📚 Year 12 CAIE Engineering: Formula & Theorem Quick Reference Handbook | Year 12 CAIE 工程：公式定理速查手册
This concise handbook gathers the essential formulas and theorems you will encounter throughout the Year 12 CAIE Engineering course. Use it as a quick revision aid to reinforce your understanding of stress analysis, electrical principles, mechanics, and fluid behaviour. Each section pairs a clear English explanation with a Chinese translation, followed by the key formulas presented in a ready-to-recall format.
这本简练的手册汇集了你在 Year 12 CAIE 工程课程中将要遇到的核心公式和定理。可将其用作快速复习工具，以巩固对应力分析、电学原理、力学和流体行为的理解。每一节都配以清晰的英文解释、对应的中文翻译，以及便于记忆的关键公式。
1. Stress and Strain | 应力与应变
Stress measures the internal resistance of a material to an applied force, expressed as force per unit area. Direct strain quantifies the deformation by comparing the change in length to the original length. These two concepts form the foundation of material behaviour analysis.
应力衡量材料对所施加载荷的内部抗力，表示为每单位面积上的力。正应变通过比较长度变化与原始长度来量化变形。这两个概念是材料行为分析的基础。
σ = F / A ε = ΔL / L₀
Here σ = direct stress (Pa or N/m²), F = applied force (N), A = cross-sectional area (m²). ε = strain (no units), ΔL = change in length (m), L₀ = original length (m).
其中 σ = 正应力（帕斯卡，Pa），F = 施加的力（N），A = 横截面积（m²）。ε = 应变（无量纲），ΔL = 长度变化（m），L₀ = 原始长度（m）。
2. Young’s Modulus | 杨氏模量
Young’s modulus (E) describes the stiffness of a material. It is the gradient of the stress–strain curve within the linear elastic region, where Hooke’s law applies. A higher value indicates a stiffer material that deforms less under load.
杨氏模量 (E) 描述材料的刚度。它是应力-应变曲线在线弹性区域内的斜率，在该区域内满足胡克定律。模量值越高，表示材料刚度越大，在载荷下变形越小。
E = σ / ε = (F L₀) / (A ΔL)
The unit is pascals (Pa) and the relationship holds only up to the limit of proportionality.
单位为帕斯卡 (Pa)，该关系仅在比例极限内成立。
3. Factor of Safety | 安全系数
Engineers incorporate a factor of safety (FoS) to ensure that structures and components operate well below their failure limits. It accounts for uncertainties in loading, material defects, and unforeseen stresses.
工程师引入安全系数 (FoS) 以确保结构和部件在远低于其失效极限的状态下工作。它考虑了载荷不确定性、材料缺陷和意外应力。
FoS = Ultimate Stress / Working Stress
It can also be expressed in terms of loads: FoS = Failure load / Design load. A typical FoS for structural steel may be around 1.5 to 2.0.
也可用载荷表示：安全系数 = 失效载荷 / 设计载荷。结构钢的典型安全系数约为 1.5 至 2.0。
4. Bending Moment and Shear Force | 弯矩与剪力
When a beam is subjected to transverse loads, internal shear forces and bending moments develop. A bending moment tends to bend the beam, while a shear force tends to slide one part of the beam relative to the other.
当梁承受横向载荷时，其内部会产生剪力和弯矩。弯矩使梁趋于弯曲，而剪力则使梁的一部分相对于另一部分产生滑移趋势。
M = F × d V = ΣF_vertical
For a simply supported beam with a central point load W and span L, the maximum bending moment occurs at the centre: M_max = WL / 4.
对于跨度为 L、中心受集中载荷 W 的简支梁，最大弯矩出现在跨中：M_max = WL / 4。
5. Linear Thermal Expansion | 线性热膨胀
Most solid materials expand when heated. The change in length is proportional to the original length and the temperature change, with the coefficient of linear expansion (α) characterizing the material.
大多数固体材料受热时会膨胀。长度的变化与原始长度及温度变化成正比，线膨胀系数 (α) 表征了材料的这一特性。
ΔL = α L₀ ΔT
ΔL = change in length (m), L₀ = original length (m), ΔT = temperature change (K or °C), α = coefficient of linear expansion (K⁻¹ or °C⁻¹).
ΔL = 长度变化 (m)，L₀ = 原始长度 (m)，ΔT = 温度变化 (K 或 °C)，α = 线膨胀系数 (K⁻¹ 或 °C⁻¹)。
6. Ohm’s Law and Electrical Power | 欧姆定律与电功率
Ohm’s law defines the relationship between voltage, current and resistance in a conductor at constant temperature. The power dissipated in a resistive element can be expressed in three useful forms.
欧姆定律定义了在恒温条件下导体中电压、电流和电阻之间的关系。电阻元件中消耗的功率可以用三种有用的形式表示。
V = I R P = I V = I² R = V² / R
V = voltage (V), I = current (A), R = resistance (Ω), P = power (W).
V = 电压（V），I = 电流（A），R = 电阻（Ω），P = 功率（W）。
7. Kirchhoff’s Laws | 基尔霍夫定律
Kirchhoff’s current law (KCL) and voltage law (KVL) are essential for analysing electrical circuits. KCL states that the total current entering a junction equals the total current leaving it. KVL states that the algebraic sum of the voltages around any closed loop is zero.
基尔霍夫电流定律 (KCL) 和电压定律 (KVL) 是分析电路的基础。KCL 指出流入一个节点的总电流等于流出该节点的总电流。KVL 指出沿任何闭合回路的电压代数和为零。
ΣI_in = ΣI_out ΣV_loop = 0
These laws enable the calculation of unknown currents and voltages in complex circuits containing multiple loops and components.
这些定律能够计算含有多个回路和元件的复杂电路中未知的电流与电压。
8. Gear Ratio and Mechanical Advantage | 齿轮比与机械效益
Gears transmit rotary motion and torque between shafts. The gear ratio determines the relationship between input and output speeds, while mechanical advantage relates the output force to the input force in a machine.
齿轮用于在轴之间传递旋转运动和转矩。齿轮比决定了输入与输出转速的关系，而机械效益则关联着机械的输出力与输入力。
Gear Ratio = N₂ / N₁ = ω₁ / ω₂ MA = F_out / F_in
Where N = number of teeth, ω = angular velocity. A gear ratio greater than 1 indicates a reduction in speed and an increase in torque.
其中 N = 齿数，ω = 角速度。齿轮比大于 1 表示减速增扭。
9. Moments and Equilibrium | 力矩与平衡
A moment is the turning effect of a force about a pivot. For an object in static equilibrium, the sum of clockwise moments about any point equals the sum of anticlockwise moments, and the resultant force is zero.
力矩是力绕支点产生的转动效应。对于处于静力平衡的物体，绕任一点的顺时针力矩之和等于逆时针力矩之和，且合力为零。
Moment = F × d⊥ ΣM_clockwise = ΣM_anticlockwise
d⊥ is the perpendicular distance from the pivot to the line of action of the force. This principle is used to analyse levers, beams, and frameworks.
d⊥ 是支点到力作用线的垂直距离。此原理用于分析杠杆、梁和框架。
10. Bernoulli’s Principle | 伯努利原理
For an incompressible, non-viscous fluid flowing steadily along a streamline, the total mechanical energy per unit volume remains constant. This principle connects pressure, kinetic energy, and gravitational potential energy.
对于沿流线稳定流动的不可压缩、无粘性流体，单位体积的总机械能保持恒定。该原理将压力、动能和重力势能联系了起来。
P + ½ρv² + ρgh = constant
P = static pressure (Pa), ρ = fluid density (kg/m³), v = flow velocity (m/s), g = gravitational acceleration (9.81 m/s²), h = height above a reference (m).
P = 静压 (Pa)，ρ = 流体密度 (kg/m³)，v = 流速 (m/s)，g = 重力加速度 (9.81 m/s²)，h = 相对于基准面的高度 (m)。
11. Work, Energy and Power | 功、能与功率
Work is done when a force moves its point of application. Energy is the capacity to do work, and power is the rate of doing work or transferring energy. Efficiency measures how much input energy is usefully converted.
当力使其作用点发生位移时即做功。能量是做功的能力，功率是做功或传递能量的速率。效率衡量输入能量中有多少被有效转换。
W = F d cosθ E_k = ½mv² E_p = mgh
P = W / t = F v η = P_out / P_in
W = work (J), F = force (N), d = displacement (m), θ = angle between force and displacement. η = efficiency (often expressed as a percentage).
W = 功 (J)，F = 力 (N)，d = 位移 (m)，θ = 力与位移的夹角。η = 效率（通常以百分比表示）。
12. Basic Trigonometry for Forces | 力的基础三角学
Forces acting at an angle to a reference system are best analysed by resolving them into perpendicular components. The resultant of several forces can then be found by vector addition.
对于与参考系成一定角度的力，最佳分析方法是将它们分解为相互垂直的分量。然后通过矢量加法可求得多个力的合力。
F_x = F cosθ F_y = F sinθ F_resultant = √(F_x² + F_y²)
This technique is essential for solving equilibrium problems on inclined planes and for determining unknown forces in pin-jointed frameworks.
这项技术对于解决斜面上的平衡问题以及确定铰接框架中的未知力至关重要。
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