Year 12 Cambridge Engineering Formula & Theorem Quick Reference | 剑桥12年级工程：公式定理速查手册

📚 Year 12 Cambridge Engineering Formula & Theorem Quick Reference | 剑桥12年级工程：公式定理速查手册

This quick reference handbook brings together the essential formulas, theorems, and key relationships required for the Year 12 Cambridge Engineering course. It is designed as a revision tool to help you recall the core physical principles and mathematical models that underpin engineering analysis. Use it alongside your class notes and practice problems to deepen your understanding of mechanics, materials, fluids, thermal physics and electrical systems.

本速查手册汇集了剑桥12年级工程课程所需的重要公式、定理和关键关系式。它是一份复习工具，旨在帮助你回忆支撑工程分析的核心物理原理和数学模型。请结合课堂笔记和练习题使用，以加深对力学、材料、流体、热学及电气系统的理解。

1. SI Units and Prefixes | 国际单位制与词头

All engineering calculations rely on a coherent system of units. The SI base units used most frequently are: metre (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for temperature, and mole (mol) for amount of substance. Derived units such as the newton (N), joule (J), watt (W) and pascal (Pa) are built from these bases.

所有工程计算都依赖一套自洽的单位制。最常用的SI基本单位是：米（m）表示长度，千克（kg）表示质量，秒（s）表示时间，安培（A）表示电流，开尔文（K）表示温度，摩尔（mol）表示物质的量。诸如牛顿（N）、焦耳（J）、瓦特（W）和帕斯卡（Pa）等导出单位均由这些基本单位构成。

1 N = 1 kg·m·s⁻²
1 J = 1 N·m = 1 kg·m²·s⁻²
1 W = 1 J·s⁻¹ = 1 kg·m²·s⁻³
1 Pa = 1 N·m⁻² = 1 kg·m⁻¹·s⁻²

Common prefixes that you must be able to convert confidently: giga (G, 10⁹), mega (M, 10⁶), kilo (k, 10³), centi (c, 10⁻²), milli (m, 10⁻³), micro (μ, 10⁻⁶), nano (n, 10⁻⁹). Always work in base units before substituting into formulas to avoid scaling errors.

必须能够熟练转换的常用词头有：吉（G, 10⁹）、兆（M, 10⁶）、千（k, 10³）、厘（c, 10⁻²）、毫（m, 10⁻³）、微（μ, 10⁻⁶）、纳（n, 10⁻⁹）。始终先转换为基本单位再代入公式，以避免数量级错误。

2. Scalar and Vector Quantities | 标量与矢量

Scalars possess magnitude only (mass, temperature, energy, speed). Vectors possess both magnitude and direction (displacement, velocity, acceleration, force, momentum). Vector addition can be performed graphically by the tip-to-tail method or analytically by resolving each vector into perpendicular components, typically horizontal (x) and vertical (y).

标量只有大小（质量、温度、能量、速率）。矢量既有大小又有方向（位移、速度、加速度、力、动量）。矢量加法可用三角形法则（首尾相接）作图完成，也可将各矢量分解为相互垂直的分量（通常为水平x和竖直y方向）进行解析计算。

For a vector A at angle θ to the horizontal:

Aₑ = A cos θ, Aᵥ = A sin θ

Resultant magnitude: R = √(Rₑ² + Rᵥ²); direction: θ = tan⁻¹(Rᵥ / Rₑ).

对于与水平方向夹角为θ的矢量A：水平分量为A cos θ，竖直分量为A sin θ。合矢量大小：R = √(Rₑ² + Rᵥ²)；方向：θ = tan⁻¹(Rᵥ / Rₑ)。

3. Moments and Equilibrium | 力矩与平衡

The moment of a force about a point is the product of the force and the perpendicular distance from the point to the line of action of the force.

力对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

For a body in static equilibrium, two conditions must be satisfied simultaneously: (i) the vector sum of all forces is zero, ΣF = 0, and (ii) the sum of clockwise moments about any point equals the sum of anticlockwise moments about that point, ΣM = 0. This principle of moments is central to beam reactions, levers, and structures.

物体处于静力平衡时，必须同时满足两个条件：(i) 所有力的矢量和为零，ΣF = 0；(ii) 对任意一点，所有顺时针力矩之和等于所有逆时针力矩之和，ΣM = 0。这一力矩原理是梁的支座反力、杠杆和结构分析的核心。

When a force acts at an angle to the distance, the perpendicular distance becomes d sin φ or d cos φ depending on geometry. Always identify the pivot clearly and draw a free-body diagram to avoid sign errors.

当力与距离成一定角度时，垂直距离需根据几何关系取d sin φ 或 d cos φ。务必明确支点并绘制受力分析图，以避免正负号错误。

4. Stress, Strain and Young Modulus | 应力、应变与杨氏模量

Tensile stress σ is the force applied per unit cross-sectional area. Tensile strain ε is the extension per unit original length. For small deformations, many engineering materials obey Hooke’s law, where stress is proportional to strain.

拉伸应力σ是指施加在单位横截面积上的力。拉伸应变ε是指单位原始长度的伸长量。在小变形条件下，许多工程材料遵循胡克定律，即应力与应变成正比。

σ = F / A, ε = ΔL / L₀

Young modulus E = σ / ε = (F L₀) / (A ΔL)

The Young modulus E is a material property measured in pascals (Pa). A stiff material has a large E. Stress–strain graphs often show a linear elastic region up to the limit of proportionality, followed by yielding, plastic deformation and ultimate fracture. The area under the graph up to fracture represents the toughness of the material.

杨氏模量E是材料的一种属性，单位为帕斯卡（Pa）。刚度大的材料E值大。应力–应变曲线通常先有一段线性弹性区直至比例极限，随后出现屈服、塑性变形和最终断裂。曲线下方直至断裂点所围的面积代表材料的韧性。

5. Pressure and Density | 压力与密度

Pressure p is defined as the normal force acting per unit area. In a fluid at rest, the pressure at a depth h below the surface is given by the hydrostatic equation, where ρ is the fluid density and g is the acceleration of free fall.

压力p定义为作用在单位面积上的正压力。在静止流体中，液面以下深度h处的压强由流体静力学方程给出，其中ρ为流体密度，g为自由落体加速度。

p = F / A

p = ρ g h (for a fluid column)

Density ρ is mass per unit volume: ρ = m / V. Upthrust on an object fully or partially immersed in a fluid is equal to the weight of fluid displaced (Archimedes’ principle). The pressure at a point in a fluid acts equally in all directions, and pressure differences drive hydraulic systems and manometer readings.

密度ρ是单位体积的质量：ρ = m / V。物体全部或部分浸没在流体中所受的浮力等于被排开流体的重量（阿基米德原理）。流体中一点的压力向各个方向等大传递，压力差驱动着液压系统并影响液柱压差计的读数。

6. Fluid Flow and Continuity | 流体流动与连续性

For an incompressible fluid flowing steadily through a pipe or duct, the volume flow rate is conserved. This is the continuity equation, which relates the cross-sectional area A and the average flow speed v.

对于不可压缩流体的稳定流动，体积流量守恒。连续性方程将横截面积A与平均流速v联系起来。

A₁ v₁ = A₂ v₂ or Q = A v (constant)

A narrower section forces a higher velocity. The mass flow rate ṁ = ρ A v is constant if density remains unchanged. These principles are fundamental when analysing pipes, nozzles, and ventilation systems.

截面越小，流速越高。若密度不变，质量流量ṁ = ρ A v 为恒定值。这些原理是分析管道、喷嘴和通风系统的基础。

7. Bernoulli’s Equation | 伯努利方程

Bernoulli’s principle expresses conservation of energy for a steady, inviscid, incompressible flow along a streamline. It states that the sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant.

伯努利原理表示在稳定、无黏、不可压缩流体沿流线流动时的能量守恒。它指出，压力能、单位体积动能与单位体积势能之和保持恒定。

p + ½ ρ v² + ρ g h = constant

This equation explains phenomena such as the pressure drop in a faster-moving fluid (Venturi effect), lift on an aerofoil, and the operation of a Pitot tube. In application, you must identify two points on the same streamline and set the Bernoulli sum equal at both, accounting for energy losses if the fluid is not ideal.

该方程解释了诸多现象，例如流速增大处压力降低（文丘里效应）、翼型上的升力以及皮托管的工作原理。应用时，必须在同一流线上选取两点，令伯努利和相等；若非理想流体，还需计入能量损失。

8. Work, Energy and Power | 功、能与功率

Work is done when a force moves its point of application in the direction of the force. In vector terms, work is the product of force and displacement in the direction of the force.

力沿着其作用方向使作用点发生移动时做功力。在矢量表达中，功等于力与沿力方向的位移的乘积。

W = F d cos θ

Kinetic energy Eₙ = ½ m v². Gravitational potential energy near Earth’s surface is ΔEₚ = m g Δh. The work–energy principle states that the net work done on an object equals its change in kinetic energy. Power P is the rate of doing work or transferring energy.

动能 Eₙ = ½ m v²。近地表面的重力势能变化为 ΔEₚ = m g Δh。功能原理指出，合外力对物体做的功等于其动能的变化量。功率P是做功或能量转移的速率。

P = W / t = F v (for constant force and velocity parallel)

Efficiency of a machine is the ratio of useful output power to total input power, usually expressed as a percentage. In engineering, energy audits and power matching are essential to minimise losses and select appropriate motors or engines.

机械效率是指有用输出功率与总输入功率之比，通常以百分数表示。在工程中，进行能量核算和功率匹配对于减少损耗和选用合适的电动机或发动机至关重要。

9. Thermal Physics: Heat and Temperature | 热学：热与温度

Temperature is a measure of the average kinetic energy of particles in a substance. Heat is the transfer of thermal energy from a region of higher temperature to one of lower temperature. The SI unit of heat is the joule (J), but you will also encounter the kilowatt-hour (1 kWh = 3.6 × 10⁶ J).

温度是物质内部粒子平均动能的量度。热量是热能从高温区域向低温区域的传递。热量的国际单位是焦耳（J），但也会遇到千瓦时（1 kWh = 3.6 × 10⁶ J）。

The absolute temperature scale (kelvin) is related to Celsius by: T(K) = θ(°C) + 273.15. Thermal expansion is an important consideration in engineering: linear expansion ΔL = α L₀ ΔT, where α is the coefficient of linear expansion. Similarly, volume expansion follows ΔV = γ V₀ ΔT, with γ ≈ 3α for isotropic solids.

绝对温标（开尔文）与摄氏度的关系为：T(K) = θ(°C) + 273.15。热膨胀是工程中的重要考量：线膨胀 ΔL = α L₀ ΔT，其中α为线膨胀系数。类似地，体膨胀公式为 ΔV = γ V₀ ΔT，对各向同性固体，γ ≈ 3α。

10. Specific Heat Capacity and Latent Heat | 比热容与潜热

When a substance is heated, its temperature rise depends on its mass and its specific heat capacity c, which is the energy required to raise the temperature of 1 kg of the material by 1 K.

物质受热时，其温度升高取决于质量与比热容c，c是指1 kg材料每升高1 K所需的热量。

Q = m c ΔT

During a change of state (melting, boiling), the temperature remains constant while the substance absorbs latent heat. The specific latent heat L is the energy per kilogram needed to change state without temperature change.

在物态变化（熔化、沸腾）过程中，温度保持不变，物质吸收潜热。比潜热L是指单位质量物质在不发生温度变化的情况下完成物态转变所需的能量。

Q = m L (L = L_f for fusion, L = L_v for vaporisation)

These equations apply to calorimetry, engine cooling system design, and energy storage calculations. For instance, the high specific latent heat of water makes it an effective coolant.

这些方程应用于量热学、发动机冷却系统设计和储能计算。例如，水的高比潜热使其成为高效的冷却剂。

11. Electrical Quantities and Ohm’s Law | 电量与欧姆定律

Charge Q, current I, voltage V, and resistance R are linked by fundamental definitions. Current is the rate of flow of charge, and potential difference is the work done per unit charge.

电荷Q、电流I、电压V和电阻R由基本定义相联系。电流是电荷流动的速率，电势差是每单位电荷所做的功。

I = ΔQ / Δt

V = W / Q

Ohm’s law states that for a metallic conductor at constant temperature, the current through it is directly proportional to the potential difference across it.

欧姆定律指出，对于恒定温度下的金属导体，通过它的电流与其两端的电势差成正比。

V = I R

Electrical power dissipated in a component can be expressed as:

P = I V = I² R = V² / R

Resistance of a uniform wire depends on its resistivity ρ, length L and cross-sectional area A: R = ρ L / A. Resistors in series add: R_total = R₁ + R₂ + … ; in parallel: 1/R_total = 1/R₁ + 1/R₂ + … . Mastering these relationships is essential for circuit analysis.

元件消耗的电功率可表示为 P = I V = I² R = V² / R。均匀导线的电阻取决于其电阻率ρ、长度L和横截面积A：R = ρ L / A。电阻串联时相加：R_total = R₁ + R₂ + …；并联时：1/R_total = 1/R₁ + 1/R₂ + …。掌握这些关系对电路分析至关重要。

12. Kirchhoff’s Laws and Circuits | 基尔霍夫定律与电路

Kirchhoff’s current law (KCL): At any junction in a circuit, the sum of currents entering the junction equals the sum of currents leaving the junction. This expresses conservation of charge.

基尔霍夫电流定律（KCL）：在电路任一节点处，流入该节点的电流之和等于流出该节点的电流之和。它表达了电荷守恒。

Σ I_in = Σ I_out

Kirchhoff’s voltage law (KVL): In any closed loop, the algebraic sum of the potential differences (voltage rises and drops) is zero. This is a consequence of energy conservation.

基尔霍夫电压定律（KVL）：在任一闭合回路中，各部分电势差（电压升与电压降）的代数和为零。这是能量守恒的结果。

Σ V = 0 (around a closed loop)

These laws enable you to solve complex networks with multiple voltage sources and resistors. The potential divider rule is a useful shortcut: for two resistors in series across a supply V_supply, the voltage across R₂ is V_out = V_supply × (R₂ / (R₁ + R₂)). In sensing circuits, thermistors and light-dependent resistors (LDRs) are combined with fixed resistors to produce a varying output voltage that can control other systems.

这些定律使你能够求解含多个电压源和电阻的复杂网络。分压器规则是一个有用的捷径：两个电阻串联于电源V_supply两端时，R₂两端的电压为 V_out = V_supply × (R₂ / (R₁ + R₂))。在传感电路中，热敏电阻和光敏电阻（LDR）与固定电阻组合，产生可变的输出电压以控制其他系统。

An ideal ammeter has zero internal resistance and is connected in series; an ideal voltmeter has infinite internal resistance and is connected in parallel. In practice, their resistances affect readings, and you should be aware of loading effects when making measurements.

理想电流表内阻为零，串联在电路中；理想电压表内阻为无穷大，并联在电路中。实际测量时它们的内阻会影响读数，应意识到测量时的负载效应。

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Year 12 Cambridge Engineering Formula & Theorem Quick Reference | 剑桥12年级工程：公式定理速查手册