Year 12 Cambridge Engineering: Formulas & Theorems Quick Reference | 剑桥工程12年级：公式定理速查手册

📚 Year 12 Cambridge Engineering: Formulas & Theorems Quick Reference | 剑桥工程12年级：公式定理速查手册

This concise quick-reference handbook covers the essential formulas, laws and theorems you will meet in Year 12 Cambridge Engineering. Use it for revision, problem-solving drills and to build a solid foundation for the engineering principles examined at AS level.

这本速查手册涵盖剑桥工程12年级（AS阶段）的核心公式、定律和定理。用它来复习、练习解题，为工程原理考试打下扎实基础。

1. Kinematics Equations | 运动学方程

For uniform acceleration along a straight line, the four ‘suvat’ equations relate displacement s, initial velocity u, final velocity v, acceleration a and time t. Always define your positive direction before substituting values.

在匀变速直线运动中，四个“suvat”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。代值前必须先规定正方向。

v = u + at

s = ut + ½at²

v² = u² + 2as

s = ½(u + v)t

When the acceleration is due to gravity g (≈9.81 m/s²) acting vertically downwards, replace a with ±g depending on your sign convention.

若加速度为重力加速度 g (≈9.81 m/s²)，方向竖直向下，则将 a 替换为 ±g，正负取决于你选定的正方向。

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2. Newton’s Laws of Motion | 牛顿运动定律

Newton’s three laws govern the relationship between force, mass and motion. They underpin almost every mechanical analysis in engineering.

牛顿三定律规定了力、质量和运动之间的关系，是几乎所有工程力学分析的基石。

First law: An object remains at rest or in uniform motion unless acted upon by a resultant force. Second law: F = ma, where F is the resultant force in newtons, m is mass in kg and a is acceleration in m/s². Third law: When one body exerts a force on another, the second exerts an equal but opposite force on the first.

第一定律：物体将保持静止或匀速直线运动状态，除非受到合力作用。第二定律：F = ma，其中 F 为合力（牛顿），m 为质量（kg），a 为加速度（m/s²）。第三定律：两个物体之间的作用力与反作用力总是大小相等、方向相反。

Weight, W = mg, is the force due to gravity. In free-body diagrams, always show all forces acting on the body.

重量 W = mg 是重力产生的力。画受力图时，务必画出所有作用在物体上的力。

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3. Work, Energy and Power | 功、能与功率

Work done is the product of force and displacement in the direction of the force. It transfers energy from one store to another.

功是力与沿力方向的位移的乘积，它实现能量的转移。

W = F d cosθ

Kinetic energy (KE) = ½mv² and gravitational potential energy (GPE) = mgh, where h is the vertical height above a reference level.

动能 (KE) = ½mv²，重力势能 (GPE) = mgh，其中 h 是相对于参考面的竖直高度。

Power is the rate of doing work or transferring energy. For a constant force acting on an object moving at velocity v, the power transmitted is P = Fv. Average power P = W/t or ΔE/t.

功率是做功或能量转移的速率。若物体以速度 v 运动且受力 F 恒定，则功率为 P = Fv。平均功率 P = W/t 或 ΔE/t。

The principle of conservation of energy states that energy cannot be created or destroyed, only changed from one form to another. Apply it to systems where friction or other dissipative forces are present by including work done against those forces.

能量守恒定律指出能量既不能凭空产生，也不能凭空消失，只能从一种形式转化为另一种形式。对于有摩擦力等耗散力作用的系统，分析时需计入克服这些力所做的功。

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4. Impulse and Momentum | 冲量与动量

Momentum is a vector quantity defined as the product of mass and velocity. Impulse is the change in momentum caused by a force acting over a time interval.

动量是质量与速度的乘积，为矢量。冲量是力在一段时间内作用而产生的动量变化量。

p = mv

Impulse = FΔt = Δp = mv − mu

In a closed system with no external resultant force, total momentum is conserved. This principle is essential for analysing collisions and explosions.

在没有外合力的封闭系统中，总动量守恒。该原理用于分析碰撞和爆炸问题。

For an elastic collision, kinetic energy is conserved alongside momentum. In an inelastic collision, momentum is conserved but kinetic energy is not.

对于弹性碰撞，动能与动量均守恒；对于非弹性碰撞，仅动量守恒，动能不守恒。

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

When a material is subjected to forces, stress and strain describe its internal resistance and deformation. Young’s modulus characterises the stiffness of a material in the elastic region.

材料受力时，应力与应变描述其内部抗力和变形。杨氏模量表征材料在弹性阶段的刚度。

σ = F / A (tensile or compressive stress)

ε = ΔL / L° (strain, where L° is original length)

E = σ / ε (Young’s modulus, unit: Pa)

The stress–strain graph for a ductile material shows an initial linear (elastic) region obeying Hooke’s law, a yield point, plastic deformation, and finally fracture. Engineering stress is calculated using the original cross-sectional area A°, unless the true stress is specified.

韧性材料的应力-应变图显示出初始线性（弹性）阶段（遵守胡克定律）、屈服点、塑性变形直至断裂。工程应力通常使用原始横截面积 A° 计算，除非要求真实应力。

For shear, shear stress τ = F/A and shear strain γ = Δx / L°. The shear modulus G = τ/γ.

对于剪切，剪应力 τ = F/A，剪应变 γ = Δx / L°，剪切模量 G = τ/γ。

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6. Moments and Equilibrium | 力矩与平衡

A moment is the turning effect of a force about a pivot. For a body to be in static equilibrium, both the resultant force and the resultant moment must be zero.

力矩是力绕某个支点的转动效应。物体静力平衡时，合外力为零且合力矩为零。

Moment M = F d (force × perpendicular distance from pivot)

The principle of moments states that for a body in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any pivot.

力矩原理：物体处于转动平衡时，绕任意支点的顺时针力矩之和等于逆时针力矩之和。

When solving engineering structures, identify reaction forces, resolve forces into components, and take moments about a convenient point to eliminate unknown forces.

求解工程结构时，需标出支反力，将力分解为分量，并对合适的点取矩以消去未知力。

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7. DC Circuit Laws | 直流电路定律

Ohm’s law and Kirchhoff’s laws form the backbone of circuit analysis. Understanding resistance combinations is essential for designing and troubleshooting circuits.

欧姆定律和基尔霍夫定律是电路分析的基石。理解电阻串并联是设计和排查电路的基础。

V = IR

Resistive power dissipation: P = IV = I²R = V²/R.

电阻产生的电功率：P = IV = I²R = V²/R。

Series combination: R_total = R₁ + R₂ + R₃ …

串联：R_total = R₁ + R₂ + R₃ …

Parallel combination: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ …; for two parallel resistors the shortcut is R_total = (R₁ × R₂) / (R₁ + R₂).

并联：1/R_total = 1/R₁ + 1/R₂ + 1/R₃ …；两个电阻并联的简式为 R_total = (R₁ × R₂) / (R₁ + R₂)。

Kirchhoff’s current law (KCL): At any junction, the sum of currents entering equals the sum of currents leaving. Kirchhoff’s voltage law (KVL): Around any closed loop, the algebraic sum of emfs and potential differences is zero.

基尔霍夫电流定律 (KCL)： 任一节点流入电流之和等于流出电流之和。基尔霍夫电压定律 (KVL)： 沿任一闭合回路，电动势和电位差的代数和为零。

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8. Electromagnetic Induction | 电磁感应

Electromagnetic induction links magnetic fields and electric circuits. Faraday’s law and Lenz’s law predict the induced emf in coils and transformers.

电磁感应将磁场与电路联系在一起。法拉第定律和楞次定律可预测线圈和变压器中的感应电动势。

Magnetic flux: Φ = B A cosθ (B: magnetic flux density, A: area, θ: angle between B and normal to A).

磁通量：Φ = B A cosθ (B: 磁通密度，A: 面积，θ: B 与面积法线的夹角)。

ε = −N dΦ/dt

The minus sign embodies Lenz’s law: the induced current flows so as to oppose the change in flux producing it.

负号体现了楞次定律：感应电流的方向总是使其产生的磁通阻碍引起感应电流的原磁通变化。

For an ideal transformer: Vₚ / Vₛ = Nₚ / Nₛ and, assuming 100% efficiency, Vₚ Iₚ = Vₛ Iₛ. Real transformers have losses due to resistance, eddy currents and hysteresis.

对于理想变压器：Vₚ / Vₛ = Nₚ / Nₛ，并假设效率为 100%，有 Vₚ Iₚ = Vₛ Iₛ。实际变压器因电阻、涡流和磁滞存在损耗。

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9. Simple Machines and Mechanical Advantage | 简单机械与机械利益

Simple machines such as levers, pulleys and gears allow a small effort to overcome a large load. Their performance is measured by mechanical advantage, velocity ratio and efficiency.

杠杆、滑轮、齿轮等简单机械能用较小的动力克服较大的负载。其性能由机械利益、速比和效率衡量。

MA = Load / Effort

VR = Distance moved by effort / Distance moved by load

η = (MA / VR) × 100%

For an ideal (frictionless) machine, MA = VR and efficiency is 100%. In practice, friction and wear reduce MA and efficiency.

对于理想（无摩擦）机械，MA = VR，效率为 100%。实际中，摩擦和磨损会降低机械利益和效率。

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10. Fluid Mechanics – Pressure and Bernoulli | 流体力学 – 压力与伯努利方程

Fluid statics and dynamics are fundamental to hydraulic systems, aerodynamics and pipe flow. Pressure at a point in a static fluid depends on depth and density.

流体静力学与动力学是液压系统、空气动力学和管道流动的基础。静态流体中某点的压力取决于深度和密度。

p = F / A (pressure definition)

p = ρ g h (hydrostatic pressure, where h is depth below free surface)

For an incompressible fluid flowing steadily, mass continuity gives A₁v₁ = A₂v₂ (constant volume flow rate Q = Av). Bernoulli’s equation applies along a streamline for an ideal, inviscid fluid:

对于稳定流动的不可压缩流体，连续性方程给出 A₁v₁ = A₂v₂（体积流量 Q = Av 不变）。伯努利方程沿流线适用于理想无黏流体：

p₁ + ½ρv₁² + ρgh₁ = p₂ + ½ρv₂² + ρgh₂

Each term represents a form of energy per unit volume: pressure energy, kinetic energy and potential energy.

每一项均代表单位体积的能量：压力能、动能和势能。

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11. Thermodynamics – Laws and Heat Engines | 热力学 – 定律与热机

The first and second laws of thermodynamics govern energy conservation and the direction of thermal processes. Engineering applications include internal combustion engines and refrigeration cycles.

热力学第一和第二定律规定了能量守恒与热过程的方向。工程应用涵盖内燃机和制冷循环。

Sensible heat transfer: Q = m c Δθ, where c is specific heat capacity. Latent heat: Q = m L.

显热传递：Q = m c Δθ，c 为比热容。潜热：Q = m L。

The thermal efficiency of a heat engine is the ratio of useful work output to heat input:

η_thermal = W_out / Q_in = 1 − Q_out / Q_in

For the ideal Carnot cycle operating between temperatures T_hot and T_cold (in kelvin), the maximum possible efficiency is:

η_Carnot = 1 − T_c / T_h

The second law states that heat cannot spontaneously flow from a colder to a hotter body, and that the entropy of an isolated system never decreases.

第二定律指出热量不能自发地从低温物体流向高温物体，且孤立系统的熵永不减少。

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12. Logic Gates and Boolean Algebra | 逻辑门与布尔代数

Digital electronics relies on logic gates to perform Boolean operations. The output depends only on the inputs’ logic levels (0 or 1).

数字电子学依赖逻辑门执行布尔运算。输出仅取决于输入的逻辑电平（0 或 1）。

Gate	Symbol / Operation	Truth Table (A, B → Out)
AND	A · B	1·1=1, else 0
OR	A + B	0+0=0, else 1
NOT	A̅ or A’	A=0 gives 1; A=1 gives 0
NAND	(A · B)̅	1·1=0, else 1
NOR	(A + B)̅	0+0=1, else 0
XOR	A ⊕ B	Same inputs = 0, different = 1

Boolean algebra provides the rules for simplifying complex logic expressions. For example, the commutative, associative and distributive laws, and De Morgan’s theorems:

布尔代数提供了化简复杂逻辑表达式的规则。例如交换律、结合律、分配律以及德摩根定理：

(A · B)̅ = A̅ + B̅

(A + B)̅ = A̅ · B̅

These relationships allow engineers to minimise the number of gates in a digital circuit, reducing cost and power consumption.

利用这些关系，工程师可以减少数字电路中的门数量，从而降低成本和功耗。

Published by TutorHao | Engineering Revision Series | aleveler.com

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