📚 KS3 OCR Engineering: Formula & Theorem Quick Reference Handbook | KS3 OCR 工程：公式定理速查手册
This quick reference handbook has been designed for KS3 OCR Engineering students. It brings together the most important formulas and theorems you will need during your design, make and evaluate projects. Understanding these key principles will help you solve real-world engineering problems with confidence, from calculating forces in mechanisms to analysing electrical circuits.
本快速参考手册专为 KS3 OCR 工程学生编写。它汇集了你在设计、制作和评估项目中所需的重要公式与定理。理解这些关键原理将帮助你自信地解决现实世界的工程问题，从计算机械中的力到分析电路。
1. Levers and Moments | 杠杆和力矩
A lever is a simple machine that uses a pivot to make a force more useful. The turning effect of a force is called a moment. In engineering, moments are crucial when designing anything that rotates or balances, such as bridges, see-saws, spanners and robotic arms.
杠杆是利用支点使力变得更有用的简单机械。力的转动效果被称为力矩。在工程中，当设计任何旋转或平衡的物体时（如桥梁、跷跷板、扳手和机械臂），力矩至关重要。
Moment (M) = Force (F) × perpendicular distance from pivot (d)
sh a spanner with a force of 20 N at a distance of 0.25 m from the centre of a bolt, the moment applied is M = 20 × 0.25 = 5 N m.
力矩以牛顿米 (N m) 为单位。距离 d 必须垂直于力的作用线测量。例如，若你用 20 N 的力在距离螺栓中心 0.25 m 处推动扳手，施加的力矩为 M = 20 × 0.25 = 5 N m。
The principle of moments states that for a lever in equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments. This is used to ensure structures are stable and to calculate unknown forces.
力矩原理指出，对于平衡的杠杆，顺时针力矩之和等于逆时针力矩之和。这用于确保结构稳定并计算未知力。
2. Mechanical Advantage | 机械效益
Mechanical advantage (MA) describes how many times a mechanism multiplies an input force. A high mechanical advantage means less effort is needed to move a large load. Cranes, pulleys and gear systems all rely on this concept.
机械效益 (MA) 描述一个机构将输入力放大了多少倍。高机械效益意味着只需较小的力就能移动大负载。起重机、滑轮和齿轮系统都依赖此概念。
Mechanical Advantage = Load (output force) ÷ Effort (input force)
four times more effective. In an ideal system with no friction, this also relates to distances moved.
如果一台起重机器用 50 N 的力举起 200 N 的负载，MA = 200 ÷ 50 = 4。该机器使力放大了四倍。在无摩擦的理想系统中，它也与移动距离相关。
For real mechanisms, friction reduces the actual mechanical advantage. Engineers must consider efficiency, which compares useful work output to total work input.
在实际机构中，摩擦会降低实际机械效益。工程师必须考虑效率，它比较有用功输出与总功输入。
3. Speed and Acceleration | 速度和加速度
Speed is a measure of how fast an object moves. In engineering, designers need to know speeds of moving parts like conveyor belts, vehicles or rotating gears to ensure safety and performance.
速度是衡量物体移动快慢的量。在工程中，设计者需要了解传送带、车辆或旋转齿轮等移动部件的速度，以确保安全与性能。
Average Speed = total distance travelled ÷ total time taken
iles per hour. If a robotic vehicle covers 6 m in 3 s, its average speed is 6 ÷ 3 = 2 m/s.
速度可用米每秒 (m/s) 或英里每小时测量。若一辆机器人车在 3 s 内行驶 6 m，其平均速度为 6 ÷ 3 = 2 m/s。
Acceleration is the rate at which velocity changes. It is important when designing machines that start and stop smoothly.
加速度是速度变化的速率。在设计平稳启停的机器时很重要。
Acceleration = change in velocity ÷ time taken
g. For an elevator that increases its speed from 0 to 4 m/s in 2 s, the acceleration is (4 – 0) ÷ 2 = 2 m/s².
单位是米每二次方秒 (m/s²)。负加速度表示减速或制动。若一部电梯在 2 s 内速度从 0 增至 4 m/s，加速度为 (4 – 0) ÷ 2 = 2 m/s²。
4. Newton's Second Law of Motion | 牛顿第二运动定律
Newton's Second Law connects force, mass and acceleration. It is fundamental to understanding how structures and vehicles respond to loads. Every engineered system that moves must obey this relationship.
牛顿第二定律将力、质量和加速度联系起来。它是理解结构和车辆如何响应负载的基础。每一个运动的工程系统都必须遵循此关系。
Force (F) = mass (m) × acceleration (a)
18 N. This law helps engineers calculate the size of motors or the strength of components.
力以牛顿 (N) 为单位。若一辆总质量 12 kg 的小车以 1.5 m/s² 加速，所需净力为 F = 12 × 1.5 = 18 N。该定律帮助工程师计算马达尺寸或部件的强度。
In rocket design, a large force is produced by expelling exhaust gases, and the mass of the rocket changes as fuel burns. The same formula governs the initial acceleration.
在火箭设计中，通过喷出废气产生很大的力，且随着燃料燃烧质量改变。相同的公式支配着初始加速度。
5. Work and Energy | 功与能量
Work is done when a force moves an object in the direction of the force. Energy is the capacity to do work. Engineering systems constantly transform energy from one form to another, for example electrical energy into kinetic energy in a motor.
当力使物体沿力的方向移动时，就做了功。能量是做功的能力。工程系统不断将能量从一种形式转换为另一种，例如电动机中电能转换为动能。
Work done (W) = Force (F) × distance moved in direction of force (d)
(J). If you push a trolley with a force of 25 N over 4 m, the work done is 25 × 4 = 100 J.
功以焦耳 (J) 为单位。若你用 25 N 的力推手推车前进 4 m，做功为 25 × 4 = 100 J。
Two common forms of mechanical energy are gravitational potential energy (GPE) and kinetic energy (KE). GPE depends on height, mass and gravitational field strength. KE depends on mass and speed.
机械能有两种常见形式：重力势能 (GPE) 和动能 (KE)。重力势能取决于高度、质量和重力场强度。动能取决于质量和速度。
GPE = m × g × h
KE = ½ × m × v²
3 m gains GPE = 2 × 10 × 3 = 60 J. A 2 kg ball moving at 4 m/s has KE = ½ × 2 × 4² = 16 J.
此处 g 为重力场强度，在地球上约为 10 N/kg。一个 2 kg 的球被抬高 3 m 获得的 GPE = 2 × 10 × 3 = 60 J。一个 2 kg 以 4 m/s 运动的球具有 KE = ½ × 2 × 4² = 16 J。
6. Power | 功率
Power measures how quickly work is done or energy is transferred. High-power machines can complete a task in a shorter time. When selecting motors, engines or even simple hand tools, power rating is a key engineering consideration.
功率衡量做功或能量转移的快慢。高功率机器能在更短时间内完成任务。在选择马达、发动机甚至简单的手动工具时，功率额定值是关键的工程考量。
Power (P) = work done (W) ÷ time taken (t)
has a power output of 240 ÷ 20 = 12 W. A more powerful drill can do the same work faster.
功率以瓦特 (W) 为单位，1 W = 1 J/s。一把电动螺丝刀在 20 s 内做 240 J 的功，其输出功率为 240 ÷ 20 = 12 W。更强大的电钻可以更快地完成同样的工作。
In mechanical systems, power can also be calculated from the force and velocity if the motion is steady.
在机械系统中，如果运动稳定，功率也可以根据力和速度计算。
Power = Force × velocity
This is particularly useful when analysing the performance of vehicles and conveyor belts.
这在分析车辆和传送带性能时尤其有用。
7. Electricity: Ohm's Law and Power | 电学：欧姆定律和电功率
Electrical circuits are everywhere in modern engineering, from simple switches to programmable microcontrollers. Ohm's Law is the fundamental rule linking voltage, current and resistance.
现代工程中电路无处不在，从简单的开关到可编程微控制器。欧姆定律是联系电压、电流和电阻的基本规律。
Voltage (V) = Current (I) × Resistance (R)
. Understanding this helps in choosing the right resistors to protect components like LEDs.
电压单位为伏特 (V)，电流单位为安培 (A)，电阻单位为欧姆 (Ω)。若一盏灯从 6 V 电池中汲取 0.4 A 电流，其电阻为 R = V ÷ I = 6 ÷ 0.4 = 15 Ω。理解这一点有助于选择合适的电阻来保护 LED 等元件。
Electrical power is the rate at which electrical energy is converted into light, heat or motion. It is the product of voltage and current.
电功率是电能转换为光、热或运动的速率。它是电压与电流的乘积。
Electrical Power (P) = Voltage (V) × Current (I)
power rating helps engineers select batteries and power supplies with sufficient capacity.
一个在 12 V 电压下工作、电流为 3 A 的电动机功率为 12 × 3 = 36 W。此功率额定值帮助工程师选择容量足够的电池和电源。
8. Density and Materials | 密度和材料
Density tells us how much mass is packed into a given volume. Choosing materials with the right density is vital in engineering – lightweight structures need low-density materials, while ballast or counterweights need high-density ones.
密度表示一定体积内包含的质量。在工程中，选择合适密度的材料至关重要——轻型结构需要低密度材料，而压载物或配重则需要高密度材料。
Density (ρ) = mass (m) ÷ volume (V)
ith mass 780 g and volume 100 cm³ has a density of 780 ÷ 100 = 7.8 g/cm³, which is much denser than aluminium (2.7 g/cm³).
符号 ρ 是希腊字母 rho。密度单位是千克每立方米 (kg/m³) 或克每立方厘米 (g/cm³)。一块质量 780 g、体积 100 cm³ 的钢密度为 780 ÷ 100 = 7.8 g/cm³，比铝 (2.7 g/cm³) 密度大得多。
When designing a product, engineers often use this formula to determine the mass of a component before it is made, or to select a material that meets weight limits while maintaining strength.
在设计产品时，工程师常用这个公式在制造前确定部件的质量，或选择满足重量限制且保持强度的材料。
9. Stress and Strain | 应力和应变
Stress and strain describe how materials respond to forces. Stress is the force applied per unit area, while strain is the resulting deformation. These concepts help ensure that bridges, frames and machine parts will not break or permanently bend during use.
应力和应变描述材料如何响应力。应力是单位面积上施加的力，应变是随之产生的变形。这些概念有助于确保桥梁、框架和机器零件在使用中不会断裂或永久弯曲。
Stress (σ) = Force (F) ÷ cross-sectional area (A)
Strain (ε) = change in length (ΔL) ÷ original length (L₀)
g stretches by 0.005 m under a tension force, the strain is 0.005 ÷ 2.00 = 0.0025 (or 0.25%).
应力以帕斯卡 (Pa) 或 N/m² 为单位。应变没有单位，因为它是比值。如果一根长 2.00 m 的钢丝在拉力下伸长 0.005 m，应变为 0.005 ÷ 2.00 = 0.0025 (即 0.25%)。
Engineers work within the elastic limit of materials, where the object returns to its original shape when the force is removed. Exceeding this limit can cause permanent deformation or failure.
工程师在材料的弹性极限内工作，此时撤去力后物体会恢复原状。超过此极限会导致永久变形或破坏。
10. Gears and Gear Ratios | 齿轮和齿轮比
Gears are toothed wheels that transmit rotational motion and torque. They are used in bicycles, car gearboxes and robotics. The gear ratio determines how many times one gear turns relative to another, affecting speed and turning force.
齿轮是传递旋转运动和扭矩的齿形轮，用于自行车、汽车变速箱和机器人。齿轮比决定一个齿轮相对于另一个齿轮转了多少圈，并影响速度和转动力。
Gear Ratio = number of teeth on driven gear ÷ number of teeth on driver gear
n gear rotates at half the speed but with increased torque, making it easier to climb hills on a bicycle.
如果一个有 15 齿的主动齿轮带动一个 30 齿的从动齿轮，齿轮比为 30 ÷ 15 = 2:1。这意味着从动齿轮以一半的速度旋转但扭矩增大，使得自行车爬坡更省力。
A ratio less than 1 increases output speed but reduces torque. Compound gear trains allow designers to achieve precise speed and force combinations in a compact space.
齿轮比小于 1 会提高输出速度但减小扭矩。复合齿轮系使设计者能在紧凑空间内实现精确的速度与力组合。
Simple machines like inclined planes and pulleys also change the relationship between force and distance, but gears provide a reliable way to control rotary motion in engineered products.
像斜面和滑轮等简单机械也可改变力与距离的关系，但齿轮为控制工程产品中的旋转运动提供了可靠方法。
11. Thermal Expansion | 热膨胀
Most materials expand when heated. Engineers must allow for thermal expansion in structures such as bridges, rail tracks and pipelines to prevent buckling or damage.
大多数材料受热时会膨胀。在桥梁、铁轨和管道等结构中，工程师必须考虑热膨胀，以防止弯曲或损坏。
h (ΔL) ≈ linear expansion coefficient (α) × original length (L₀) × temperature change (ΔT)
The symbol α (alpha) is the material' ughly 0.000012 /°C, then ΔL = 0.000012 × 10 × 15 = 0.0018 m (1.8 mm). Expansion gaps are left to absorb this.
符号 α（阿尔法）是材料的线膨胀系数。一根 10 m 的钢轨温度升高 15 °C，会膨胀一个虽小但显著的量。若钢的 α 约为 0.000012 /°C，则 ΔL = 0.000012 × 10 × 15 = 0.0018 m (1.8 mm)。为此会留有伸缩缝。
Different materials expand at different rates, which is crucial when joining them in composite designs, such as windows with metal and glass.
不同材料膨胀速率不同，这在复合设计中（如包含金属和玻璃的窗户）将它们连接在一起时至关重要。
12. Simple Machines: Inclined Plane and Pulleys | 简单机械：斜面和滑轮
An inclined plane reduces the effort needed to lift a load by increasing the distance over which the force acts. The ideal mechanical advantage of an inclined plane is the length of the slope divided by its height.
斜面通过增加力作用距离来减少提升负载所需的力。斜面的理想机械效益是斜面长度除以高度。
Ideal MA for inclined plane = slope length ÷ height
If a ramp is 5 m long and rises 1 m, the ideal MA is 5. In practice, friction reduces this, but the ramp still makes it easier to push a heavy trolley upwards.
如果一条坡道长 5 m、升高 1 m，理想 MA 为 5。实际上摩擦会降低此值，但坡道仍使向上推动重手推车更省力。
Pulley systems multiply force by using multiple rope sections to support a load. The number of supporting rope strands determines the ideal mechanical advantage. A single movable pulley gives an ideal MA of 2, halving the effort needed.
滑轮系统通过使用多段绳索支撑负载来放大拉力。支撑绳索的股数决定了理想机械效益。一个动滑轮提供理想 MA 为 2，使所需拉力减半。
Ideal MA for pulley system ≈ number of rope sections supporting the load
These principles are applied in cranes, lifts and theatre flying systems, demonstrating how combining simple machines leads to effective engineering solutions.
这些原理应用于起重机、电梯和剧场吊杆系统，展示了如何通过组合简单机械实现高效的工程解决方案。
Published by TutorHao | Engineering Revision Series | aleveler.com
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