📚 AS CAIE Engineering: Quick Reference Handbook of Formulas & Principles | AS CAIE 工程：公式定理速查手册
In AS Level CAIE Engineering, mastering essential formulas and principles is crucial for solving problems across mechanics, materials, thermodynamics, and electrical systems. This concise handbook provides a structured reference for every key formula you need, explained with clarity and precision.
在 AS Level CAIE 工程课程中，掌握力学、材料、热力学和电气系统等领域的核心公式和原理是解题的关键。本手册为你整理了所有重要公式的结构化参考，内容清晰准确，方便速查。
1. Mechanics Fundamentals | 力学基础
The core equations of linear motion are the foundation of dynamics and statics analysis.
线性运动的核心方程是动力学和静力学分析的基础。
- v = u + at — Final velocity v equals initial velocity u plus acceleration a multiplied by time t.
- v = u + at — 末速度 v 等于初速度 u 加上加速度 a 乘以时间 t。
- s = ut + ½at² — Displacement s equals initial velocity times time plus half the acceleration times the square of time.
- s = ut + ½at² — 位移 s 等于初速度乘以时间加上加速度乘以时间平方的一半。
- v² = u² + 2as — The square of final velocity equals the square of initial velocity plus twice the product of acceleration and displacement.
- v² = u² + 2as — 末速度的平方等于初速度的平方加上加速度与位移乘积的两倍。
- F = ma — Newton's Second Law states that net force F equals mass m multiplied by acceleration a.
- F = ma — 牛顿第二定律指出，合力 F 等于质量 m 乘以加速度 a。
2. Moments and Equilibrium | 力矩与平衡
For a body in static equilibrium, both the sum of forces and the sum of moments must equal zero about any point.
对于处于静力平衡的物体，合力和对任意点的合力矩都必须为零。
- ΣF = 0 — The vector sum of all forces acting on a body in equilibrium is zero.
- ΣF = 0 — 作用于平衡物体的所有力的矢量和为零。
- ΣM = 0 — The sum of clockwise moments equals the sum of anticlockwise moments about any pivot.
- ΣM = 0 — 绕任意支点的顺时针力矩之和等于逆时针力矩之和。
- M = F × d — Moment M is the product of force F and the perpendicular distance d from the pivot.
- M = F × d — 力矩 M 等于力 F 与支点到力作用线的垂直距离 d 的乘积。
3. Stress and Strain | 应力与应变
Material behaviour under load is described by stress-strain relationships, which define strength and elasticity.
材料在载荷下的行为通过应力-应变关系来描述，这决定了材料的强度和弹性。
- σ = F / A — Normal stress σ (sigma) equals applied force F divided by cross-sectional area A.
- σ = F / A — 正应力 σ 等于施加的力 F 除以截面积 A。
- ε = ΔL / L₀ — Strain ε (epsilon) is the change in length ΔL divided by the original length L₀.
- ε = ΔL / L₀ — 应变 ε 等于长度变化 ΔL 除以原始长度 L₀。
- E = σ / ε — Young's modulus E measures stiffness as the ratio of stress to strain in the elastic region.
- E = σ / ε — 杨氏模量 E 衡量刚度，是在弹性区域内应力与应变的比值。
4. Pressure in Fluids | 流体中的压强
Fluid pressure acts equally in all directions and increases with depth due to the weight of the fluid above.
流体压强在各个方向上均匀作用，并随深度增加，因为上方流体的重量增大。
- p = F / A — Pressure p is the force F applied perpendicular to a surface divided by its area A.
- p = F / A — 压强 p 等于垂直于表面的力 F 除以面积 A。
- p = ρgh — Hydrostatic pressure in a fluid column equals density ρ times gravitational field strength g times height h.
- p = ρgh — 液柱静压强等于流体密度 ρ 乘以重力场强度 g 再乘以液柱高度 h。
- p₁v₁ = p₂v₂ — Boyle's Law for an ideal gas at constant temperature states that pressure and volume are inversely proportional.
- p₁v₁ = p₂v₂ — 玻义耳定律适用于恒温理想气体，压强与体积成反比。
5. Thermodynamics and Energy | 热力学与能量
Energy cannot be created or destroyed, only converted between forms; thermal energy transfer follows specific rules.
能量不能被创造或消灭，只能在形式之间转化；热能传递遵循特定规律。
- Q = mcΔθ — Heat energy Q equals mass m times specific heat capacity c times temperature change Δθ.
- Q = mcΔθ — 热量 Q 等于质量 m 乘以比热容 c 再乘以温度变化 Δθ。
- Q = mL — Latent heat Q for a phase change equals mass m times specific latent heat L.
- Q = mL — 相变潜热 Q 等于质量 m 乘以比潜热 L。
- E = P × t — Electrical energy transferred equals power P multiplied by time t.
- E = P × t — 电能传递量等于功率 P 乘以时间 t。
- η = Useful Output / Total Input — Efficiency η measures the ratio of useful output energy or power to total input.
- η = 有用输出 / 总输入 — 效率 η 衡量有用输出能量或功率与总输入之比。
6. Electrical Circuit Principles | 电路基本原理
Ohm's Law and Kirchhoff's laws form the backbone of electrical circuit analysis in engineering.
欧姆定律和基尔霍夫定律是工程学电路分析的基础。
- V = IR — Ohm's Law states that potential difference V equals current I multiplied by resistance R.
- V = IR — 欧姆定律指出，电势差 V 等于电流 I 乘以电阻 R。
- P = IV = I²R = V²/R — Electrical power P can be expressed in three equivalent forms using current, voltage, and resistance.
- P = IV = I²R = V²/R — 电功率 P 可用电流、电压和电阻的三种等效形式表达。
- Rₜ = R₁ + R₂ + … — Total resistance for resistors in series is the sum of individual resistances.
- Rₜ = R₁ + R₂ + … — 串联电阻的总电阻等于各电阻值之和。
- 1/Rₜ = 1/R₁ + 1/R₂ + … — For resistors in parallel, the reciprocal of total resistance equals the sum of reciprocals.
- 1/Rₜ = 1/R₁ + 1/R₂ + … — 并联电阻的总电阻倒数等于各电阻倒数之和。
7. Work, Power, and Mechanical Advantage | 功、功率与机械利益
Machines convert input work into useful output work, with mechanical advantage describing force amplification.
机械将输入功转化为有用输出功，机械利益描述了力的放大程度。
- W = Fd cosθ — Work W is the force F times displacement d times the cosine of the angle θ between them.
- W = Fd cosθ — 功 W 等于力 F 乘以位移 d 再乘以两者夹角 θ 的余弦。
- P = W / t — Power P is the rate of doing work, equal to work done divided by time taken.
- P = W / t — 功率 P 是做功的速率，等于所做的功除以所用时间。
- MA = Load / Effort — Mechanical Advantage (MA) is the ratio of output load force to input effort force.
- MA = 负载 / 施力 — 机械利益（MA）是输出负载力与输入施力之比。
8. Triangle of Forces and Vector Resolution | 力的三角形法则与矢量分解
Forces are vector quantities and can be resolved or combined using trigonometry and graphical methods.
力是矢量，可以通过三角学和图解法进行分解或合成。
- Fₓ = F cosθ, Fᵧ = F sinθ — A force F can be resolved into horizontal component Fₓ and vertical component Fᵧ.
- Fₓ = F cosθ, Fᵧ = F sinθ — 一个力 F 可分解为水平分量 Fₓ 和竖直分量 Fᵧ。
- F = √Fₓ² + Fᵧ² — The magnitude of a resultant force is found via Pythagoras' theorem.
- F = √Fₓ² + Fᵧ² — 合力的大小通过勾股定理求得。
- sinθ = Opposite/Hypotenuse, cosθ = Adjacent/Hypotenuse — Basic trigonometric ratios for resolving vectors.
- sinθ = 对边/斜边, cosθ = 邻边/斜边 — 用于分解矢量的基本三角比。
9. Dimensional Analysis and Units | 量纲分析与单位
All engineering equations must be dimensionally homogeneous; checking base units verifies correctness.
所有工程方程必须是量纲齐次的；检查基本单位可以验证方程的正确性。
| Quantity 物理量 | SI Unit 国际单位 | Symbol 符号 |
|---|---|---|
| Mass 质量 | kilogram 千克 | kg |
| Length 长度 | metre 米 | m |
| Time 时间 | second 秒 | s |
| Current 电流 | ampere 安培 | A |
Force is expressed in newtons (N), equivalent to kg·m/s²; pressure in pascals (Pa) is N/m².
力以牛顿 (N) 表示，等效单位为 kg·m/s²；压强以帕斯卡 (Pa) 表示，即 N/m²。
Power is measured in watts (W), equivalent to J/s or N·m/s; energy in joules (J) is N·m.
功率以瓦特 (W) 衡量，等效单位为 J/s 或 N·m/s；能量以焦耳 (J) 计，即 N·m。
10. Factor of Safety and Design Stress | 安全系数与设计应力
Engineers apply a factor of safety to account for uncertainties in loading, material properties, and manufacturing.
工程师采用安全系数来应对载荷、材料特性及制造过程中的不确定性。
- Factor of Safety = Ultimate Stress / Allowable Stress — It ensures components operate well below failure limits.
- 安全系数 = 极限应力 / 许用应力 — 它确保零件的工作应力远低于破坏极限。
- σ_allowable = σ_yield / n — Allowable design stress is the yield stress divided by the chosen factor of safety n.
- σ_许用 = σ_屈服 / n — 许用设计应力等于屈服应力除以所选安全系数 n。
- Typical n values range from 1.5 to 10, depending on application criticality and material variability.
- 典型的 n 值范围为 1.5 至 10，取决于应用的临界性和材料的变异性。
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