure of stiffness: E = σ/ε, valid within the linear elastic region. Typical units: GPa. The stress-strain graph shows the limit of proportionality, elastic limit, yield point, and ultimate tensile strength.
杨氏模量是刚度的量度：E = σ/ε，在弹性限度内有效。典型单位为 GPa。应力-应变曲线显示出比例极限、弹性极限、屈服点和抗拉强度。
Shear stress τ = F/A (parallel force) and shear strain γ = x/h lead to the shear modulus G = τ/γ. Poisson's ratio ν relates lateral strain to axial strain: ν = -ε_lat / ε_ax.
剪切应力 τ = F/A（平行力）和剪切应变 γ = x/h 导出剪切模量 G = τ/γ。泊松比 ν 关联横向应变与轴向应变：ν = -ε_lat / ε_ax。
llowable stress. Ductile materials show large plastic deformation; brittle materials fracture with little plastic strain.
安全系数 = 极限应力 / 许用应力。韧性材料表现出较大的塑性变形；脆性材料在微小塑性应变下即断裂。
7. Fluid Mechanics | 流体力学
Density: ρ = m/V. Pressure: p = F/A, unit pascal. Fluid pressure at a depth h in a static fluid: p = ρgh, plus atmospheric pressure if the surface is open.
密度：ρ = m/V。压强：p = F/A，单位帕斯卡。静止流体中深度 h 处的流体压强：p = ρgh，若表面敞开则需加上大气压强。
Archimedes' principle: the upthrust (buoyant force) equals the weight of fluid displaced: Fb = ρ_fluid g V_submerged. Floatation occurs when upthrust equals weight.
阿基米德原理：浮力等于排开流体的重量：Fb = ρ_fluid g V_submerged。当浮力等于物体重量时物体漂浮。
Continuity equation for an incompressible fluid: A₁v₁ = A₂v₂, where A is cross‑sectional area and v flow speed. Bernoulli's equation for steady, inviscid, incompressible flow: p + ½ρv² + ρgh = constant along a streamline.
不可压缩流体的连续性方程：A₁v₁ = A₂v₂，其中 A 为横截面积，v 为流速。伯努利方程适用于稳定、无粘、不可压缩流动：沿流线满足 p + ½ρv² + ρgh = constant。
Reynolds number predicts flow regime: Re = ρvL/μ, where L is characteristic length and μ dynamic viscosity. Low Re → laminar; high Re → turbulent.
雷诺数预测流态：Re = ρvL/μ，其中 L 为特征长度，μ 为动力粘度。低雷诺数为层流，高雷诺数为湍流。
8. Thermodynamics | 热力学
The Zeroth Law establishes thermal equilibrium and temperature measurement. The First Law: ΔU = Q - W, where ΔU is change in internal energy, Q heat supplied to the system, and W work done by the system.
第零定律定义了热平衡和温度测量。第一定律：ΔU = Q - W，其中 ΔU 为内能变化，Q 为供给系统的热量，W 为系统对外做的功。
Heat required to change temperature: Q = mcΔθ, where c is specific heat capacity. Latent heat for phase change: Q = mL, where L is specific latent heat.
改变温度所需的热量：Q = mcΔθ，其中 c 是比热容。相变潜热：Q = mL，其中 L 是比潜热。
Ideal gas equation: pV = nRT or pV = NkT, with R = 8.31 J/(mol·K) and k = 1.38×10⁻²³ J/K. For an adiabatic process of an ideal gas: pV^γ = constant, where γ = cₚ/cᵥ.
理想气体状态方程：pV = nRT 或 pV = NkT，其中 R = 8.31 J/(mol·K)，k = 1.38×10⁻²³ J/K。对于理想气体的绝热过程：pV^γ = 常数，其中 γ = cₚ/cᵥ。
The maximum efficiency of a heat engine operating between temperatures TH and TC is the Carnot efficiency: η = 1 - TC/TH (temperatures in kelvin).
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