Formula & Theorem Quick Reference Handbook for A-Level Cambridge Engineering | A-Level Cambridge 工程：公式定理速查手册

📚 Formula & Theorem Quick Reference Handbook for A-Level Cambridge Engineering | A-Level Cambridge 工程：公式定理速查手册

This handbook collects the essential formulas and theorems you will encounter in the Cambridge International A-Level Engineering syllabus. It is designed as a rapid revision tool, covering mechanics, materials, fluids, thermodynamics, electricity, and fundamental mathematics. Each section presents key relationships with brief explanations, helping you to recall and apply them confidently during problem-solving. Keep this guide close for last-minute review before your examination.

本手册汇集了剑桥国际 A-Level 工程课程中你将遇到的核心公式与定理，旨在用作快速复习工具，涵盖力学、材料、流体、热力学、电学以及基础数学。每个小节以简要说明呈现关键关系，帮助你自信地回忆和运用这些知识进行解题。考前冲刺时，可随时翻阅本指南。

1. Kinematics and Mechanics | 运动学与力学

The equations of uniformly accelerated motion relate displacement, initial velocity, final velocity, acceleration and time. They assume constant acceleration along a straight line.

匀加速运动方程将位移、初速度、末速度、加速度和时间联系起来。它们假设加速度恒定且沿直线运动。

v = u + a t

v = u + at

Newton’s second law states that the net force acting on an object is equal to the product of its mass and acceleration.

牛顿第二定律指出，作用在物体上的合力等于其质量与加速度的乘积。

F = m a

Momentum is the product of mass and velocity. The impulse delivered by a net force over a time interval equals the change in momentum.

动量是质量与速度的乘积。合力的冲量等于动量的变化量。

p = m v, F Δt = Δp = m(v − u)

Work done by a constant force is the product of the force and the displacement in the direction of the force. Kinetic energy depends on mass and the square of speed, while gravitational potential energy depends on mass, height and gravitational field strength.

恒力做的功等于力与沿力方向的位移的乘积。动能取决于质量和速度的平方，而重力势能取决于质量、高度和重力场强。

W = F s cosθ, KE = ½m v², GPE = m g h

Power is the rate of doing work. For an object moving at constant velocity under a force parallel to the motion, power is the product of force and velocity.

功率是做功的快慢。对于在力与运动方向平行且以恒定速度运动的物体，功率等于力与速度的乘积。

P = W / t, P = F v

2. Statics and Equilibrium | 静力学与平衡

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

力对某点的力矩等于力的大小乘以力的作用线到该点的垂直距离。

M = F d

For a body to be in static equilibrium, both the resultant force and the resultant moment must be zero.

物体处于静力平衡时，合外力和合力矩均须为零。

ΣF = 0, ΣM = 0

When a uniform regular object is supported at its centre of mass, the weight produces no turning effect. The centre of mass of a composite body can be found using moments about a chosen axis.

当均匀规则物体在其质心处被支撑时，重力不产生转动效果。组合体的质心可通过相对于选定轴的力矩来求得。

3. Stress and Strain | 应力与应变

Tensile or compressive stress is the internal force per unit cross-sectional area. Strain is the fractional change in length.

拉伸或压缩应力是单位横截面积上的内力。应变是长度的相对变化量。

σ = F / A, ε = ΔL / L₀

Young’s modulus is a measure of the stiffness of a material in the elastic region, where stress is proportional to strain (Hooke’s law).

杨氏模量衡量材料在弹性范围内的刚度，在此范围内应力与应变成正比（胡克定律）。

E = σ / ε

Shear stress and shear strain occur under parallel opposing forces. The shear modulus characterises the material’s resistance to shape change.

剪切应力和剪切应变由平行反向力引起。剪切模量表征材料抵抗形状改变的能力。

τ = F / A, γ = Δx / L, G = τ / γ

Poisson’s ratio is the negative ratio of lateral strain to axial strain for a material under uniaxial loading.

泊松比是材料在单轴加载下横向应变与轴向应变之比的负值。

ν = − ε_lateral / ε_axial

The factor of safety is the ratio of the material’s ultimate stress to the allowable working stress.

安全系数是材料极限应力与许用工作应力之比。

FoS = σ_ult / σ_allow

4. Fluid Mechanics | 流体力学

For an incompressible fluid flowing through a closed pipe, the mass flow rate is constant. This principle leads to the continuity equation.

对于流经封闭管道的不可压缩流体，质量流量恒定。该原理导出连续性方程。

A₁ v₁ = A₂ v₂

Bernoulli’s equation expresses the conservation of energy along a streamline for steady, inviscid, incompressible flow. It relates pressure, kinetic energy per unit volume and potential energy per unit volume.

伯努利方程表达了定常、无粘、不可压缩流沿流线的能量守恒。它将压力、单位体积动能和单位体积势能联系起来。

p + ½ρv² + ρgh = constant

The Reynolds number is a dimensionless quantity that predicts the transition between laminar and turbulent flow regimes.

雷诺数是一个无量纲量，用于预测层流与湍流流态的转变。

Re = ρ v L / μ

Volumetric flow rate is the product of cross-sectional area and mean flow velocity. It is a fundamental parameter in hydraulic and pneumatic systems.

体积流量是横截面积与平均流速的乘积，是液压与气动系统中的基本参数。

Q = A v

5. Thermodynamics | 热力学

The first law of thermodynamics states that the change in internal energy of a system equals the net heat added minus the net work done by the system.

热力学第一定律指出，系统内能的变化等于净加热量减去系统对外做的净功。

ΔU = Q − W

The ideal gas equation relates pressure, volume, amount of substance and absolute temperature. It is essential for analysing gaseous systems.

理想气体状态方程将压力、体积、物质的量和绝对温度联系起来。它是分析气体系统的基础。

pV = n R T

Thermal efficiency of a heat engine is the ratio of net work output to heat input. The Carnot efficiency represents the theoretical maximum for a reversible engine operating between two reservoirs.

热机的热效率是净输出功与输入热量之比。卡诺效率代表在高温热源和低温热源之间运行的可逆热机的理论上限。

η = W_net / Q_in, η_Carnot = 1 − T_cold / T_hot

Heat energy required to change the temperature of a substance depends on its mass, specific heat capacity, and temperature change. During a phase change, latent heat governs energy transfer at constant temperature.

改变物体温度所需的热量取决于物体的质量、比热容和温度变化。在相变过程中，潜热决定了恒温下的能量传递。

Q = m c ΔT, Q = m L

6. Electricity and Circuits | 电学与电路

Ohm’s law describes the linear relationship between potential difference across a conductor and the current flowing through it, provided temperature remains constant.

欧姆定律描述了在温度不变的条件下，导体两端电势差与流过电流的线性关系。

V = I R

Electrical power dissipated in a resistor can be expressed in three equivalent forms, useful for different circuit analysis scenarios.

电阻上消耗的电功率可用三种等价形式表达，适用于不同的电路分析场景。

P = I V = I² R = V² / R

Kirchhoff’s current law states that the algebraic sum of currents entering a node is zero. Kirchhoff’s voltage law states that the sum of emfs around any closed loop equals the sum of potential drops.

基尔霍夫电流定律指出，流入节点的电流代数和为零。基尔霍夫电压定律指出，沿任何闭合回路的电动势之和等于电势降之和。

Σ I_in = Σ I_out, Σ V = 0 (around loop)

The equivalent resistance for resistors in series is the sum of individual resistances. For parallel resistors, the reciprocal of the equivalent resistance equals the sum of the reciprocals.

串联电阻的等效电阻等于各个电阻之和。并联电阻的等效电阻的倒数等于各个电阻倒数之和。

R_series = R₁ + R₂ + …, 1/R_parallel = 1/R₁ + 1/R₂ + …

7. Electromagnetism and Induction | 电磁学与感应

A current-carrying conductor placed in a magnetic field experiences a force perpendicular to both the field and the current direction.

放置在磁场中的载流导体会受到一个垂直于磁场和电流方向的力。

F = B I L sinθ

Faraday’s law of electromagnetic induction states that the magnitude of induced emf is proportional to the rate of change of magnetic flux linkage. Lenz’s law determines the direction of the induced current.

法拉第电磁感应定律指出，感应电动势的大小与磁通量链的变化率成正比。楞次定律决定了感应电流的方向。

ε = − N dΦ / dt

An ideal transformer transfers electrical energy from one circuit to another through mutual induction, with voltage and current ratios determined by the turns ratio.

理想变压器通过互感将电能从一个电路传递到另一个电路，电压比和电流比由匝数比决定。

V_s / V_p = N_s / N_p, I_p V_p = I_s V_s

8. Engineering Mathematics Essentials | 工程数学基础

Trigonometric ratios in a right-angled triangle relate side lengths to an acute angle. The identity sin²θ + cos²θ = 1 is fundamental in vector resolution and harmonic analysis.

直角三角形中的三角比将边长与锐角联系起来。恒等式 sin²θ + cos²θ = 1 在矢量分解和谐波分析中至关重要。

sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent

The derivative of a power function is a basic tool for finding gradients, rates of change and optimising engineering designs.

幂函数的导数是求斜率、变化率和进行工程设计优化的基本工具。

d(xⁿ)/dx = n xⁿ⁻¹, d(sin x)/dx = cos x, d(cos x)/dx = − sin x

The indefinite integral is the reverse process of differentiation. It is used to compute areas, volumes and work done by variable forces.

不定积分是微分的逆运算，用于计算面积、体积和变力做功。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, ∫ sin x dx = − cos x + C

The scalar (dot) product of two vectors gives a measure of how much they point in the same direction, useful in calculating work and projections.

两个向量的标量积（点积）可衡量它们指向同一方向的程度，常用于计算功和投影。

A · B = |A||B| cosθ

9. Energy, Work and Power Systems | 能量、功与功率系统

Efficiency quantifies how well a system converts input energy into useful output. In mechanical systems, it compares useful work out to total energy input.

效率量化了系统将输入能量转化为有用输出的程度。在机械系统中，它比较有用输出功与总输入能量。

Efficiency = (Useful energy output / Total energy input) × 100%

The principle of conservation of energy states that energy cannot be created or destroyed, only transferred from one form to another. This underlies all energy analysis.

能量守恒定律指出，能量不能被创造或消灭，只能从一种形式转换为另一种形式。这是所有能量分析的基础。

Mechanical advantage (MA) is the ratio of load to effort for a simple machine. Velocity ratio (VR) is the ratio of distance moved by the effort to the distance moved by the load. Efficiency = MA / VR.

机械效益（MA）是简单机械中负载与作用力之比。速度比（VR）是作用力移动距离与负载移动距离之比。效率 = MA / VR。

MA = Load / Effort, VR = Distance_effort / Distance_load, η = MA / VR

10. Motion in a Circle | 圆周运动

An object moving in a circle at constant speed experiences a centripetal acceleration directed towards the centre. This acceleration results from a centripetal force.

以恒定速率做圆周运动的物体具有指向圆心的向心加速度，该加速度由向心力产生。

a = v² / r = ω² r, F = m v² / r = m ω² r

Angular velocity relates the angle swept per unit time. For uniform circular motion, period and frequency are linked to angular velocity.

角速度表示单位时间内扫过的角度。对于匀速圆周运动，周期和频率与角速度相关联。

ω = Δθ / Δt = 2π f = 2π / T

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Formula & Theorem Quick Reference Handbook for A-Level Cambridge Engineering | A-Level Cambridge 工程：公式定理速查手册