📚 Pre-U AQA Engineering: Quick Reference Formula & Theorem Handbook | Pre-U AQA 工程：公式定理速查手册
This handbook provides a concise collection of essential formulas, laws, and theorems for the AQA Pre-U Engineering course. Use it for quick revision and problem-solving, covering mechanics, materials, fluids, electronics and thermodynamics.
本手册汇集了AQA Pre-U工程课程的核心公式、定律与定理，涵盖力学、材料、流体、电子和热力学，适合快速复习和解题参考。
1. Equations of Motion | 运动学方程
For an object moving with constant acceleration a, the SUVAT equations relate displacement s, initial velocity u, final velocity v, acceleration a, and time t.
对于匀加速度 a 运动的物体，SUVAT 方程关联位移 s、初速度 u、末速度 v、加速度 a 和时间 t。
v = u + at — directly links final velocity to time when acceleration is constant.
v = u + at —— 当加速度恒定时，直接关联末速度与时间。
s = ut + ½at² — displacement when initial velocity and constant acceleration are known.
s = ut + ½at² —— 已知初速度和恒定加速度时的位移公式。
v² = u² + 2as — useful when time is not given.
v² = u² + 2as —— 当时间未给出时非常实用。
s = ½(u + v)t — average velocity multiplied by time.
s = ½(u + v)t —— 平均速度乘以时间。
2. Newton’s Laws of Motion | 牛顿运动定律
First Law: An object remains at rest or in uniform motion unless acted upon by a resultant force.
第一定律： 除非受到合力作用，物体将保持静止或匀速直线运动状态。
Second Law: F = ma — resultant force equals mass times acceleration.
第二定律： F = ma —— 合力等于质量乘以加速度。
Third Law: Forces occur in equal and opposite action–reaction pairs.
第三定律： 力以大小相等、方向相反的作用力与反作用力对出现。
On an inclined plane, the component of weight along the slope is mg sinθ and the normal reaction is mg cosθ.
在斜面上，重力沿斜面的分量为 mg sinθ，法向反作用力为 mg cosθ。
3. Work, Energy & Power | 功、能与功率
Work done: W = Fd cosθ, where θ is the angle between force and displacement.
功： W = Fd cosθ，其中 θ 为力与位移之间的夹角。
Kinetic energy: KE = ½mv² — energy due to motion.
动能： KE = ½mv² —— 由运动产生的能量。
Gravitational potential energy: GPE = mgh, where h is the vertical height.
重力势能： GPE = mgh，其中 h 为垂直高度。
Principle of conservation of energy: Total energy in an isolated system remains constant.
能量守恒原理： 孤立系统的总能量保持不变。
Power: P = W/t = Fv for a constant force moving at speed v.
功率： P = W/t = Fv，适用于恒力以速度 v 运动的情况。
Efficiency: η = (useful output energy / total input energy) × 100%.
效率： η = (有用输出能量 / 总输入能量) × 100%。
4. Momentum & Impulse | 动量与冲量
Momentum: p = mv, a vector quantity.
动量： p = mv，为矢量。
Impulse: J = FΔt = Δp = m(v − u) — equal to the area under a force–time graph.
冲量： J = FΔt = Δp = m(v − u) —— 等于力–时间图下的面积。
Conservation of momentum: In a closed system, total momentum before collision equals total momentum after collision: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
动量守恒： 在封闭系统中，碰撞前的总动量等于碰撞后的总动量：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。
For elastic collisions, kinetic energy is conserved; for inelastic collisions, only momentum is conserved.
弹性碰撞中动能守恒；非弹性碰撞中仅动量守恒。
5. Circular Motion | 圆周运动
Angular velocity: ω = θ/t = 2πf, measured in rad s⁻¹.
角速度： ω = θ/t = 2πf，单位为弧度每秒。
Linear speed: v = ωr.
线速度： v = ωr。
Centripetal acceleration: a = v²/r = ω²r — always directed towards the centre.
向心加速度： a = v²/r = ω²r —— 总指向圆心。
Centripetal force: F = mv²/r = mω²r — required to keep an object in circular motion.
向心力： F = mv²/r = mω²r —— 维持物体圆周运动所需的力。
6. Stress, Strain & Young’s Modulus | 应力、应变与杨氏模量
Tensile stress: σ = F/A, where F is the applied force and A is the cross‑sectional area.
拉应力： σ = F/A，其中 F 为施加的力，A 为横截面积。
Tensile strain: ε = ΔL/L₀, the ratio of extension to original length.
拉应变： ε = ΔL/L₀，伸长量与原长之比。
Young’s modulus: E = σ/ε — a measure of material stiffness, valid within the proportional limit.
杨氏模量： E = σ/ε —— 衡量材料刚度的量，在比例极限内有效。
Hooke’s law: F = kx, where k is the spring constant and x is the extension.
胡克定律： F = kx，其中 k 为弹簧常数，x 为伸长量。
Elastic strain energy: U = ½Fx = ½kx².
弹性应变能： U = ½Fx = ½kx²。
7. Fluid Mechanics | 流体力学
Pressure due to a liquid column: P = ρgh, where ρ is density, g is gravitational field strength, h is height.
液柱产生的压强： P = ρgh，ρ 为密度，g 为重力场强，h 为高度。
Archimedes’ principle: Upthrust = weight of displaced fluid = ρgV.
阿基米德原理： 浮力 = 排开流体的重量 = ρgV。
Continuity equation: For an incompressible fluid, A₁v₁ = A₂v₂.
连续性方程： 对于不可压缩流体，A₁v₁ = A₂v₂。
Bernoulli’s equation (simplified): P + ½ρv² + ρgh = constant along a streamline.
简化伯努利方程： 沿流线，P + ½ρv² + ρgh = 常数。
8. Electric Circuits | 电路
Ohm’s law: V = IR, where V is potential difference, I is current, R is resistance.
欧姆定律： V = IR，V 为电势差，I 为电流，R 为电阻。
Kirchhoff’s current law: ΣI entering a junction = ΣI leaving a junction.
基尔霍夫电流定律： 进入节点的电流总和 = 离开节点的电流总和。
Kirchhoff’s voltage law: In any closed loop, ΣEMF = Σpotential drops.
基尔霍夫电压定律： 在任意闭合回路中，电动势总和 = 电势降落总和。
Resistors in series: Rₜₒₜₐₗ = R₁ + R₂ + …
串联电阻： Rₜₒₜₐₗ = R₁ + R₂ + …
Resistors in parallel: 1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + …
并联电阻： 1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + …
Electrical power: P = IV = I²R = V²/R.
电功率： P = IV = I²R = V²/R。
Capacitance: C = Q/V; time constant τ = RC for RC circuits.
电容： C = Q/V；RC 电路的时间常数 τ = RC。
9. Thermodynamics & Heat Transfer | 热力学与热传递
Ideal gas law: pV = nRT, where p is pressure, V volume, n number of moles, R molar gas constant, T temperature in kelvin.
理想气体状态方程： pV = nRT，p 为压强，V 为体积，n 为摩尔数，R 为摩尔气体常数，T 为开尔文温度。
Heat energy: Q = mcΔθ, where c is specific heat capacity.
热量： Q = mcΔθ，c 为比热容。
Conduction rate: Q/t = kA(ΔT/d), where k is thermal conductivity, A area, d thickness.
热传导速率： Q/t = kA(ΔT/d)，k 为导热系数，A 为面积，d 为厚度。
Thermal efficiency: η = (work output / heat input); maximum Carnot efficiency = 1 − T_cold/T_hot.
热效率： η = (输出功 / 输入热量)；卡诺最大效率 = 1 − T_c/T_h。
10. Engineering Mathematics | 工程数学常用公式
Trigonometric identities: sin²θ + cos²θ = 1; tanθ = sinθ / cosθ.
三角恒等式： sin²θ + cos²θ = 1；tanθ = sinθ / cosθ。
Sine rule: a/sin A = b/sin B = c/sin C.
正弦定理： a/sin A = b/sin B = c/sin C。
Cosine rule: a² = b² + c² − 2bc cos A.
余弦定理： a² = b² + c² − 2bc cos A。
Area of a triangle: ½ab sin C.
三角形面积： ½ab sin C。
Vector dot product: a·b = |a||b| cosθ, useful for work calculations.
向量点积： a·b = |a||b| cosθ，常用于功的计算。
Differentiation basics: d/dx (xⁿ) = nxⁿ⁻¹; d/dx (sin x) = cos x.
微分基础： d/dx (xⁿ) = nxⁿ⁻¹；d/dx (sin x) = cos x。
Integration basics: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C; ∫ cos x dx = sin x + C.
积分基础： ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C；∫ cos x dx = sin x + C。
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