📚 Year 13 Edexcel Engineering: Formula & Theorem Quick Reference Handbook | Edexcel工程13年级：公式定理速查手册
This handbook brings together the essential formulas, theorems, and relationships required for Year 13 Edexcel Engineering. It is designed as a rapid revision tool, covering mechanics, materials, fluids, thermodynamics, electrical principles, and more. Each section presents key equations with concise bilingual explanations to reinforce understanding and exam readiness.
本手册汇集了Edexcel工程13年级必备的公式、定理和基本关系，旨在作为快速复习工具，涵盖力学、材料、流体力学、热力学、电学原理等核心内容。每个小节提供关键方程，并配有简洁的双语解释，以加深理解、帮助备考。
1. Linear Motion & Dynamics | 直线运动与动力学
For constant acceleration, the SUVAT equations give relationships between displacement s, initial velocity u, final velocity v, acceleration a, and time t.
匀加速运动中，SUVAT方程组给出位移 s、初速度 u、末速度 v、加速度 a 和时间 t 之间的关系。
v = u + a t
v = u + a t
s = ½ (u + v) t
s = ½ (u + v) t
s = u t + ½ a t²
s = u t + ½ a t²
v² = u² + 2 a s
v² = u² + 2 a s
Newton’s second law defines the resultant force as the product of mass and acceleration.
牛顿第二定律将合外力定义为质量与加速度的乘积。
F = m a
F = m a
Momentum p is a vector quantity conserved in the absence of external resultant forces.
动量 p 是矢量，当无合外力作用时动量守恒。
p = m v
p = m v
Impulse = Δp = F Δt
冲量 = 动量变化量 = F Δt
2. Work, Energy & Power | 功、能与功率
Work done by a constant force acting at an angle θ to the displacement d.
恒力做功，力与位移夹角为 θ。
W = F d cos θ
W = F d cos θ
Kinetic energy and gravitational potential energy near the Earth’s surface.
动能与地球表面附近的重力势能。
KE = ½ m v²
KE = ½ m v²
GPE = m g h
GPE = m g h
Work–energy principle: net work done equals the change in kinetic energy.
功能原理：合外力做功等于动能的变化。
Wnet = ΔKE
Wnet = ΔKE
Mechanical power is the rate of doing work; it can also be expressed as force times velocity.
机械功率是做功的快慢程度；也可表示为力乘以速度。
P = W / t = F v
P = W / t = F v
3. Stress and Strain | 应力与应变
Direct (normal) stress σ and direct strain ε.
正应力 σ 和正应变 ε。
σ = F / A
σ = F / A
ε = ΔL / L₀
ε = ΔL / L₀
Young’s modulus E quantifies material stiffness in the linear elastic region.
杨氏模量 E 衡量材料在线弹性区域的刚度。
E = σ / ε
E = σ / ε
Poisson’s ratio ν relates lateral strain to axial strain.
泊松比 ν 将横向应变与轴向应变联系起来。
ν = −εlateral / εaxial
ν = −εlateral / εaxial
Factor of safety is the ratio of ultimate (or yield) stress to allowable working stress.
安全系数是极限应力（或屈服应力）与许用工作应力之比。
FoS = σult / σallow
FoS = σult / σallow
4. Beam Bending & Torsion | 梁的弯曲与扭转
The simple bending equation relates bending moment M, second moment of area I, bending stress σ, distance from the neutral axis y, Young’s modulus E, and radius of curvature R.
简单弯曲方程将弯矩 M、截面二次矩 I、弯曲应力 σ、到中性轴的距离 y、杨氏模量 E 和曲率半径 R 联系起来。
M / I = σ / y = E / R
M / I = σ / y = E / R
For a rectangular cross-section of breadth b and depth d, I = b d³ / 12 about the centroidal axis. For a solid circular shaft, I = π d⁴ / 64.
对于宽度 b、高度 d 的矩形截面，绕形心轴的二次矩 I = b d³ / 12。对于实心圆轴，I = π d⁴ / 64。
The torsion equation for a circular shaft connects torque T, polar second moment of area J, shear stress τ, radius r, modulus of rigidity G, and angle of twist per unit length θ/L.
圆轴的扭转方程连接扭矩 T、极截面二次矩 J、剪切应力 τ、半径 r、刚性模量 G 以及单位长度扭转角 θ/L。
T / J = τ / r = G θ / L
T / J = τ / r = G θ / L
For a solid circular shaft, J = π d⁴ / 32. For a hollow shaft of inner and outer diameters dᵢ and dₒ, J = π (dₒ⁴ − dᵢ⁴) / 32.
实心圆轴 J = π d⁴ / 32；对于内径 dᵢ、外径 dₒ 的空心轴，J = π (dₒ⁴ − dᵢ⁴) / 32。
5. Fluid Mechanics | 流体力学
The continuity equation for an incompressible fluid states that the volume flow rate is constant along a streamline.
不可压缩流体的连续性方程表明，沿流线的体积流量保持恒定。
A₁ v₁ = A₂ v₂
A₁ v₁ = A₂ v₂
Bernoulli’s equation expresses conservation of energy for a steady, inviscid, incompressible flow along a streamline.
伯努利方程表达了沿流线稳态、无黏、不可压缩流动的能量守恒。
p₁ + ½ ρ v₁² + ρ g h₁ = p₂ + ½ ρ v₂² + ρ g h₂
p₁ + ½ ρ v₁² + ρ g h₁ = p₂ + ½ ρ v₂² + ρ g h₂
The Reynolds number Re indicates whether flow is laminar or turbulent.
雷诺数 Re 用于判断流动是层流还是湍流。
Re = ρ v d / μ
Re = ρ v d / μ
Re = v d / ν (where kinematic viscosity ν = μ / ρ)
Re = v d / ν（运动粘度 ν = μ / ρ）
For pipe flow, the Darcy–Weisbach equation gives head loss hf = f (L / d) (v² / 2g), where f is the friction factor.
对于管流，达西–魏斯巴赫公式给出水头损失 hf = f (L / d) (v² / 2g)，其中 f 为摩擦系数。
6. Thermodynamics | 热力学
The first law of thermodynamics: the change in internal energy ΔU equals heat added to the system Q minus work done by the system W.
热力学第一定律：内能变化量 ΔU 等于系统吸收的热量 Q 减去系统对外做的功 W。
ΔU = Q − W
ΔU = Q − W
Ideal gas equation of state links pressure p, volume V, amount n, and absolute temperature T.
理想气体状态方程关联压强 p、体积 V、物质的量 n 和绝对温度 T。
p V = n R T
p V = n R T
p₁ V₁ / T₁ = p₂ V₂ / T₂ (for a fixed mass of gas)
p₁ V₁ / T₁ = p₂ V₂ / T₂（适用于质量固定的气体）
Specific heat capacity: the energy required to raise the temperature of a unit mass by one degree.
比热容：使单位质量物质温度升高一度所需的能量。
Q = m c ΔT
Q = m c ΔT
For gases, molar heat capacities at constant pressure Cₚ and constant volume Cᵥ are related by Cₚ − Cᵥ = R.
对于气体，定压摩尔热容 Cₚ 与定容摩尔热容 Cᵥ 满足 Cₚ − Cᵥ = R。
The thermal efficiency of a heat engine: η = Wnet / Qin = 1 − (Qout / Qin).
热机热效率：η = Wnet / Qin = 1 − (Qout / Qin)。
7. Electrical Fundamentals | 电路基础
Ohm’s law relates voltage V, current I, and resistance R.
欧姆定律关联电压 V、电流 I 和电阻 R。
V = I R
V = I R
Electrical power can be expressed in several useful forms.
电功率有多种实用的表达形式。
P = I V = I² R = V² / R
P = I V = I² R = V² / R
Resistors in series and parallel: Rtotal = R₁ + R₂ + … and 1/Rtotal = 1/R₁ + 1/R₂ + … .
串联与并联电阻：Rtotal = R₁ + R₂ + …；1/Rtotal = 1/R₁ + 1/R₂ + …。
Kirchhoff’s current law (KCL): the algebraic sum of currents entering a node is zero.
基尔霍夫电流定律（KCL）：流入节点的电流代数和为零。
Σ Iin = Σ Iout
Σ Iin = Σ Iout
Kirchhoff’s voltage law (KVL): around any closed loop, the algebraic sum of voltages is zero.
基尔霍夫电压定律（KVL）：沿任一闭合回路，电压代数和为零。
Σ V = 0
Σ V = 0
For a capacitor, charge Q = C V and energy stored = ½ C V². For an inductor, V = L (dI/dt) and energy stored = ½ L I².
电容：电荷 Q = C V，储能 = ½ C V²；电感：V = L (dI/dt)，储能 = ½ L I²。
8. Motors & Transformers | 电机与变压器
For a DC motor, the back EMF Eb is proportional to the product of motor constant K, magnetic flux Φ, and angular velocity ω.
对于直流电机，反电动势 Eb 与电机常数 K、磁通量 Φ 以及角速度 ω 的乘积成正比。
Eb = K Φ ω
Eb = K Φ ω
The torque T produced by the motor is proportional to the same constant, the flux, and the armature current Ia.
电机产生的扭矩 T 与此常数、磁通量以及电枢电流 Ia 成正比。
T = K Φ Ia
T = K Φ Ia
Armature circuit equation: supply voltage V = Eb + Ia Ra, where Ra is armature resistance.
电枢电路方程：电源电压 V = Eb + Ia Ra，其中 Ra 为电枢电阻。
For an ideal transformer, the voltage ratio equals the turns ratio, and the current ratio is the inverse.
对于理想变压器，电压比等于匝数比，电流比为其倒数。
Vp / Vs = Np / Ns
Vp / Vs = Np / Ns
Ip / Is = Ns / Np
Ip / Is = Ns / Np
Power input to an ideal transformer equals power output: Pp = Ps. Transformer efficiency η = Pout / Pin.
理想变压器输入功率等于输出功率：Pp = Ps。变压器效率 η = Pout / Pin。
9. Statics & Moments | 静力学与力矩
For a body in static equilibrium, the vector sum of all forces and the sum of all moments about any point must be zero.
刚体处于静力平衡时，所有力的矢量和以及绕任一点的所有力矩之和必须为零。
Σ F = 0
Σ F = 0
Σ M = 0
Σ M = 0
The moment of a force about a point is the product of the force magnitude and the perpendicular distance from the point to the line of action.
力对某一点的力矩等于力的大小乘以该点到力作用线的垂直距离。
M = F d
M = F d
For a couple (pair of equal, opposite parallel forces), the moment is F × perpendicular distance between the forces.
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