📚 Formula & Theorem Quick Reference for Year 13 Cambridge Engineering | 剑桥工程十三年级公式定理速查手册
This quick reference handbook compiles essential formulas, theorems, and key concepts for Year 13 Cambridge Engineering students, covering mechanics, materials, thermodynamics, fluid dynamics, electrical circuits, and engineering mathematics. It serves as a handy revision tool for mastering the core principles.
这本速查手册汇编了剑桥工程十三年级学生必备的核心公式、定理和关键概念，涵盖力学、材料、热力学、流体力学、电路和工程数学等领域，是掌握核心原理的便捷复习工具。
1. Mechanics: Kinematics & Dynamics | 力学：运动学与动力学
Kinematics describes motion without considering forces, while dynamics links forces to motion via Newton’s laws. The SUVAT equations are fundamental for constant acceleration problems.
运动学在不考虑力的情况下描述运动，而动力学通过牛顿定律将力与运动联系起来。匀加速运动的 SUVAT 方程是解决恒加速度问题的基础。
The velocity-time relation is given by v = u + at, where u is initial velocity, v final velocity, a constant acceleration and t time.
速度-时间关系由 v = u + at 给出，其中 u 为初速度，v 为末速度，a 为恒定加速度，t 为时间。
v = u + at
Displacement in terms of initial velocity, acceleration and time is s = ut + ½at².
位移用初速度、加速度和时间表示为 s = ut + ½at²。
s = ut + ½at²
The equation linking velocities and displacement without time: v² = u² + 2as.
不显含时间的速度-位移关系：v² = u² + 2as。
v² = u² + 2as
Average velocity can simplify displacement: s = ½(u + v)t.
利用平均速度可简化位移计算：s = ½(u + v)t。
s = ½(u + v)t
2. Newton’s Laws, Momentum & Impulse | 牛顿定律、动量与冲量
Newton’s laws form the bedrock of classical mechanics. The second law is most often applied in its momentum form or as F = ma for constant mass.
牛顿定律是经典力学的基石。第二定律通常以动量形式应用，或在质量不变时写作 F = ma。
Resultant force equals rate of change of momentum: F = dp/dt; for constant mass it simplifies to F = ma.
合力等于动量的变化率：F = dp/dt；对恒质量简化为 F = ma。
F = ma
Linear momentum is a vector: p = mv. Impulse J delivered by a constant force is J = FΔt = Δp = mv – mu.
线动量是矢量：p = mv。恒力的冲量 J = FΔt = Δp = mv – mu。
p = mv, J = FΔt = Δp
In a closed system, total momentum is conserved. For a collision: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
在封闭系统中总动量守恒。对于碰撞：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Newton’s third law: every action has an equal and opposite reaction, crucial for free-body diagrams.
牛顿第三定律：作用力与反作用力等大反向，对受力分析至关重要。
3. Work, Energy & Power | 功、能与功率
Work done by a constant force is the product of the force component in the direction of motion and displacement.
恒力所做的功等于力在运动方向上的分量与位移的乘积。
Work W = Fs cos θ, where θ is the angle between force and displacement vectors.
功 W = Fs cos θ，θ 是力与位移矢量间的夹角。
W = Fs cos θ
Kinetic energy: KE = ½mv². Gravitational potential energy near Earth’s surface: GPE = mgh.
动能：KE = ½mv²。地表附近的重力势能：GPE = mgh。
KE = ½mv², GPE = mgh
The work–energy theorem states that net work equals change in kinetic energy: W_net = ΔKE.
动能定理指出，合力功等于动能变化：W_net = ΔKE。
W_net = ΔKE
Power is the rate of doing work: P = W/t = Fv, where v is instantaneous velocity in the direction of force.
功率是做功的快慢：P = W/t = Fv，v 为力方向上的瞬时速度。
P = Fv
Efficiency of a mechanical system: η = (useful output power) / (input power) × 100%.
机械系统效率：η = （有用输出功率）/（输入功率）× 100%.
4. Circular Motion | 圆周运动
An object moving in a circle at constant speed experiences a centripetal acceleration directed towards the centre.
匀速圆周运动的物体具有指向圆心的向心加速度。
Centripetal acceleration magnitude: a_c = v²/r = ω²r, where r is radius, v tangential speed and ω angular speed.
向心加速度大小：a_c = v²/r = ω²r，r 为半径，v 为切向速率，ω 为角速度。
a_c = v²/r = ω²r
Centripetal force is the net force causing this acceleration: F_c = mv²/r = mω²r.
向心力是产生该加速度的合力：F_c = mv²/r = mω²r。
F_c = mv²/r
Angular velocity relates to period T and frequency f: ω = 2π/T = 2πf. Tangential speed v = ωr.
角速度与周期 T 和频率 f 的关系：ω = 2π/T = 2πf。切向速率 v = ωr。
ω = 2π/T, v = ωr
For vertical circular motion, tension or normal force varies with position, combining weight and centripetal requirement.
竖直圆周运动中，拉力或法向力随位置变化，结合重力与向心需求。
5. Statics, Moments & Equilibrium | 静力学、力矩与平衡
A rigid body is in static equilibrium when both the resultant force and resultant moment about any point are zero.
当刚体所受合力及对任一点的合力矩均为零时，达到静力平衡。
Equilibrium conditions: ΣF_x = 0, ΣF_y = 0, ΣM = 0. Moments are taken as force × perpendicular distance.
平衡条件：ΣF_x = 0、ΣF_y = 0、ΣM = 0。力矩计算为力 × 垂直距离。
ΣF = 0, ΣM = 0
Moment of a force: M = Fd, where d is the perpendicular distance from the pivot to the line of action.
力矩：M = Fd，d 为支点到力作用线的垂直距离。
M = Fd
For a couple, the moment is C = F × separation, independent of pivot choice. Beam reactions can be found by taking moments about supports.
力偶矩：C = 力 × 间距，与支点选取无关。梁的支反力可通过对支点取矩求解。
Centre of gravity: the point through which the entire weight appears to act. For composite shapes, use moments of areas or masses.
重心：可视为全部重量作用的位置。对于组合形状，使用面积矩或质量矩计算。
6. Materials: Stress, Strain & Young’s Modulus | 材料：应力、应变与杨氏模量
Stress σ is defined as force per unit area; strain ε is the fractional change in length.
应力 σ 定义为单位面积上的力；应变 ε 为长度的变化分数。
Tensile stress: σ = F/A (A is original cross-sectional area). Tensile strain: ε = ΔL/L₀.
拉应力：σ = F/A（A 为原截面积）。拉应变：ε = ΔL/L₀。
σ = F/A, ε = ΔL/L₀
Young’s modulus E characterises stiffness in the elastic region: E = σ/ε, valid while Hooke’s law holds.
杨氏模量 E 描述弹性区的刚度：E = σ/ε，在胡克定律成立范围内有效。
E = σ/ε
Stress-strain curves show key points: limit of proportionality, elastic limit, yield stress, ultimate tensile stress (UTS), and fracture.
应力-应变曲线显示关键点：比例极限、弹性极限、屈服应力、极限抗拉强度 (UTS) 和断裂。
Elastic strain energy per unit volume = ½ σε = ½ E ε². Stiffness k of a uniform bar: k = EA/L₀.
单位体积的弹性应变能 = ½ σε = ½ E ε²。均匀杆的刚度 k = EA/L₀。
Strain energy per volume = ½ E ε²
Factor of safety = UTS / permissible working stress. Ductile materials show plastic deformation; brittle materials fail with little strain.
安全系数 = 极限抗拉强度 / 许用工作应力。延性材料呈现塑性变形；脆性材料在极小应变下失效。
7. Thermodynamics & Heat Engines | 热力学与热机
The first law of thermodynamics is an energy conservation statement: ΔU = Q – W, where ΔU is change in internal energy, Q heat added to the system, and W work done by the system.
热力学第一定律是能量守恒表述：ΔU = Q – W，ΔU 为内能变化，Q 为系统吸热，W 为系统对外做功。
ΔU = Q – W
For a heat engine, thermal efficiency η_th = W_net / Q_in = 1 – Q_out / Q_in. The ideal Carnot efficiency depends only on reservoir temperatures: η_Carnot = 1 – T_cold / T_hot (Kelvin).
热机热效率 η_th = W_net / Q_in = 1 – Q_out / Q_in。理想卡诺效率仅取决于热源温度：η_Carnot = 1 – T_cold / T_hot（开尔文温度）。
η_Carnot = 1 – T₂/T₁
Kinetic theory: average translational kinetic energy of a gas molecule = (3/2)kT, where k is Boltzmann constant. Ideal gas law: PV = nRT.
分子动理论：气体分子平均平移动能 = (3/2)kT，k 为玻尔兹曼常数。理想气体定律：PV = nRT。
PV = nRT
Conduction rate: Fourier’s law Q/t = kA(ΔT/d). Convection and radiation (Stefan-Boltzmann: P = εσAT⁴) are also key.
导热速率：傅里叶定律 Q/t = kA(ΔT/d)。对流和辐射（斯特藩-玻尔兹曼：P = εσAT⁴）同样重要。
P = εσAT⁴
8. Fluid Mechanics: Bernoulli & Continuity | 流体力学：伯努利与连续性
For an incompressible, inviscid fluid in steady flow, the continuity equation states that mass flow rate is constant along a streamline: A₁v₁ = A₂v₂.
对于不可压缩、无黏流体的定常流动，连续性方程指出沿流线的质量流量恒定：A₁v₁ = A₂v₂。
A₁v₁ = A₂v₂
Bernoulli’s equation expresses conservation of energy per unit volume: p + ½ρv² + ρgh = constant, where p is static pressure, ρ density, v flow speed, and h height.
伯努利方程表达了单位体积的能量守恒：p + ½ρv² + ρgh = 常数，p 为静压，ρ 为密度，v 为流速，h 为高度。
p + ½ρv² + ρgh = constant
Volume flow rate Q = Av. Dynamic pressure is ½ρv²; stagnation pressure = static + dynamic. Venturi meter and pitot tube rely on these principles.
体积流量 Q = Av。动压为 ½ρv²；总压 = 静压 + 动压。文丘里管和皮托管均基于这些原理。
Reynolds number Re = ρvd/µ, used to predict flow regime: laminar (Re < 2000) or turbulent (Re > 4000).
雷诺数 Re = ρvd/µ，用于判断流态：层流 (Re < 2000) 或湍流 (Re > 4000)。
Re = ρvd/µ
For a fluid at rest, pressure increases with depth: p = p₀ + ρgh (hydrostatic pressure).
静止流体中，压力随深度增加：p = p₀ + ρgh（静水压力）。
9. Electrical Circuits: Ohm’s Law, Kirchhoff & AC | 电路：欧姆定律、基尔霍夫与交流电
Ohm’s law for a resistor at constant temperature: V = IR, linking voltage, current and resistance.
恒定温度下电阻的欧姆定律：V = IR，连接电压、电流和电阻。
V = IR
Electrical power: P = IV = I²R = V²/R. Energy dissipated as heat: E = Pt = I²Rt.
电功率：P = IV = I²R = V²/R。焦耳热能量：E = Pt = I²Rt。
P = IV, E = I²Rt
Kirchhoff’s current law (KCL): sum of currents entering a junction equals sum leaving (ΣI_in = ΣI_out). Kirchhoff’s voltage law (KVL): sum of voltages around any
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