📚 KS3 AQA Engineering: Formula & Theorem Quick Reference Guide | KS3 AQA 工程：公式定理速查手册
Engineering at KS3 AQA involves understanding how systems work, from simple machines to electrical circuits. Mastering the fundamental formulas and theorems is essential for designing, analysing and solving real-world problems. This quick reference guide brings together the key equations every young engineer needs to know, with clear explanations in both English and Chinese to support bilingual learners.
KS3 AQA 工程学涉及理解各类系统的工作原理，从简单机械到电路。掌握基本公式和定理对于设计、分析和解决实际问题至关重要。本速查手册汇集了每位年轻工程师都需要掌握的关键方程式，并提供清晰的中英双语解释，方便双语学习者使用。
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, as long as the temperature remains constant. This fundamental law allows engineers to calculate any one of the three quantities if the other two are known.
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，只要温度保持不变，流过导体的电流与导体两端的电压成正比。这个基本定律使工程师能够在已知其中两个量的情况下计算出第三个量。
The formula is:
公式为：
V = I × R
Where: V represents voltage measured in volts (V), I represents current measured in amperes (A), and R represents resistance measured in ohms (Ω).
其中：V 代表电压，单位伏特 (V)；I 代表电流，单位安培 (A)；R 代表电阻，单位欧姆 (Ω)。
Typical applications include sizing resistors for LED circuits or checking the current in a simple series circuit. In engineering design, Ohm’s Law is the cornerstone of electronic system analysis.
典型应用包括为 LED 电路选择合适的电阻值，或检查简单串联电路中的电流。在工程设计中，欧姆定律是电子系统分析的基石。
2. Electrical Power | 电功率
Power is the rate at which electrical energy is transferred by a circuit. The power formula relates power, voltage and current. Knowing how to calculate power helps engineers determine energy consumption and select appropriate components like power supplies.
功率是电路传输电能的速率。功率公式将功率、电压和电流联系起来。懂得如何计算功率有助于工程师确定能量消耗并选择合适的元件，如电源。
The standard equation is:
标准方程为：
P = V × I
Where P is power in watts (W), V is voltage in volts (V), and I is current in amperes (A). This means that power equals voltage multiplied by current.
其中 P 为功率，单位瓦特 (W)；V 为电压，单位伏特 (V)；I 为电流，单位安培 (A)。这意味着功率等于电压乘以电流。
Combining Ohm’s Law with this formula, engineers can also derive other useful forms: P = I²R and P = V²/R, which are used when only resistance and current or voltage are known.
将这个公式与欧姆定律结合，工程师还可以推导出其他有用的形式：P = I²R 和 P = V²/R，这些用于仅知道电阻和电流或电压的情况。
3. Mechanical Advantage (Levers) | 机械利益（杠杆）
Mechanical advantage (MA) is a measure of the force amplification achieved by using a tool, mechanical device or machine system. For levers, it is the ratio of the output force (load) to the input force (effort). A mechanical advantage greater than 1 means the machine multiplies the effort force.
机械利益 (MA) 是衡量使用工具、机械设备或机器系统所实现的力放大效果的指标。对于杠杆而言，它是输出力（负载）与输入力（作用力）的比值。机械利益大于 1 表示机器放大了作用力。
The formula for mechanical advantage is:
机械利益的计算公式为：
MA = Load (L) ÷ Effort (E)
In an ideal frictionless system, MA can also be calculated from the distances from the pivot: MA = Effort arm length ÷ Load arm length. This principle is used to design levers such as crowbars, scissors and seesaws.
在理想无摩擦系统中，MA 也可以通过距支点的距离来计算：MA = 动力臂长度 ÷ 阻力臂长度。这一原理用于设计撬棍、剪刀和跷跷板等杠杆。
4. Gear Ratio | 齿轮比
Gears are toothed wheels that transmit rotary motion and torque. The gear ratio determines how the rotational speed and torque change between two meshing gears. A high gear ratio reduces speed but increases torque, and vice versa.
齿轮是传递旋转运动和扭矩的带齿轮子。齿轮比决定了两个啮合齿轮之间旋转速度和扭矩的变化。高齿轮比会降低转速但增加扭矩，反之亦然。
The gear ratio is found by:
齿轮比的计算方法为：
Gear Ratio = Number of teeth on driven gear ÷ Number of teeth on driver gear
If the driven gear has more teeth, the output speed is slower but torque is higher (a speed reducer). Engineers use gear trains in bicycles, car gearboxes and lifting mechanisms.
如果从动齿轮的齿数更多，输出速度就会更慢但扭矩更高（减速装置）。工程师在自行车、汽车变速箱和起重机构中使用齿轮传动系统。
5. Principle of Moments | 力矩原理
The principle of moments states that for an object to be in equilibrium, the sum of the clockwise moments about any pivot must equal the sum of the anticlockwise moments. This principle is widely used in structural engineering and when analysing levers and balances.
力矩原理指出，对于一个处于平衡状态的物体，绕任何支点的顺时针力矩之和必须等于逆时针力矩之和。这个原理广泛应用于结构工程以及分析杠杆和天平。
The moment of a force is calculated by:
力的力矩计算如下：
Moment = Force (F) × Perpendicular distance from pivot (d)
M = F × d
The unit of moment is the newton-metre (Nm). When designing bridges or beams, engineers ensure that all moments balance so the structure does not rotate.
力矩的单位是牛顿·米 (Nm)。在设计桥梁或梁时，工程师确保所有力矩平衡，这样结构才不会发生转动。
6. Work Done and Energy | 功和能量
In engineering, work is done when a force moves an object. Energy is the capacity to do work. The work done equation links force, distance and energy transfer, making it fundamental in mechanical systems and efficiency calculations.
在工程学中，当力使物体移动时，就做了功。能量是做功的能力。功的方程式将力、距离和能量传递联系起来，使其成为机械系统和效率计算的基础。
The formula for work done is:
功的计算公式为：
Work Done (W) = Force (F) × Distance moved in the direction of the force (d)
W = F × d
Work is measured in joules (J), where 1 J = 1 N × 1 m. When lifting an object against gravity, the work done equals the gravitational potential energy gained. This principle is used to size motors and actuators.
功以焦耳 (J) 为单位，1 J = 1 N × 1 m。当克服重力举起物体时，所做的功等于获得的引力势能。该原理用于选择电机和执行器的规格。
7. Pressure in Fluids and Solids | 流体与固体中的压力
Pressure describes how concentrated a force is over an area. Hydraulic systems, pneumatic systems and structural supports all rely on the pressure formula to function safely and effectively.
压力描述了力在面积上的集中程度。液压系统、气动系统和结构支撑都依赖压力公式来安全有效地运行。
The basic pressure equation is:
基本的压力方程为：
Pressure (P) = Force (F) ÷ Area (A)
Pressure is measured in pascals (Pa) or N/m². A small force applied to a large area produces low pressure, while the same force on a small area gives high pressure. This concept explains why sharp tools cut more easily and how hydraulic presses multiply force.
压力以帕斯卡 (Pa) 或 N/m² 为单位。施加在大面积上的小力产生低压，而相同的力作用在小面积上则产生高压。这个概念解释了为什么锋利的工具更容易切割，以及液压机如何放大力的作用。
8. Speed, Distance and Time | 速度、距离和时间
Calculating speed is essential for analysing the motion of vehicles, production line conveyors, and robotic components. The relationship between speed, distance and time is one of the most frequently used formulas in engineering kinematics.
计算速度对于分析车辆、生产线传送带和机器人部件的运动至关重要。速度、距离和时间之间的关系是工程运动学中最常用的公式之一。
The formula is:
公式为：
Speed (s) = Distance (d) ÷ Time (t)
Alternatively, distance = speed × time and time = distance ÷ speed. Engineers use these variations to plan travel times, determine production rates, or set the speed of conveyor belts.
或者，距离 = 速度 × 时间，时间 = 距离 ÷ 速度。工程师使用这些变体来规划行程时间、确定生产速率或设定传送带的速度。
9. Density | 密度
Density links the mass of a material to its volume. Understanding density is crucial when selecting materials for a design—lightweight but strong materials like aluminium alloys are chosen for aircraft, while denser materials may be used for ballast or radiation shielding.
密度将材料的质量与其体积联系起来。在设计选材时了解密度至关重要——飞机会选择像铝合金这样轻质高强的材料，而较重的材料可能用于压载或辐射屏蔽。
The density equation is:
密度方程为：
Density (ρ) = Mass (m) ÷ Volume (V)
The unit of density is kilograms per cubic metre (kg/m³) or grams per cubic centimetre (g/cm³). For example, water has a density of 1000 kg/m³ at 4°C. In engineering, density values are used to calculate weight, buoyancy and material costs.
密度的单位是千克每立方米 (kg/m³) 或克每立方厘米 (g/cm³)。例如，水在 4°C 时的密度为 1000 kg/m³。在工程中，密度值用于计算重量、浮力和材料成本。
10. Efficiency | 效率
No machine is perfect; some input energy is always lost to friction, heat or sound. Efficiency measures how well a system converts input energy into useful output energy. Improving efficiency is a key goal in sustainable engineering design.
没有机器是完美的；一些输入能量总是会以摩擦、热或声音的形式
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