📚 Year 8 CCEA Engineering: Quick Reference Formula & Theorem Handbook | Year 8 CCEA 工程：公式定理速查手册
This handbook brings together all the essential formulas and theorems you need for Year 8 CCEA Engineering. Use it to refresh your knowledge of forces, electronics, structures and mechanical systems quickly and accurately.
本手册汇集了 Year 8 CCEA 工程课程所需的所有关键公式与定理。你可以利用它快速、准确地重温力学、电子学、结构和机械系统方面的知识。
1. Hooke’s Law and Elastic Extension | 胡克定律与弹性伸长
Hooke’s Law explains how elastic objects like a spring behave when a stretching or compressing force is applied.
胡克定律解释了弹簧等弹性物体在受到拉伸或压缩力时的行为。
It states that the extension (e) is directly proportional to the force (F) as long as the limit of proportionality is not exceeded. The constant of proportionality is the spring constant k, expressed in newtons per metre (N/m).
定律指出，只要不超过比例极限，伸长量 (e) 与所施加的力 (F) 成正比。比例常数为弹簧常数 k，单位为牛顿每米 (N/m)。
F = k × e
A spring with k = 40 N/m stretches by 0.15 m. The force applied is F = 40 × 0.15 = 6.0 N.
示例：一根弹簧的 k = 40 N/m，伸长 0.15 m，则所施加的力 F = 40 × 0.15 = 6.0 N。
2. Ohm’s Law and Basic Circuit Rules | 欧姆定律与基本电路规则
Ohm’s Law relates the voltage across a conductor, the current flowing through it and its resistance.
欧姆定律描述了导体两端的电压、流过导体的电流及其电阻之间的关系。
ge (volts, V), I is current (amperes, A) and R is resistance (ohms, Ω). In a series circuit the current is the same everywhere, and the total resistance is the sum of the individual resistances.
公式为 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）。在串联电路中，各处电流相同，总电阻为各个电阻之和。
V = I × R Rtotal = R₁ + R₂ + R₃ + …
a 5 Ω resistor, the total resistance is 8 Ω and the current is I = V / R = 12 / 8 = 1.5 A.
若一个 12 V 的电池为一个 3 Ω 和一个 5 Ω 的串联电阻供电，总电阻为 8 Ω，电流 I = V / R = 12 / 8 = 1.5 A。
3. Density and Mass-Volume Relationships | 密度与质量体积关系
Density measures how much mass is packed into a given volume. It lets engineers choose the right material for a job.
密度衡量单位体积内所含质量的大小，可帮助工程师选择合适的材料。
Density (ρ) is calculated by dividing mass (m) by volume (V). Common units are kilograms per cubic metre (kg/m³) or grams per cubic centimetre (g/cm³).
密度 (ρ) 等于质量 (m) 除以体积 (V)。常用单位为千克每立方米 (kg/m³) 或克每立方厘米 (g/cm³)。
ρ = m ÷ V
ck of aluminium with mass 540 g and volume 200 cm³ has a density of 540 ÷ 200 = 2.7 g/cm³.
一块质量为 540 g、体积为 200 cm³ 的铝块，其密度为 540 ÷ 200 = 2.7 g/cm³。
4. Speed and Motion | 速度与运动
Speed describes how fast an object moves, without considering direction. It is a scalar quantity essential for timing mechanisms.
速度描述物体运动的快慢，不考虑方向。它是一个标量，对计时机构至关重要。
Average speed is found from the total distance travelled divided by the time taken. The triangle formula helps rearrange the variables.
平均速度等于运动的总距离除以所用时间。利用公式三角形可对变量进行重新整理。
v = d / t d = v × t t = d / v
If a small robot travels 3.6 m in 12 seconds, its average speed is v = 3.6 / 12 = 0.3 m/s.
如果一个小型机器人在 12 秒内移动了 3.6 m，其平均速度 v = 3.6 / 12 = 0.3 m/s。
5. Moments and Equilibrium | 力矩与平衡
A moment is the turning effect of a force about a pivot. Understanding moments is key to levers, seesaws and structural design.
力矩是力对支点的转动效应。理解力矩是设计杠杆、跷跷板和结构的关键。
The moment of a force equals the force multiplied by the perpendicular distance from the pivot to the line of action of the force. For a system in equilibrium the total clockwise moment equals the total anticlockwise moment.
力矩等于力与支点到力作用线的垂直距离的乘积。要使系统平衡，总顺时针力矩必须等于总逆时针力矩。
Moment = F × d Σ Clockwise moments = Σ Anticlockwise moments
If a 15 N force acts 0.4 m from a pivot, the moment is 15 × 0.4 = 6 N m.
若一个 15 N 的力作用在距支点 0.4 m 处，力矩为 15 × 0.4 = 6 N m。
6. Mechanical Advantage and Simple Machines | 机械优势与简单机械
Simple machines such as levers and pulleys make work easier by multiplying force or changing its direction.
杠杆和滑轮等简单机械通过放大作用力或改变其方向使工作更轻松。
Mechanical advantage (MA) is the ratio of load to effort. Velocity ratio (VR) compares the distance moved by the effort to the distance moved by the load. In an ideal machine MA equals VR, but friction always reduces efficiency.
机械优势 (MA) 是负载与施加力的比值。速度比 (VR) 比较施加力移动的距离与负载移动的距离。理想机械中 MA 等于 VR，但摩擦总会降低效率。
MA = Load / Effort VR = Distanceeffort / Distanceload
.5 m, VR = 2 / 0.5 = 4. If the load is 200 N and the effort is 60 N, MA = 200 / 60 ≈ 3.33.
用一个滑轮组，拉动 2 m 绳索将负载提升 0.5 m，VR = 2 / 0.5 = 4。若负载为 200 N，施加力为 60 N，则 MA = 200 / 60 ≈ 3.33。
7. Work, Energy and Power | 功、能与功率
Work is done when a force moves an object in the direction of the force. Energy is the ability to do work.
当力使物体沿力的方向移动时，便做了功。能量是做功的能力。
Work done W equals force F times distance d moved in the direction of the force, measured in joules (J). Power P is the rate of doing work, measured in watts (W).
功 W 等于力 F 乘以物体在力的方向上移动的距离 d，单位为焦耳 (J)。功率 P 是做功的速率，单位为瓦特 (W)。
W = F × d P = W / t P = V × I
× 1.2 = 60 J of work. If the lift takes 3 seconds, the power output is P = 60 / 3 = 20 W.
将一个 50 N 的工具箱提升 1.2 m 需要做功 W = 50 × 1.2 = 60 J。若提升用时 3 秒，则输出功率为 P = 60 / 3 = 20 W。
8. Triangle Law and Structural Stability | 三角形法则与结构稳定性
Triangles are the only polygons that will not change shape unless the side lengths change. This property is exploited in trusses and frameworks.
三角形是唯一一种只要边长不变就不会改变形状的多边形。桁架和框架结构正是利用了三角形的这一特性。
Adding diagonal braces to form triangles turns a flimsy rectangle into a rigid structure. This geometric rule is fundamental in bridge and tower design.
通过添加对角支撑形成三角形，可将不牢固的矩形转变为刚性结构。这一几何法则是桥梁和塔架设计的基础。
Where three forces act on a point in equilibrium, the force vectors can be arranged head-to-tail to form a closed triangle; the ratio of side lengths is proportional to the magnitudes of the forces.
当三个力作用于一点并达到平衡时，可将力矢量首尾相连形成闭合三角形；其边长比与力的大小成正比。
9. Pythagorean Theorem in Engineering Design | 工程设计中的勾股定理
Right-angled triangles appear everywhere in engineering, from diagonal braces to ramp calculations. The Pythagorean theorem links the three sides.
直角三角形在工程中随处可见，从对角支撑到斜坡计算。勾股定理将直角三角形的三条边联系起来。
heorem states a² + b² = c². Use it to find an unknown length when the other two are known.
对于直角边为 a 和 b、斜边为 c 的直角三角形，定理为 a² + b² = c²。已知任意两边即可求出第三边的长度。
c² = a² + b² c = √a² + b²
A diagonal brace must reach between two points 1.5 m apart horizontally and 0.8 m vertically. The brace length is √1.5² + 0.8² = √2.25 + 0.64 = √2.89 = 1.7 m.
一根对角支撑需连接水平距离 1.5 m、垂直距离 0.8 m 的两点。支撑长度 = √1.5² + 0.8² = √2.25 + 0.64 = √2.89 = 1.7 m。
10. Gear and Pulley Ratios | 齿轮与滑轮传动比
Gears and pulleys transmit rotary motion and can change speed, torque or direction.
齿轮和滑轮传递旋转运动，并可改变转速、扭矩或方向。
The gear ratio is defined as the number of teeth on the driven gear divided by the number of teeth on the driver gear. A ratio greater than 1 reduces speed but increases torque; a ratio less than 1 increases speed.
齿轮比定义为从动轮齿数除以主动轮齿数。齿轮比大于 1 会降低转速但增大扭矩；小于 1 则提高转速。
Gear ratio = Teethdriven / Teethdriver
For pulley systems the velocity ratio is found from the number of rope segments supporting the load. In a simple movable pulley the VR is 2.
对于滑轮系统，速度比由承载负载的绳索段数决定。单个动滑轮的速度比为 2。
3. The driven gear rotates at one-third the speed but with three times the turning force.
示例：一个 15 齿的主动齿轮带动一个 45 齿的从动齿轮，齿轮比 = 45/15 = 3。从动轮以主动轮三分之一的速度旋转，但扭矩增大为三倍。
11. Efficiency of Energy Systems | 能量系统的效率
No real machine converts all input energy into useful output energy; some is always lost, often as heat due to friction.
现实中没有机器能将输入能量完全转化为有用输出能量；总会有一部分损耗，通常是由于摩擦而转化为热量。
Efficiency is calculated as the useful output energy (or power) divided by the total input energy (or power), multiplied by 100 to give a percentage. It can also be expressed using MA and VR.
效率等于有用的输出能量（或功率）除以总输入能量（或功率），再乘以 100%，以百分比表示。也可用 MA 和 VR 表示。
Efficiency = (Useful output / Total input) × 100% Efficiency = (MA / VR) × 100%
s a weight by 350 J of gravitational potential energy, the efficiency is (350 / 500) × 100% = 70%.
如果一台电动机消耗了 500 J 的电能，但仅将重物提升了 350 J 的重力势能，则效率为 (350 / 500) × 100% = 70%。
Published by TutorHao | Engineering Revision Series | aleveler.com
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