Year 10 CAIE Engineering: Formula & Theorem Quick Reference Handbook

📚 Year 10 CAIE Engineering: Formula & Theorem Quick Reference Handbook | 公式定理速查手册

Engineering at Year 10 builds the foundation for analysing real-world systems – from mechanical levers to electronic circuits. This quick reference handbook brings together the essential formulas and theorems you will encounter throughout the CAIE course. Each entry is paired with a concise explanation, the standard units to use, and when the principle applies. Use it for revision, problem solving, and strengthening your numerical confidence before assessments.

十年级工程学为分析真实世界的系统（从机械杠杆到电子电路）打下基础。本速查手册汇集了你在CAIE课程中将会遇到的所有重要公式与定理。每一条目都配有简洁的说明、标准单位以及适用条件，可用于复习、解题和考前强化你的计算信心。

1. Force, Mass and Acceleration | 力、质量与加速度

Newton’s second law relates unbalanced force, mass and acceleration in a straight line: F = m × a. Force is measured in newtons (N), mass in kilograms (kg) and acceleration in metres per second squared (m/s²). Always check that the force you use is the net force acting on the object. This relationship is the starting point for analysing structures, vehicles and moving parts in mechanical systems.

牛顿第二定律将不平衡力、质量和直线加速度联系起来：F = m × a。力的单位是牛顿（N），质量单位是千克（kg），加速度单位是米每二次方秒（m/s²）。务必确保所用的力是作用在物体上的合力。该关系是分析结构、车辆和机械运动部件的基础。

2. Moments and Equilibrium | 力矩与平衡

A moment is the turning effect of a force: M = F × d, where d is the perpendicular distance from the pivot to the line of action of the force. The unit is newton metre (N·m). For a system to be in equilibrium, the sum of clockwise moments must equal the sum of anticlockwise moments about any pivot: Σ M_clockwise = Σ M_{anticlockwise}. This principle is used extensively when designing levers, bridges and crane jibs.

力矩是力的转动效应：M = F × d，其中 d 是从支点到力作用线的垂直距离，单位是牛顿·米（N·m）。系统要处于平衡状态，绕任意支点的顺时针力矩总和必须等于逆时针力矩总和：Σ M_顺时针 = Σ M_逆时针。该原理广泛应用于设计杠杆、桥梁和起重机臂架。

3. Stress, Strain and the Young Modulus | 应力、应变与杨氏模量

Stress measures how much internal force a material experiences per unit area: σ = F / A, where σ is the tensile or compressive stress in pascals (Pa), F is the force in newtons, and A is the cross‑sectional area in m². Strain measures the relative deformation: ε = ΔL / L₀, which is a ratio with no units. For a material obeying Hooke’s law within its elastic limit, the Young modulus links stress and strain: E = σ / ε. The Young modulus has units of pascals (Pa) and describes a material’s stiffness.

应力衡量材料单位面积所承受的内力：σ = F / A，其中 σ 为拉伸或压缩应力，单位帕斯卡（Pa）；F 为力（N）；A 为横截面积（m²）。应变衡量相对变形：ε = ΔL / L₀，为一个比值，无单位。对于在弹性极限内遵循胡克定律的材料，杨氏模量将应力与应变联系起来：E = σ / ε。杨氏模量单位为帕斯卡（Pa），它描述材料的刚度。

4. Mechanical Advantage, Velocity Ratio and Efficiency | 机械利益、速度比与效率

For any simple machine, mechanical advantage (MA) = load / effort. The load is the output force the machine overcomes, and the effort is the input force applied. Velocity ratio (VR) is a theoretical measure of how far the effort moves compared with the load: VR = distance moved by effort / distance moved by load. The efficiency of a machine compares useful output work to input work: η (%) = (MA / VR) × 100%. Real machines always have an efficiency less than 100% due to friction.

对于任何简单机械，机械利益 (MA) = 负载 / 动力。负载是机械克服的输出力，动力是施加的输入力。速度比 (VR) 是理论上衡量输入力移动距离与负载移动距离之比：VR = 动力移动的距离 / 负载移动的距离。机械效率比较有用输出功与输入功：η (%) = (MA / VR) × 100%。由于摩擦，实际机械的效率始终低于100%。

5. Work, Energy and Power | 功、能与功率

Work done (W) is energy transferred when a force moves its point of application: W = F × d, with work in joules (J) when force is in newtons and distance in metres. Kinetic energy of a moving object is KE = ½ m v² (m in kg, v in m/s). Gravitational potential energy gained when lifting an object is GPE = m g h, where g is the gravitational field strength (9.8 m/s² on Earth). Power is the rate of doing work: P = W / t or P = E / t, measured in watts (W).

功 (W) 是力使其作用点移动时所传递的能量：W = F × d，功的单位是焦耳（J），力为牛顿，距离为米。运动物体的动能为 KE = ½ m v²（m 单位 kg，v 单位 m/s）。将物体抬高所获得的重力势能为 GPE = m g h，其中 g 是重力场强度（地球表面约为 9.8 m/s²）。功率是做功的快慢：P = W / t 或 P = E / t，单位是瓦特（W）。

6. Ohm’s Law and Resistance | 欧姆定律与电阻

For a metallic conductor at constant temperature, the current through it is directly proportional to the potential difference across it: V = I × R, where V is voltage in volts (V), I is current in amperes (A), and R is resistance in ohms (Ω). Resistance depends on the material’s resistivity (ρ), length (L) and cross‑sectional area (A): R = ρ L / A. This law is fundamental to all electronic and electrical engineering calculations.

对于温度不变的金属导体，通过它的电流与其两端的电位差成正比：V = I × R，其中 V 为电压（伏特，V），I 为电流（安培，A），R 为电阻（欧姆，Ω）。电阻取决于材料的电阻率 (ρ)、长度 (L) 和横截面积 (A)：R = ρ L / A。该定律是所有电子与电气工程计算的基础。

7. Resistors in Series and Parallel | 串并联电阻

When resistors are connected end‑to‑end (in series), the total resistance is the sum of the individual resistances: R_total = R₁ + R₂ + R₃ + …. The same current flows through each resistor, but the potential difference divides. For resistors connected side‑by‑side (in parallel), the reciprocal of the total resistance equals the sum of the reciprocals: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …. The voltage across each parallel branch is the same.

当电阻器依次首尾相接（串联）时，总电阻等于各个电阻之和：R_总 = R₁ + R₂ + R₃ + …。流过每个电阻器的电流相同，但电压会按比例分配。当电阻器并排连接（并联）时，总电阻的倒数等于各电阻倒数之和：1/R_总 = 1/R₁ + 1/R₂ + 1/R₃ + …。各并联支路两端的电压相同。

8. Electrical Power and Energy Dissipation | 电功率与能量耗散

Electrical power can be calculated using three equivalent expressions, easily derived from Ohm’s law: P = V × I, P = I² R and P = V² / R. Power is measured in watts (W). The energy consumed by a device is found from E = P × t, giving energy in joules (J) when power is in watts and time in seconds. In practical engineering, energy is often expressed in kilowatt‑hours (kWh). For a resistive component, all electrical energy is converted into heat.

电功率可以通过三个等效公式计算（均可由欧姆定律导出）：P = V × I、P = I² R 和 P = V² / R。功率单位为瓦特（W）。器件消耗的电能用 E = P × t 计算，功率单位用瓦特、时间单位用秒时，能量单位为焦耳（J）。在实际工程中，电能常用于千瓦时（kWh）表示。对于纯电阻元件，所有电能都转化为热量。

9. Thermal Expansion | 热膨胀

Most materials expand when heated. For a solid bar of original length L₀, the change in length ΔL caused by a temperature change ΔT is: ΔL = α L₀ ΔT, where α is the coefficient of linear expansion of the material, measured in per kelvin (/K) or per degree Celsius (/°C). This formula is vital when designing bridges, railway tracks and pipelines to allow for expansion gaps. Neglecting thermal expansion can lead to buckling or structural failure.

大多数材料受热时会膨胀。对于原始长度为 L₀ 的固体棒材，温度变化 ΔT 引起的长度变化 ΔL 为：ΔL = α L₀ ΔT，其中 α 为材料线膨胀系数，单位为每开尔文（/K）或每摄氏度（/°C）。在设计桥梁、铁轨和管道时，该公式对于预留伸缩缝至关重要。忽视热膨胀可能导致屈曲或结构失效。

10. Fluid Pressure | 流体压力

Pressure in a fluid at rest increases with depth because of the weight of the fluid above. The gauge pressure at a depth h is given by: p = ρ g h, where ρ is the density of the fluid (kg/m³), g is the gravitational field strength (N/kg) and h is the depth (m). The total pressure at that depth is the sum of atmospheric pressure and p. This principle explains hydraulic lift systems: a small force applied to a small area can generate a large force on a larger area, as pressure is transmitted equally throughout a confined fluid (Pascal’s principle).

静止流体的压强随深度增加，原因是上方流体的重量。深度 h 处的计示压强为：p = ρ g h，其中 ρ 为流体密度（kg/m³），g 为重力场强度（N/kg），h 为深度（m）。该深度处的总压强等于大气压强加上 p。这一原理解释了液压提升系统：由于在密闭流体中压强等值传递（帕斯卡原理），施加在小面积上的小力可以在大面积上产生大力。

11. Logic Gates and Truth Tables | 逻辑门与真值表

In digital control systems, logic gates process binary signals (0 and 1, or LOW and HIGH). The three fundamental gates are: AND (output 1 only if both inputs are 1), OR (output 1 if at least one input is 1) and NOT (inverter, output is the opposite of the single input). Truth tables record all possible input combinations and the resulting output. When designing safety circuits or automated manufacturing cells, engineers combine these gates to create the required control logic.

在数字控制系统中，逻辑门处理二进制信号（0 和 1，或低电平和高电平）。三种基本门是：与门 (AND)（仅当两个输入均为 1 时输出 1）、或门 (OR)（至少一个输入为 1 时输出 1）以及非门 (NOT)（反相器，输出与单一输入相反）。真值表记录所有可能的输入组合及其对应的输出。在设计安全电路或自动化生产单元时，工程师组合使用这些门来构建所需的控制逻辑。

12. Density and Material Properties | 密度与材料性能

Density links mass and volume: ρ = m / V, where m is mass (kg) and V is volume (m³). The SI unit is kg/m³, although g/cm³ is also common. Understanding density is essential when selecting materials for lightweight structures (low density) or for applications requiring high inertia (high density). Other key engineering properties include hardness, toughness, ductility and electrical conductivity, but density often provides the first indication of a material’s suitability for a given task.

密度将质量与体积联系起来：ρ = m / V，其中 m 为质量（kg），V 为体积（m³）。国际单位制单位为 kg/m³，不过 g/cm³ 也常用。在选择轻质结构材料（低密度）或需要高惯性的应用（高密度）时，理解密度至关重要。其他关键的工程性能还包括硬度、韧性、延展性和导电性，但密度往往是判断材料是否适合特定任务的首要指标。

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Year 10 CAIE Engineering: Formula & Theorem Quick Reference Handbook | 公式定理速查手册