📚 AS CIE Engineering: Formula & Theorem Quick Reference Guide | AS CIE 工程：公式定理速查手册
This quick reference guide covers the essential formulas, laws, and theorems required for the AS Level CIE Engineering (9706) syllabus. It provides a concise, topic-by-topic summary to support problem-solving and revision. Each section pairs crucial equations with their practical context, ensuring you can recall and apply them effectively during the exam.
本速查手册涵盖了 AS Level CIE 工程 (9706) 大纲要求的所有关键公式、定律与定理。手册以主题为单位提供精炼总结，帮助解题与复习。每个小节都将重要方程式与其实际应用场景对应，确保你在考试中能够高效回忆并准确运用。
1. Fundamentals of Mechanics and Vectors | 力学基础与矢量
Understanding vector quantities and their resolution is the foundation of engineering mechanics. Force, displacement, velocity, and acceleration are all vectors and must be treated with both magnitude and direction.
理解矢量及其分解是工程力学的基础。力、位移、速度和加速度均为矢量，处理时必须同时考虑大小和方向。
the horizontal, the components are Fₓ = F cos θ (horizontal) and Fᵧ = F sin θ (vertical).
矢量可以利用三角函数分解为两个互相垂直的分量。若力 F 与水平方向夹角为 θ，其分量为 Fₓ = F cos θ（水平分量）和 Fᵧ = F sin θ（垂直分量）。
The resultant of two or more vectors is found by summing components, then using Pythagoras' theorem: R = √Fₓ² + Fᵧ². The direction is given by θ = tan⁻¹(Fᵧ / Fₓ).
两个或多个矢量的合力可通过分量求和，再利用勾股定理求得：R = √Fₓ² + Fᵧ²。合力的方向由 θ = tan⁻¹(Fᵧ / Fₓ) 确定。
.81 m s⁻² on Earth. Note that mass is a scalar, while weight is a vector acting downwards.
重力是由引力产生的力，计算公式为 W = m g，其中地球上的 g = 9.81 m s⁻²。注意质量是标量，而重力是方向向下的矢量。
2. Moments and Equilibrium | 力矩与平衡
distance from the line of action of the force to the pivot. The SI unit is newton metre (N m).
力矩是力绕支点产生的转动效应，计算公式为 M = F d，其中 d 是从力的作用线到支点的垂直距离。国际单位制单位为牛·米 (N m)。
bout any pivot equals the sum of anticlockwise moments: Σ M_clockwise = Σ M_anticlockwise.
力矩原理指出，物体处于转动平衡时，对于任意支点，顺时针力矩之和等于逆时针力矩之和：Σ M_顺时针 = Σ M_逆时针。
torque of a couple is T = F d, where d is the perpendicular separation between the forces.
力偶由大小相等、方向相反且不共线的一对力组成，力偶的转矩为 T = F d，其中 d 为两力之间的垂直距离。
both translational equilibrium (Σ Fₓ = 0, Σ Fᵧ = 0) and rotational equilibrium (Σ M = 0).
刚体处于静力平衡时，必须同时满足平动平衡 (Σ Fₓ = 0, Σ Fᵧ = 0) 和转动平衡 (Σ M = 0)。
Centre of gravity is the point where the entire weight of a body appears to act. Its position can be determined experimentally or by calculation in composite shapes using Σ Wᵢ xᵢ / Σ Wᵢ.
重心是物体全部重力似乎汇集的点，可通过实验测定，或通过组合形状公式 Σ Wᵢ xᵢ / Σ Wᵢ 计算位置。
3. Kinematics | 运动学
Kinematics describes motion without considering forces. The four SUVAT equations apply to uniform acceleration in a straight line.
运动学在不涉及力的情况下描述运动。四个 SUVAT 方程适用于匀加速度直线运动。
v = u + a t — relates final velocity (v), initial velocity (u), acceleration (a), and time (t).
v = u + a t ——关联末速度 (v)、初速度 (u)、加速度 (a) 和时间 (t)。
s = (u + v)/2 × t — displacement equals average velocity multiplied by time.
s = (u + v)/2 × t ——位移等于平均速度乘以时间。
s = u t + ½ a t² — displacement with initial velocity and acceleration.
s = u t + ½ a t² ——含有初速度和加速度的位移公式。
v² = u² + 2 a s — velocity-displacement relation without time.
v² = u² + 2 a s ——不含时间的速度-位移关系式。
For motion under gravity near the Earth's surface, a = g = 9.81 m s⁻² downwards. Take upward as positive, then a = -9.81 m s⁻².
在地表附近的抛体运动中，加速度 a = g = 9.81 m s⁻²，方向向下。若取向上为正，则 a = -9.81 m s⁻²。
components. Horizontal velocity is constant, while vertical motion uses SUVAT with a = -g.
抛体运动通过分解水平与竖直分量进行分析。水平方向速度恒定，竖直方向采用 a = -g 的 SUVAT 方程。
4. Dynamics and Newton's Laws | 动力学与牛顿定律
Newton's first law: an object remains at rest or in uniform motion unless acted upon by a resultant external force.
牛顿第一定律：除非受到合外力的作用，物体将保持静止或匀速直线运动状态。
Newton' ectly proportional to the resultant force and inversely proportional to its mass. F = m a.
牛顿第二定律：物体的加速度与所受合外力成正比，与其质量成反比，即 F = m a。
Newton's third law: if body A exerts a force on body B, body B exerts an equal and opposite force on body A; action and reaction act on different objects.
牛顿第三定律：若物体 A 对物体 B 施加力，则物体 B 同时对物体 A 施加大小相等、方向相反的力；作用力与反作用力作用在不同物体上。
Friction: static friction Ff ≤ μ_s R and kinetic friction Ff = μ_k R, where R is the normal reaction force and μ is the coefficient of friction.
摩擦力：静摩擦力 Ff ≤ μ_s R，滑动摩擦力 Ff = μ_k R，其中 R 是法向反作用力，μ 是摩擦系数。
Momentum p = m v is a vector. Newton' ed as F = Δp / Δt, which leads to the impulse-momentum principle: impulse F t = m v - m u.
动量 p = m v 是矢量。牛顿第二定律可表达为 F = Δp / Δt，由此导出冲量-动量原理：冲量 F t = m v - m u。
al momentum before a collision equals total momentum after: m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂.
动量守恒定律：在无外力作用下，碰撞前的总动量等于碰撞后的总动量：m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂。
5. Work, Energy and Power | 功、能与功率
s W = F s cos θ, where θ is the angle between the force and displacement. Unit: joule (J).
恒力做功的计算公式为 W = F s cos θ，其中 θ 是力与位移之间的夹角，单位为焦耳 (J)。
Gravitational potential energy change: ΔEp = m g Δh, where Δh is the change in vertical height.
重力势能的变化量：ΔEp = m g Δh，其中 Δh 是竖直高度的变化量。
Kinetic energy of a moving object: Ek = ½ m v².
运动物体的动能：Ek = ½ m v²。
The work-energy principle: the net work done on a body equals the change in its kinetic energy: Wnet = ΔEk.
功能原理：作用于物体的净功等于其动能的变化量：Wnet = ΔEk。
Conservation of mechanical energy: in the absence of non-conservative forces, total mechanical energy (Ek + Ep) remains constant.
机械能守恒：在无非保守力做功的情况下，总机械能 (Ek + Ep) 保持恒定。
doing work: P = W / t. For a constant force moving at velocity v, P = F v. Unit: watt (W).
功率是做功的速率：P = W / t。若恒力作用下的物体以速度 v 运动，则 P = F v，单位为瓦特 (W)。
Efficiency = (useful output power / input power) × 100%, always less than 100% due to energy losses.
效率 = (有用输出功率 / 输入功率) × 100%，由于存在能量损耗，效率恒小于 100%。
6. Materials: Stress, Strain and Elastic Moduli | 材料力学：应力、应变与弹性模量
/ A. Tensile stress is positive, compressive stress negative. Unit: pascal (Pa) or N m⁻².
正应力 (σ) 是单位截面积上所受的力：σ = F / A。拉伸应力为正，压缩应力为负，单位为帕斯卡 (Pa) 或 N m⁻²。
it original length: ε = ΔL / L₀. Strain is dimensionless, often expressed as a percentage.
正应变 (ε) 是单位原长的伸长量：ε = ΔL / L₀。应变为无量纲量，常用百分比表示。
Hooke' y proportional to strain. The constant of proportionality is the Young modulus E: σ = E ε.
胡克定律：在比例极限内，许多材料的应力与应变成正比。比例常数即为杨氏模量 E：σ = E ε。
/ (A ΔL). Its unit is Pa. A steeper stress-strain graph indicates a larger Young modulus.
杨氏模量 E = (F L₀) / (A ΔL)，单位为 Pa。应力-应变曲线越陡，表示杨氏模量越大。
energy: the work done in deforming a material. For a linear elastic material obeying Hooke's law, the elastic strain energy U = ½ F ΔL = ½ k (ΔL)².
应变能：使材料形变所做的功。对于遵循胡克定律的线弹性材料，弹性应变能 U = ½ F ΔL = ½ k (ΔL)²。
7. Factor of Safety and Material Properties | 安全因数与材料性能
ss / maximum allowable working stress. Typical values range from 1.5 to 10 depending on application.
安全因数 (FoS) 是设计上的一种衡量指标：FoS = 极限拉伸应力 / 最大许用工作应力。根据应用场合，典型值在 1.5 至 10 之间。
Ductile materials show large plastic deformation before fracture, while brittle materials fracture with little plastic strain. Toughness is the energy absorbed up to fracture.
韧性材料在断裂前表现出较大的塑性变形，而脆性材料在断裂时几乎无塑性应变。韧性是指材料断裂前吸收的能量。
Hardness is resistance to indentation or scratching. Stiffness relates to Young modulus, while strength refers to the maximum stress a material can withstand.
硬度指抵抗压痕或划痕的能力。刚度与杨氏模量有关，强度则指材料能承受的最大应力。
Creep is slow, continuous deformation under constant load at high temperatures. Fatigue is failure due to repeated cyclic loading, often at stresses below the yield strength.
蠕变是在高温和恒定载荷下缓慢、持续的变形。疲劳是由反复循环载荷引起的失效，其应力通常低于屈服强度。
8. DC Circuit Fundamentals | 直流电路基础
charge: I = Q / t. Unit: ampere (A). Conventional current flows from positive to negative.
电流 I 是电荷流动的速率：I = Q / t，单位为安培 (A)。规定电流方向为正电荷移动方向。
Ohm' erence V across it is proportional to the current I: V = I R. Resistance R is in ohms (Ω).
欧姆定律：在温度不变的条件下，欧姆导体两端的电势差 V 与通过其的电流 I 成正比：V = I R。电阻 R 的单位为欧姆 (Ω)。
l property: R = ρ L / A, where L is length and A is cross-sectional area. The unit is Ω m.
电阻率 ρ 是材料的固有属性：R = ρ L / A，其中 L 为长度，A 为截面积，单位为 Ω m。
Resistors in series: Rtotal = R₁ + R₂ + R₃ + ... The same current flows through all components.
串联电阻：R_总 = R₁ + R₂ + R₃ + ... 通过每个元件的电流相等。
Resistors in parallel: 1/Rtotal = 1/R₁ + 1/R₂ + 1/R₃ + ... The same voltage appears across each branch.
并联电阻：1/R_总 = 1/R₁ + 1/R₂ + 1/R₃ + ... 各支路两端的电压相等。
Conductance G = 1/R, unit siemens (S). It is useful in parallel circuits: Gtotal = G₁ + G₂ + G₃.
电导 G = 1/R，单位为西门子 (S)，在并联电路中便于使用：G_总 = G₁ + G₂ + G₃。
9. Resistance, Resistivity and Potential Dividers | 电阻、电阻率与分压器
The resistance of a metallic conductor increases with temperature because increased lattice vibrations impede electron flow. For a thermistor, resistance typically decreases as temperature rises.
金属导体的电阻随温度升高而增大，因为晶格振动加剧阻碍了电子流动。热敏电阻的阻值通常随温度升高而减小。
A potential divider uses two resistors in series to produce a fraction of the input voltage: Vout = Vin × R₂ / (R₁ + R₂).
分压器利用两个串联电阻获得输入电压的一部分：Vout = Vin × R₂ / (R₁ + R₂)。
When one resistor is replaced by a sensor (LDR, thermistor), the output voltage changes with physical conditions. This principle is used in sensing and control circuits.
当其中一个电阻替换为传感器（如光敏电阻、热敏电阻）时，输出电压会随外界条件变化，这一原理常用于传感与控制电路。
A variable potential divider (potentiometer) allows continuous voltage adjustment and can be used to compare emfs without drawing current at balance.
可变分压器（电位器）可实现连续电压调节，并能在平衡状态下不取电流地比较电动势。
10. Kirchhoff's Laws and Circuit Analysis | 基尔霍夫定律与电路分析
Kirchhoff's first law (current law): at any junction, the total current entering equals the total current leaving: Σ Iin = Σ Iout. This arises from conservation of charge.
基尔霍夫第一定律（电流定律）：任一节点处，流入的电流总和等于流出的电流总和：Σ Iin = Σ Iout。这源自电荷守恒。
Kirchhoff' the algebraic sum of emfs equals the algebraic sum of potential differences: Σ ε = Σ I R.
基尔霍夫第二定律（电压定律）：在电路的任一闭合回路中，电动势的代数和等于各元件电势差的代数和：Σ ε = Σ I R。
These laws are used to solve complex circuits with multiple loops and components. Systematic sign conventions must be adopted for emf and p.d.
这两条定律用于求解含有多个回路和元件的复杂电路，求解时须对电动势和电势差采用系统的符号约定。
terminal p.d. V to drop under load: V = ε - I r. The maximum current (short-circuit) is Imax = ε / r.
电源内阻 r 会导致有负载时端电压 V 下降：V = ε - I r。最大电流（短路）为 Imax = ε / r。
11. Electrical Power and Energy | 电功率与电能
Electrical power P is the rate at which energy is transferred: P = I V. Using Ohm's law, it can also be expressed as P = I² R or P = V² / R. Unit: watt (W).
电功率 P 是能量传递的速率：P = I V。结合欧姆定律，亦可表示为 P = I² R 或 P = V² / R，单位为瓦特 (W)。
e: E = P t = I V t, measured in joules (J) or kilowatt-hours (kW h). 1 kW h = 3.6 × 10⁶ J.
一段时间内消耗的电能：E = P t = I V t，单位为焦耳 (J) 或千瓦时 (kW h)。1 kW h = 3.6 × 10⁶ J。
Heating effect of current (Joule heating) is described by Pheat This is used in fuses and heating elements; excessive heating can cause component damage.
电流热效应（焦耳热）由 Pheat = I² R 描述，应用于保险丝和加热元件；过度发热会损坏元件。
In a purely resistive AC circuit, root-mean-square (rms) values relate to peak values: Vrms = V₀ / √2, Irms = I₀ / √2. Power is P = Irms Vrms.
在纯电阻交流电路中，有效值（rms）与峰值的关系为：Vrms = V₀ / √2，Irms = I₀ / √2。功率为 P = Irms Vrms。
12. Fluid Statics and Manometry | 流体静力学与压力计
it: pascal (Pa). 1 Pa = 1 N m⁻². Fluid pressure acts equally in all directions at a point.
压强 p 是单位面积上的作用力：p = F / A，单位为帕斯卡 (Pa)，1 Pa = 1 N m⁻²。流体中某点的压强各向相等。
id of density ρ is given by p = ρ g h. This assumes uniform density and incompressibility.
密度为 ρ 的液体中，深度 h 处的静水压强为 p = ρ g h，公式假设密度均匀且液体不可压缩。
Pascal's principle: any change in pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid. This is the basis of hydraulic systems.
帕斯卡原理：封闭流体中任一点压强的变化将大小不变地传递至流体各处。这一原理是液压系统的基础。
tio of piston areas: F₂ / F₁ = A₂ / A₁. The displacement of the smaller piston is greater.
在液压机中，力的放大倍数等于活塞面积之比：F₂ / F₁ = A₂ / A₁。小活塞的位移相对较大。
g Δh. A U-tube manometer compares unknown pressure with atmospheric or reference pressure.
压力计利用液柱测量压强差：Δp = ρ g Δh。U 形管压力计将未知压强与大气压或参考压强进行比较。
Published by TutorHao | Engineering Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图