📚 Year 7 WJEC Engineering Formula & Theorem Quick Reference Handbook | 威尔士联合教育委员会七年级工程学公式定理速查手册
Welcome to your essential quick reference guide for Year 7 WJEC Engineering. This handbook brings together all the key formulas, theorems, and principles you need to master during your first year of engineering studies. Keep this guide handy for homework, revision, and classroom activities. Understanding these fundamentals now will build a strong foundation for your future in engineering, whether you dream of designing bridges, programming robots, or creating sustainable energy solutions.
欢迎使用你的七年级 WJEC 工程学必备速查指南。本手册汇集了你在工程学习第一年需要掌握的所有关键公式、定理和原理。请将本指南放在手边，用于家庭作业、复习和课堂活动。现在理解这些基础知识将为你未来的工程学学习打下坚实的基础，无论你梦想设计桥梁、编程机器人还是创造可持续能源解决方案。
1. SI Base Units | 国际单位制基本单位
In engineering, we use the International System of Units (SI) to ensure everyone measures quantities in the same way. The seven base units form the foundation for all other derived units you will encounter. Memorising these base units is essential because they appear in every calculation and design specification throughout your engineering journey.
在工程学中，我们使用国际单位制 (SI) 来确保每个人以相同的方式测量数量。这七个基本单位构成了你将遇到的所有其他导出单位的基础。记住这些基本单位至关重要，因为它们会出现在你工程学习旅程中的每一个计算和设计规范中。
| Quantity (English) | 量 (中文) | Unit | Symbol |
|---|---|---|---|
| Length | 长度 | metre | m |
| Mass | 质量 | kilogram | kg |
| Time | 时间 | second | s |
| Electric Current | 电流 | ampere | A |
| Temperature | 温度 | kelvin | K |
| Amount of Substance | 物质的量 | mole | mol |
| Luminous Intensity | 发光强度 | candela | cd |
2. Ohm’s Law | 欧姆定律
Ohm’s Law describes the relationship between voltage, current, and resistance in an electrical circuit. This is one of the most important principles in electrical engineering. When you know any two of these three quantities, you can calculate the third. Think of voltage as the ‘push’ that moves electrons, current as the flow of those electrons, and resistance as anything that opposes that flow.
欧姆定律描述了电路中电压、电流和电阻之间的关系。这是电气工程中最重要的原理之一。当你知道这三个量中的任意两个时，你就可以计算第三个。把电压想象成推动电子运动的”推力”，电流是这些电子的流动，而电阻则是任何阻碍这种流动的东西。
V = I × R
- V: Voltage measured in volts (V) — 电压，以伏特 (V) 为单位测量
- I: Current measured in amperes (A) — 电流，以安培 (A) 为单位测量
- R: Resistance measured in ohms (Ω) — 电阻，以欧姆 (Ω) 为单位测量
For example, if a circuit has a current of 3 A flowing through a resistance of 4 Ω, the voltage across it is V = 3 × 4 = 12 V. Understanding this relationship helps engineers design safe circuits that deliver exactly the right amount of power to components without overheating or failing.
例如，如果一个电路有 3 A 的电流流过 4 Ω 的电阻，则其两端的电压为 V = 3 × 4 = 12 V。理解这种关系有助于工程师设计安全的电路，为元器件提供恰好合适的功率，而不会过热或故障。
| Rearranged Form | Use When… |
|---|---|
| I = V ÷ R | You know voltage and resistance, need current |
| R = V ÷ I | You know voltage and current, need resistance |
3. Gravitational Potential Energy | 重力势能
Gravitational potential energy (GPE) is the energy stored in an object because of its height above the ground. Whenever you lift something against the force of gravity, you transfer energy into it. This stored energy can be released when the object falls. Engineers consider GPE when designing everything from roller coasters to water dams, as the conversion between potential and kinetic energy drives many mechanical systems.
重力势能 (GPE) 是物体因其离地高度而储存的能量。每当你克服重力举起物体时，你就将能量传递给了它。当物体下落时，这种储存的能量可以被释放出来。工程师在设计过山车到水坝等各种事物时都要考虑重力势能，因为势能和动能之间的转换驱动着许多机械系统。
Eₚ = m × g × h
- Eₚ: Gravitational potential energy in joules (J) — 重力势能，以焦耳 (J) 为单位
- m: Mass in kilograms (kg) — 质量，以千克 (kg) 为单位
- g: Gravitational field strength (on Earth, approximately 9.8 N/kg, often rounded to 10 N/kg for Year 7 calculations) — 重力场强度 (在地球上约为 9.8 N/kg，七年级计算中通常四舍五入为 10 N/kg)
- h: Height in metres (m) — 高度，以米 (m) 为单位
Consider lifting a 5 kg toolbox onto a shelf 2 metres high. The gravitational potential energy gained is Eₚ = 5 × 10 × 2 = 100 J. This means 100 joules of work were done against gravity to raise the toolbox, and that energy is now stored, ready to be converted back if the toolbox falls.
考虑将一个 5 kg 的工具箱举到 2 米高的架子上。获得的重力势能为 Eₚ = 5 × 10 × 2 = 100 J。这意味着克服重力做了 100 焦耳的功将工具箱举起，而该能量现在被储存起来，一旦工具箱掉落就可以转换回来。
4. Force, Mass, and Acceleration (Newton’s Second Law) | 力、质量和加速度 (牛顿第二定律)
Newton’s Second Law of Motion tells us exactly how the motion of an object changes when a force is applied. The greater the force applied to an object, the greater its acceleration. However, objects with more mass accelerate less for the same force — this is why a heavy lorry needs a much more powerful engine than a small car to achieve the same acceleration. This principle is fundamental to all mechanical and structural engineering.
牛顿第二运动定律准确地告诉我们，当力作用于物体时，物体的运动如何变化。施加在物体上的力越大，其加速度就越大。然而，质量较大的物体在受到相同力时加速度较小——这就是为什么重型卡车需要比小型汽车强大得多的发动机才能达到相同的加速度。这一原理是所有机械和结构工程的基础。
F = m × a
- F: Resultant force in newtons (N) — 合力，以牛顿 (N) 为单位
- m: Mass in kilograms (kg) — 质量，以千克 (kg) 为单位
- a: Acceleration in metres per second squared (m/s²) — 加速度，以米每二次方秒 (m/s²) 为单位
If a car with a mass of 1200 kg accelerates at 3 m/s², the engine must provide a resultant force of F = 1200 × 3 = 3600 N. Engineers use this formula to determine the power requirements for vehicles, the forces acting on structures during earthquakes, and the thrust needed for rockets to escape Earth’s gravity.
如果一辆质量为 1200 kg 的汽车以 3 m/s² 加速，发动机必须提供 F = 1200 × 3 = 3600 N 的合力。工程师使用这个公式来确定车辆的动力需求、地震期间作用在结构上的力，以及火箭逃离地球引力所需的推力。
5. Speed, Distance, and Time | 速度、距离和时间
The relationship between speed, distance, and time is one of the most practical formulas you will use in engineering. Whether you are calculating the speed of a conveyor belt in a factory, the time needed for a signal to travel through a cable, or the distance a robot arm moves in one second, this formula is essential. Speed describes how fast something is moving without considering its direction.
速度、距离和时间之间的关系是你在工程学中会用到的最实用的公式之一。无论你是在计算工厂传送带的速度、信号通过电缆所需的时间，还是机械臂在一秒内移动的距离，这个公式都是必不可少的。速度描述物体运动快慢的程度，而不考虑其方向。
v = d ÷ t
- v: Speed in metres per second (m/s) — 速度，以米每秒 (m/s) 为单位
- d: Distance in metres (m) — 距离，以米 (m) 为单位
- t: Time in seconds (s) — 时间，以秒 (s) 为单位
A delivery drone flies 300 metres in 25 seconds. Its speed is v = 300 ÷ 25 = 12 m/s. Rearranging the formula allows engineers to predict how far a vehicle will travel in a given time (d = v × t) or how long a journey will take (t = d ÷ v). These calculations are critical for designing automated systems and planning efficient transport routes.
一架送货无人机在 25 秒内飞行了 300 米。其速度为 v = 300 ÷ 25 = 12 m/s。重新排列公式可以让工程师预测车辆在给定时间内行驶多远 (d = v × t) 或行程需要多长时间 (t = d ÷ v)。这些计算对于设计自动化系统和规划高效运输路线至关重要。
6. Electrical Power | 电功率
Electrical power tells us how quickly electrical energy is converted into another form, such as light, heat, or motion. A high-power device uses more energy each second than a low-power device. Understanding power ratings helps engineers select appropriate components and ensure circuits are not overloaded. The power rating printed on every electrical appliance, from phone chargers to industrial motors, comes from this fundamental relationship.
电功率告诉我们电能转化为另一种形式（如光、热或运动）的速度有多快。高功率设备每秒比低功率设备消耗更多能量。理解功率额定值有助于工程师选择合适的元器件并确保电路不过载。从手机充电器到工业电机，每个电器上印有的功率额定值都源自这一基本关系。
P = V × I
- P: Power in watts (W) — 功率，以瓦特 (W) 为单位
- V: Voltage in volts (V) — 电压，以伏特 (V) 为单位
- I: Current in amperes (A) — 电流，以安培 (A) 为单位
A lamp operating at 230 V with a current of 0.26 A has a power rating of P = 230 × 0.26 ≈ 60 W. Engineers must consider power when designing any electrical system because components have maximum power limits. Exceeding these limits can cause overheating and failure, which is why fuses and circuit breakers are essential safety devices.
一盏在 230 V 电压下工作、电流为 0.26 A 的灯，其功率额定值为 P = 230 × 0.26 ≈ 60 W。工程师在设计任何电气系统时都必须考虑功率，因为元器件有最大功率限制。超过这些限制会导致过热和故障，这就是为什么保险丝和断路器是必不可少的安全装置。
7. Work Done | 做功
In engineering, ‘work’ has a precise scientific meaning: work is done whenever a force moves an object through a distance in the direction of the force. If you push against a wall and it does not move, no work is done in the scientific sense, even though you might feel tired. This concept is crucial for understanding energy transfer in mechanical systems, from simple levers to complex engines.
在工程学中，”功”具有精确的科学含义：每当一个力使物体沿力的方向移动一段距离时，就做了功。如果你推一堵墙而它没有移动，从科学意义上讲就没有做功，即使你可能感到累。这个概念对于理解从简单杠杆到复杂发动机的机械系统中的能量传递至关重要。
W = F × d
- W: Work done in joules (J) — 做功，以焦耳 (J) 为单位
- F: Force applied in newtons (N) — 施加的力，以牛顿 (N) 为单位
- d: Distance moved in the direction of the force in metres (m) — 沿力的方向移动的距离，以米 (m) 为单位
If a construction worker uses a force of 200 N to push a wheelbarrow 15 metres across a site, the work done is W = 200 × 15 = 3000 J. This is equivalent to 3 kilojoules (kJ) of energy transferred. The same formula applies whether you are calculating the work done by a hydraulic press, a crane lifting steel beams, or the force exerted by wind on a turbine blade.
如果一名建筑工人用 200 N 的力将手推车推过工地 15 米，则做的功为 W = 200 × 15 = 3000 J。这相当于传递了 3 千焦 (kJ) 的能量。无论是计算液压机做的功、起重机吊装钢梁做的功，还是风对涡轮叶片施加的力，都适用相同的公式。
8. Efficiency | 效率
No engineering system is perfect — some energy is always ‘lost’ (usually as heat or sound) during any energy conversion. Efficiency measures how well a system converts input energy into useful output energy. A highly efficient machine wastes very little energy, making it cheaper to run and better for the environment. Engineers constantly strive to improve the efficiency of everything from car engines to solar panels.
没有一个工程系统是完美的——在任何能量转换过程中，总会有一些能量”损失”（通常以热或声的形式）。效率衡量系统将输入能量转化为有用输出能量的程度。高效率的机器浪费的能量很少，使其运行成本更低，对环境也更好。工程师不断努力提高从汽车发动机到太阳能电池板等各种事物的效率。
Efficiency (%) = (Useful Energy Output ÷ Total Energy Input) × 100
An electric motor receives 500 J of electrical energy and produces 425 J of useful kinetic energy, with the remaining 75 J lost as heat and sound. Its efficiency is (425 ÷ 500) × 100 = 85%. This is considered good for an electric motor. Efficiency can never exceed 100% because of the law of conservation of energy — you cannot get more energy out than you put in.
一台电动机接收 500 J 的电能并产生 425 J 的有用动能，其余 75 J 以热和声的形式损失。其效率为 (425 ÷ 500) × 100 = 85%。这对电动机来说算是好的。由于能量守恒定律，效率永远不会超过 100%——你输出的能量不可能超过输入的能量。
9. The Principle of Moments | 力矩原理
The principle of moments explains why seesaws balance and how levers allow us to lift heavy loads with relatively little effort. A moment is the turning effect of a force around a pivot point. When a system is in equilibrium (balanced), the total clockwise moments equal the total anticlockwise moments. This principle is used in countless engineering applications, from bridge design to the operation of spanners and scissors.
力矩原理解释了为什么跷跷板能平衡，以及杠杆如何能让我们用相对较小的力举起重物。力矩是力围绕支点产生的转动效应。当系统处于平衡状态时，顺时针力矩的总和等于逆时针力矩的总和。这一原理被用于无数的工程应用中，从桥梁设计到扳手和剪刀的操作。
M = F × d
- M: Moment in newton-metres (Nm) — 力矩，以牛顿·米 (Nm) 为单位
- F: Force in newtons (N) — 力，以牛顿 (N) 为单位
- d: Perpendicular distance from the pivot in metres (m) — 到支点的垂直距离，以米 (m) 为单位
For equilibrium: Sum of clockwise moments = Sum of anticlockwise moments. If a 40 kg child sits 1.5 m from the centre of a seesaw, their moment is 400 N × 1.5 m = 600 Nm clockwise. To balance this, a 60 kg adult (600 N) must sit 1.0 m on the opposite side, creating a moment of 600 N × 1.0 m = 600 Nm anticlockwise.
对于平衡状态：顺时针力矩总和 = 逆时针力矩总和。如果一个 40 kg 的儿童坐在距离跷跷板中心 1.5 m 处，其力矩为 400 N × 1.5 m = 600 Nm 顺时针。为平衡此力矩，一个 60 kg 的成人 (600 N) 必须坐在对面 1.0 m 处，产生 600 N × 1.0 m = 600 Nm 的逆时针力矩。
10. Stress and Strain | 应力和应变
When engineers design structures and components, they must understand how materials behave under load. Stress measures the internal force distributed over a cross-sectional area, while strain measures how much a material deforms relative to its original length. These concepts help engineers choose the right materials for bridges, aircraft wings, and even smartphone cases to ensure they are strong enough without being excessively heavy or expensive.
当工程师设计结构和构件时，他们必须了解材料在载荷下的行为。应力衡量分布在横截面上的内力，而应变衡量材料相对于其原始长度的变形程度。这些概念帮助工程师为桥梁、飞机机翼甚至手机壳选择合适的材料，确保它们足够坚固而不会过于沉重或昂贵。
Stress: σ = F ÷ A
- σ (sigma): Stress in pascals (Pa) or N/m² — 应力，以帕斯卡 (Pa) 或 N/m² 为单位
- F: Force in newtons (N) — 力，以牛顿 (N) 为单位
- A: Cross-sectional area in square metres (m²) — 横截面积，以平方米 (m²) 为单位
Strain: ε = ΔL ÷ L
- ε (epsilon): Strain (no units, it is a ratio) — 应变 (无单位，是一个比率)
- ΔL: Change in length in metres (m) — 长度变化，以米 (m) 为单位
- L: Original length in metres (m) — 原始长度，以米 (m) 为单位
A steel cable with a cross-sectional area of 0.0001 m² experiences a pulling force of 5000 N. The stress on the cable is σ = 5000 ÷ 0.0001 = 50,000,000 Pa, or 50 MPa (megapascals). If the cable was originally 2 metres long and stretches by 0.5 mm, the strain is ε = 0.0005 ÷ 2 = 0.00025. This very small strain shows that steel is stiff and resists deformation well.
一根横截面积为 0.0001 m² 的钢缆承受 5000 N 的拉力。钢缆上的应力为 σ = 5000 ÷ 0.0001 = 50,000,000 Pa，即 50 MPa（兆帕）。如果钢缆原长 2 米，拉伸了 0.5 mm，则应变为 ε = 0.0005 ÷ 2 = 0.00025。这个非常小的应变表明钢材刚度大，能很好地抵抗变形。
11. Hooke’s Law | 胡克定律
Hooke’s Law describes the behaviour of springs and many other elastic materials. It states that the extension of a spring is directly proportional to the force applied, provided the material’s elastic limit is not exceeded. This means if you double the force, you double the extension. Engineers use this principle when designing suspension systems, measuring devices, and any application where predictable spring behaviour is required.
胡克定律描述了弹簧和许多其他弹性材料的行为。它指出，只要不超过材料的弹性极限，弹簧的伸长量与施加的力成正比。这意味着如果你将力加倍，伸长量也会加倍。工程师在设计悬挂系统、测量设备以及任何需要可预测弹簧行为的应用时都会使用这一原理。
F = k × x
- F: Force applied in newtons (N) — 施加的力，以牛顿 (N) 为单位
- k: Spring constant (stiffness) in newtons per metre (N/m) — 弹簧常数 (刚度)，以牛顿每米 (N/m) 为单位
- x: Extension (or compression) in metres (m) — 伸长量 (或压缩量)，以米 (m) 为单位
A spring with a spring constant of 150 N/m extends by 0.04 m when a force is applied. Using Hooke’s Law, F = 150 × 0.04 = 6 N. A stiffer spring has a higher k value, meaning more force is needed to produce the same extension. The elastic limit is crucial — beyond this point, the spring will not return to its original length and becomes permanently deformed.
一根弹簧常数为 150 N/m 的弹簧在施加力时伸长了 0.04 m。使用胡克定律，F = 150 × 0.04 = 6 N。较硬的弹簧具有较高的 k 值，这意味着需要更大的力才能产生相同的伸长量。弹性极限至关重要——超过此点，弹簧将无法恢复到原始长度并发生永久变形。
12. Cost and Efficiency in Engineering | 工程中的成本和效率
Engineering is not just about technical calculations — it also involves making smart economic decisions. When comparing design options, engineers must consider both the purchase cost and the long-term running costs. An energy-efficient device might cost more initially but save money over its lifetime through lower energy bills. This applies to everything from household appliances to industrial machinery and transport systems.
工程学不仅仅涉及技术计算——它还涉及做出明智的经济决策。在比较设计方案时，工程师必须同时考虑购买成本和长期运行成本。一个节能设备最初可能成本更高，但通过降低能源账单在其使用寿命内节省资金。这适用于从家用电器到工业机械和运输系统的各个方面。
Energy Used (kWh) = Power (kW) × Time (hours)
Cost = Energy Used (kWh) × Price per kWh
A 2 kW electric heater runs for 5 hours. Energy used = 2 kW × 5 h = 10 kWh. If electricity costs £0.25 per kWh, the cost is 10 × £0.25 = £2.50. When comparing two appliances with different efficiencies, engineers calculate the total lifetime cost, which includes purchase price plus all energy costs over the expected years of use. This approach ensures that engineering solutions are both technically sound and economically viable.
一个 2 kW 的电暖器运行了 5 小时。使用的能量 = 2 kW × 5 h = 10 kWh。如果电费为每千瓦时 £0.25，则成本为 10 × £0.25 = £2.50。在比较两个效率不同的电器时，工程师计算总生命周期成本，包括购买价格加上预期使用年限内的所有能源成本。这种方法确保工程解决方案在技术上合理且经济上可行。
13. Density | 密度
Density links the mass and volume of a material, telling us how tightly matter is packed together. A material with high density, like steel, has a lot of mass in a small volume. A material with low density, like polystyrene foam, has very little mass in the same volume. Engineers choose materials based on their density for different purposes — dense materials for strong structures and lightweight materials for aircraft and vehicles where fuel efficiency is critical.
密度将材料的质量和体积联系起来，告诉我们物质聚集的紧密程度。高密度材料（如钢）在很小的体积内就有很大的质量。低密度材料（如聚苯乙烯泡沫）在相同体积内质量很小。工程师根据密度为不同目的选择材料——致密材料用于坚固的结构，轻质材料用于燃油效率至关重要的飞机和车辆。
ρ = m ÷ V
- ρ (rho): Density in kilograms per cubic metre (kg/m³) — 密度，以千克每立方米 (kg/m³) 为单位
- m: Mass in kilograms (kg) — 质量，以千克 (kg) 为单位
- V: Volume in cubic metres (m³) — 体积，以立方米 (m³) 为单位
Aluminium has a density of approximately 2700 kg/m³, while steel has a density of about 7850 kg/m³. This means a steel component weighs nearly three times as much as an aluminium component of the same size. Knowing this, an aerospace engineer might choose aluminium for parts where weight is critical, while a civil engineer might specify steel for a building frame where strength is the priority.
铝的密度约为 2700 kg/m³，而钢的密度约为 7850 kg/m³。这意味着相同尺寸的钢质部件重量几乎是铝质部件的三倍。了解到这一点，航空工程师可能在重量关键的部件上选择铝，而土木工程师可能在强度优先的建筑框架上指定使用钢材。
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