📚 Year 8 CCEA Engineering: Formula & Theorem Quick Reference Handbook | 八年级CCEA工程：公式定理速查手册
This handbook brings together the essential formulas and theorems you will encounter in Year 8 CCEA Engineering. Keep it close when tackling design challenges, investigating mechanical systems or prototyping electrical circuits. Each entry is presented with a clear statement, a worked interpretation and the correct SI units so you can apply it with confidence.
本手册汇编了八年级CCEA工程课程中必须掌握的核心公式与定理。在面对设计挑战、研究机械系统或制作电路原型时，可以随时查阅。每个条目都配有明确陈述、解释说明和正确的国际单位，帮助你自信运用。
1. Force and Motion | 力与运动
Newton’s Second Law tells us that the net force acting on an object equals its mass multiplied by its acceleration. This is the foundation for understanding how vehicles, mechanisms and structures respond to loads.
牛顿第二定律指出，作用在物体上的净力等于物体的质量乘以加速度。这是理解车辆、机构和结构如何响应载荷的基础。
F = m × a
In many engineering scenarios the dominant force is weight. Weight is the pull of gravity on a mass and is calculated using gravitational field strength g, which is approximately 10 N/kg on the surface of the Earth.
在许多工程场景中，主要的作用力是重量。重量是重力对质量的吸引，可以用重力场强度 g 计算，地球表面 g 约为 10 牛每千克。
W = m × g
For constant straight-line motion, speed is simply the distance travelled divided by the time taken. Acceleration measures how quickly speed changes; it is the change in velocity divided by time.
对于匀速直线运动，速率就是经过的距离除以所用时间。加速度衡量速度变化的快慢，等于速度变化量除以时间。
v = d ÷ t and a = (v − u) ÷ t
2. Work, Energy and Power | 功、能与功率
When a force moves an object through a distance along the line of action, we say that work is done. Work transfers energy, measured in joules. In simple lifting machines this tells us the energy input needed to raise a load.
当一个力沿作用线方向使物体移动一段距离时，我们称做了功。功转移能量，单位是焦耳。在简单起重机械中，这告诉我们提升载荷所需的能量输入。
W = F × d
Power is the rate at which work is done or energy is converted. A powerful motor delivers the same amount of work in a shorter time. The standard unit of power is the watt (W).
功率是做功或能量转换的速率。大功率电机会在更短的时间内完成同样的功。功率的标准单位是瓦特。
P = W ÷ t or P = E ÷ t
3. Mechanical Advantage | 机械利益
Mechanical advantage (MA) describes how much a mechanism multiplies an applied effort force. It is the ratio of the load output to the effort input. An MA greater than 1 means the machine makes the task feel easier.
机械利益表示机构能将施加的作用力放大多少倍。它是输出载荷与输入作用力的比值。机械利益大于 1 表示机器让操作感觉更省力。
MA = Load ÷ Effort
The ideal mechanical advantage (IMA) assumes no friction and is calculated from the distances moved by the effort and the load. The actual MA is always lower because of energy losses.
理想机械利益假设无摩擦，由作用力移动距离与载荷移动距离的比值计算。实际机械利益因能量损耗而总是更小。
IMA = deffort ÷ dload
4. Moments and Levers | 力矩与杠杆
A moment is the turning effect of a force about a pivot. It depends on how large the force is and how far it is applied from the pivot. Engineers use moments to design levers, seesaws and balanced structures.
力矩是力对支点的转动效应，取决于力的大小以及作用点到支点的垂直距离。工程师利用力矩设计杠杆、跷跷板和平衡结构。
M = F × d⟂
For a lever in equilibrium, the sum of the clockwise moments equals the sum of the anticlockwise moments. This principle lets us calculate unknown forces in linkages and beam balances.
对于处于平衡状态的杠杆，顺时针力矩之和等于逆时针力矩之和。这一原理可用于计算连杆和梁式天平中的未知力。
F₁ × d₁ = F₂ × d₂
Levers are classified by the relative positions of fulcrum, load and effort. Class 1 has the fulcrum in the middle, class 2 has the load in the middle, and class 3 has the effort in the middle. Each class trades distance for force in a different way.
杠杆按支点、载荷和作用力的相对位置分为三类：一类杠杆支点在中间，二类杠杆载荷在中间，三类杠杆作用力在中间。每一类都以不同方式用距离换取力。
5. Pressure | 压力
Pressure measures how concentrated a force is over a surface. A sharp blade exerts higher pressure than a blunt one because the same force acts on a smaller area. This concept is crucial for cutting tools, foundations and hydraulic systems.
压力度量力在表面上的集中程度。锋利的刀刃比钝刀产生更大的压力，因为同样的力作用在更小的面积上。这一概念对切割工具、地基和液压系统至关重要。
P = F ÷ A
Pascal's principle states that a change in pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid. This is why a small force on a narrow piston can lift a heavy load on a wide piston in a hydraulic press.
帕斯卡原理指出，施加在封闭流体上的压强变化会大小不变地向流体各处传递。这就是液压机中窄活塞上的小力能举起宽活塞上重物的原因。
F₁ ÷ A₁ = F₂ ÷ A₂
6. Electricity Basics | 电学基础
Ohm’s Law links voltage, current and resistance in a simple circuit. It helps engineers choose the right components so that LEDs, buzzers and motors work within safe limits.
欧姆定律给出了简单电路中电压、电流和电阻的关系，帮助工程师选择合适的元件，使LED、蜂鸣器和电机在安全范围内工作。
V = I × R
In a series circuit, the total resistance is the sum of individual resistances. In a parallel circuit, the reciprocal of the total resistance equals the sum of the reciprocals of the branches.
在串联电路中，总电阻等于各个电阻之和。在并联电路中，总电阻的倒数等于各支路电阻倒数之和。
Rtotal = R₁ + R₂ (series)
1 ÷ Rtotal = 1 ÷ R₁ + 1 ÷ R₂ (parallel)
The electrical power converted by a component is the product of the voltage across it and the current through it. This formula allows you to size batteries and check whether a resistor will overheat.
元件转换的电功率等于其两端电压与通过电流的乘积。用这个公式可以估算电池容量，并检查电阻是否会过热。
P = V × I
7. Density | 密度
Density links the mass of a material to the volume it occupies. Knowing the density helps engineers select light but strong materials for aerospace components or heavy, stable ones for foundations.
密度将材料的质量与其占据的体积联系起来。了解密度有助于工程师为航空航天部件选择轻而强的材料，或为地基选择重而稳固的材料。
ρ = m ÷ V
³) and water (1000 kg/m³). Always check units: if mass is in grams and volume in cm³, density will be in g/cm³.
八年级项目中可能用到的常见材料密度有铝（约 2700 kg/m³）、钢（约 7800 kg/m³）和水（1000 kg/m³）。务必检查单位：若质量用克、体积用立方厘米，则密度单位为 g/cm³。
8. Efficiency | 效率
Efficiency tells us how well a machine converts input energy into useful output. No real machine is 100% efficient because some energy is always dissipated as heat or sound.
效率反映机器将输入能量转化为有用输出能量的程度。没有真实的机器能达到 100% 效率，因为总有一部分能量以热或声的形式散失。
η = (Useful output energy ÷ Total input energy) × 100%
ncy can also be worked out with power: η = (Useful power output ÷ Total power input) × 100% riction low, lubricating moving parts and using streamlined shapes all improve efficiency.
效率也可用功率计算：η = (有用输出功率 ÷ 总输入功率) × 100%。减少摩擦、润滑运动部件和使用流线型外形都能提高效率。
9. Buoyancy and Archimedes' Principle | 浮力与阿基米德原理
Archimedes' principle states that an object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. This principle explains why ships float and why hot-air balloons rise.
阿基米德原理指出，浸在流体中的物体会受到向上的浮力，其大小等于它排开流体的重量。这一原理解释了轮船为什么能漂浮、热气球为什么会升空。
Fb = ρfluid × Vdisplaced × g
If the buoyant force exceeds the object’s weight, it rises; if it is less, it sinks. Designers use this relationship to plan buoyancy aids, submarines and even floating platforms for offshore engineering.
当浮力大于物体重量时，物体上浮；小于物体重量时，物体下沉。设计人员利用这一关系设计浮标、潜水艇，甚至海上工程的浮动平台。
10. Springs and Hooke's Law | 弹簧与胡克定律
Hooke's Law describes the behaviour of many elastic materials. The extension of a spring is directly proportional to the force applied to it, provided the material’s elastic limit is not exceeded. This is a key principle behind force meters and suspension systems.
胡克定律描述了许多弹性材料的行为：只要不超过材料的弹性极限，弹簧的伸长量与其受到的作用力成正比。这是弹簧测力计和悬挂系统背后的关键原理。
F = k × e
The spring constant k indicates how stiff the spring is. A large k means a strong, difficult-to-stretch spring; a small k describes a soft, easily extended one. In engineering, spring constants guide the selection of shock absorbers, switches and clamps.
弹簧常数 k 表示弹簧的刚度。k 值大说明弹簧硬、难拉伸；k 值小说明弹簧软、易伸长。在工程中，弹簧常数指导减震器、开关和夹具的选择。
11. Gear Ratios and Speed | 齿轮比与速度
Gears transfer rotary motion from one shaft to another. The gear ratio relates the number of teeth on the driven gear to the number on the driver gear. A ratio greater than 1 reduces speed but multiplies torque, which is useful for climbing hills or lifting loads.
齿轮将旋转运动从一根轴传递到另一根轴。齿轮比是被动齿轮齿数与主动齿轮齿数之比。齿轮比大于 1 时会降低转速但增大力矩，适用于爬坡或举升重物。
Gear Ratio = Tdriven ÷ Tdriver
If you know the input speed and the gear ratio, you can find the output speed. A simple formula relates rotational speeds to tooth counts.
如果已知输入转速和齿轮比，就可以求出输出转速。旋转速度与齿数之间有简单的换算关系。
ωdriver ÷ ωdriven = Tdriven ÷ Tdriver
Compound gear trains combine several pairs to achieve large speed reductions or increases in a compact space. They appear in clocks, bicycle derailleurs and robot arm joints.
复合齿轮系将多对齿轮组合在一起，可在紧凑的空间内实现大幅的减速或增速，常见于钟表、自行车变速器和机器人手臂关节。
12. Units and Conversions | 单位与换算
Using consistent SI units prevents mistakes. The table below lists the base and derived quantities you will use most often, together with their symbols and units.
使用一致的国际单位可以避免错误。下表列出了你最常用的基本量和导出量及其符号和单位。
| Quantity / 物理量 | Symbol / 符号 | SI Unit / 国际单位 | Unit Symbol |
|---|---|---|---|
| Length / 长度 | d, s | metre / 米 | m |
| Mass / 质量 | m | kilogram / 千克 | kg |
| Time / 时间 | t | second / 秒 | s |
| Force / 力 | F, W | newton / 牛 | N |
| Pressure / 压力 | P | pascal / 帕 | Pa |
| Work & Energy / 功与能 | W, E | joule / 焦 | J |
| Power / 功率 | P | watt / 瓦 | W |
| Voltage / 电压 | V | volt / 伏 | V |
| Current / 电流 | I | ampere / 安 | A |
| Resistance / 电阻 | R | ohm / 欧姆 | Ω |
m² = 0.0001 m². Always convert to base SI units before substituting numbers into formulas.
常用换算：1 千牛 = 1000 牛，1 兆帕 = 1,000,000 帕，1 吨 = 1000 千克，1 平方厘米 = 0.0001 平方米。代入公式计算前务必先换算为基本国际单位。
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