📚 Year 10 CAIE Engineering: Formula & Theorem Quick Reference Handbook | Year 10 剑桥工程：公式定理速查手册
This handbook is a curated collection of the most important formulas and theorems you will encounter in Year 10 CAIE Engineering. It covers mechanics, materials, electricity, and manufacturing – all aligned with the CAIE IGCSE Engineering syllabus. Use it for quick revision and problem-solving practice.
本手册精选了 Year 10 CAIE 工程课程中最重要的公式与定理，涵盖力学、材料、电学与制造工艺，完全贴合 CAIE IGCSE 工程教学大纲。可用作快速复习和解题训练的速查工具。
1. Forces and Equilibrium | 力与平衡
Newton’s Second Law states that the resultant force acting on an object is equal to its mass multiplied by its acceleration. For objects near the Earth’s surface, weight is the force due to gravity. An object is in equilibrium when both the resultant force and the resultant moment acting on it are zero.
F = ma W = mg
牛顿第二定律指出，物体所受的合外力等于质量与加速度的乘积。在地球表面附近，重量是引力产生的力。当物体所受的合外力和合力矩均为零时，物体处于平衡状态。
A body will remain at rest or in uniform motion unless acted upon by a net external force (Newton’s First Law). For every action, there is an equal and opposite reaction (Newton’s Third Law).
若不受净外力作用，物体将保持静止或匀速直线运动（牛顿第一定律）。每一个作用力都有一个大小相等、方向相反的反作用力（牛顿第三定律）。
2. Moments and Levers | 力矩与杠杆
The moment of a force about a pivot is the product of the force and the perpendicular distance from the pivot to the line of action of the force. The Principle of Moments states that for a system in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments.
Moment = F × d
力对支点的力矩等于力的大小乘以支点到力作用线的垂直距离。力矩原理指出，对于处于转动平衡的系统，顺时针力矩总和等于逆时针力矩总和。
Levers are classified into three orders depending on the relative positions of the fulcrum, load and effort. The moment of a force can be increased by increasing the force or the distance from the pivot.
根据支点、负载和施力点的相对位置，杠杆可分为三类。增大作用力或增加力臂长度均可增大力矩。
3. Work, Energy and Power | 功、能与功率
Work is done when a force moves an object in the direction of the force. Kinetic energy depends on mass and speed, while gravitational potential energy depends on mass, height and gravitational field strength. Power is the rate of doing work.
W = Fd Eₖ = ½mv² Eₚ = mgh P = W / t
力使物体沿力的方向移动时做功。动能取决于质量和速度，重力势能取决于质量、高度和重力场强度。功率是做功的快慢。
Efficiency is the ratio of useful output work (or power) to total input work (or power), expressed as a percentage.
Efficiency = (useful output / total input) × 100%
效率是有用输出功（或功率）与总输入功（或功率）之比，以百分数表示。
4. Simple Machines: MA, VR and Efficiency | 简单机械：机械利益、速比与效率
e effort to the distance moved by the load. In an ideal machine with no friction, MA = VR.
MA = Load / Effort VR = Distance moved by effort / Distance moved by load
机械利益（MA）表明机器放大施力的倍数。速比（VR）是施力移动距离与负载移动距离的比值。在没有摩擦的理想机器中，MA = VR。
The efficiency of a simple machine can also be calculated using MA and VR. Friction causes the MA to be less than VR, reducing efficiency.
Efficiency = (MA / VR) × 100%
简单机械的效率也可通过 MA 和 VR 计算。摩擦会使 MA 小于 VR，从而降低效率。
5. Stress and Strain | 应力与应变
Stress quantifies the internal force per unit area within a material. Strain measures the deformation relative to the original dimension. Young’s modulus describes the stiffness of a material within its elastic limit.
Stress (σ) = F / A Strain (ε) = ΔL / L₀ E = σ / ε
应力衡量材料内部单位面积上的内力。应变是变形量相对于原尺寸的度量。杨氏模量描述材料在弹性限度内的刚度。
The factor of safety is used in design to ensure components can withstand loads beyond the expected maximum.
Factor of Safety = Ultimate tensile strength / Allowable working stress
安全系数用于设计中，确保构件能承受超出预期最大载荷的力。
6. Basic Electricity: Ohm’s Law | 电学基础：欧姆定律
Ohm’s Law states that the current through a conductor is directly proportional to the potential difference across it, provided the temperature remains constant. Electrical power can be expressed in three useful forms combining voltage, current and resistance.
V = IR P = IV P = I²R P = V² / R
欧姆定律指出，在温度不变的条件下，通过导体的电流与导体两端的电势差成正比。电功率有三种组合电压、电流和电阻的表达形式。
The unit of power is the watt (W), which is equivalent to one joule per second. These relationships underpin all circuit analysis in the CAIE Engineering course.
功率的单位是瓦特（W），相当于每秒一焦耳。这些关系式为 CAIE 工程课程中所有电路分析奠定了基础。
7. Resistance and Resistivity | 电阻与电阻率
The resistance of a conductor depends on its length, cross-sectional area, and the resistivity of the material. Resistivity is a material property that indicates how strongly it opposes the flow of electric current.
R = ρL / A
导体的电阻取决于其长度、横截面积和材料的电阻率。电阻率是衡量材料阻碍电流能力强弱的属性。
For a given material, doubling the length doubles the resistance, while doubling the cross-sectional area halves the resistance. Temperature changes also affect resistance; for metals, resistance increases with temperature.
对于给定材料，长度加倍则电阻加倍，横截面积加倍则电阻减半。温度变化也会影响电阻；对金属而言，电阻随温度升高而增大。
8. Series and Parallel Circuits | 串联与并联电路
When resistors are connected in series, the total resistance is the sum of the individual resistances, and the same current flows through each resistor. In parallel, the reciprocal of the total resistance equals the sum of the reciprocals of each resistance, and the voltage across each branch is the same.
Rtotal(series) = R₁ + R₂ + ... 1/Rtotal(parallel) = 1/R₁ + 1/R₂ + ...
电阻串联时，总电阻等于各个电阻之和，通过每个电阻的电流相同。电阻并联时，总电阻的倒数等于各电阻倒数之和，各支路两端的电压相同。
Understanding these rules helps in calculating equivalent resistance, current division and voltage division in mixed circuits, which are common in practical engineering problems.
理解这些规则有助于计算混联电路中的等效电阻、分流和分压，这在工程实际应用中十分常见。
9. Manufacturing Calculations | 制造工艺计算
In machining operations like turning and milling, cutting speed is the speed at which the cutting edge passes over the workpiece. It is calculated from spindle speed and workpiece diameter. Feed rate and machining time are essential for production planning.
Cutting speed (Vc, m/min) = (π × D × N) / 1000
在车削、铣削等加工中，切削速度是切削刃相对于工件表面的线速度，由主轴转速和工件直径计算得出。进给速度和加工时间是生产计划的重要参数。
| Symbol / 符号 | Meaning / 含义 | Unit / 单位 |
|---|---|---|
| D | Workpiece or tool diameter / 工件或刀具直径 | mm |
| N | Spindle speed / 主轴转速 | rpm |
| f | Feed per revolution / 每转进给量 | mm/rev |
| L | Length of cut / 切削长度 | mm |
Feed rate (F, mm/min) = f × N. Machining time (T, min) = L / F.
进给速度 (F, mm/min) = f × N。加工时间 (T, min) = L / F。
10. Fluid Pressure and Hydraulics | 流体压力与液压
Pressure is defined as the force acting perpendicularly per unit area. In a static liquid, pressure increases with depth and depends on the density of the liquid and gravitational field strength. Pascal’s principle explains force multiplication in hydraulic systems.
P = F / A P (in a liquid) = ρgh F₁/A₁ = F₂/A₂
压强定义为单位面积上承受的正压力。在静止液体中，压强随深度增加，且与液体密度和重力场强度相关。帕斯卡原理解释了液压系统的力放大效应。
Hydraulic presses and braking systems use the fact that a pressure applied to an enclosed fluid is transmitted undiminished to all parts of the fluid. A small force on a small piston can balance a larger force on a larger piston, achieving mechanical advantage.
液压机和制动系统利用封闭流体传递压强不变的特性。在小活塞上施加较小的力，可以平衡大活塞上较大的力，从而获得机械利益。
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