📚 GCSE AQA Engineering: Formula & Theorem Quick Reference Guide | GCSE AQA 工程：公式定理速查手册
This handbook brings together the fundamental formulas, relationships, and theorems you need for the AQA GCSE Engineering (8852) specification. It is designed as a fast-reference tool to support your revision, classroom work, and problem-solving tasks. All equations are presented with clear symbols and practical examples to help you apply them confidently in the exam.
本手册汇集了 AQA GCSE 工程（8852）课程所需的核心公式、关系和定理，可作为快速查阅工具，辅助你的复习、课堂学习与解题实践。所有公式均使用清晰的符号并结合实际应用，帮助你自信应对考试。
1. Mechanical Advantage & Velocity Ratio | 机械效益与速度比
Mechanical Advantage (MA) measures how much a mechanism multiplies the effort force. It is defined as the ratio of the load force to the effort force: MA = Fₗ ÷ Fₑ, where Fₗ is the load (output force) and Fₑ is the effort (input force). A larger MA means less effort is needed to move a given load.
机械效益（MA）衡量机构将输入力放大的倍数，定义为负载力与施力之比：MA = Fₗ ÷ Fₑ，其中 Fₗ 为负载力（输出力），Fₑ 为施力（输入力）。MA 越大，移动同一负载所需的输入力越小。
Velocity Ratio (VR) describes the ratio of the distance moved by the effort to the distance moved by the load in the same time: VR = dₑ ÷ dₗ. It is a purely geometric quantity determined by the mechanism’s design, unaffected by friction or wear.
速度比（VR）表示在相同时间内施力移动的距离与负载移动距离之比：VR = dₑ ÷ dₗ。它是一个纯几何量，由机构设计决定，不受摩擦或磨损影响。
In all real machines, friction causes the actual MA to be lower than the VR. When friction is negligible, the ideal MA equals the VR. Understanding both quantities helps engineers predict performance and identify energy losses.
在所有真实机械中，摩擦导致实际 MA 低于 VR。当摩擦可忽略时，理想 MA 等于 VR。理解这两个量有助于工程师预测性能并识别能量损失。
2. Efficiency | 效率
Efficiency (η) expresses how well a machine converts the input work into useful output work. It is calculated as a percentage: η = (MA ÷ VR) × 100%. This formula links the mechanical advantage and the velocity ratio, making it a key indicator of a mechanism’s energy performance.
效率（η）表示机器将输入功转化为有用输出功的程度，以百分比计算：η = (MA ÷ VR) × 100%。该式将机械效益与速度比联系起来，是衡量机构能量性能的关键指标。
An ideal machine would have 100% efficiency, but friction, air resistance, and material deformation always cause losses. For example, a simple pulley system might have an efficiency around 80–90%, while a worm gear system often falls well below 70%.
理想机器的效率为 100%，但摩擦、空气阻力和材料变形总会造成损失。例如，一个简单滑轮系统的效率约在 80–90%，而蜗轮蜗杆系统通常远低于 70%。
Efficiency can also be written using work or power: η = (useful output energy ÷ total input energy) × 100%, or in terms of power, η = (P_out ÷ P_in) × 100%. You must be comfortable converting between these forms in exam questions.
效率也可用功或功率表示：η =（有用输出能量 ÷ 总输入能量）× 100%，或用功率表示，η =（P_out ÷ P_in）× 100%。你需在考题中熟练转换这些形式。
3. Levers and Moments | 杠杆与力矩
A lever is a rigid bar that rotates around a pivot (fulcrum). The principle of moments states that for a lever in equilibrium, the clockwise moments equal the anticlockwise moments: F₁ × d₁ = F₂ × d₂, where d is the perpendicular distance from the pivot to the line of action of the force.
杠杆是一根绕支点（转轴）转动的刚性杆。力矩原理指出，处于平衡状态的杠杆，顺时针力矩等于逆时针力矩：F₁ × d₁ = F₂ × d₂，其中 d 为支点到力作用线的垂直距离。
There are three classes of levers determined by the relative positions of effort, load, and fulcrum. In a Class 1 lever (e.g., crowbar), the fulcrum lies between effort and load; in Class 2 (e.g., wheelbarrow), the load is between; in Class 3 (e.g., tweezers), the effort is in the middle. The MA can be greater or less than 1 depending on the lever class.
根据施力、负载和支点的相对位置，杠杆分为三类。第一类杠杆（如撬棍）支点在中间；第二类杠杆（如手推车）负载在中间；第三类杠杆（如镊子）施力在中间。MA 可大于或小于 1，取决于杠杆类别。
Moments are measured in newton-metres (Nm). A larger force or a greater distance from the pivot increases the turning effect. Engineers use the moment equation to size levers, gear selectors, and brake pedals.
力矩的单位是牛顿·米（Nm）。更大的力或距支点更远的距离会增大转动效果。工程师利用力矩方程确定杠杆、换挡机构和刹车踏板的尺寸。
4. Gear and Pulley Ratios | 齿轮比与滑轮比
Gear systems transmit rotational motion and torque. The gear ratio (GR) is given by the number of teeth on the driven gear divided by the number of teeth on the driver gear: GR = T_driven ÷ T_driver. If the driven gear has more teeth, speed reduces and torque increases; the opposite gives a speed increase with reduced torque.
齿轮系统传递旋转运动和扭矩。齿轮比（GR）为从动轮齿数除以主动轮齿数：GR = T_driven ÷ T_driver。若从动轮齿数更多，则转速降低、扭矩增大；反之则转速提高、扭矩减小。
For a simple pulley system, the Velocity Ratio equals the number of rope sections supporting the load. A 4-pulley block and tackle has VR = 4, meaning the effort must move 4 m to lift the load 1 m, although the MA may be slightly lower due to friction.
对于简单滑轮系统，速度比等于承载负载的绳索段数。4 轮滑轮组的 VR = 4，即施力需移动 4 米才将负载提升 1 米，但由于摩擦，实际 MA 可能略低。
In belt and chain drives, the ratio is based on the diameters or numbers of teeth of the pulleys/sprockets: VR = diameter of driven ÷ diameter of driver. These systems are widely used in bicycles, conveyor belts, and engine camshaft drives.
在带传动和链传动中，速度比基于带轮/链轮的直径或齿数：VR = 从动轮直径 ÷ 主动轮直径。这类系统广泛用于自行车、传送带和发动机凸轮轴驱动。
5. Ohm’s Law and Resistance | 欧姆定律与电阻
Ohm’s Law links voltage (V), current (I), and resistance (R) in a circuit: V = I × R. This fundamental relationship allows you to calculate any one of the three quantities when the other two are known.
欧姆定律将电路中的电压（V）、电流（I）和电阻（R）联系起来：V = I × R。这个基本关系使你在已知两个量时能够计算第三个量。
Resistors in series add directly: R_total = R₁ + R₂ + R₃ + …. For resistors in parallel, the total resistance is found using the reciprocal formula: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …. The total resistance in parallel is always less than the smallest individual resistor.
串联电阻直接相加：R 总 = R₁ + R₂ + R₃ + …。对于并联电阻，总电阻通过倒数公式计算：1/R 总 = 1/R₁ + 1/R₂ + 1/R₃ + …。并联后的总电阻始终小于其中最小的单个电阻。
In engineering applications, Ohm’s Law governs the design of sensors, actuators, wiring looms, and microcontroller interfaces. Understanding how resistance changes with temperature in thermistors or with light in LDRs is essential for building automated systems.
在工程应用中，欧姆定律支配着传感器、执行器、线束和微控制器接口的设计。理解热敏电阻随温度或光敏电阻随光照变化的电阻规律，对构建自动化系统至关重要。
6. Electrical Power and Energy | 电功率与电能
Electrical power (P) is the rate at which electrical energy is transferred. It can be calculated with P = I × V, or by combining with Ohm’s Law: P = I² × R and P = V² ÷ R. Choose the most convenient form based on the quantities given.
电功率（P）是电能传递的速率。可用 P = I × V 计算，或结合欧姆定律得出 P = I² × R 与 P = V² ÷ R。根据已知量选择最方便的形式。
Energy (E) is power multiplied by time: E = P × t. In the SI system, energy is measured in joules (J), power in watts (W), and time in seconds (s). For household electricity, kilowatt-hours (kWh) are commonly used: 1 kWh = 3,600,000 J.
电能（E）等于功率乘以时间：E = P × t。SI 单位制中，能量的单位是焦耳（J），功率为瓦（W），时间为秒（s）。家庭用电常用千瓦时（kWh）：1 kWh = 3,600,000 J。
Engineers must size components such as fuses, cables, and batteries by calculating the expected current and power requirements. Overlooking these calculations can lead to overheating, voltage drops, or complete system failure.
工程师必须通过计算预期电流和功率需求来选用保险丝、电缆和电池等元件。忽略这些计算可能导致过热、电压跌落甚至整个系统失效。
7. Stress and Strain | 应力与应变
Stress (σ) is the internal force per unit area within a material when an external load is applied: σ = F ÷ A. It is measured in pascals (Pa) or megapascals (MPa). Tensile stress stretches the material; compressive stress shortens it.
应力（σ）是材料受到外部载荷时，单位面积上的内力：σ = F ÷ A，单位为帕斯卡（Pa）或兆帕（MPa）。拉伸应力会使材料伸长，压缩应力则使其缩短。
Strain (ε) is the deformation per unit original length: ε = ΔL ÷ L₀. Because it is a ratio of two lengths, strain has no units. It is often expressed as a percentage in engineering contexts.
应变（ε）是单位原始长度的变形量：ε = ΔL ÷ L₀。由于是两个长度的比值，应变没有单位，在工程中常以百分比表示。
Stress-strain data help engineers choose materials that will not permanently deform or break under expected loads. The elastic limit marks the point beyond which a material no longer returns to its original shape.
应力–应变数据帮助工程师选择在预期载荷下不会发生永久变形或断裂的材料。弹性极限标志着材料不再恢复原状的转折点。
8. Young’s Modulus and Material Stiffness | 杨氏模量与材料刚度
Young’s Modulus (E) describes the stiffness of a solid material in the elastic region. It is defined as the ratio of stress to strain: E = σ ÷ ε, and has units of pascals (Pa). A high Young’s modulus means the material is very stiff and resists deformation.
杨氏模量（E）表征固体材料在弹性区内的刚度，定义为应力与应变之比：E = σ ÷ ε，单位为帕斯卡（Pa）。杨氏模量高意味着材料刚度大，难以变形。
For many metals, Young’s modulus is constant up to the elastic limit. Steel has a much higher E than aluminium, which is why steel beams are used in structures that need to carry heavy loads without bending excessively.
许多金属的杨氏模量在弹性极限内保持恒定。钢的 E 值远高于铝，因此钢梁用于需要承受重载且不能过度弯曲的结构中。
In the AQA GCSE course, you may not be required to use the term ‘Young’s modulus’ explicitly, but you must understand that stiffness relates to how much a material stretches or compresses under a given force. The concept underpins the design of bridges, bicycle frames, and engine parts.
在 AQA GCSE 课程中，可能不要求明确使用“杨氏模量”一词，但你必须理解刚度反映了材料在给定力下伸长或压缩的程度。这一概念支撑着桥梁、自行车车架和发动机零件的设计。
9. Work, Energy and Power | 功、能量与功率
Mechanical work (W) is done when a force moves an object over a distance: W = F × d. The force must act in the direction of motion. Work is measured in joules (J), where 1 J = 1 N × 1 m.
当一个力使物体沿其作用方向移动一段距离时，就做了机械功：W = F × d。功的单位是焦耳（J），1 J = 1 N × 1 m。
Gravitational potential energy (Eₚ) gained by an object raised vertically is: Eₚ = m × g × h, where m is mass, g is gravitational field strength (9.8 m/s² on Earth), and h is height. Kinetic energy (Eₖ) is given by Eₖ = ½ × m × v².
物体竖直升高获得的重力势能（Eₚ）为：Eₚ = m × g × h，其中 m 为质量，g 为重力场强（地球取 9.8 m/s²），h 为高度。动能（Eₖ）公式为 Eₖ = ½ × m × v²。
Power (P) is the rate of doing work or transferring energy: P = W ÷ t (or P = E ÷ t). In mechanical systems, power is often stated in watts (W) or kilowatts (kW). Remember that 1 W = 1 J/s.
功率（P）是做功或传递能量的速率：P = W ÷ t（或 P = E ÷ t）。在机械系统中，功率常以瓦（W）或千瓦（kW）表示。记住 1 W = 1 J/s。
10. Formula Summary Table | 公式汇总表
The table below provides a quick reference to the main formulas covered. Use it as a checklist during revision to ensure you can recall each equation and its units with confidence.
下表汇总了上述主要公式，供你快速查阅。复习时可将其作为核对清单，确保能自信地回忆每个方程及其单位。
| Quantity (量) | Symbol | Formula (公式) | SI Unit (国际单位) |
|---|---|---|---|
| Mechanical Advantage | MA | Fₗ ÷ Fₑ | (no unit) |
| Velocity Ratio | VR | dₑ ÷ dₗ | (no unit) |
| Efficiency | η | (MA ÷ VR) × 100% | % |
| Moment | M | F × d | Nm |
| Gear Ratio | GR | T_driven ÷ T_driver | (no unit) |
| Ohm’s Law | – | V = I × R | V, A, Ω |
| Resistors in Series | Rₜₒₜ | R₁ + R₂ + … | Ω |
| Resistors in Parallel | Rₜₒₜ | 1/Rₜₒₜ = 1/R₁ + 1/R₂ + … | Ω |
| Electrical Power | P | I × V; I²R; V²/R | W |
| Electrical Energy | E | P × t | J |
| Stress | σ | F ÷ A | Pa |
| Strain | ε | ΔL ÷ L₀ | (no unit) |
| Young’s Modulus | E | σ ÷ ε | Pa |
| Work Done | W | Published by TutorHao | GCSE 工程 Revision Series | aleveler.com
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