📚 GCSE Eduqas Engineering: Formula Theorem Quick Reference Guide | GCSE Eduqas 工程：公式定理速查手册
This quick reference guide compiles the essential formulas and theorems required for the GCSE Eduqas Engineering specification. From fundamental mechanics and materials science to electrical principles, each section provides a clear breakdown of the relationship, its units, and a concise explanation. Use this handbook to reinforce your understanding, solve numerical problems efficiently, and prepare confidently for your examinations.
本速查手册汇集了 GCSE Eduqas 工程学科所必需的核心公式与定理。从基础力学和材料科学到电学原理，每个部分都清晰拆解了公式关系、单位及简明解释。利用本手册巩固理解、高效解答数值问题，自信备战考试。
1. Hooke’s Law and Spring Constant | 胡克定律与弹簧常数
Hooke’s law describes the extension of an elastic object, such as a spring, when a force is applied. The extension is directly proportional to the force, provided the elastic limit is not exceeded. The formula is expressed as F = k × Δx, where F is the applied force in newtons (N), k is the spring constant in newtons per metre (N/m), and Δx is the extension in metres (m). A stiffer spring has a larger k value. This law is foundational for understanding elastic potential energy and material behaviour under load.
胡克定律描述了弹性物体（如弹簧）在受力时的伸长情况。只要不超过弹性极限，伸长量与施加的力成正比。公式表示为 F = k × Δx，其中 F 为施加的力（牛顿，N），k 为弹簧常数（牛顿每米，N/m），Δx 为伸长量（米，m）。弹簧越硬，k 值越大。该定律是理解弹性势能及材料在载荷下行为的基础。
F = k × Δx
2. Stress, Strain and Young’s Modulus | 应力、应变与杨氏模量
Stress (σ) is the internal force per unit area within a material, calculated as σ = F / A, where F is force (N) and A is cross-sectional area (m²). Its unit is the pascal (Pa). Strain (ε) is the fractional change in length, given by ε = ΔL / L, where ΔL is change in length and L is original length; strain is dimensionless. Young’s modulus (E) measures material stiffness: E = σ / ε, expressed in pascals (Pa). This relationship applies within the linear-elastic region of a stress-strain graph and helps engineers select materials with appropriate rigidity.
应力（σ）是材料内部单位面积上的内力，计算公式为 σ = F / A，其中 F 为力（N），A 为横截面积（m²）。单位为帕斯卡（Pa）。应变（ε）是长度的相对变化量，由 ε = ΔL / L 给出，其中 ΔL 为长度变化量，L 为原始长度；应变无量纲。杨氏模量（E）衡量材料的刚度：E = σ / ε，单位为帕斯卡（Pa）。该关系适用于应力-应变图的线弹性区域，帮助工程师选择具有合适刚度的材料。
σ = F / A ε = ΔL / L E = σ / ε
3. Density | 密度
Density (ρ) relates the mass of a material to its volume and is a critical property in engineering for weight estimation and material selection. The formula is ρ = m / V, where ρ is density (kg/m³), m is mass (kg), and V is volume (m³). Regular solids use geometric formulas for volume; irregular objects can be measured via displacement. Knowing density allows conversion between mass and volume and helps in buoyancy calculations and composite material design.
密度（ρ）将材料的质量与其体积联系起来，是工程中用于重量估算和材料选择的关键属性。公式为 ρ = m / V，其中 ρ 为密度（kg/m³），m 为质量（kg），V 为体积（m³）。规则固体的体积使用几何公式计算；不规则物体可通过排水法测量。了解密度可在质量与体积之间进行换算，并有助于浮力计算和复合材料设计。
ρ = m / V
4. Moments and Levers | 力矩与杠杆
A moment is the turning effect of a force about a pivot. The principle of moments states that for a system in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments. The moment (M) is calculated as M = F × d, where F is the force (N) and d is the perpendicular distance from the pivot to the line of action of the force (m). The unit is the newton-metre (Nm). This principle is essential for analysing levers, beams, and structural supports, allowing engineers to determine required forces or distances to achieve balance.
力矩是力绕支点产生的转动效应。力矩原理指出，对于处于转动平衡的系统，顺时针力矩之和等于逆时针力矩之和。力矩（M）计算公式为 M = F × d，其中 F 为力（N），d 为从支点到力作用线的垂直距离（m）。单位为牛顿米（Nm）。该原理对于分析杠杆、梁和结构支撑至关重要，使工程师能够确定实现平衡所需的力或距离。
M = F × d (Σ clockwise moments = Σ anticlockwise moments)
5. Work, Energy and Power | 功、能与功率
Work done (W) transfers energy when a force moves an object. It is given by W = F × d, with force in newtons (N) and distance moved in the direction of the force in metres (m); the unit is the joule (J). Gravitational potential energy (GPE) stored when elevating an object is GPE = m × g × h, where m is mass (kg), g is gravitational field strength (≈9.8 N/kg on Earth), and h is height (m). Kinetic energy (KE) of a moving body is KE = ½ × m × v², with v in m/s. Power (P) measures the rate of doing work: P = W / t or P = E / t, in watts (W). These equations are central to energy conservation and efficiency analyses.
功（W）是力使物体移动时传递的能量。公式为 W = F × d，力以牛顿（N）为单位，沿力方向移动的距离以米（m）为单位；单位为焦耳（J）。提升物体时储存的重力势能（GPE）为 GPE = m × g × h，其中 m 为质量（kg），g 为重力场强度（地球上约 9.8 N/kg），h 为高度（m）。运动物体的动能（KE）为 KE = ½ × m × v²，v 单位为 m/s。功率（P）衡量做功的快慢：P = W / t 或 P = E / t，单位为瓦特（W）。这些方程是能量守恒和效率分析的核心。
W = F × d GPE = mgh KE = ½mv² P = W / t
6. Efficiency | 效率
In any engineering system, some input energy is dissipated, usually as heat or sound. Efficiency (η) compares useful output to total input, expressed as a percentage: Efficiency (%) = (Useful output energy / Total input energy) × 100. The same formula can use power: Efficiency = (Useful power output / Total power input) × 100. No machine is 100% efficient due to friction and other losses. Evaluating efficiency guides design improvements aimed at reducing waste, making systems more sustainable and cost-effective.
在任何工程系统中，部分输入能量会被耗散，通常以热或声的形式。效率（η）比较有用输出与总输入，表示为百分比：效率 (%) = (有用输出能量 / 总输入能量) × 100。功率形式相同：效率 = (有用功率输出 / 总功率输入) × 100。由于摩擦和其他损耗，没有机器具有 100% 的效率。评估效率可引导设计改进，以减少浪费，使系统更可持续且经济高效。
η = (Eout / Ein) × 100% or η = (Pout / Pin) × 100%
7. Ohm’s Law and Resistance | 欧姆定律与电阻
Ohm’s law states that for a metallic conductor at constant temperature, the current (I) through it is directly proportional to the potential difference (V) across it. The relationship is V = I × R, where V is in volts (V), I in amperes (A), and R is resistance in ohms (Ω). Resistance depends on material, length, cross-sectional area, and temperature. This law is fundamental for analysing and designing DC circuits, calculating unknown values, and predicting component behaviour under steady conditions.
欧姆定律指出，对于恒温下的金属导体，通过它的电流（I）与其两端的电位差（V）成正比。关系式为 V = I × R，其中 V 单位为伏特（V），I 为安培（A），R 为电阻（欧姆，Ω）。电阻取决于材料、长度、横截面积和温度。该定律是分析和设计直流电路、计算未知量以及预测稳态下元件行为的基础。
V = I × R
8. Resistors in Series and Parallel | 电阻的串联与并联
When resistors are connected end-to-end (in series), the total resistance is simply the sum: Rtotal = R₁ + R₂ + R₃ + …. For resistors joined so that the current splits (in parallel), the total resistance is found using the reciprocal formula: 1 / Rtotal = 1 / R₁ + 1 / R₂ + 1 / R₃ + …. For two parallel resistors, a convenient rearrangement is Rtotal = (R₁ × R₂) / (R₁ + R₂). These rules enable the simplification of complex networks and the calculation of current and voltage distribution across components.
当电阻首尾相连（串联）时，总电阻为各电阻之和：R总 = R₁ + R₂ + R₃ + …。对于并联连接的电阻（电流分流），总电阻使用倒数公式求得：1 / R总 = 1 / R₁ + 1 / R₂ + 1 / R₃ + …。对于两个并联电阻，常见的变换形式为 R总 = (R₁ × R₂) / (R₁ + R₂)。这些规则可用于简化复杂网络，并计算各元件上的电流和电压分配。
Series: Rt = R₁ + R₂ + …
Parallel: 1/Rt = 1/R₁ + 1/R₂ + … (two: Rt = R₁R₂/(R₁+R₂))
9. Electrical Power | 电功率
Electrical power is the rate at which electrical energy is transferred by a circuit. The core equation is P = I × V, where P is power in watts (W), I is current (A), and V is voltage (V). Combining with Ohm’s law yields two alternative forms: P = I² × R and P = V² / R. These formulas help determine the energy consumption of components, the required ratings for fuses, and the heat dissipation in resistive elements. In engineering, managing electrical power is crucial for safety and efficiency.
电功率是电路传输电能的速率。核心方程为 P = I × V，其中 P 为功率（瓦特，W），I 为电流（A），V 为电压（V）。结合欧姆定律可得到两种替代形式：P = I² × R 和 P = V² / R。这些公式有助于确定元件的能耗、所需保险丝额定值以及电阻元件的散热情况。在工程中，管理电功率对安全与效率至关重要。
P = I × V P = I²R P = V² / R
10. Newton’s Second Law and Motion | 牛顿第二定律与运动
Newton’s second law relates resultant force, mass, and acceleration: F = m × a, where F is the unbalanced force (N), m is mass (kg), and a is acceleration (m/s²). For uniform acceleration, key motion equations link initial velocity (u), final velocity (v), acceleration (a), displacement (s), and time (t): v = u + at ; s = ut + ½at² ; and v² = u² + 2as. These tools allow engineers to predict how vehicles, projectiles, or moving machinery parts will behave under forces, essential for dynamics and structural load analysis.
牛顿第二定律将合力、质量和加速度联系起来：F = m × a，其中 F 为不平衡力（N），m 为质量（kg），a 为加速度（m/s²）。对于匀加速运动，关键运动学方程关联初速度（u）、末速度（v）、加速度（a）、位移（s）和时间（t）：v = u + at；s = ut + ½at²；以及 v² = u² + 2as。这些工具使工程师能够预测车辆、抛射体或运动机械部件在力作用下的行为，对动力学和结构载荷分析不可或缺。
F = m × a
v = u + at s = ut + ½at² v² = u² + 2as
11. Gear Ratio and Mechanical Advantage | 齿轮比与机械效益
Gears transmit rotary motion and torque between shafts. The gear ratio is determined by the number of teeth: Gear ratio = Number of teeth on driven gear / Number of teeth on driver gear. A ratio greater than 1 produces an increase in torque and a decrease in speed (gear down); a ratio less than 1 increases speed and reduces torque (gear up). Mechanical advantage (MA) reflects the force amplification of simple machines: MA = Load / Effort. In an ideal system without friction, MA equals the velocity ratio (VR). Understanding these ratios is vital for designing transmissions in bicycles, lifting mechanisms, and robotic actuators.
齿轮在轴之间传递旋转运动和扭矩。齿轮比由齿数决定：齿轮比 = 从动齿轮齿数 / 主动齿轮齿数。齿轮比大于 1 会增加扭矩并降低速度（减速）；齿轮比小于 1 则增加速度并减少扭矩（增速）。机械效益（MA）反映简单机械的力放大作用：MA = 负载 / 动力。在无摩擦的理想系统中，MA 等于速度比（VR）。理解这些比率对于设计自行车传动、提升机构和机器人执行器至关重要。
Gear Ratio = Teethdriven / Teethdriver
MA = Load / Effort
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