📚 KS3 CIE Engineering Formula & Theorem Quick Reference Handbook | KS3 CIE 工程公式定理速查手册
This handbook provides a concise summary of essential formulas and theorems for KS3 CIE Engineering, covering mechanics, electricity, materials, and simple machines. Use it for quick revision and practical problem-solving in design and technology as well as physics-based engineering contexts.
本手册简要总结了 KS3 CIE 工程课程的核心公式与定理，涵盖力学、电学、材料和简单机械，供快速复习与实际问题解决使用，适用于设计技术及物理工程相关场景。
1. Basic Mechanics: Speed, Acceleration and Forces | 基础力学：速度、加速度与力
he distance travelled per unit time: v = s / t, where s is distance (m) and t is time (s).
速度（方向不变时可视为速率）是单位时间内移动的距离：v = s / t，其中 s 为距离（米），t 为时间（秒）。
ial velocity, v is final velocity, and t is the time taken for the change. The unit is m/s².
加速度是速度的变化率：a = (v - u) / t，u 为初速度，v 为末速度，t 为变化所用时间，单位为 m/s²。
Newton' : F = m a. Force is measured in newtons (N), mass in kilograms (kg), acceleration in m/s².
牛顿第二定律联系合力、质量与加速度：F = m a。力以牛顿（N）为单位，质量以千克（kg）为单位，加速度以 m/s² 为单位。
m g. On Earth, g is approximately 9.8 m/s², often rounded to 10 m/s² for KS3 calculations.
重力为物体所受地球引力：W = m g。在地球上，g 约为 9.8 m/s²，KS3 计算中常取 10 m/s²。
2. Pressure | 压强
ed perpendicularly per unit area: P = F / A. The unit is pascal (Pa), where 1 Pa = 1 N/m².
压强是垂直作用在单位面积上的力：P = F / A。单位为帕斯卡（Pa），1 Pa = 1 N/m²。
g/m³), g is gravitational field strength, h is depth (m). This formula is used in designing dams and hydraulic systems.
液体压强随深度增加：P = ρ g h，ρ 为液体密度（kg/m³），g 为重力场强度，h 为深度（米）。该公式用于水坝及液压系统设计。
Atmospheric pressure at sea level is approximately 101,325 Pa. In weather systems and suction devices, pressure differences drive flow.
海平面大气压约为 101,325 Pa。在天气系统和抽吸装置中，压差驱动流动。
3. Work, Energy and Power | 功、能与功率
ject in its direction: W = F × d. Work is measured in joules (J), where 1 J = 1 N m.
当力使物体沿其方向移动时做功：W = F × d。功的单位是焦耳（J），1 J = 1 N m。
Gravitational potential energy gained by lifting an object: Eₐ = m g h. This energy can be transformed into kinetic energy as the object falls.
举起物体获得的重力势能：Eₐ = m g h。物体下落时此能量可转化为动能。
Kinetic energy of a moving object: E˰ = ½ m v²;. The faster an object moves, the greater its kinetic energy.
运动物体的动能：E˰ = ½ m v²;。速度越大，动能越大。
measured in watts (W); 1 W = 1 J/s. Engineering machines are rated by their power output.
功率是做功或能量转移的速率：P = W / t。功率单位为瓦特（W）；1 W = 1 J/s。工程机械按输出功率进行标定。
seful output to total input: η = (useful work output / total energy input) × 100%. No machine can be 100% efficient due to friction and other losses.
效率是有用输出与总输入的比值：η =（有用功输出 / 总能量输入）× 100%。由于摩擦等损耗，任何机械都无法达到 100% 效率。
4. Levers and Moments | 杠杆与力矩
The moment of a force about a pivot is force × perpendicular distance from the pivot. Units: N m.
力对支点的力矩 = 力 × 力到支点的垂直距离，单位为牛·米（N m）。
r in equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments: F₁ d₁ = F₂ d₂.
杠杆平衡时，顺时针力矩之和等于逆时针力矩之和：F₁ d₁ = F₂ d₂。
a lever = load / effort. In an ideal (frictionless) lever, MA = effort arm / load arm = d₁ / d₂.
杠杆的机械效益（MA）= 负载 / 作用力。理想（无摩擦）杠杆中，MA = 作用力臂 / 负载臂 = d₁ / d₂。
When the effort arm is longer than the load arm, the lever acts as a force multiplier. Conversely, it can act as a distance multiplier.
作用力臂大于负载臂时，杠杆为省力杠杆；反之则为省距离杠杆（速度杠杆）。
5. Pulleys and Wheel & Axle | 滑轮与轮轴
he load. For example, a system with two supporting ropes gives MA = 2 (ignoring friction).
在简单滑轮组中，机械效益等于支撑负载的绳索段数。例如两段绳索支撑负载时 MA = 2（忽略摩擦）。
e. A larger wheel provides a greater mechanical advantage but requires more input rope or chain travel.
轮轴是一种旋转机械：MA = 轮半径 / 轴半径。轮半径越大，机械效益越高，但需要更长的绳索或链条行程。
ratio of distance moved by effort to distance moved by load. For ideal machines, MA = VR.
这两种系统的速比（VR）均为作用力移动距离与负载移动距离之比。理想机械中 MA = VR。
6. Gears and Transmission | 齿轮与传动
meshing gears, angular velocity is inversely proportional to the number of teeth: ω˰ / ωᵥ = Nᵥ / N˰. Here ω is angular speed, N is teeth number. (A = driver, B = driven)
对两个啮合齿轮，角速度与齿数成反比：ω˰ / ωᵥ = Nᵥ / N˰，其中 ω 为角速度，N 为齿数（A 为主动轮，B 为从动轮）。
Gear ratio is defined as teeth on driven / teeth on driver. A gear ratio greater than 1 means speed reduction and torque increase.
齿轮比定义为从动轮齿数 / 主动轮齿数。齿轮比大于 1 时减速增扭，小于 1 时增速减扭。
Torque ratio is the inverse of speed ratio: τᵥ / τ˰ = Nᵥ / N˰. Torque multiplication is essential in gearboxes of vehicles and lifts.
扭矩比是速比的反比：τᵥ / τ˰ = Nᵥ / N˰。扭矩放大在车辆变速箱和升降机中必不可少。
Compound gear trains multiply individual gear ratios to achieve large speed reductions or increases in complex engineering systems.
复合齿轮系通过各级齿轮比的乘积实现大型变速，应用于复杂工程系统中。
7. Electricity: Ohm's Law and Resistance | 电学：欧姆定律与电阻
Ohm' , I is current (amperes, A), R is resistance (ohms, Ω). It applies to ohmic conductors at constant temperature.
欧姆定律联系电压、电流与电阻：V = I R。V 为电压（伏特），I 为电流（安培），R 为电阻（欧姆）。适用于恒温下的欧姆导体。
Resistors in series: R₎ᵞᵐ₿ = R₁ + R₂ + R₃ + ... The current is the same through each, but voltage divides.
串联电阻：R₎ᵞᵐ
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