📚 Year 10 Edexcel Engineering Formula & Theorem Quick Reference Handbook | 公式定理速查手册
This handbook provides a condensed reference of essential formulas and theorems for Year 10 Edexcel Engineering. It is designed to support quick revision, problem-solving, and understanding of core principles across mechanics, electronics, and materials. Each formula is presented with its standard notation, units, and a brief explanation to reinforce application in engineering contexts.
本手册为 Year 10 Edexcel 工程课程浓缩整理了核心公式与定理，旨在辅助快速复习、解题并理解涵盖力学、电子与材料的关键原理。每则公式均注明标准符号、单位，并附简要说明，以强化工程情境中的应用。
1. Hooke's Law and Elasticity | 胡克定律与弹性
Hooke's Law states that the extension of a spring is directly proportional to the force applied, provided the elastic limit is not exceeded.
胡克定律指出，在弹性限度内，弹簧的伸长量与施加的力成正比。
F = k × x
F is the force applied (N). | F 为施加的力（牛）。
k is the spring constant (N/m). | k 为弹簧常数（牛/米）。
x is the extension (m). | x 为伸长量（米）。
A stiffer spring has a larger k value, requiring more force for the same extension. The relationship remains linear until permanent deformation occurs.
弹簧刚度越大，k 值越大，产生相同伸长量需要更大的力。这一线性关系保持到永久变形发生之前。
2. Stress and Strain | 应力与应变
Stress measures the internal resistance of a material to an external force, while strain measures the deformation relative to original dimensions.
应力度量材料抵抗外力的内部阻力，应变则度量相对于原始尺寸的变形。
Stress (σ) = F / A
Strain (ε) = ΔL / L₀
F is force (N), A is cross-sectional area (m²). | F 为力（牛），A 为截面积（米²）。
ΔL is change in length (m), L₀ is original length (m). | ΔL 为长度变化量（米），L₀ 为原始长度（米）。
Stress unit: Pascal (Pa) or N/m². | 应力单位：帕斯卡（Pa）或牛/米²。
Strain has no units; it is a ratio. | 应变无单位，为比率。
Engineers use stress-strain curves to determine yield strength, ultimate tensile strength, and ductility of materials.
工程师利用应力-应变曲线确定材料的屈服强度、极限抗拉强度和延展性。
3. Young's Modulus | 杨氏模量
Young's modulus quantifies the stiffness of a material within its elastic region, linking stress and strain linearly.
杨氏模量量化材料在弹性区的刚度，将应力与应变线性关联。
E = σ / ε
E is Young's modulus (Pa). | E 为杨氏模量（帕）。
σ is stress (Pa), ε is strain. | σ 为应力（帕），ε 为应变。
A high Young's modulus indicates a stiff material that resists deformation. For example, steel has a much higher E than rubber.
高杨氏模量表示材料坚硬、抗变形。例如，钢的 E 值远高于橡胶。
This formula is only valid up to the limit of proportionality.
该公式仅在比例极限以内有效。
4. Moments and Levers | 力矩与杠杆
The moment of a force about a pivot is the turning effect, calculated by multiplying force by the perpendicular distance from the pivot.
力对支点的力矩是转动效果，通过力与力臂的乘积计算。
M = F × d
M is the moment (Nm). | M 为力矩（牛·米）。
F is force (N). | F 为力（牛）。
d is perpendicular distance from the pivot (m). | d 为从支点起的垂直距离（米）。
For a lever in equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments.
杠杆平衡时，顺时针力矩之和等于逆时针力矩之和。
Levers are classified into three classes based on the relative positions of load, effort, and fulcrum, affecting mechanical advantage.
根据负载、施力点和支点的相对位置，杠杆分为三类，影响其机械效益。
5. Velocity and Acceleration | 速度与加速度
These equations describe linear motion, fundamental for analysing mechanisms and vehicle dynamics.
这些方程描述直线运动，是分析机构与车辆动力学的基础。
v = s / t
a = (v − u) / t
v = final velocity (m/s), u = initial velocity (m/s), s = displacement (m), t = time (s). | v 为末速度（米/秒），u 为初速度（米/秒），s 为位移（米），t 为时间（秒）。
a is acceleration (m/s²). | a 为加速度（米/秒²）。
ation is constant, motion can be analysed using SUVAT equations. In uniform motion, a = 0.
如果加速度恒定，运动可用 SUVAT 方程分析。匀速运动时 a = 0。
Typical applications include conveyor belt speeds, vehicle stopping distances, and lift mechanisms.
典型应用包括传送带速度、车辆制动距离和升降机构。
6. Work, Energy and Power | 功、能与功率
Work is done when a force moves an object, transferring energy. Power is the rate at which work is done or energy is transferred.
力使物体移动时就做了功，传递能量。功率是做功或能量传递的速率。
W = F × d
P = W / t = E / t
P = F × v (for constant velocity and force in same direction)
W is work done (J). | W 为做功（焦）。
P is power (W). | P 为功率（瓦）。
d is distance moved in direction of force (m). | d 为在力的方向上移动的距离（米）。
E is energy transferred (J). | E 为传递的能量（焦）。
Understanding energy conservation and power ratings is essential for selecting motors, engines, and actuators in engineered systems.
理解能量守恒与功率等级对工程系统中电机、发动机和致动器的选型至关重要。
7. Ohm's Law and Electrical Resistance | 欧姆定律与电阻
Ohm's Law relates voltage, current, and resistance in an electrical circuit, forming the foundation for circuit analysis.
欧姆定律关联电路中的电压、电流与电阻，构成电路分析的基础。
V = I × R
V is voltage (V). | V 为电压（伏）。
I is current (A). | I 为电流（安）。
R is resistance (Ω). | R 为电阻（欧）。
For a fixed resistor, doubling the voltage doubles the current. Resistance depends on material, length, cross-sectional area, and temperature.
对固定电阻而言，电压加倍则电流加倍。电阻取决于材料、长度、截面积和温度。
Resistivity formula: R = ρ × (L / A), where ρ is resistivity (Ω·m).
电阻率公式：R = ρ × (L / A)，其中 ρ 为电阻率（欧·米）。
8. Electrical Power and Energy | 电功率与电能
Electrical power determines how quickly a device converts electrical energy into other forms. Energy consumption is calculated for cost and efficiency.
电功率决定设备将电能转化为其他形式能量的快慢。能耗用于计算成本与效率。
P = I × V
P = I² × R
E = P × t
P is power (W), I is current (A), V is voltage (V), R is resistance (Ω). | P 为功率（瓦），I 为电流（安），V 为电压（伏），R 为电阻（欧）。
E is energy (J or kWh), t is time (s or h). | E 为能量（焦 或 千瓦时），t 为时间（秒 或 小时）。
In practical engineering, energy is often measured in kilowatt-hours (kWh) for household and industrial billing.
在实际工程中，家用电费和工业计费常用千瓦时（kWh）度量能量。
9. Gear Ratio and Mechanical Advantage | 齿轮比与机械效益
Gears transmit rotary motion and torque. The gear ratio determines the relationship between input and output speeds and torques.
齿轮传递旋转运动和扭矩。齿轮比决定输入与输出转速和扭矩的关系。
Gear Ratio = Number of teeth on driven gear / Number of teeth on driver gear
A ratio greater than 1 reduces speed and increases torque; less than 1 increases speed and reduces torque.
齿轮比大于 1 则减速增扭；小于 1 则增速减扭。
Mechanical Advantage (MA) = Load / Effort
For an ideal gear system (no friction), MA equals the gear ratio. In real systems, efficiency must be considered.
对于理想齿轮系统（无摩擦），机械效益等于齿轮比。实际系统中需考虑效率。
Compound gear trains multiply individual ratios to achieve very high or very low overall ratios.
复合齿轮系将各级齿轮比相乘，以实现极高或极低的总传动比。
10. Fluid Pressure | 流体压力
Pressure in fluids is caused by molecular collisions and acts equally in all directions. It is fundamental in hydraulic and pneumatic systems.
流体压力由分子碰撞产生，向所有方向均等作用，是液压与气动系统的基础。
P = F / A
P is pressure (Pa or N/m²). | P 为压力（帕 或 牛/米²）。
F is force normal to surface (N), A is area (m²). | F 为垂直于表面的力（牛），A 为面积（米²）。
Hydraulic systems use Pascal's principle: pressure applied to a confined fluid is transmitted undiminished. This allows force multiplication.
液压系统运用帕斯卡原理：施加于封闭流体的压强将无衰减地传递。这得以实现力的倍增。
For a hydraulic press, F₁/A₁ = F₂/A₂, enabling a small force to lift a heavy load.
对于液压机，F₁/A₁ = F₂/A₂，使得小力可举升重载。
11. Thermal Expansion | 热膨胀
Most materials expand when heated and contract when cooled. Linear expansion is crucial when designing structures with tight tolerances.
多数材料受热膨胀，冷却收缩。线膨胀在公差严格的的结构设计中至关重要。
ΔL = α × L₀ × ΔT
ΔL is change in length (m). | ΔL 为长度变化量（米）。
α is linear expansion coefficient (1/°C or 1/K). | α 为线膨胀系数（1/°C 或 1/K）。
L₀ is original length (m). | L₀ 为原始长度（米）。
ΔT is change in temperature (°C or K). | ΔT 为温度变化量（°C 或 K）。
Engineers must account for expansion gaps in bridges, railways, and pipelines to prevent buckling.
工程师必须在桥梁、铁路和管道中预留伸缩缝，以防屈曲。
12. Efficiency | 效率
Efficiency measures how well an engineered system converts input energy or power into useful output. It is always less than 100% due to losses.
效率衡量工程系统将输入能量或功率转换为有用输出的程度。因存在损耗，效率始终低于 100%。
Efficiency (η) = (Useful output energy or power / Total input energy or power) × 100%
η is efficiency (no units or %). | η 为效率（无单位或百分比）。
Useful output and total input must be in the same unit (J or W). | 有用输出与总输入必须单位一致（焦 或 瓦）。
Common losses include friction, heat, sound, and vibration. Improving efficiency saves energy and reduces operating costs.
常见损耗包括摩擦、热量、声音和振动。提高效率可节约能源并降低运行成本。
For a motor, Pout = mechanical power; Pin = electrical power. Efficiency helps assess performance and sizing.
对电动机而言，Pout 为机械功率，Pin 为电功率。效率有助于评估性能和选型。
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