📚 Formulas and Theorems Quick Reference | 工程公式定理速查手册
Welcome to the Cambridge IGCSE Engineering Quick Reference Handbook. This guide consolidates the essential formulas, theorems, and principles you need to master for your Year 11 assessments. From mechanics and materials to electricity and thermodynamics, these equations form the backbone of engineering problem-solving. Use this handbook as a revision aid to reinforce your understanding and boost your exam confidence.
欢迎使用剑桥 IGCSE 工程速查手册。本指南汇总了您在十一年级评估中需要掌握的基本公式、定理和原理。从力学和材料学到电学和热力学，这些方程是解决工程问题的核心。请将本手册作为复习工具，巩固知识并增强考试信心。
1. Newton's Laws of Motion | 牛顿运动定律
Newton' niform motion in a straight line unless acted upon by a net external force. The Second Law quantifies this: F = m × a, where F is the resultant force in newtons (N), m is mass in kilograms (kg), and a is acceleration in metres per second squared (m/s²). The Third Law declares that for every action, there is an equal and opposite reaction; forces always occur in pairs acting on different bodies.
牛顿三大定律阐述了物体与作用力之间的关系。第一定律（惯性定律）指出，除非受到净外力的作用，否则物体将保持静止或匀速直线运动状态。第二定律将其量化：F = m × a，其中 F 为合力（牛顿，N），m 为质量（千克，kg），a 为加速度（米每二次方秒，m/s²）。第三定律表明，每一个作用力都有一个大小相等、方向相反的反作用力；力总是成对出现，并作用在不同的物体上。
s the gravitational field strength (approx. 9.81 N/kg on Earth). The direction of weight is always towards the centre of the Earth. Remember that mass is a scalar quantity, while weight is a vector.
重力是第二定律的一个具体应用：W = m × g，其中 g 为重力场强度（地球表面约为 9.81 N/kg）。重力的方向总是指向地心。请记住，质量是标量，而重力是矢量。
2. Moments and Equilibrium | 力矩与平衡
the force (N), and d is the perpendicular distance from the pivot to the line of action of the force (m). For an object to be in rotational equilibrium, the Principle of Moments must hold: sum of clockwise moments = sum of anticlockwise moments about any pivot.
力矩是力绕支点产生的转动效应。其计算公式为：M = F × d，其中 M 为力矩（牛顿·米，Nm），F 为力（N），d 为从支点到力作用线的垂直距离（米）。要使物体处于转动平衡，必须满足力矩原理：绕任何支点的顺时针力矩之和等于逆时针力矩之和。
tant force in any direction must be zero: ΣF = 0. When solving problems, always draw a free-body diagram and clearly label forces and distances.
除了转动平衡外，要完全达到静态平衡，物体还必须处于平动平衡状态。这意味着任何方向上的合力必须为零：ΣF = 0。解题时，务必画出受力图，并清晰标示各个力和距离。
3. Stress, Strain, and Young's Modulus | 应力、应变与杨氏模量
ma; = F / A, where σ (sigma) is stress in pascals (Pa) or N/m², F is the applied force (N), and A is the original cross-sectional area (m²). Strain is a dimensionless measure of deformation: ε = ΔL / L₀, where ε (epsilon) is strain, ΔL is the change in length, and L₀ is the original length.
应力衡量材料在受到外部载荷时单位面积上的内力。拉伸或压缩应力的公式为：σ = F / A，其中 σ 为应力（帕斯卡，Pa 或 N/m²），F 为施加的力（N），A 为原始横截面积（m²）。应变是描述变形的无量纲量：ε = ΔL / L₀，其中 ε 为应变，ΔL 为长度变化量，L₀ 为原始长度。
Within the elastic limit, stress is directly proportional to strain (Hooke's Law). The constant of proportionality is Young' s, E: E = σ / ε. The unit of E is pascals (Pa). A material with a high Young' s flexibility. The gradient of the linear portion of a stress-strain graph gives the Young's Modulus.
在弹性限度内，应力与应变成正比（胡克定律）。比例常数即为杨氏模量 E：E = σ / ε。E 的单位也是帕斯卡（Pa）。杨氏模量高的材料刚度大、不易变形，低值则代表柔韧性好。应力-应变曲线线性部分的斜率即为杨氏模量。
4. Velocity, Acceleration, and Equations of Motion | 速度、加速度与运动方程
final velocity (m/s), a = acceleration (m/s²), s = displacement (m), t = time (s). The five key equations are: v = u + a t; s = ((u + v)/2) × t; s = u t + ½ a t²; s = v t - ½ a t²; and v² = u² + 2 a s.
对于沿直线做匀加速运动的物体，可运用以下 SUVAT 方程。符号定义：u = 初速度（m/s），v = 末速度（m/s），a = 加速度（m/s²），s = 位移（m），t = 时间（s）。五个核心方程为：v = u + a t；s = ((u + v)/2) × t；s = u t + ½ a t²;；s = v t - ½ a t²;；以及 v² = u² + 2 a s。
time graph, the gradient gives acceleration, and the area under the graph gives displacement. When an object falls freely under gravity, a is replaced by g (9.81 m/s² downwards).
加速度是速度的变化率：a = (v - u) / t。在速度-时间图中，斜率表示加速度，图线下的面积表示位移。当物体在重力作用下自由下落时，a 取为 g（9.81 m/s²，方向向下）。
5. Work, Energy, and Power | 功、能量与功率
rgy is the capacity to do work. Kinetic energy (KE) = ½ m v²; gravitational potential energy (GPE) = m g h, where h is the vertical height. The principle of conservation of energy states that energy cannot be created or destroyed, only transferred or converted from one form to another.
功是力与在力方向上移动距离的乘积：W = F × d。功的单位是焦耳（J），1 J = 1 N m。能量是做功的本领。动能（KE）= ½ m v²;；重力势能（GPE）= m g h，其中 h 为垂直高度。能量守恒定律指出，能量既不能凭空产生，也不能凭空消失，只能从一种形式转移或转化为另一种形式。
y speed v, power can also be expressed as P = F × v. In electrical contexts, power is calculated via P = I V, which will be covered later.
功率是做功或能量转移的速率：P = W / t，单位为瓦特（W，相当于 J/s）。对于以恒定速度 v 运动的恒力，功率也可表示为 P = F × v。在电学情境下，功率通过 P = I V 计算，稍后将作介绍。
6. Mechanical Advantage and Efficiency | 机械利益与效率
ad / Effort. It is a dimensionless number indicating how many times the machine amplifies the input force. Velocity Ratio (VR) is the ratio of the distance moved by the effort to the distance moved by the load in the same time: VR = distance moved by effort / distance moved by load.
机器的设计目的在于放大力量或改变力的方向。机械利益（MA）是负载力与施力之比：MA = 负载 / 施力。这是一个无量纲数值，表示机器将输入力放大了多少倍。速度比（VR）是在相同时间内施力移动距离与负载移动距离之比：VR = 施力移动距离 / 负载移动距离。
ne converts input work to output work: Efficiency = (Work output / Work input) × 100%. It can also be expressed as Efficiency = (MA / VR) × 100%. An ideal machine has 100% efficiency, but practical machines are always below this.
由于摩擦和其他损耗，实际 MA 始终小于理想 VR。效率衡量机器将输入功转化为输出功的程度：效率 = (输出功 / 输入功) × 100%。它也可表示为：效率 = (MA / VR) × 100%。理想机器的效率为 100%，但实际机器总是低于此值。
7. Simple Machines: Levers and Pulleys | 简单机械：杠杆与滑轮
rum. For all levers in equilibrium, Load × distance from fulcrum to load = Effort × distance from fulcrum to effort. Therefore, MA = effort arm / load arm. By changing the lever arm lengths, a large mechanical advantage can be achieved.
杠杆是绕支点转动的刚性杆。根据负载、施力和支点的相对位置，杠杆分为三类。对于所有处于平衡状态的杠杆，负载 × 支点到负载的距离 = 施力 × 支点到施力的距离。因此，MA = 施力臂 / 负载臂。通过改变力臂长度，可以获得较大的机械利益。
Pulley systems use ropes and grooved wheels to change the direction of force and achieve mechanical advantage. The velocity ratio (VR) of a simple pulley system is equal to the number of rope segments that support the moving load (ignoring friction). For example, a system with three supporting ropes has a VR of 3. The theoretical MA equals the VR if the system is frictionless.
滑轮系统利用绳索和带槽的轮子来改变力的方向并获得机械利益。简单滑轮系统的速度比（VR）等于支撑移动负载的绳段数（忽略摩擦）。例如，若有三段绳索支撑负载，VR 即为 3。若系统无摩擦，理论 MA 等于 VR。
8. Gear Systems and Velocity Ratio | 齿轮系统与速度比
th: VR = number of teeth on driven gear / number of teeth on driver gear. This is also the ratio of their angular speeds: VR = speed of driver / speed of driven. A small driver gear turning a larger driven gear produces a speed reduction but a torque increase.
齿轮用于在轴间传递旋转运动和转矩。一对啮合齿轮的速度比（VR）是它们齿数的反比：VR = 从动轮齿数 / 主动轮齿数。这也是它们角速度之比：VR = 主动轮转速 / 从动轮转速。小主动轮驱动大从动轮会降低转速但增大转矩。
For compound gear trains, the overall VR is the product of the VRs of each pair. Torque transmitted relates to the gear ratio; if VR > 1 (speed reduction), the output torque is approximately input torque × VR, assuming 100% efficiency. In practice, efficiency is reduced by friction between gear teeth.
对于复合齿轮系，总 VR 是每一对齿轮 VR 的乘积。传递的转矩与齿轮比相关；若 VR > 1（减速），在 100% 效率假设下，输出转矩约等于输入转矩 × VR。实际上，轮齿间的摩擦会降低效率。
9. Ohm's Law and Electrical Power | 欧姆定律与电功率
Ohm' is the potential difference in volts (V), I is current in amperes (A), and R is resistance in ohms (Ω). Components that obey this linear relationship are called ohmic conductors. Diodes, filament lamps, and thermistors are non-ohmic because their resistance changes with conditions.
欧姆定律描述了在恒温条件下金属导体中电压、电流和电阻之间的关系：V = I × R，其中 V 为电势差（伏特，V），I 为电流（安培，A），R 为电阻（欧姆，Ω）。遵守这种线性关系的元件称为欧姆导体。二极管、白炽灯和热敏电阻是非欧姆的，因为它们的电阻随条件而变化。
p2; / R. The unit of power is the watt (W). Energy consumed by an appliance is E = P × t, measured in joules (J) or kilowatt-hours (kWh) for practical electricity metering.
电功率是电能转换的速率。有三个等效表达式：P = I × V；P = I²; × R；以及 P = V² / R。功率的单位是瓦特（W）。用电器消耗的电能为 E = P × t，单位为焦耳（J），实际电表计量常用千瓦时（kWh）。
10. Series and Parallel Circuits | 串联与并联电路
In a series circuit, components are connected end to end, so the same current flows through each: Itotal = I₁ = I₂ = ... . The total potential difference is the sum of individual p.d.s: Vtotal = V₁ + V₂ + ... . The equivalent resistance is Rtotal = R₁ + R₂ + ... . If one component fails, the circuit breaks.
在串联电路中，各元件首尾相连，因此流过每个元件的电流相同：Itotal = I₁ = I₂ = ... 。总电势差等于各分电势差之和：Vtotal = V₁ + V₂ + ... 。等效电阻为 Rtotal = R₁ + R₂ + ... 。若一个元件损坏，整个电路断开。
In a parallel circuit, components are connected across the same two points, so the p.d. across each branch is the same: Vtotal = V₁ = V₂ = ... . The total current is the sum of branch currents: Itotal = I₁ + I₂ + ... . The reciprocal of total resistance is the sum of reciprocals: 1/Rtotal = 1/R₁ + 1/R₂ + ... . For two resistors, Rtotal = (R₁ × R₂) / (R₁ + R₂). Parallel circuits allow independent operation of branches.
在并联电路中，各元件连接在相同的两个节点之间，因此各支路两端的电势差相同：Vtotal = V₁ = V₂ = ... 。总电流等于各支路电流之和：Itotal = I₁ + I₂ + ... 。总电阻的倒数为各支路电阻倒数之和：1/Rtotal = 1/R₁ + 1/R₂ + ... 。对于两个电阻，Rtotal = (R₁ × R₂) / (R₁ + R₂)。并联电路允许各支路独立工作。
11. Pressure and Fluid Mechanics | 压强与流体力学
. In fluids, pressure increases with depth due to the weight of the fluid above: P = ρ g h, where ρ (rho) is the fluid density (kg/m³), g is gravitational field strength, and h is the depth (m). This hydrostatic pressure acts equally in all directions at a given depth.
压强定义为作用于单位面积上的力：P = F / A，单位为帕斯卡（Pa）。相同的力作用在较小的面积上会产生更大的压强。在流体中，由于上方流体的重量，压强随深度增加：P = ρ g h，其中 ρ 为流体密度（kg/m³），g 为重力场强度，h 为深度（m）。在给定深度处，这种流体静压强沿各个方向等值作用。
Pascal' f its container. This is the basis for hydraulic systems: a small force applied on a small-area piston generates a large pressure, which transmits to a large-area piston to produce a large output force. Mathematically, F₁ / A₁ = F₂ / A₂, giving MA = A₂ / A₁.
帕斯卡原理指出，施加在密闭流体上的压强变化会大小不变地向流体各个部分和容器壁传递。这是液压系统的基础：作用在小面积活塞上的小力产生大压强，传递到大面积活塞上，从而产生大输出力。数学表达为 F₁ / A₁ = F₂ / A₂，从而 MA = A₂ / A₁。
12. Thermal Expansion and Heat Transfer | 热膨胀与热传递
ses expand when heated. The linear expansion of a solid is given by: ΔL = α L&# change in length, α (alpha) is the coefficient of linear expansion (K⁻¹ or °C⁻¹), L&# , and ΔT is the temperature change. The volumetric expansion is ΔV = β V&# ids. Engineers must accommodate thermal expansion in structures like bridges and railways.
大多数固体、液体和气体受热时会膨胀。固体的线性膨胀公式为：ΔL = α L₀ ΔT，其中 ΔL 为长度变化量，α 为线膨胀系数（K⁻¹ 或 °C⁻¹），L₀ 为原始长度，ΔT 为温度变化量。体积膨胀为 ΔV = β V₀ ΔT，对于各向同性固体，β ≈ 3α。工程师必须在桥梁和铁路等结构中考虑热膨胀的影响。
lta;T / d), where k is thermal conductivity (W/m·K), A is cross-sectional area, ΔT is temperature difference, and d is thickness. For thermal design, materials with low conductivity (e.g., insulation) reduce heat loss, while high conductivity materials (e.g., metals) are used for heat sinks.
热量可以通过传导、对流和辐射传递。通过均匀材料的导热速率公式为：Q / t = k A (ΔT / d)，其中 k 为导热系数（W/m·K），A 为横截面积，ΔT 为温差，d 为厚度。在热设计中，低导热性材料（如绝缘体）可减少热量损失，而高导热性材料（如金属）则用于散热器。
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