📚 KS3 WJEC Engineering: Formula & Theorem Quick Reference Handbook | KS3 WJEC 工程：公式定理速查手册
This quick reference handbook covers the essential formulas, theorems and key relationships you need to master for KS3 WJEC Engineering. From motion and forces to electricity and simple machines, each section provides clear definitions, worked formula displays and bilingual explanations to support revision and classroom learning.
本速查手册涵盖了 KS3 WJEC 工程学所需掌握的基本公式、定理和关键关系。从运动与力到电学与简单机械，每个小节都提供清晰的定义、公式展示及双语解释，助力复习与课堂学习。
1. Speed, Distance and Time | 速度、距离和时间
v = s / t or speed = distance ÷ time
Speed is a measure of how quickly an object moves. The average speed (v) is calculated by dividing the total distance travelled (s) by the total time taken (t). The standard unit for speed is metres per second (m/s), but kilometres per hour (km/h) is also common.
速度是衡量物体运动快慢的物理量。平均速度 (v) 可由总距离 (s) 除以总时间 (t) 得出。速度的标准单位是米每秒 (m/s)，千米每小时 (km/h) 也常用。
ways ensure the units are consistent; for example, if speed is in m/s and time in seconds, distance will be in metres. A handy tool is the 'DST triangle' where you cover the quantity you need.
可变形公式求距离 (s = v × t) 或时间 (t = s / v)。务必保持单位一致；例如，若速度用 m/s，时间用秒，距离则为米。可以借助“DST 三角形”快速变形。
2. Force, Mass and Acceleration | 力、质量和加速度
F = m × a
Newton’s Second Law links unbalanced force (F), mass (m) and acceleration (a). The resultant force acting on an object is equal to the mass of the object multiplied by its acceleration. Force is measured in newtons (N), mass in kilograms (kg) and acceleration in metres per second squared (m/s²).
牛顿第二定律将不平衡力 (F)、质量 (m) 和加速度 (a) 联系起来。作用在物体上的合力等于物体的质量乘以其加速度。力的单位是牛顿 (N)，质量用千克 (kg)，加速度用米每二次方秒 (m/s²)。
If multiple forces act along a straight line, find the resultant force first. An object will accelerate in the direction of the resultant force. When forces are balanced, acceleration is zero and the object moves at constant speed or stays at rest.
若多个力沿同一直线作用，需先求合力。物体沿合力方向加速。当力平衡时，加速度为零，物体保持静止或匀速直线运动。
3. Pressure and Area | 压强与面积
P = F / A
force over a large area reduces pressure. The SI unit of pressure is the pascal (Pa), where 1 Pa = 1 N/m².
压强定义为单位面积上所受的垂直作用力。较小面积上施加较大的力会产生高压强，而将力分散在大面积上则可降低压强。压强的国际单位是帕斯卡 (Pa)，1 Pa = 1 N/m²。
sure). When using the formula, make sure force is in newtons and area in square metres to obtain pressure in pascals.
实际应用如锋利的刀具（小面积 → 高压强）和雪地鞋（大面积 → 低压强）。使用公式时，确保力的单位是牛顿，面积单位是平方米，从而得到压强单位为帕斯卡。
4. Moments and Levers | 力矩与杠杆
M = F × d
A moment is the turning effect of a force. It depends on the size of the force (F) and the perpendicular distance (d) from the pivot to the line of action of the force. Moments are measured in newton metres (Nm).
力矩是力的转动效应，取决于力的大小 (F) 以及从支点到力作用线的垂直距离 (d)。力矩的单位是牛顿米 (Nm)。
used in seesaws, spanners and many tools. Mathematically: F₁ × d₁ = F₂ × d₂ when balanced.
杠杆平衡时，顺时针力矩之和等于逆时针力矩之和。该原理应用于跷跷板、扳手和许多工具。数学表达式：平衡时 F₁ × d₁ = F₂ × d₂。
Levers are classified into three classes depending on the relative positions of force, load and pivot. The mechanical advantage gained helps us design machines that magnify force or distance.
根据力、负载和支点的相对位置，杠杆可分为三类。所获得的机械效益有助于我们设计出能放大力量或距离的机器。
5. Work, Energy and Power | 功、能量与功率
W = F × d and P = W / t and E = P × t
Work is done when a force moves an object in the direction of the force. Work (W) equals force (F) multiplied by the distance (d) moved in the direction of the force. The unit of work and energy is the joule (J).
力使物体沿力的方向移动时即做功。功 (W) 等于力 (F) 乘以沿力方向移动的距离 (d)。功和能量的单位是焦耳 (J)。
nergy transferred can also be found from E = P × t, often used to calculate electrical energy consumption in kilowatt-hours (kWh).
功率是做功或能量转移的速率。功率 (P) = 所做的功 (W) / 所用时间 (t)。功率单位是瓦特 (W)，1 W = 1 J/s。能量转移也可由 E = P × t 求得，常用于计算以千瓦时 (kWh) 为单位的电能消耗。
In engineering contexts, understanding power helps in selecting motors and comparing machine performance. Always convert time to seconds when using standard units.
在工程背景中，理解功率有助于选择电机和比较机器性能。使用国际单位时，务必将时间转换为秒。
6. Density and Materials | 密度与材料
ρ = m / V
Density (ρ, the Greek letter rho) describes how much mass is packed into a given volume. It is calculated by dividing mass (m) by volume (V). The standard unit is kilograms per cubic metre (kg/m³), though g/cm³ is also used for solids and liquids.
密度 (ρ，希腊字母) 描述单位体积内所含物质的质量。由质量 (m) 除以体积 (V) 得出。标准单位是千克每立方米 (kg/m³)，固体和液体中也常用 g/cm³。
Materials with low density (e.g. aluminium, foam) are suitable for lightweight engineering components, while high-density materials (e.g. steel) are used where strength and mass are needed. Density determines whether an object floats: objects with density less than water (1000 kg/m³) will float.
低密度材料（如铝、泡沫）适用于轻质工程构件，而高密度材料（如钢）用于需要强度和质量的场合。密度决定物体是否漂浮：密度小于水 (1000 kg/m³) 的物体将浮起。
h. Accurate knowledge of density is essential in material selection and structural design.
测量不规则形状的体积时，采用排水法。对于规则固体，V = 长 × 宽 × 高。准确掌握密度对材料选择和结构设计至关重要。
7. Ohm's Law and Electrical Circuits | 欧姆定律与电路
V = I × R
Ohm’s Law states that the voltage (V) across a conductor is directly proportional to the current (I) flowing through it, provided temperature remains constant. Resistance (R) is the constant of proportionality. Voltage is measured in volts (V), current in amperes (A) and resistance in ohms (Ω).
欧姆定律指出，温度不变时，导体两端的电压 (V) 与流过导体的电流 (I) 成正比。电阻 (R) 是比例常数。电压单位是伏特 (V)，电流安培 (A)，电阻欧姆 (Ω)。
Electrical power is given by P = I × V . Energy consumed is E = P × t, often used to determine battery life or electricity bills.
电功率由 P = I × V 给出。结合欧姆定律可导出 P = I² × R 和 P = V² / R。消耗的电能 E = P × t，常用于确定电池续航或电费。
In series circuits, the total resistance Rtotal = R₁ + R₂ + R₃ + … rrent is the same everywhere. In parallel circuits, the total resistance is found from 1/Rtotal = 1/R₁ + 1/R₂ + … and the voltage across each branch is the same.
串联电路总电阻 Rₜₒₜₐₗ = R₁ + R₂ + R₃ + …，各处电流相同。并联电路总电阻满足 1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + …，各支路两端电压相同。
8. Mechanical Advantage and Efficiency | 机械效益与效率
MA = load / effort VR = distance moved by effort / distance moved by load
Mechanical advantage (MA) tells us how much a machine multiplies the input force. It is the ratio of load (output force) to effort (input force). Velocity ratio (VR) is the ratio of the distance moved by the effort to the distance moved by the load in the same time.
机械效益 (MA) 表明机器将输入力放大的倍数，它是负载（输出力）与施力（输入力）之比。速比 (VR) 是同一时间内施力移动距离与负载移动距离之比。
Efficiency is a measure of how well a machine converts input work into useful output work. It is expressed as a percentage: Efficiency (%) = (MA / VR) × 100. No real machine is 100% efficient due to friction and other losses.
效率衡量机器将输入功转化为有用输出功的程度，以百分比表示：效率 (%) = (MA / VR) × 100。由于摩擦等损耗，实际机器的效率永远达不到 100%。
Understanding MA and efficiency helps engineers design better gear systems, pulleys and levers to minimise energy waste and improve performance. Low efficiency means more energy is lost as heat or sound.
理解 MA 与效率有助于工程师设计更佳的齿轮系统、滑轮和杠杆，以减少能量浪费、提升性能。低效率意味着更多能量以热或声音形式散失。
9. Gears and Pulleys | 齿轮与滑轮
Speed ratio = driven teeth / driver teeth or diameter ratio
ith 20 teeth turns a driven gear with 40 teeth, the speed ratio is 40/20 = 2. The driven gear rotates at half the speed but with increased torque.
齿轮传递旋转运动，可改变速度、扭矩和方向。齿轮比由齿数决定：若 20 齿的主动齿轮带动 40 齿的从动齿轮，速比为 40/20 = 2，从动齿轮转速减半但扭矩增大。
Pulley systems use belts and sheaves. The velocity ratio for a pulley drive is the ratio of the diameters of the driven pulley to the driver pulley, assuming no slip. Multiple pulley blocks can yield large mechanical advantages for lifting heavy loads.
带轮系统依靠皮带与轮槽传递运动。无滑动时，带轮传动的速比等于从动轮直径与主动轮直径之比。多组滑轮组能提供很大的机械效益，用于起吊重物。
When analysing gear trains, multiply individual ratios to find overall ratio. Idler gears change direction but do not affect the overall ratio. Correct meshing and lubrication are essential for efficient transmission.
分析齿轮系时，将各级比相乘得到总传动比。惰轮改变转向但不影响总比。正确啮合与润滑对高效传动至关重要。
10. Hooke's Law and Springs | 胡克定律与弹簧
F = k × x
Hooke's Law states that the extension (x) of a spring is directly proportional to the applied force (F), provided the elastic limit is not exceeded. The spring constant (k) is a measure of stiffness, expressed in newtons per metre (N/m).
胡克定律指出，在不超过弹性限度时，弹簧的伸长量 (x) 与施加的力 (F) 成正比。劲度系数 (k) 反映弹簧的刚度，单位为牛顿每米 (N/m)。
The extension x is the difference between the stretched length and the original length. If the elastic limit is exceeded, the spring becomes permanently deformed and Hooke's Law no longer applies. The area under a force-extension graph represents the work done (elastic potential energy stored).
伸长量 x 是拉伸后长度与原长之差。若超过弹性限度，弹簧将发生永久变形，胡克定律不再适用。力-伸长量图下的面积代表所做的功（储存的弹性势能）。
Springs are used in engineering for suspension systems, force measurement (spring balances) and energy storage. Combining springs in series gives a lower overall spring constant, while in parallel they act stiffer.
弹簧在工程中用于悬挂系统、测力（弹簧秤）和储能。弹簧串联时总劲度系数减小，并联时整体刚度增大。
11. Forces in Equilibrium | 力的平衡
ΣF = 0 and ΣM = 0
An object is in equilibrium when the resultant force and the resultant moment acting on it are both zero. For static structures, this means all upward forces balance all downward forces and all clockwise moments balance anticlockwise moments.
当物体所受合外力与合力矩均为零时，物体处于平衡状态。对于静定结构，意味着所有向上的力与向下的力平衡，所有顺时针力矩与逆时针力矩平衡。
Free-body diagrams are used to isolate an object and show all forces acting on it. Resolving forces into horizontal and vertical components helps solve problems involving angled forces. Triangles of forces can represent three coplanar forces in equilibrium.
受力分析图用于隔离对象并展示其受到的所有力。将力分解为水平和竖直分量有助于求解角度力问题。三力平衡时，力矢量可构成闭合的力三角形。
Equilibrium principles are fundamental in bridge and frame design. The analysis ensures that structures can support loads safely without toppling or collapsing. Engineers use these conditions to calculate reaction forces at supports.
平衡原理在桥梁和框架设计中十分基础。分析可确保结构安全承载而不会倾覆或坍塌。工程师利用这些条件计算支座反力。
12. Simple Machines and Their Principles | 简单机械及其原理
Input work = output work (for an ideal machine)
Simple machines such as inclined planes, pulleys, levers, wheels and axles, and screws, make work easier by either changing the direction of a force or reducing the effort needed. In an ideal machine without friction, input work equals output work – energy is conserved.
斜面、滑轮、杠杆、轮轴和螺旋等简单机械通过改变力的方向或减小所需施力而使工作更轻松。在无摩擦的理想机器中，输入功等于输出功，能量守恒。
The mechanical advantage of an inclined plane is the ratio of the length of the slope to its height. For pulleys, counting the number of supporting rope strands gives the ideal MA. In practice, friction reduces efficiency, but the principle helps in designing lifting systems.
斜面的机械效益为斜面长与高之比。对于滑轮，承重绳索的股数即为其理想 MA。实际中摩擦会降低效率，但仍可依据此原理设计起重系统。
Understanding these ancient yet powerful principles enables modern engineers to create complex machinery from basic building blocks. They remain essential in linkages, mechanisms and material handling devices.
理解这些古老而强大的原理，使现代工程师能够从基本构件出发设计复杂机械。它们在连杆机构、机构学和物料搬运设备中仍至关重要。
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