Year 13 CIE Engineering: Quick Reference Handbook of Formulas and Theorems | Year 13 CIE 工程：公式定理速查手册

📚 Year 13 CIE Engineering: Quick Reference Handbook of Formulas and Theorems | Year 13 CIE 工程：公式定理速查手册

This comprehensive quick-reference handbook covers the essential formulas, theorems, and principles required for the Year 13 CIE Engineering syllabus. It is designed as a portable revision tool that organises key concepts from mechanics, materials, thermodynamics, fluid dynamics, electrical theory, and essential mathematics into a single, easily navigable document. Each section presents formulas in a clear central display followed by concise explanations of the variables and their typical units, helping you to reinforce understanding and improve recall during the final stages of exam preparation.

这份全面的公式定理速查手册涵盖了 Year 13 CIE 工程课程所必需的核心公式、定理与原理。它被设计为一本便携式复习工具，将力学、材料、热力学、流体力学、电气理论以及基础数学中的关键知识点整合在一份易于查阅的文件中。每个章节都以清晰的居中展示公式，并附有对变量及常见单位的简要解释，帮助你在备考的最后阶段强化理解、提升记忆效率。

1. Fundamental Concepts and Units | 基本概念与单位

All engineering calculations rely on a consistent system of units. The International System (SI) uses base units for mass (kilogram, kg), length (metre, m), time (second, s), electric current (ampere, A), temperature (kelvin, K), and amount of substance (mole, mol). Derived units such as the newton (N) for force and the pascal (Pa) for pressure are built from these bases.

所有工程计算都依赖于统一的单位制。国际单位制（SI）采用质量（千克，kg）、长度（米，m）、时间（秒，s）、电流（安培，A）、温度（开尔文，K）和物质的量（摩尔，mol）作为基本单位。导出单位如力的牛顿（N）和压强的帕斯卡（Pa）均由这些基本单位组合而成。

Quantity	SI Unit	Symbol
Force	newton	N
Pressure	pascal	Pa
Energy, work	joule	J
Power	watt	W
Voltage	volt	V
Resistance	ohm	Ω

Dimensional analysis is a powerful technique to verify equations and derive unknown relationships. By expressing each quantity in terms of mass (M), length (L) and time (T), you can check the homogeneity of a physical equation. For example, the left-hand side and right-hand side of an energy balance must both have dimensions M L² T⁻².

量纲分析是验证方程和推导未知关系的有效技术。将每个物理量用质量（M）、长度（L）和时间（T）的基本量纲表示，即可检验物理方程式的齐次性。例如，能量平衡等式的左侧和右侧必须同具量纲 M L² T⁻²。

2. Statics and Force Analysis | 静力学与受力分析

For a body in static equilibrium, the vector sum of all forces and the sum of all moments about any point must both be zero. This yields two essential equations that govern the solution of trusses, frames and balanced structures.

对于处于静力平衡的物体，所有力的矢量和以及对于任意点的所有力矩之和都必须为零。由此得出两个基本方程，用于求解桁架、框架和平衡结构。

ΣF = 0 and ΣM = 0

When resolving forces on an inclined plane, the weight mg is split into components parallel to the slope (mg sinθ) and perpendicular to the slope (mg cosθ). The normal reaction N often cancels the perpendicular component, while friction f acts up or down the slope depending on the direction of impending motion.

在斜面上分解力时，重力 mg 被分解为平行于斜面的分量（mg sinθ）和垂直于斜面的分量（mg cosθ）。法向反力 N 通常与垂直分量平衡，而摩擦力 f 则根据即将运动的方向沿斜面向上或向下作用。

F_parallel = mg sinθ, F_perpendicular = mg cosθ

The moment of a force about a pivot is the product of the force and the perpendicular distance from the pivot to the line of action of the force. A couple consists of two equal, opposite and parallel forces that produce pure rotation; its moment is given by one force times the perpendicular distance between the lines of action.

力对支点的力矩等于力的大小乘以支点到力作用线的垂直距离。力偶由两个大小相等、方向相反且平行的力组成，产生纯转动效果；其力矩等于其中一个力乘以两力作用线之间的垂直距离。

Moment = F × d

3. Stress, Strain and Elasticity | 应力、应变与弹性

Normal stress σ is the internal force per unit area resisting an applied tensile or compressive load. Tensile strain ε is the extension per unit original length. For many engineering materials, stress is proportional to strain up to the limit of proportionality, as stated by Hooke’s law.

正应力 σ 是抵抗拉伸或压缩载荷的内部单位面积力。拉伸应变 ε 是单位原长的伸长量。对许多工程材料而言，在比例极限以下，应力与应变成正比，正如胡克定律所述。

σ = F / A and ε = ΔL / L₀

The constant of proportionality between stress and strain in the linear region is the Young modulus E, a measure of material stiffness. Its value is obtained from the gradient of the stress–strain graph and is independent of the specimen’s dimensions.

在线性区域中，应力与应变的比值常数即为杨氏模量 E，它是衡量材料刚度的指标。其值可通过应力–应变图的斜率求得，且与试样的尺寸无关。

E = σ / ε

Poisson’s ratio ν compares lateral contraction to longitudinal extension. For most metals, ν is around 0.3. Ductile materials exhibit significant plastic deformation before fracture, while brittle materials fail with little warning. Ultimate tensile strength (UTS) is the maximum stress a material can withstand.

泊松比 ν 对照横向收缩与纵向伸长。对于大多数金属，ν 约为 0.3。延性材料在断裂前表现出显著的塑性变形，而脆性材料则在几乎没有预警的情况下失效。极限抗拉强度（UTS）是材料能承受的最大应力。

ν = – ε_lateral / ε_axial

4. Shear and Torsion | 剪切与扭转

Shear stress τ acts parallel to a surface and is produced by a force that tends to make one part of a body slide over another. For a single rivet or bolt in a lap joint, the average shear stress is the applied force divided by the cross-sectional area in shear. Shear strain γ is the angular distortion in radians.

剪切应力 τ 平行于表面作用，由使物体一部分相对于另一部分产生滑动的力所引起。对于搭接接头中的单个铆钉或螺栓，平均剪切应力等于所施加的力除以受剪的横截面积。剪应变 γ 是以弧度表示的角变形。

τ = F / A_shear

The shear modulus (or modulus of rigidity) G relates shear stress to shear strain within the elastic range. It is analogous to Young’s modulus but for shear loading. For a shaft under torsion, the angle of twist θ is proportional to the applied torque T and the shaft length L, and inversely proportional to the polar second moment of area J and the shear modulus G.

剪切模量（刚性模量）G 在弹性范围内将剪切应力与剪应变联系起来。它类似于杨氏模量，但用于剪切载荷。对于受扭转的轴，扭转角 θ 与施加的扭矩 T 和轴长 L 成正比，与截面极惯性矩 J 和剪切模量 G 成反比。

θ = T L / (G J)

For a solid circular shaft, the polar second moment of area J is given by πd⁴/32, where d is the diameter. The maximum shear stress in a twisted circular shaft occurs at the outermost surface and is calculated from τ_max = T r / J, where r is the outer radius.

对于实心圆轴，截面极惯性矩 J 为 πd⁴/32，其中 d 为直径。受扭圆轴的最大剪切应力出现在最外表面，其计算公式为 τ_max = T r / J，其中 r 为外半径。

J = πd⁴ / 32 and τ_max = T r / J

5. Dynamics and Kinematics | 动力学与运动学

The equations of motion for constant linear acceleration link displacement s, initial velocity u, final velocity v, acceleration a and time t. These four SUVAT equations are the cornerstones of kinematic analysis in one dimension.

用于恒定线性加速度的运动方程将位移 s、初速度 u、末速度 v、加速度 a 和时间 t 联系起来。这四个 SUVAT 方程是一维运动学分析的基石。

v = u + a t

s = u t + ½ a t²

s = (u + v) t / 2

v² = u² + 2 a s

Newton’s second law states that the net force acting on an object equals the rate of change of its momentum. For constant mass, this reduces to F = m a. The momentum p of a body is the product of its mass and velocity, and is conserved in a closed system when no external resultant force acts.

牛顿第二定律指出，作用在物体上的净力等于其动量的变化率。对于质量恒定的情况，它简化为 F = m a。物体的动量 p 是其质量与速度的乘积，并且当没有外部合力作用时，动量在封闭系统中守恒。

F = m a and p = m v

For rotational motion, analogous equations apply. Torque T replaces force, moment of inertia I replaces mass, and angular acceleration α replaces linear acceleration. The fundamental rotational equation is T = I α.

对于转动运动，可应用类比方程。扭矩 T 替代力，转动惯量 I 替代质量，角加速度 α 替代线加速度。基本的转动方程为 T = I α。

T = I α

6. Work, Energy and Power | 功、能与功率

Work done by a constant force is the product of the force and the displacement in the direction of the force. When the force is not parallel to the displacement, the scalar product is used. The work–energy principle states that the net work done on an object equals its change in kinetic energy.

恒力所做的功等于力的大小乘以沿力方向的位移。当力与位移不平行时，应使用标量积。功能原理指出，对物体所做的净功等于其动能的变化量。

W = F d cosθ or W = F · s

Kinetic energy KE is the energy an object possesses by virtue of its motion, while gravitational potential energy GPE is the energy stored due to its position in a gravitational field. In a conservative system, the sum of KE and GPE remains constant if no external work is done.

动能 KE 是物体因运动而具有的能量，而重力势能 GPE 是因其在重力场中的位置而储存的能量。在保守系统中，若不施加外部功，动能与重力势能的总和保持不变。

KE = ½ m v² and GPE = m g h

Power is the rate at which work is done or energy is transferred. The mechanical power output of an engine can be expressed as the product of force and velocity, or as torque times angular velocity for rotating machinery.

功率是做功或传递能量的速率。发动机的机械功率输出可表示为力与速度的乘积，或对旋转机械表示为扭矩与角速度的乘积。

P = W / t = F v and P = T ω

7. Fluid Mechanics Basics | 流体力学基础

Hydrostatic pressure at a depth h in a fluid of density ρ is given by p = ρ g h, where g is the gravitational field strength. This pressure acts equally in all directions and is independent of the shape of the container.

在密度为 ρ 的流体中，深度 h 处的流体静压由 p = ρ g h 给出，其中 g 为重力场强度。该压强在各个方向上均等作用，且与容器形状无关。

p = ρ g h

The continuity equation expresses mass conservation in a flowing fluid. For an incompressible fluid in a closed pipe, the volume flow rate Q is constant, so the product of cross-sectional area A and flow velocity v remains the same at any two points along the streamline.

连续性方程表达了流动流体中的质量守恒。对于封闭管道中的不可压缩流体，体积流量 Q 恒定，因此横截面积 A 与流速 v 的乘积在沿流线的任意两点保持不变。

A₁ v₁ = A₂ v₂

Bernoulli’s principle relates pressure, velocity and elevation along a streamline for an ideal, inviscid, steady flow. The sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant when frictional losses can be ignored.

伯努利原理将理想、无黏性、稳定流动中沿流线的压强、速度与高度关联起来。当摩擦损失可忽略时，压强能、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

8. Thermodynamics and Heat Transfer | 热力学与传热

The first law of thermodynamics is a statement of energy conservation applied to thermal systems. The increase in internal energy of a system ΔU equals the net heat supplied Q minus the net work done by the system W. For a closed cycle, the net change in internal energy is zero.

热力学第一定律是能量守恒应用于热系统的表述。系统内能的增量 ΔU 等于净供给的热量 Q 减去系统对外做的净功 W。对于封闭循环，内能的净变化为零。

ΔU = Q – W

For an ideal gas, the relationship between pressure, volume and temperature is given by the universal gas law. The gas constant R is 8.31 J mol⁻¹ K⁻¹, and n is the number of moles. Isothermal processes occur at constant temperature (pV = constant), while adiabatic processes involve no heat exchange (pV^γ = constant for an ideal gas).

对于理想气体，压强、体积和温度之间的关系由通用气体定律给出。气体常数 R 为 8.31 J mol⁻¹ K⁻¹，n 为摩尔数。等温过程发生在恒温条件下（pV = 常数），而绝热过程不涉及热量交换（对理想气体有 pV^γ = 常数）。

p V = n R T

The efficiency of a heat engine is the fraction of heat input that is converted into useful work. For an ideal Carnot engine operating between a hot reservoir at temperature T_hot and a cold reservoir at T_cold (in kelvin), the maximum possible efficiency is determined solely by the two temperatures.

热机的效率为输入热量中转化为有用功的比例。对于工作在高温热源 T_hot 和低温热源 T_cold（开尔文温度）之间的理想卡诺热机，其最大可能效率仅由这两个温度决定。

η = W_out / Q_in = 1 – T_cold / T_hot

Heat transfer by conduction through a slab of material follows Fourier’s law. The rate of heat flow ḳ (or Q/t) is proportional to the thermal conductivity k, the cross-sectional area A, and the temperature gradient ΔT/Δx.

通过材料平板的导热遵循傅里叶定律。热流量率 ḳ（或 Q/t）与导热系数 k、横截面积 A 以及温度梯度 ΔT/Δx 成正比。

Q̇ = –k A (ΔT / Δx)

9. Electrical and Electronic Principles | 电气与电子原理

Ohm’s law states that the current I through a resistor is directly proportional to the potential difference V across it, provided temperature and other physical conditions remain constant. Resistance R is the ratio of voltage to current.

欧姆定律指出，在温度和其他物理条件保持不变时，通过电阻器的电流 I 与它两端的电势差 V 成正比。电阻 R 是电压与电流的比值。

V = I R

Kirchhoff’s current law (KCL) requires that the algebraic sum of currents entering a node is zero, meaning total current in equals total current out. Kirchhoff’s voltage law (KVL) states that the sum of electromotive forces and potential differences around any closed loop is zero, reflecting energy conservation.

基尔霍夫电流定律（KCL）要求流入一个节点的电流代数和为零，即流入的总电流等于流出的总电流。基尔霍夫电压定律（KVL）指出，沿任一闭合回路的电动势与电势差之和为零，这体现了能量守恒。

ΣI_in = ΣI_out and ΣV_loop = 0

Electrical power P dissipated in a resistor can be expressed in three useful forms by combining Ohm’s law. The total resistance of resistors in series is the sum of individual resistances; for resistors in parallel, the reciprocal of the total resistance is the sum of the reciprocals.

电阻器中消耗的电功率 P 可通过与欧姆定律结合，表示为三种实用形式。串联电阻的总电阻为各电阻值之和；对于并联电阻，总电阻的倒数等于各电阻倒数之和。

P = I V = I² R = V² / R

Series: R_total = R₁ + R₂ + …

Parallel: 1/R_total = 1/R₁ + 1/R₂ + …

The operational amplifier (op-amp) is a fundamental building block used in many analogue circuits. In a negative-feedback configuration, the closed-loop gain for an inverting amplifier is set by the ratio of two external resistors. The virtual-earth principle assumes the inverting input is at nearly zero voltage while drawing negligible current.

运算放大器（运放）是许多模拟电路中使用的基础构件。在负反馈配置中，反相放大器的闭环增益由两个外部电阻的比值设定。虚地原理假设反相输入端电压近乎为零，且吸收的电流可以忽略不计。

Gain (inverting) = –R_f / R_in

10. Engineering Mathematics Toolkit | 工程数学工具箱

Trigonometric identities are used extensively in resolving forces, analysing alternating currents and modelling mechanical vibrations. The sine rule and cosine rule are indispensable for non-right-angled triangles in vector addition and structural geometry.

三角恒等式被广泛用于力的分解、交流电分析以及机械振动建模。正弦定理和余弦定理对于矢量加法及结构几何中的非直角三角形计算不可或缺。

a / sinA = b / sinB = c / sinC

a² = b² + c² – 2bc cosA

Differentiation gives the rate of change of a function. In kinematics, velocity is the derivative of displacement with respect to time, and acceleration is the derivative of velocity. Integration recovers a quantity from its rate of change, e.g. finding displacement from a velocity–time graph.

微分给出函数的变化率。在运动学中，速度是位移对时间的导数，加速度是速度对时间的导数。积分可从变化率恢复原量，例如从速度–时间图中求位移。

v = ds/dt, a = dv/dt

Definite integration is used to calculate areas under curves, centroids of areas and moments of inertia. For a function y = f(x), the area between the curve and the x-axis from x = a to x = b is given by the definite integral. The trapezium rule provides an approximate numerical method when analytical integration is difficult.

定积分用于计算曲线下的面积、面的形心以及惯性矩。对于函数 y = f(x)，从 x = a 到 x = b 之间曲线与 x 轴围成的面积由定积分给出。当解析积分困难时，梯形法则提供了一种近似的数值方法。

Area = ∫ₐᵇ f(x) dx

Trapezium rule: Area ≈ (h/2)[y₀ + yₙ + 2(y₁ + y₂ + … )]

Complex numbers simplify the analysis of AC circuits and mechanical oscillations. A phasor can be represented in rectangular form z = a + jb or in polar form z = r∠θ, where r = √(a² + b²) and θ = tan⁻¹(b/a). The magnitude r gives the amplitude and θ gives the phase.

复数简化了交流电路与机械振荡的分析。相量既可用直角坐标形式 z = a + jb 表示，也可用极坐标形式 z = r∠θ 表示，其中 r = √(a² + b²)，θ = tan⁻¹(b/a)。幅值 r 给出振幅，而 θ 给出相位。

z = r (cosθ + j sinθ) = r e^(jθ)

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Year 13 CIE Engineering: Quick Reference Handbook of Formulas and Theorems | Year 13 CIE 工程：公式定理速查手册