📚 Year 9 WJEC Engineering: Formula & Theorem Quick Reference Handbook | Year 9 WJEC 工程：公式定理速查手册
This handbook provides a concise collection of essential formulas and theorems for Year 9 WJEC Engineering students. Use it as a quick revision guide to reinforce your understanding of core principles from mechanics, electronics, materials and applied mathematics. Each section presents the key relationships you need to recall for problem‑solving and design tasks.
本手册为 Year 9 WJEC 工程学生汇集了核心公式与定理，是快速复习的理想工具。内容涵盖力学、电子、材料与应用数学中必须掌握的基本关系式，帮助你在解题和设计任务中随时查阅关键知识点。
1. Pythagoras' Theorem and Right‑Angle Triangles | 勾股定理与直角三角形
In any right‑angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This is essential for calculating lengths in structures, linkages and force diagrams.
在任何直角三角形中，斜边的平方等于两直角边平方之和。在结构、连杆和力图中用来计算长度至关重要。
c² = a² + b²
Where c is the hypotenuse and a, b are the other sides. You can also find a shorter side with a² = c² − b².
其中 c 为斜边，a、b 为直角边。求短边可用 a² = c² − b²。
Engineers use this to determine diagonal braces, belt lengths on pulley systems and the resultant of perpendicular forces.
工程师用它确定斜撑、带传动系统中的皮带长度以及垂直力的合力。
2. Basic Trigonometry: SOH CAH TOA | 基础三角函数：SOH CAH TOA
Trigonometry relates angles to side ratios in right‑angled triangles. It is fundamental for resolving forces, calculating angles of lift and designing ramps or linkages.
三角函数将角度与直角三角形边长的比值联系起来，是分解力、计算举升角度和设计坡道或连杆的基础。
sin θ = opposite / hypotenuse | sin θ = 对边/斜边
cos θ = adjacent / hypotenuse | cos θ = 邻边/斜边
tan θ = opposite / adjacent | tan θ = 对边/邻边
tions: θ = sin⁻¹(O/H), etc. Always check whether your calculator is in degrees or radians.
求角度时使用反函数：θ = sin⁻¹(对边/斜边) 等等。务必检查计算器是角度模式还是弧度模式。
3. Forces and Newton’s Second Law | 力与牛顿第二定律
The relationship between force, mass and acceleration is the cornerstone of dynamics. It explains how objects move when a resultant force acts on them.
力、质量和加速度之间的关系是动力学的基石，解释了合外力作用下物体如何运动。
F = m × a
- F = force (newtons, N)
- m = mass (kilograms, kg)
- a = acceleration (metres per second squared, m/s²)
- F = 力（牛顿，N）
- m = 质量（千克，kg）
- a = 加速度（米每二次方秒，m/s²）
For objects on Earth, weight is the force due to gravity: W = m × g, where g ≈ 9.8 m/s². In design, always consider the weight of components.
地球上物体所受重力为：W = m × g，g 约为 9.8 m/s²。设计中始终要考虑零部件的重量。
4. Moments and Levers | 力矩与杠杆
A moment is the turning effect of a force about a pivot. Levers use moments to magnify effort and are found in many mechanical systems.
力矩是力对支点产生的转动效应。杠杆利用力矩放大作用力，广泛存在于机械系统中。
M = F × d
- M = moment (newton‑metres, N·m)
- F = force (N)
- d = perpendicular distance from pivot to line of action (m)
- M = 力矩（牛顿·米，N·m）
- F = 力（N）
- d = 支点到力作用线的垂直距离（m）
, total clockwise moments = total anticlockwise moments. This is the principle of moments.
杠杆平衡时，顺时针力矩之和等于逆时针力矩之和，即力矩原理。
5. Pressure and Hydraulic Systems | 压强与液压系统
Pressure is force distributed over an area. Fluid systems use pressure to transmit and multiply force, as in hydraulic brakes and jacks.
压强是力分布在面积上的大小。液压系统利用压强传递并放大力，如液压制动和千斤顶。
p = F / A
- p = pressure (pascals, Pa; 1 Pa = 1 N/m²)
- F = force (N)
- A = area (m²)
- p = 压强（帕斯卡，Pa；1 Pa = 1 N/m²）
- F = 力（N）
- A = 面积（m²）
o F₁/A₁ = F₂/A₂. A small force on a small piston can lift a large load on a larger piston.
在封闭流体系统中，压强处处相等：p₁ = p₂，即 F₁/A₁ = F₂/A₂。小活塞上的小力可以在大活塞上举起大载荷。
6. Work, Energy and Power | 功、能量与功率
These concepts describe how machines do work, convert energy and how quickly they operate. They are crucial for motor and engine selection.
这些概念描述机器如何做功、转换能量以及运作快慢，对电机和发动机选型至关重要。
Work done = F × d (force × distance moved in direction of force)
做功 = F × d（力 × 力方向上移动的距离）
Power = Work done / time or P = E / t
功率 = 做功 / 时间 或 P = E / t
- Work and energy in joules (J)
- Power in watts (W; 1 W = 1 J/s)
- 功与能量单位焦耳（J）
- 功率单位瓦特（W；1 W = 1 J/s）
Efficiency = (useful output energy / total input energy) × 100%. No machine is 100% efficient; energy is lost as heat, sound or friction.
效率 =（有用输出能量 / 总输入能量）× 100%。所有机械都有能量损失，如热、声音或摩擦，效率不可能达到 100%。
7. Ohm's Law and Electrical Circuits | 欧姆定律与电路
Ohm's Law defines the relationship between voltage, current and resistance. It underpins all simple circuit analysis.
欧姆定律定义了电压、电流和电阻之间的关系，是所有简单电路分析的基础。
V = I × R
- V = voltage (volts, V)
- I = current (amperes, A)
- R = resistance (ohms, Ω)
- V = 电压（伏特，V）
- I = 电流（安培，A）
- R = 电阻（欧姆，Ω）
Electrical power: P = V × I or P = I² × R. Energy transferred: E = V × I × t.
电功率：P = V × I 或 P = I² × R。电能：E = V × I × t。
In series circuits, current is the same everywhere, voltages add. In parallel circuits, voltage is the same across branches, currents add.
串联电路中电流处处相等，电压相加；并联电路中各支路电压相等，电流相加。
8. Stress, Strain and Young's Modulus | 应力、应变与杨氏模量
Engineers must understand how materials behave under load. Stress and strain describe the internal response to external forces.
工程师必须理解材料在载荷下的行为。应力和应变描述了材料对外力的内部响应。
Stress = F / A (units: Pa or N/m²)
应力 = F / A（单位：Pa 或 N/m²）
Strain = ΔL / L₀ (change in length / original length; no units)
应变 = ΔL / L₀（伸长量 / 原长；无单位）
Young's Modulus E = stress / strain (Pa)
杨氏模量 E = 应力 / 应变（Pa）
Within the elastic limit, materials return to their original shape. Beyond the yield point, plastic deformation occurs. Young's modulus measures stiffness.
在弹性极限内卸载后材料恢复原状；超过屈服点则发生塑性变形。杨氏模量衡量材料的刚度。
9. Gears, Pulleys and Speed Ratios | 齿轮、带轮与转速比
Simple machines alter speed, torque and direction. The speed ratio is key to analysing gear trains and pulley systems.
简单机械改变转速、扭矩和方向。转速比是分析齿轮系和带轮系统的关键。
Speed ratio = number of teeth on driven gear / number of teeth on driver gear
转速比 = 从动轮齿数 / 主动轮齿数
Output speed = input speed × (teeth on driver / teeth on driven)
输出转速 = 输入转速 × (主动轮齿数 / 从动轮齿数)
For pulleys, replace tooth count with diameters. A ratio greater than 1 gives speed increase (gearing up), less than 1 gives torque increase (gearing down).
对于带轮，用直径代替齿数。比值大于 1 为增速，小于 1 为提高扭矩（减速）。
10. Density, Mass and Volume | 密度、质量与体积
Density links the mass and volume of a material. It is essential for weight calculations and material selection in design.
密度关联材料的质量和体积，是重量计算和设计中材料选择的基础。
ρ = m / V
- ρ (rho) = density (kg/m³)
- m = mass (kg)
- V = volume (m³)
- ρ（密度）= 质量 / 体积（kg/m³）
- m = 质量（kg）
- V = 体积（m³）
teel ≈ 7800 kg/m³. For regular shapes use volume formulas (e.g. cube a³, cylinder π×r²×h).
常见密度：水 1000 kg/m³，铝约 2700 kg/m³，钢约 7800 kg/m³。规则形状的体积公式如立方体 a³、圆柱体 π×r²×h。
11. Mechanical Advantage and Velocity Ratio | 机械效益与速比
These terms describe how much a machine multiplies force or changes speed. They are used alongside efficiency to evaluate mechanisms.
这两个术语描述机器放大力的倍数或改变速度的程度，与效率一起用来评估机构性能。
Mechanical Advantage (MA) = load / effort
机械效益 (MA) = 负载 / 施力
Velocity Ratio (VR) = distance moved by effort / distance moved by load
速比 (VR) = 施力移动距离 / 负载移动距离
Efficiency = (MA / VR) × 100%
效率 = (MA / VR) × 100%
ces efficiency, so MA is always less than VR. A perfect machine would have MA = VR and 100% efficiency.
摩擦降低效率，因此 MA 始终小于 VR。理想机器的 MA = VR，效率为 100%。
12. Safety Factor and Design Stress | 安全系数与设计应力
Engineers build in a safety margin to account for uncertainties. The safety factor ensures structures and components do not fail under unexpected loads.
工程师会加入安全裕度以应对不确定性。安全系数确保结构和部件在意外载荷下不会失效。
Safety Factor = ultimate stress / allowable working stress
安全系数 = 极限应力 / 许用工作应力
Typical safety factors range from 1.5 for static predictable loads to 5 or more for dynamic or passenger‑carrying applications. Always refer to design codes for your material.
典型安全系数从静态可预期载荷的 1.5 到动态或载人应用的 5 或更高。设计时应始终参考相关材料规范。
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