📚 KS3 Edexcel Engineering: Quick-Reference Formula & Theorem Handbook | KS3 Edexcel 工程：公式定理速查手册
Engineering at Key Stage 3 introduces you to the core principles that shape the world around us – from structures and mechanisms to electrical systems and material behaviour. This handbook gathers every essential formula and theorem you need to revise efficiently, with each rule explained in plain English and paired with a Chinese translation so you can build understanding in both languages. Keep it handy as you tackle practical challenges, design tasks, and end‑of‑topic tests.
KS3 工程课程带你认识塑造世界的基本原理——从结构与机械到电气系统和材料行为。这本手册汇集了你需要高效复习的所有关键公式和定理，每条规则都用通俗的英文解释，并配有中文翻译，帮助你在双语环境中建立理解。在解决实际挑战、设计任务和单元测试时，随时翻阅它。
1. The Principle of Moments | 力矩原理
For a lever to be in equilibrium, the sum of the clockwise moments about any pivot must equal the sum of the anticlockwise moments. The moment of a force is calculated as force multiplied by the perpendicular distance from the pivot: Moment = F × d.
杠杆平衡时，对于任意支点，顺时针力矩的总和必须等于逆时针力矩的总和。力矩的计算方法是力乘以力臂（力到支点的垂直距离）：力矩 = F × d。
A moment is measured in newton metres (Nm) when force is in newtons (N) and distance in metres (m). This rule is used to design see‑saws, cranes, and any system where a load must be balanced by an effort.
当力以牛顿（N）为单位、距离以米（m）为单位时，力矩的单位是牛米（Nm）。这一规则用于设计跷跷板、起重机以及任何需要用力来平衡负载的系统。
2. Force, Mass and Acceleration (Newton’s Second Law) | 力、质量与加速度（牛顿第二定律）
The net force acting on an object is equal to its mass multiplied by its acceleration: F = m × a. Here, F is force in newtons (N), m is mass in kilograms (kg), and a is acceleration in metres per second squared (m/s²).
作用在物体上的合力等于其质量乘以加速度：F = m × a。其中 F 是力（N），m 是质量（kg），a 是加速度（m/s²）。
This law explains why a lighter car accelerates more quickly with the same engine force, and why heavier objects are harder to start or stop. In engineering, it helps to size motors, brakes, and actuators.
这一定律解释了为什么在相同发动机力作用下，较轻的汽车加速更快，以及为什么较重的物体更难启动或停止。在工程中，它有助于确定电机、制动器和执行器的规格。
3. Work Done and Energy Transfers | 做功与能量转移
Work is done when a force moves an object. The formula is Work = F × d, where work is measured in joules (J). One joule is the work done when a force of one newton moves an object one metre in the direction of the force.
力使物体移动时便做了功。公式为 功 = F × d，功的单位是焦耳（J）。1 焦耳相当于 1 牛顿的力使物体沿力的方向移动 1 米所做的功。
In all machines, the work output is never greater than the work input because of friction and heat losses. Understanding work leads directly to the conservation of energy: energy cannot be created or destroyed, only changed from one form to another.
在所有机器中，由于摩擦和热损失，输出的功永远不会大于输入的功。理解功的概念直接引向能量守恒定律：能量既不能被创造也不能被消灭，只能从一种形式转化为另一种形式。
4. Mechanical Advantage and Velocity Ratio | 机械利益与速度比
Mechanical advantage (MA) tells you how much a machine multiplies the effort force. It is given by MA = Load ÷ Effort. A crowbar lifting a heavy load with a small effort has a high MA.
机械利益（MA）表示机器将输入力放大了多少倍。公式为 MA = 负载 ÷ 动力。用较小的力撬起重物的撬棍具有较高的 MA。
Velocity ratio (VR) compares the distance moved by the effort to the distance moved by the load: VR = distance moved by effort ÷ distance moved by load. Efficiency can then be found from Efficiency = (MA ÷ VR) × 100%.
速度比（VR）比较动力移动的距离与负载移动的距离：VR = 动力移动距离 ÷ 负载移动距离。然后可以通过 效率 = (MA ÷ VR) × 100% 求得效率。
- For a pulley system with 4 supporting ropes, VR = 4.
- 对于有 4 根承载绳的滑轮系统，VR = 4。
- For a lever, VR = effort arm length ÷ load arm length.
- 对于杠杆，VR = 动力臂长 ÷ 阻力臂长。
5. Gear and Pulley Ratios | 齿轮与皮带轮传动比
The speed ratio of two meshing gears depends on the number of teeth: Speed of driven gear = (teeth on driver ÷ teeth on driven) × speed of driver. If a 20‑tooth gear drives a 40‑tooth gear, the driven gear turns at half the speed but with twice the torque (ignoring losses).
两个啮合齿轮的转速比取决于齿数：从动齿轮转速 = (主动齿轮齿数 ÷ 从动齿轮齿数) × 主动齿轮转速。如果一个 20 齿的齿轮带动一个 40 齿的齿轮，从动齿轮的转速是主动齿轮的一半，但扭矩加倍（忽略损失）。
For belt and pulley drives, the ratio is based on pulley diameters: Speed of driven pulley = (diameter of driver ÷ diameter of driven) × speed of driver. This principle governs bicycle gears, conveyor belts, and engine timing systems.
对于带轮传动，传动比基于皮带轮直径：从动轮转速 = (主动轮直径 ÷ 从动轮直径) × 主动轮转速。这一原理支配着自行车变速、传送带和发动机正时系统。
6. Ohm’s Law and Electrical Power | 欧姆定律与电功率
Ohm’s Law relates voltage, current, and resistance in a circuit: V = I × R. Here V is voltage in volts (V), I is current in amperes (A), and R is resistance in ohms (Ω).
欧姆定律描述了电路中电压、电流和电阻之间的关系：V = I × R。其中 V 是电压（V），I 是电流（A），R 是电阻（Ω）。
Electrical power is the rate of energy transfer: P = V × I. Power P is measured in watts (W). In engineering, you also use P = I² × R to calculate heat generation in wires.
电功率是能量转移的速率：P = V × I。功率 P 的单位是瓦特（W）。在工程中，你也会用 P = I² × R 来计算导线中的发热。
| Quantity/量 | Symbol/符号 | Unit/单位 |
| Voltage / 电压 | V | Volt (V) |
| Current / 电流 | I | Ampere (A) |
| Resistance / 电阻 | R | Ohm (Ω) |
| Power / 功率 | P | Watt (W) |
7. Series and Parallel Circuits | 串联与并联电路
In a series circuit, the current is the same everywhere, and the total resistance is the sum of individual resistances: Rtotal = R₁ + R₂ + R₃ + …. The supply voltage splits across components in proportion to their resistances.
在串联电路中，各处电流相等，总电阻等于各个电阻之和：R总 = R₁ + R₂ + R₃ + …。电源电压按电阻比例分配到各个元器件上。
In a parallel circuit, the voltage across each branch is the same, and the total resistance is lower than the smallest individual resistance. For two resistors in parallel, you can use: 1/Rtotal = 1/R₁ + 1/R₂.
在并联电路中，各支路两端电压相同，总电阻小于最小的单个电阻。对于两个并联电阻，可以使用：1/R总 = 1/R₁ + 1/R₂。
Understanding these rules is vital for designing circuits that correctly distribute power to sensors, motors, and lights in engineering projects.
理解这些规则对于在工程项目中正确分配传感器、电机和灯具的电能至关重要。
8. Pressure in Fluids and Solid Contacts | 流体与固体接触中的压强
Pressure is defined as force per unit area: P = F ÷ A. Pressure P is in pascals (Pa), where 1 Pa = 1 N/m². A sharp knife cuts well because the small area gives a high pressure for the same force.
压强定义为单位面积上所受的力：P = F ÷ A。压强的单位是帕斯卡（Pa），1 Pa = 1 N/m²。锋利的刀容易切割，是因为在同样的力下，较小的面积能产生较大的压强。
In a liquid, pressure increases with depth and acts equally in all directions (Pascal’s principle). The pressure at a depth h in a liquid of density ρ is: P = ρ × g × h, where g is the gravitational field strength (10 N/kg on Earth).
在液体中，压强随深度增加，并且向各个方向同等作用（帕斯卡原理）。在密度为 ρ 的液体中，深度 h 处的压强为：P = ρ × g × h，其中 g 是重力场强度（地球上约为 10 N/kg）。
Hydraulic systems use this: a small force on a small piston can create a larger force on a bigger piston because pressure is constant throughout the enclosed fluid.
液压系统就利用了这一原理：由于封闭液体中的压强处处相等，在小活塞上施加的小力可以在大活塞上产生放大的力。
9. Stress, Strain and Young’s Modulus | 应力、应变与杨氏模量
When a material is loaded, we measure stress as the internal force per cross‑sectional area: Stress (σ) = F ÷ A. Strain measures how much the material deforms relative to its original length: Strain (ε) = change in length ÷ original length.
当材料承受载荷时，我们将应力定义为单位横截面积上的内力：应力 (σ) = F ÷ A。应变衡量材料相对于原始长度的变形程度：应变 (ε) = 长度变化 ÷ 原始长度。
Young’s modulus E describes the stiffness of a material in the elastic region: E = σ ÷ ε. A high Young’s modulus (e.g. steel) means the material is stiff; a low modulus (e.g. rubber) means it stretches easily.
杨氏模量 E 描述材料在弹性范围内的刚度：E = σ ÷ ε。高杨氏模量（如钢）意味着材料较刚硬；低模量（如橡胶）则容易拉伸。
At KS3 you mainly explore Hooke’s Law for springs: the extension is proportional to the applied force, F = k × e, where k is the spring constant (N/m) and e is extension (m). This holds until the elastic limit.
在 KS3 阶段，你主要探索弹簧的胡克定律：伸长量与施加的力成正比，F = k × e，其中 k 是弹簧常数（N/m），e 是伸长量（m）。这一关系在弹性极限内成立。
10. Density and Specific Gravity | 密度与比重
Density is the mass per unit volume of a substance: ρ = m ÷ V. The Greek letter rho (ρ) is used, and the unit is kilograms per cubic metre (kg/m³) or grams per cubic centimetre (g/cm³). Water has a density of 1000 kg/m³.
密度是物质单位体积的质量：ρ = m ÷ V。符号使用希腊字母 ρ，单位为千克每立方米（kg/m³）或克每立方厘米（g/cm³）。水的密度为 1000 kg/m³。
Engineers regularly calculate density to choose materials for buoyancy, lightweight structures, or heavy‑duty foundations. An object will float if its average density is less than the density of the fluid it is placed in.
工程师经常计算密度，以选择用于浮力、轻量化结构或重型基础的材料。如果物体的平均密度小于它所置流体的密度，它就会漂浮。
Specific gravity (relative density) is the ratio of a material’s density to that of water: a specific gravity less than 1 means the material floats.
比重（相对密度）是材料密度与水的密度之比：比重小于 1 意味着材料浮于水。
11. Linear Motion Equations (Constant Acceleration) | 匀加速直线运动公式
When an object moves in a straight line with constant acceleration, three key equations link its initial speed (u), final speed (v), acceleration (a), displacement (s), and time (t). These are often called SUVAT equations.
当物体沿直线做匀加速运动时，有三个关键方程将初速度 (u)、末速度 (v)、加速度 (a)、位移 (s) 和时间 (t) 联系起来，常被称为 SUVAT 方程。
- v = u + a × t
- s = (u + v) ÷ 2 × t
- v² = u² + 2 × a × s
For example, a model car accelerating from rest (u = 0) at 2 m/s² for 5 seconds will reach a speed of 10 m/s and travel a distance of 25 m.
例如，一辆模型车从静止（u = 0）以 2 m/s² 加速 5 秒，将达到 10 m/s 的速度，行驶 25 m 的距离。
In engineering, these equations help to design safe braking distances, conveyor belt speeds, and launch mechanisms for small‑scale projects.
在工程中，这些方程有助于设计安全的制动距离、传送带速度以及小型项目的发射机构。
12. Conservation of Energy in Engineering Systems | 工程系统中的能量守恒
The law of conservation of energy states that energy can be transferred usefully, stored, or dissipated, but it cannot be created or destroyed. In a perfect machine, the input energy equals the useful output energy plus wasted energy.
能量守恒定律指出，能量可以被有效转移、储存或耗散，但无法被创造或消灭。在理想机器中，输入能量等于有用的输出能量加上浪费的能量。
Common energy stores include kinetic energy (KE = ½ × m × v²) and gravitational potential energy (GPE = m × g × h). When a roller‑coaster car descends, GPE converts to KE; the sum remains constant ignoring friction.
常见的能量储存形式包括动能（KE = ½ × m × v²）和重力势能（GPE = m × g × h）。当过山车下降时，重力势能转化为动能；不计摩擦时总和保持不变。
Efficiency calculations show how well a system converts input into useful output: Efficiency = (useful output energy ÷ total input energy) × 100%. Real‑world devices always have efficiency less than 100% due to friction, air resistance, and heat.
效率计算展示系统将输入转化为有用输出的程度：效率 = (有用输出能量 ÷ 总输入能量) × 100%。现实设备因摩擦、空气阻力和热量，效率总是低于 100%。
This principle is the backbone of all energy‑aware design, from wind turbines to LED lighting, and reinforces why engineers persistently seek to minimise waste.
这一原理是所有关注能耗的设计的基础——从风力涡轮机到 LED 照明——也强化了工程师为何不断寻求减少浪费。
Published by TutorHao | Engineering Revision Series | aleveler.com
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