📚 Year 12 CAIE Engineering: Formula & Theorem Quick Reference Handbook | A Level工程公式定理速查手册
This handbook provides a comprehensive summary of key formulas, principles, and theorems covered in the Year 12 CAIE Engineering syllabus. It is designed as a quick-reference revision tool, grouping essential knowledge by topic area, including mechanics, materials, thermodynamics, and electrical circuits. Each section pairs concise English explanations with Chinese translations to support bilingual learners.
本手册汇总了CAIE 工程学科12年级（AS阶段）大纲所涵盖的核心公式、原理和定理。它可作为快速查阅的复习工具，按照力学、材料学、热力学和电路等专题领域归集关键知识点。每个小节均配有简洁的英文解释和对应的中文翻译，以帮助双语学习者。
1. Linear Motion and Kinematics | 直线运动与运动学
For an object moving with uniform acceleration along a straight line, four fundamental equations of motion apply. Each equation links displacement, initial velocity, final velocity, acceleration, and time.
对于沿直线做匀加速运动的物体，有四个基本运动学方程。每个方程都将位移、初速度、末速度、加速度和时间联系起来。
v = u + at
This equation gives the final velocity (v) after time t when initial velocity (u) and constant acceleration (a) are known.
已知初速度(u)和恒定加速度(a)时，该方程给出经过时间t后的末速度(v)。
s = ut + ½at²
Displacement (s) can be calculated using initial velocity, time, and acceleration. The term ½at² accounts for the additional displacement due to acceleration.
位移(s)可通过初速度、时间和加速度计算。½at² 项表示由于加速度产生的额外位移。
s = ½(u + v)t
This shows that displacement equals the average velocity multiplied by time. It is useful when acceleration is not explicitly needed.
该式表明位移等于平均速度乘以时间。当不需要直接使用加速度时非常有用。
v² = u² + 2as
This equation links final velocity directly to displacement and acceleration, omitting time. It is particularly useful in problems where time is not given or required.
此方程将末速度直接与位移和加速度关联起来，省去了时间。在未给出或不需求解时间的问题中尤其有用。
Key terms: s – displacement (m); u – initial velocity (m/s); v – final velocity (m/s); a – acceleration (m/s²); t – time (s). All quantities are vector quantities, so sign conventions must be applied consistently for direction.
关键术语： s – 位移 (m)；u – 初速度 (m/s)；v – 末速度 (m/s)；a – 加速度 (m/s²)；t – 时间 (s)。所有物理量均为矢量，因此必须一致地应用方向的正负符号约定。
2. Forces and Newton’s Laws | 力与牛顿定律
Newton’s three laws of motion form the foundation of classical mechanics. They describe how forces influence the motion of objects and are essential for analysing static and dynamic engineering systems.
牛顿运动三定律构成了经典力学的基础。它们描述了力如何影响物体的运动，对于分析静态和动态工程系统至关重要。
Newton’s First Law: An object remains at rest or in uniform motion in a straight line unless acted upon by a net external force.
牛顿第一定律： 任何物体都要保持静止或匀速直线运动状态，直到有净外力迫使它改变这种状态为止。
Newton’s Second Law: The net force acting on an object is equal to the product of its mass and acceleration.
牛顿第二定律： 作用在物体上的净外力等于物体的质量乘以加速度。
F = ma
This relationship is vectorial: the acceleration is in the same direction as the net force. For systems with multiple forces, resolve them into perpendicular components (often horizontal and vertical) using free-body diagrams.
此关系为矢量式：加速度方向与净力方向一致。对于多力系统，应利用受力图将力沿相互垂直的方向（通常是水平和竖直方向）进行分解。
Newton’s Third Law: Whenever one body exerts a force on a second body, the second body exerts an equal and opposite force on the first.
牛顿第三定律： 每当一个物体对第二个物体施加一个力，第二个物体同时对第一个物体施加一个大小相等、方向相反的力。
Common forces encountered in engineering include weight (W = mg), tension, normal reaction, friction, and applied loads. The coefficient of friction μ relates the frictional force to the normal reaction for dry surfaces: Ff ≤ μR, with Ff = μR at the point of sliding.
工程中常见的力包括重力（W = mg）、拉力、法向反力、摩擦力和外加载荷。对于干接触面，摩擦系数 μ 将摩擦力与法向反力联系起来：Ff ≤ μR，即将滑动时 Ff = μR。
3. Work, Energy, and Power | 功、能量和功率
Work is done when a force causes displacement of its point of application in the direction of the force. Energy is the capacity to do work, and power is the rate at which work is done.
当一个力使其作用点沿力的方向发生位移时，力就对物体做了功。能量是做功的本领，功率则是做功的快慢程度。
W = F s cos θ
Here θ is the angle between the force vector and the displacement vector. When force and displacement are in the same direction, W = F s.
式中 θ 为力矢量与位移矢量之间的夹角。当力与位移方向相同时，W = F s。
Kinetic energy (KE) is the energy of an object due to its motion:
动能 (KE) 是物体由于运动而具有的能量：
KE = ½ m v²
Gravitational potential energy (GPE) near the Earth’s surface is given by:
靠近地球表面的重力势能 (GPE) 由下式给出：
GPE = m g h
where h is the vertical height above a chosen reference level. The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In the absence of non-conservative forces (such as friction), total mechanical energy (KE + GPE) remains constant.
式中 h 为相对于所选参考面的竖直高度。能量守恒定律指出，能量既不会凭空产生也不会凭空消失，只会从一种形式转化为另一种形式。在没有非保守力（如摩擦力）的情况下，系统的总机械能（动能+重力势能）保持不变。
Power is defined as work done per unit time:
功率 定义为单位时间内所做的功：
P = ΔW / Δt
For a constant force acting in the direction of motion, power can also be expressed as P = F v, where v is the instantaneous velocity. The SI unit of power is the watt (W), equivalent to 1 J/s.
对于沿运动方向作用的恒力，功率也可表示为 P = F v，其中 v 为瞬时速度。功率的国际单位是瓦特 (W)，相当于 1 J/s。
4. Momentum and Impulse | 动量与冲量
Linear momentum is a vector quantity defined as the product of an object’s mass and velocity. The principle of conservation of momentum is fundamental for analysing collisions and explosions.
线动量是一个矢量，定义为物体质量与速度的乘积。动量守恒定律是分析碰撞和爆炸问题的基本定律。
p = m v
In an isolated system (no external forces), the total momentum before an interaction equals the total momentum after the interaction: Σpbefore = Σpafter.
在一个孤立系统（无外力作用）中，相互作用前的总动量等于相互作用后的总动量：Σp前 = Σp后。
Impulse is the product of the net force and the time interval during which it acts, and it equals the change in momentum:
冲量 是净力与其作用时间间隔的乘积，且等于动量的变化量：
F Δt = Δp = m v – m u
The area under a force–time graph represents the impulse. In collisions, impulse can be used to determine the average force experienced over a very short time, such as in crash testing. Elastic collisions conserve kinetic energy as well as momentum; inelastic collisions conserve momentum but not kinetic energy. Perfectly inelastic collisions result in the bodies sticking together.
力–时间图线下的面积代表冲量。在碰撞问题中，利用冲量可求得极短时间内的平均作用力，例如在碰撞测试中。弹性碰撞既守恒动量也守恒动能；非弹性碰撞则只守恒动量而不守恒动能。完全非弹性碰撞会导致两物体粘合在一起。
5. Rotational Dynamics | 转动动力学
When a force causes an object to rotate about a fixed point or axis, the turning effect is described by torque (or moment). The rotational analogues of linear motion quantities are essential for engineering design of shafts, gears, and rotating machinery.
当力使物体绕一固定点或固定轴转动时，其转动效应用力矩来描述。与线运动量相对应的转动量对于轴、齿轮和旋转机械的工程设计至关重要。
τ = F r sin θ
Torque (τ) is the product of the force (F), the distance from the pivot (r), and the sine of the angle between the force vector and the line from the pivot to the point of application. When the force is perpendicular to the radius, τ = F r.
力矩 (τ) 是力 (F)、力臂 (r) 以及力矢量与径矢夹角的正弦之积。当力垂直于径矢时，τ = F r。
For a system in rotational equilibrium (no angular acceleration), the sum of clockwise torques about any point equals the sum of counter‑clockwise torques: Στcw = Στccw.
对于处于转动平衡（无角加速度）的系统，绕任一点的所有顺时针力矩之和等于所有逆时针力矩之和：Στ顺 = Στ逆。
Angular displacement, velocity, and acceleration are related through:
角位移、角速度和角加速度 之间的关系如下：
ω = Δθ / Δt
α = Δω / Δt
For uniform angular acceleration, analogous kinematic equations apply, replacing s with θ, u with ω₀, v with ω, and a with α.
对于匀角加速转动，类似的运动学方程同样适用，只需将 s 替换为 θ，u 替换为 ω₀，v 替换为 ω，a 替换为 α。
Moment of inertia (I) is a measure of an object’s resistance to changes in its rotational motion. The kinetic energy of rotation is KErot = ½ I ω². The rotational analogue of F = ma is τ = I α.
转动惯量 (I) 是衡量物体抵抗转动运动变化能力的量度。转动的动能为 KErot = ½ I ω²。F = ma 对应的转动形式为 τ = I α。
6. Stress, Strain, and Young’s Modulus | 应力、应变与杨氏模量
When a material is subjected to external forces, it deforms. The concepts of stress and strain quantify internal resistance and deformation, enabling engineers to select appropriate materials for structures and components.
当材料受到外力作用时会发生变形。应力和应变的概念量化了材料内部的抵抗力和形变程度，使工程师能够为结构和构件选择合适的材料。
Tensile stress (σ) is the force applied per unit cross‑sectional area:
拉应力 (σ) 是单位横截面积上所承受的力：
σ = F / A
Tensile strain (ε) is the ratio of extension to the original length:
拉应变 (ε) 是伸长量与原长之比：
ε = ΔL / L₀
Young’s modulus (E) describes the stiffness of a material in the linear elastic region:
杨氏模量 (E) 描述材料在线弹性范围内的刚度：
E = σ / ε
Young’s modulus is a constant for a given material up to the limit of proportionality, provided Hooke’s law holds. The stress–strain curve for a ductile material typically includes a linear region, a yield point, plastic deformation, and ultimate tensile strength before fracture.
在比例极限内且符合胡克定律的前提下，杨氏模量对于给定材料是一个常数。延性材料的应力–应变曲线通常包括线性区、屈服点、塑性变形阶段以及断裂前的极限抗拉强度。
Poisson’s ratio (ν) may also be introduced at Year 12: ν = – (lateral strain) / (longitudinal strain). It is a dimensionless measure of the lateral contraction that accompanies longitudinal extension.
在12年级也可能引入泊松比 (ν)：ν = –（横向应变）/（纵向应变）。它是一个无量纲量，用以衡量伴随纵向伸长而发生的横向收缩。
7. Thermodynamics and Heat Transfer | 热力学与热传导
Thermodynamic principles govern energy conversion, heating, and cooling processes. The first law of thermodynamics and calculations of heat energy are fundamental in engineering applications such as engines and heating systems.
热力学原理支配着能量转换、加热和冷却过程。热力学第一定律以及热量计算在发动机和供暖系统等工程应用中十分基础。
Sensible heat: The energy required to change the temperature of a substance without a phase change is given by:
显热： 在不发生相变的情况下，改变物质温度所需的能量由下式给出：
Q = m c ΔT
where c is the specific heat capacity (J/(kg·K)). Different materials have different specific heat capacities.
式中 c 为比热容（J/(kg·K)）。不同材料的比热容各不相同。
Latent heat: The energy required for a phase change at constant temperature is:
潜热： 在恒定温度下发生相变所需的能量为：
Q = m L
where L is the specific latent heat (J/kg) – latent heat of fusion for melting/freezing, and latent heat of vaporisation for boiling/condensing.
式中 L 为比潜热（J/kg）—— 熔化/凝固时用熔解潜热，沸腾/凝结时用汽化潜热。
First Law of Thermodynamics: The change in internal energy (ΔU) of a system equals the heat added to the system (Q) minus the work done by the system (W): ΔU = Q – W. This is a statement of energy conservation for thermodynamic processes.
热力学第一定律： 系统内能的变化量 (ΔU) 等于系统吸收的热量 (Q) 减去系统对外做的功 (W)：ΔU = Q – W。这是热力学过程能量守恒的表述。
Heat transfer occurs via conduction, convection, and radiation. For steady‑state conduction through a uniform material, Fourier’s law applies: Q/t = k A (ΔT / d), where k is thermal conductivity.
热量通过传导、对流和辐射三种方式传递。对于通过均匀材料的稳态热传导，适用傅里叶定律：Q/t = k A (ΔT / d)，其中 k 为热导率。
8. Electrical Circuits and Components | 电路与元器件
Electrical theory underpins a vast range of engineering systems. Ohm’s law, Kirchhoff’s laws, and resistance combinations are core topics in Year 12 circuit analysis.
电学理论是众多工程系统的基础。欧姆定律、基尔霍夫定律以及电阻的串并联是12年级电路分析的核心内容。
Ohm’s Law: For a metallic conductor at constant temperature, the current through it is directly proportional to the potential difference across it:
欧姆定律： 对于恒定温度下的金属导体，通过导体的电流与其两端的电位差成正比：
V = I R
Resistance and resistivity: The resistance of a uniform wire depends on its length (L), cross‑sectional area (A), and the material’s resistivity (ρ):
电阻与电阻率： 一段均匀导线的电阻取决于其长度 (L)、横截面积 (A) 以及材料的电阻率 (ρ)：
R = ρ L / A
Resistivity is a material property and varies with temperature; for most metals, ρ increases with temperature.
电阻率是材料属性，随温度变化；对于大多数金属，ρ 随温度升高而增大。
Resistors in series: Rtotal = R₁ + R₂ + R₃ + …
串联电阻： R总 = R₁ + R₂ + R₃ + …
Resistors in parallel: 1/Rtotal = 1/R₁ + 1/R₂ + 1/R₃ + …
并联电阻： 1/R总 = 1/R₁ + 1/R₂ + 1/R₃ + …
Kirchhoff’s Current Law (KCL): The algebraic sum of currents at any node in a circuit is zero: ΣIin = ΣIout.
基尔霍夫电流定律 (KCL)： 在电路的任一节点上，流入和流出该节点的电流的代数和为零：ΣI入 = ΣI出。
Kirchhoff’s Voltage Law (KVL): The algebraic sum of the potential differences around any closed loop is zero: ΣV = 0.
基尔霍夫电压定律 (KVL)： 沿任一闭合回路，各部分电位差的代数和为零：ΣV = 0。
Power in electrical circuits: P = V I = I² R = V² / R. These formulas are used to calculate energy dissipation in resistors and power ratings of components.
电路中的功率： P = V I = I² R = V² / R。这些公式用于计算电阻上的能量耗散和元器件的功率定额。
9. Engineering Materials and Properties | 工程材料与性能
Selecting the right material requires understanding key mechanical and physical properties beyond just strength. Density, hardness, toughness, and ductility all influence design choices.
选择合适的材料需要了解除强度以外的关键力学与物理性能。密度、硬度、韧性和延展性都会影响设计决策。
Density (ρ): mass per unit volume:
密度 (ρ)： 单位体积的质量：
ρ = m / V
Density is critical for weight calculations and buoyancy considerations in structures and vehicles.
密度对于结构和运载工具的重量计算及浮力考虑至关重要。
Hardness is the resistance to indentation or scratching; it is often measured using the Brinell, Vickers, or Rockwell scales. Hardness correlates roughly with wear resistance but not always with toughness.
硬度 是材料抵抗压入或刮擦的能力；通常用布氏、维氏或洛氏硬度标度测量。硬度与耐磨性大致相关，但并不总与韧性一致。
Toughness is the ability to absorb energy and plastically deform without fracturing; it corresponds to the total area under the stress–strain curve.
韧性 是材料在断裂前吸收能量并发生塑性变形的能力；它对应于应力–应变曲线下的总面积。
Ductility refers to the extent of plastic deformation before fracture, often measured by percentage elongation or reduction in area. Ductile materials, like copper and mild steel, show considerable necking before failure, while brittle materials, like cast iron and glass, show little or no plastic deformation.
延展性 指断裂前塑性变形的程度，通常用延伸率或断面收缩率衡量。延性材料（如铜和低碳钢）在破坏前表现出明显的颈缩，而脆性材料（如铸铁和玻璃）几乎无塑性变形。
Stiffness is resistance to elastic deformation and is quantified by Young’s modulus; high stiffness does not necessarily imply high strength.
刚度 是抵抗弹性变形的能力，由杨氏模量量化；高刚度并不一定意味着高强度。
Engineers also consider factors like fatigue, creep, thermal expansion coefficient (α), and electrical conductivity when specifying materials.
工程师在指定材料时还需要考虑疲劳、蠕变、热膨胀系数 (α) 以及导电性等因素。
10. Basic Fluid Mechanics | 流体力学基础
Fluid statics and simple flow principles are introduced at AS level to analyse forces on submerged surfaces, buoyancy, and fluid pressure.
在AS阶段引入流体静力学和简单流动原理，用以分析浸没表面受力、浮力以及流体压强。
Pressure in a fluid at rest: The pressure due to a fluid column of height h is given by:
静止流体中的压强： 高度为 h 的液柱产生的压强为：
p = ρ g h
At any point in a static fluid, pressure acts equally in all directions. The total pressure at a depth also includes atmospheric pressure on the surface: ptotal = patm + ρ g h.
在静止流体中的任一点，压强沿各个方向大小相等。某一深度的总压强还要加上液面处的大气压：p总 = patm + ρ g h。
Archimedes’ principle: The upward buoyant force on a submerged object equals the weight of the fluid displaced by the object. This determines whether an object floats or sinks.
阿基米德原理： 浸没在流体中的物体受到的向上浮力等于物体排开的流体的重量。这决定了物体的沉浮条件。
Continuity equation for an ideal incompressible fluid: For steady flow, the mass flow rate is constant:
不可压缩理想流体的连续性方程： 对于定常流动，质量流量守恒：
A₁ v₁ = A₂ v₂
where A is the cross‑sectional area and v is the flow velocity. This implies that fluid speed increases where the pipe narrows.
式中 A 为横截面积，v 为流速。该式表明，在管道变窄处流体速度增加。
Bernoulli’s principle (often introduced conceptually at AS): In a streamline flow of an ideal fluid, the sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant.
伯努利原理（AS阶段通常作为概念引入）：在理想流体的流线流动中，单位体积流体的压力能、动能与势能之和保持恒定。
These fundamentals prepare students for more advanced topics in fluid dynamics and hydraulic systems.
这些基础为学生进一步学习流体动力学和液压系统的高级专题做好了准备。
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