📚 Year 9 CAIE Engineering: Formula & Theorem Quick Reference Handbook | Year 9 CAIE 工程：公式定理速查手册
Welcome to your Year 9 CAIE Engineering Quick Reference Handbook. This guide brings together the most important formulas, laws, and theorems you will encounter in your engineering studies. Each chapter gives a clear statement of the principle, the standard equation written in proper notation, and practical examples to help you apply the theory. Keep this revision tool close – whether you are checking a calculation, preparing for a test, or solving a design problem.
欢迎使用 Year 9 CAIE 工程速查手册。本书汇集了工程学习中最重要的公式、定律和定理。每一章都给出了原理的清晰表述、规范书写的标准方程以及帮助你应用理论的实际例子。无论你是在核对计算、备考还是解决设计问题，都可以随时查阅这本复习工具。
1. SI Units and Common Conversions | 国际单位制与常用换算
m (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for temperature and mole (mol) for amount of substance. From these base units we derive force (newton, N = kg m s⁻²), pressure (pascal, Pa = N m⁻²), energy (joule, J = N m) and power (watt, W = J s⁻¹). Before performing any calculation, convert all values to base SI units.
所有工程量都必须用统一的单位制表示。国际单位制（SI）用米（m）表示长度，千克（kg）表示质量，秒（s）表示时间，安培（A）表示电流，开尔文（K）表示温度，摩尔（mol）表示物质的量。由这些基本单位导出力的单位牛顿（N = kg m s⁻²）、压强单位帕斯卡（Pa = N m⁻²）、能量单位焦耳（J = N m）和功率单位瓦特（W = J s⁻¹）。进行任何计算之前，请将所有数值换算为基本 SI 单位。
| Quantity | Unit name | Symbol | Common conversion |
|---|---|---|---|
| Length | metre | m | 1 cm = 0.01 m; 1 mm = 0.001 m |
| Mass | kilogram | kg | 1 g = 0.001 kg; 1 tonne = 1000 kg |
| Time | second | s | 1 min = 60 s; 1 h = 3600 s |
| Force | newton | N | 1 N = 1 kg m s⁻² |
| Pressure | pascal | Pa | 1 Pa = 1 N m⁻² |
上表总结了工程学中常用物理量的单位和换算关系。计算前务必确认所有数据已转换为合适的基本单位，这样可以避免单位错误导致的失误。
2. Newton's First Law of Motion | 牛顿第一运动定律
An object at rest will remain at rest, and an object in motion will continue to move at constant velocity in a straight line, unless acted upon by a resultant external force. This property of matter is called inertia. The larger the mass of an object, the greater its inertia, and the more force is needed to change its state of motion.
一个静止的物体将保持静止，一个运动的物体将沿直线匀速运动，除非受到一个合外力的作用。物质的这种特性叫作惯性。物体的质量越大，惯性越大，改变其运动状态所需要的力也越大。
In engineering, we apply this law when analysing balanced forces. If the forces on a stationary bridge or a vehicle moving at constant speed are balanced, the net force is zero and the motion does not change. Seat belts in cars use inertia to lock the belt when the car decelerates suddenly.
在工程中，我们分析平衡力时会应用这一定律。如果静止的桥梁或匀速行驶的车辆所受的力平衡，则合外力为零，运动状态保持不变。汽车内的安全带利用惯性在车辆突然减速时锁住织带。
3. Newton's Second Law of Motion | 牛顿第二运动定律
The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is the heart of dynamics.
物体的加速度与其所受的合外力成正比，与其质量成反比。这一关系是动力学的核心。
F = m × a
e, if a 2000 kg car accelerates at 3 m s⁻², the required driving force is F = 2000 × 3 = 6000 N. Always use the net force when several forces act together.
其中 F 是合外力，单位牛顿（N）；m 是质量，单位千克（kg）；a 是加速度，单位米每二次方秒（m s⁻²）。例如，一辆 2000 kg 的汽车以 3 m s⁻² 加速，所需推力为 F = 2000 × 3 = 6000 N。当多个力同时作用时，必须使用合力。
4. Newton's Third Law of Motion | 牛顿第三运动定律
For every action force there is an equal and opposite reaction force. These two forces act on different bodies and are of the same type. It is because of this law that rockets can accelerate in space: the engine pushes exhaust gases backwards, and the gases push the rocket forwards with an equal force.
每一个作用力都有一个大小相等、方向相反的反作用力。这两个力作用在不同的物体上且属于同种类型。正是由于这一定律，火箭能在太空中加速：发动机把燃气向后推，燃气以大小相等的力把火箭向前推。
When you push against a wall, the wall pushes back against you. In structures, supports provide reaction forces that balance the loads. Recognising action–reaction pairs is essential when drawing free-body diagrams and designing joints.
当你推墙时，墙也会对你施加一个反方向的力。在结构中，支座提供与荷载平衡的反作用力。绘制受力图和设计连接点时，识别作用力与反作用力对至关重要。
5. Hooke's Law | 胡克定律
Many engineering materials, especially springs under light loads, obey Hooke's Law: the extension of a spring is directly proportional to the load applied, provided the elastic limit is not exceeded.
许多工程材料，尤其是轻载下的弹簧，都遵循胡克定律：只要不超过弹性极限，弹簧的伸长量与所施加的载荷成正比。
F = k × Δx
Here F is the applied force (N), Δx is the extension (m), and k is the spring constant (N m⁻¹). A stiff spring has a large k. Once the elastic limit is passed, the material deforms plastically and the law no longer holds. This limit is crucial when selecting materials for load-bearing components.
式中 F 为施加的力（N），Δx 为伸长量（m），k 为弹簧常数（N m⁻¹）。弹簧越硬，k 值越大。一旦超过弹性极限，材料发生塑性变形，胡克定律不再适用。对于承力部件的材料选择，这一极限至关重要。
6. Moments and Equilibrium | 力矩与平衡
The turning effect of a force about a pivot is called a moment. It depends on both the size of the force and the perpendicular distance from the pivot to the line of action.
力绕支点的转动效应称为力矩。它取决于力的大小以及支点到力作用线的垂直距离。
M = F × d
tal clockwise moment must equal the total anticlockwise moment: Σ M_clockwise = Σ M_anticlockwise. Engineers use this principle in everything from weighing scales to crane design.
M 为力矩，单位牛顿米（N m）；F 为力，单位牛顿（N）；d 为垂直距离，单位米（m）。要使杠杆平衡，顺时针力矩之和必须等于逆时针力矩之和：Σ M_顺时针 = Σ M_逆时针。工程师将这一原理应用于从台秤到起重机设计的方方面面。
7. Work, Energy and Power | 功、能量与功率
Work is done when a force moves an object in the direction of the force. The amount of work is equal to the product of the force and the distance moved.
当力使物体沿力的方向运动时，力做了功。功的大小等于力与移动距离的乘积。
W = F × d
Work W is measured in joules (J). Energy is the capacity to do work and exists in many forms. Gravitational potential energy gained by raising a mass through a height h is given by:
功 W 以焦耳（J）为单位。能量是做功的能力，有多种形式。将质量为 m 的物体举高 h 所获得的重力势能为：
Eₚ = m × g × h
A moving object possesses kinetic energy:
运动的物体具有动能：
Eₖ = ½ m v²
In any closed system energy is conserved: it may transfer between stores, but the total amount remains constant. Power is the rate at which work is done or energy is transferred.
在任何封闭系统中能量都是守恒的：能量可以在不同储存形式之间转移，但总量保持不变。功率是做功或能量转移的速率。
P = W / t or P = E / t
= F × v. Understanding energy and power helps in motor selection and efficiency analysis.
功率 P 的单位是瓦特（W）。若物体以恒定速度 v 克服恒定力 F 运动，功率也可表示为 P = F × v。理解能量和功率有助于选择电机和进行效率分析。
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