📚 Year 8 AQA Engineering: Quick Reference Formula & Theorem Handbook | 八年级AQA工程：公式定理速查手册
Welcome to your handy quick-reference handbook for Year 8 AQA Engineering. This guide brings together the essential formulas and theorems you will encounter across topics such as electronics, mechanics, and mechanical systems. Keep it close when revising or solving problems, and use it to strengthen your understanding of the fundamental relationships that govern engineering principles. Every formula is presented with clear variable definitions and paired explanations in both English and Chinese to support your bilingual learning journey.
欢迎使用这本八年级AQA工程速查手册。本手册汇集了电子学、力学和机械系统等主题中最重要的公式和定理。你可以在复习和解题时随时查阅，并通过这些核心关系加深对工程原理的理解。每个公式都配有清晰的变量定义和中英双语解释，帮助你完成双语学习。
1. Ohm’s Law | 欧姆定律
Ohm’s Law describes the relationship between voltage, current, and resistance in an electrical circuit. It states that the current flowing through a conductor is directly proportional to the voltage across it, provided the temperature remains constant. The formula is written as:
欧姆定律描述了电路中电压、电流和电阻之间的关系。它指出，在温度保持不变的条件下，通过导体的电流与导体两端的电压成正比。公式为：
V = I × R
Where:
- V = voltage (volts, V)
- I = current (amperes, A)
- R = resistance (ohms, Ω)
其中：
- V = 电压（伏特，V）
- I = 电流（安培，A）
- R = 电阻（欧姆，Ω）
This law applies to ohmic conductors and is fundamental for analysing simple circuits and selecting components like resistors.
该定律适用于欧姆导体，是分析简单电路和选择电阻器等元件的基础。
2. Electrical Power | 电功率
Electrical power is the rate at which electrical energy is transferred by a circuit. The power dissipated in a component can be calculated when the voltage across it and the current through it are known. The formula is:
电功率是电路传输电能的速率。已知元件两端的电压和流过的电流，即可计算其消耗的功率。公式为：
P = I × V
Where:
- P = power (watts, W)
- I = current (amperes, A)
- V = voltage (volts, V)
其中：
- P = 功率（瓦特，W）
- I = 电流（安培，A）
- V = 电压（伏特，V）
If the resistance is known, you can combine this with Ohm’s Law to obtain P = I²R or P = V²/R. Understanding power ratings helps in choosing safe and efficient components.
若已知电阻，可结合欧姆定律得到 P = I²R 或 P = V²/R。了解功率额定值有助于选择安全高效的元件。
3. Resistors in Series and Parallel | 串联与并联电阻
When resistors are connected in series, the total resistance is the sum of the individual resistances. The same current flows through each resistor. The formula for series circuits is:
当电阻串联时，总电阻等于各个电阻值之和。电流在各个电阻上相同。串联电路的公式为：
Rₜₒₜₐₗ = R₁ + R₂ + R₃ + …
For resistors in parallel, the reciprocal of the total resistance equals the sum of the reciprocals of the individual resistances. The voltage across each branch is the same. The formula is:
对于并联电阻，总电阻的倒数等于各个电阻倒数之和。各支路两端的电压相同。公式为：
1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + 1/R₃ + …
- Rₜₒₜₐₗ = total resistance (ohms, Ω)
- R₁, R₂, R₃ = individual resistances (Ω)
Series circuits divide voltage, while parallel circuits divide current. These rules are essential for designing sensor networks and controlling loads.
串联电路分压，并联电路分流。这些法则对于设计传感器网络和控制负载至关重要。
4. Force, Mass and Acceleration (Newton’s Second Law) | 力、质量与加速度（牛顿第二定律）
Newton’s second law explains how the motion of an object changes when a resultant force acts on it. The acceleration produced is directly proportional to the resultant force and inversely proportional to the mass of the object.
牛顿第二定律解释了物体受合力作用时运动状态的变化。产生的加速度与合力大小成正比，与物体的质量成反比。
F = m × a
Where:
- F = resultant force (newtons, N)
- m = mass (kilograms, kg)
- a = acceleration (metres per second squared, m/s²)
其中：
- F = 合力（牛顿，N）
- m = 质量（千克，kg）
- a = 加速度（米每二次方秒，m/s²）
This formula is used when analysing mechanisms, vehicle dynamics, and structural loads. Remember that the force is the unbalanced force acting in the direction of acceleration.
该公式用于分析机构、车辆动力学和结构负载。请记住，这个力是作用在加速度方向上的不平衡力。
5. Work Done | 做功
Work is done when a force moves an object through a distance. In engineering, work transfers energy from one store to another. The work done can be calculated from the force applied and the distance moved in the direction of the force.
当力使物体移动一段距离时，就做了功。在工程中，功将能量从一种储存形式转移到另一种。已知作用力和沿力方向移动的距离，即可计算功的大小。
W = F × d
Where:
- W = work done (joules, J)
- F = force (newtons, N)
- d = distance moved in the direction of the force (metres, m)
其中：
- W = 做功（焦耳，J）
- F = 力（牛顿，N）
- d = 沿力方向移动的距离（米，m）
If the force is applied at an angle, only the component of force in the direction of motion does work. This concept is key when calculating energy requirements in machines.
如果力以某一角度施加，只有沿运动方向的分力做功。该概念是计算机器能量需求的关键。
6. Power (Mechanical) | 机械功率
Mechanical power is the rate at which work is done or energy is transferred. It tells us how quickly a machine can perform a task. The basic relationship is:
机械功率是做功或能量转移的速率。它表明了机器执行任务的快慢。基本关系为：
P = W / t
Where:
- P = power (watts, W)
- W = work done or energy transferred (joules, J)
- t = time taken (seconds, s)
其中：
- P = 功率（瓦特，W）
- W = 做功或转移的能量（焦耳，J）
- t = 所用时间（秒，s）
If a constant force moves an object at constant speed, power can also be found using P = F × v, where v is the velocity. Power calculations help compare motors, engines and other prime movers.
若一个恒力使物体匀速运动，也可用 P = F × v 计算功率，其中 v 为速度。功率计算有助于比较电动机、发动机和其他原动机。
7. Efficiency | 效率
Efficiency measures how well a system converts input energy into useful output energy. No real machine is 100% efficient because some energy is always lost, usually as heat due to friction. Efficiency is expressed as a percentage.
效率衡量系统将输入能量转换为有用输出能量的程度。现实中没有任何机器能达到100%的效率，因为总会有能量损失，通常是因摩擦生热。效率以百分比表示。
Efficiency = ( Useful output energy / Total input energy ) × 100%
Alternatively, using power:
Efficiency = ( Useful power output / Total power input ) × 100%
也可用功率表示：
Efficiency = ( 有用输出功率 / 总输入功率 ) × 100%
Where energies are in joules (J) and powers in watts (W). High efficiency is desirable for reducing running costs and environmental impact in engineered systems.
其中能量单位为焦耳（J），功率单位为瓦特（W）。在工程系统中，高效率有利于降低运行成本和环境影响。
8. Pressure | 压强
Pressure is a measure of how concentrated a force is over a given area. It is an important concept in hydraulics, pneumatics, and material selection. The formula relates the force applied perpendicular to a surface and the area of that surface.
压强衡量力在给定面积上的集中程度。它是液压、气动和材料选择中的重要概念。该公式将垂直施加于表面的力与表面面积联系起来。
P = F / A
Where:
- P = pressure (pascals, Pa or N/m²)
- F = force acting perpendicular to the surface (newtons, N)
- A = area of the surface over which the force acts (square metres, m²)
其中：
- P = 压强（帕斯卡，Pa 或 N/m²）
- F = 垂直作用于表面的力（牛顿，N）
- A = 力作用的表面面积（平方米，m²）
A small area under a given force experiences high pressure, which explains why sharp tools cut effectively and why wide tyres reduce ground pressure on soft soil.
在给定的力下，面积越小，压强越大。这解释了为何锋利工具切割效果好、宽轮胎能减小对松软路面的压强。
9. Moment of a Force | 力矩
A moment is the turning effect of a force about a pivot. The size of the moment depends on the magnitude of the force and the perpendicular distance from the pivot to the line of action of the force.
力矩是力绕支点产生的转动效应。力矩的大小取决于力的大小和支点到力作用线的垂直距离。
M = F × d
Where:
- M = moment (newton-metres, Nm)
- F = force (newtons, N)
- d = perpendicular distance from the pivot to the line of action of the force (metres, m)
其中：
- M = 力矩（牛顿米，Nm）
- F = 力（牛顿，
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