📚 Year 12 Edexcel Engineering: Formula and Theorem Quick Reference Guide | Edexcel 12年级工程公式定理速查手册
This guide brings together the essential formulas, laws and theorems you will encounter in the Year 12 Edexcel Engineering course. Use it alongside your class notes and past papers to reinforce your understanding of the underlying principles of engineering science.
本手册汇集了 12 年级 Edexcel 工程课程中必须掌握的关键公式、定律和定理。请结合课堂笔记与历年真题使用，以巩固对工程科学基本原理的理解。
1. Newton’s Laws of Motion | 牛顿运动定律
Newton’s First Law states that an object remains at rest or in uniform straight-line motion unless acted upon by a net external force.
牛顿第一定律指出，除非受到合外力作用，物体将保持静止或匀速直线运动状态。
ΣF = 0 ⇌ equilibrium
Newton’s Second Law links force, mass and acceleration: the net force on an object equals its mass multiplied by its acceleration.
牛顿第二定律将力、质量和加速度联系在一起：物体所受合外力等于质量乘以加速度。
F = m a
Force is measured in newtons (N), where 1 N = 1 kg m s⁻². Acceleration is in m s⁻².
力的单位是牛顿 (N)，1 N = 1 kg·m/s²，加速度的单位是 m/s²。
Newton’s Third Law: for every action there is an equal and opposite reaction; these forces act on different bodies.
牛顿第三定律：每一个力都有一个大小相等、方向相反的反作用力，且作用在不同物体上。
Linear momentum p is defined as mass × velocity. The principle of conservation of momentum applies when no external forces act.
线动量 p 定义为质量乘以速度。在没有外力作用时，动量守恒定律成立。
p = m v and Σp_before = Σp_after
2. Equilibrium and Moments | 平衡与力矩
For a body to be in static equilibrium, the resultant force in any direction must be zero and the sum of moments about any point must be zero.
刚体处于静力平衡的条件是：任一方向的合力为零，且对任一点的合力矩为零。
ΣF_x = 0, ΣF_y = 0, ΣM = 0
A moment is the turning effect of a force. It is calculated as the product of the force and the perpendicular distance from the pivot to the line of action of the force.
力矩是力产生的转动效应，等于力的大小乘以转轴到力作用线的垂直距离。
M = F d_perpendicular
The principle of moments states that for a lever in equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments.
力矩原理：杠杆平衡时，顺时针力矩之和等于逆时针力矩之和。
3. Stress, Strain and Young’s Modulus | 应力、应变与杨氏模量
Stress σ (sigma) is the internal force per unit cross-sectional area a material experiences. Strain ε (epsilon) is the extension per unit original length.
应力 σ 是材料内部单位截面积所受的力；应变 ε 是单位原长的伸长量。
σ = F / A ε = ΔL / L₀
Young’s modulus E is a measure of a material’s stiffness. It is the ratio of tensile stress to tensile strain within the elastic limit.
杨氏模量 E 是衡量材料刚度的指标，是拉伸应力与拉伸应变在弹性限度内的比值。
E = σ / ε
The area under the stress-strain curve up to the yield point represents the material’s resilience. Hooke’s law applies in the linear region: F = k x for springs, where k is the spring constant.
应力-应变曲线在屈服点以下所围面积代表材料的回弹能。在线性阶段，胡克定律适用于弹簧：F = k x，其中 k 为劲度系数。
4. Linear Motion (SUVAT Equations) | 直线运动公式
For constant acceleration in a straight line, the four SUVAT equations link initial velocity u, final velocity v, acceleration a, time t and displacement s.
在匀加速直线运动中，四个 SUVAT 方程将初速度 u、末速度 v、加速度 a、时间 t 和位移 s 联系起来。
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
s = ½ (u + v) t
Always define a positive direction and assign consistent signs to u, v, a and s. These equations only work when acceleration is uniform.
务必规定正方向并赋予 u、v、a、s 一致的符号。这些方程仅适用于匀加速运动。
5. Work, Energy and Power | 功、能与功率
Work done by a constant force is the product of the force and the displacement in the direction of the force. When the force and displacement are at an angle θ, use the component of force along the displacement.
恒力做的功等于力与沿力方向位移的乘积。当力与位移夹角为 θ 时，应取力在位移方向的分量。
W = F s cos θ
Kinetic energy is the energy possessed by a moving object. Gravitational potential energy is the energy an object has due to its height above a reference level.
动能是物体因运动而具有的能量；重力势能是物体因被举高而具有的能量。
KE = ½ m v² PE = m g h
The work–energy principle states that the net work done on an object equals its change in kinetic energy.
功能原理：合外力对物体做的功等于其动能的变化量。
Power is the rate of doing work. It can also be expressed as the product of force and velocity for an object moving at constant speed.
功率是做功的快慢，对于匀速运动的物体，功率也等于力与速度的乘积。
P = W / t P = F v (constant speed)
6. Mechanical Efficiency and Mechanical Advantage | 机械效率与机械利益
Mechanical advantage (MA) is the ratio of the load overcome to the effort applied. Velocity ratio (VR) is the ratio of the distance moved by the effort to the distance moved by the load in the same time.
机械利益 (MA) 是负载与输入作用力的比值。速比 (VR) 是输入作用力移动的距离与负载移动的距离在同一时间内的比值。
MA = Load / Effort VR = d_effort / d_load
Efficiency of a machine compares the useful work output to the total work input. It can also be found from MA and VR.
机械效率是对有用输出功与总输入功的比较，也可通过 MA 和 VR 求得。
η = (Useful work output / Total work input) × 100%
η = (MA / VR) × 100%
Friction always causes MA < VR, so efficiency is less than 100% in real machines.
摩擦导致实际机械中 MA < VR，因此效率总是小于 100%。
7. Basic Electrical Principles | 基本电学原理
Ohm’s law states that the potential difference V across a conductor is directly proportional to the current I flowing through it, provided temperature remains constant.
欧姆定律指出，在温度不变的条件下，导体两端的电压 V 与流过导体的电流 I 成正比。
V = I R
Resistance R is measured in ohms (Ω). Electrical power P is the rate at which electrical energy is transferred. Three equivalent forms are commonly used.
电阻 R 的单位是欧姆 (Ω)。电功率 P 是电能转换的速率，常用的三种等价形式为：
P = I V P = I² R P = V² / R
Charge Q is current multiplied by time. Energy E supplied by a circuit is power multiplied by time.
电荷量 Q 等于电流乘以时间。电路提供的能量 E 等于功率乘以时间。
Q = I t E = P t = I V t
8. Resistor Networks and Kirchhoff’s Laws | 电阻网络与基尔霍夫定律
For resistors in series, the total resistance is the sum of the individual resistances. The same current flows through each resistor.
串联电阻的总电阻为各电阻之和，通过每个电阻的电流相等。
R_total = 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.
并联电阻的总电导（电阻倒数）等于各支路电导之和，各支路电压相同。
1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …
For two resistors in parallel, the product-over-sum shortcut can be used:
对于两个并联电阻，可用积比和公式快速计算：
R_total = (R₁ R₂) / (R₁ + R₂)
Kirchhoff’s current law (KCL): the total current entering a junction equals the total current leaving it.
基尔霍夫电流定律 (KCL)：流入节点的电流之和等于流出节点的电流之和。
Kirchhoff’s voltage law (KVL): the sum of the electromotive forces (e.m.f.) around any closed loop equals the sum of the potential drops.
基尔霍夫电压定律 (KVL)：任意闭合回路中，电动势的代数和等于电压降的代数和。
9. Digital Logic | 数字逻辑基础
Digital systems use binary levels 0 and 1. The three fundamental logic gates are AND, OR and NOT. A Boolean expression describes the output in terms of the inputs.
数字系统使用二进制 0 和 1。三种基本逻辑门为与门 (AND)、或门 (OR) 和非门 (NOT)。布尔表达式用输入变量描述输出。
AND gate: Output is 1 only when all inputs are 1. Boolean: Q = A · B
与门：仅当所有输入均为 1 时输出为 1。布尔表达式：Q = A·B
OR gate: Output is 1 if at least one input is 1. Boolean: Q = A + B
或门：只要至少一个输入为 1，输出即为 1。布尔表达式：Q = A + B
NOT gate: Output is the inverse of the input. Boolean: Q = Ā (A with a bar)
非门：输出是输入的反相。布尔表达式：Q = Ā (A 上划线)
Truth tables for these gates are:
基本门电路的真值表如下：
| AND | ||
| A | B | Q |
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
| OR | ||
| A | B | Q |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
| NOT | |
| A | Q |
| 0 | 1 |
| 1 | 0 |
NAND and NOR gates are universal gates formed by combining AND or OR with a NOT. Their outputs are the inverse of AND and OR respectively.
与非门 (NAND) 和或非门 (NOR) 是通用门，分别由与门、或门和非门组合而成，输出分别是与门和或门输出的反相。
10. Thermal Expansion and Heat Transfer | 热膨胀与热传递
When a solid is heated, it expands linearly. The change in length depends on the original length, temperature change and the coefficient of linear expansion α.
固体受热时会发生线性膨胀，伸长量取决于原长、温度变化以及线膨胀系数 α。
ΔL = α L₀ Δθ
The heat energy Q absorbed or released by a substance when its temperature changes is given by the specific heat capacity equation.
当物质温度变化时，吸收或放出的热量 Q 由比热容公式给出。
Q = m c Δθ
The specific heat capacity c is the energy needed to raise the temperature of 1 kg of the material by 1 °C (or 1 K).
比热容 c 是使 1 kg
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