📚 A-Level Edexcel Engineering: Formula & Theorem Quick Reference | A-Level Edexcel 工程：公式定理速查手册
This quick-reference handbook consolidates the essential formulas, theorems, and relationships required for the Edexcel A-Level Engineering specification. Each section provides the core principle, the relevant equation(s), and a brief note on application – presented first in English and then in Chinese – to support revision and practical problem-solving.
本速查手册整理汇总了 Edexcel A-Level 工程课程所要求的核心公式、定理和关系式。每一节先给出基本原理与方程，再提供应用说明，英文与中文对照排列，便于复习与解题。
1. Static Equilibrium | 静力平衡
For a body in static equilibrium, the vector sum of all forces is zero and the sum of moments about any point is zero.
物体处于静力平衡时，所有力的矢量和为零，且对任一点的力矩之和为零。
ΣF = 0, ΣM = 0
When resolving forces into components, use Fx = F cos θ and Fy = F sin θ, where θ is measured from the horizontal axis.
分解力时使用 Fx = F cos θ 和 Fy = F sin θ，θ 为力与水平轴的夹角。
2. Moments and Couples | 力矩与力偶
The moment of a force about a pivot is given by the product of the force and the perpendicular distance from the pivot to the line of action.
力对支点的力矩等于力的大小乘以支点到力作用线的垂直距离。
M = F × d (where d is perpendicular to F; 其中 d 垂直于 F)
g a pure turning effect: C = F × s, where s is the perpendicular separation of the forces.
力偶由一对大小相等、方向相反且平行的力组成，产生纯转动效应：C = F × s，s 为两力作用线之间的垂直距离。
3. Kinematics of Linear Motion | 直线运动学
For constant acceleration, the four SUVAT equations link displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
对于匀加速直线运动，四个 SUVAT 方程关联位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t)。
v = u + a t
s = u t + ½ a t²
s = ½ (u + v) t
v² = u² + 2 a s
Select the equation that includes the three known quantities and the desired unknown.
选择包含三个已知量和一个所求未知量的方程。
4. Newton’s Laws of Motion | 牛顿运动定律
First Law: An object remains at rest or in uniform motion unless acted upon by a resultant force.
第一定律：如果没有合力作用，物体将保持静止或匀速直线运动状态。
force on an object equals the rate of change of its momentum; for constant mass, F = m a.
第二定律：物体所受合力等于其动量的变化率；质量恒定时，F = m a。
Third Law: When two bodies interact, the forces they exert on each other are equal in magnitude and opposite in direction.
第三定律：两物体相互作用时，彼此施加的力大小相等、方向相反，作用在不同物体上。
Impulse = change in momentum: J = Δp = m(v – u).
冲量等于动量的变化：J = Δp = m(v – u)。
5. Work, Energy and Power | 功、能量与功率
Work done by a constant force acting in the direction of displacement: W = F s.
恒力沿位移方向所做的功：W = F s。
If the force acts at an angle θ to the displacement, W = F s cos θ.
若力与位移的夹角为 θ，则 W = F s cos θ。
Kinetic energy: Ek = ½ m v². Gravitational potential energy: Ep = m g h.
动能：Ek = ½ m v²；重力势能：Ep = m g h。
the rate of doing work: P = W / t. For constant velocity under a driving force F, P = F v.
功率是做功的快慢：P = W / t；在驱动力 F 下匀速运动时，P = F v。
Principle of Conservation of Energy: Energy cannot be created or destroyed, only transferred from one form to another.
能量守恒定律：能量既不能创生也不能消灭，只能从一种形式转化为另一种形式。
6. Stress and Strain | 应力与应变
pressive) stress: σ = F / A, where F is the axial force and A is the cross-sectional area.
正应力（拉或压）：σ = F / A，其中 F 为轴向力，A 为横截面积。
strain: ε = ΔL / L₀, the change in length divided by the original length (dimensionless).
正应变：ε = ΔL / L₀，即长度变化除以原始长度（无量纲）。
y, the ratio of transverse displacement to the perpendicular distance from the fixed face.
剪应力：τ = F / A，力平行于作用面积；剪应变：γ = x / y，横向位移与距固定面垂直距离之比。
7. Young’s Modulus and Hooke’s Law | 杨氏模量与胡克定律
nal limit, stress is directly proportional to strain: σ = E ε, where E is Young’s modulus.
在比例极限内，应力与应变成正比：σ = E ε，E 为杨氏模量。
Hence, E = (F / A) / (ΔL / L₀) = (F L₀) / (A ΔL).
因此，E = (F / A) / (ΔL / L₀) = (F L₀) / (A ΔL)。
lical spring, Hooke’s law: F = k x, where k is the spring constant and x is the extension.
对螺旋弹簧，胡克定律：F = k x，k 为弹簧劲度系数，x 为伸长量。
Elastic potential energy stored in a spring: E_elastic = ½ k x² = ½ F x.
弹簧储存的弹性势能：E_elastic = ½ k x² = ½ F x。
8. Basic DC Circuit Laws | 直流电路基本定律
law: V = I R, where V is potential difference (V), I is current (A), R is resistance (Ω).
欧姆定律：V = I R，V 为电压 (V)，I 为电流 (A)，R 为电阻 (Ω)。
R = ρ L / A, where ρ is resistivity, L is conductor length, and A is cross-sectional area.
电阻定律：R = ρ L / A，ρ 为电阻率，L 为导体长度，A 为横截面积。
Power in electrical circuits: P = V I = I² R = V² / R.
电路中的电功率：P = V I = I² R = V² / R。
Resistors in series: Rtotal = R₁ + R₂ + ...; in parallel: 1/Rtotal = 1/R₁ + 1/R₂ + ...
串联总电阻：Rtotal = R₁ + R₂ + ...；并联：1/Rtotal = 1/R₁ + 1/R₂ + ...
9. Kirchhoff’s Laws | 基尔霍夫定律
Kirchhoff’s current law (KCL): The sum of currents entering a junction equals the sum of currents leaving the junction. ΣIin = ΣIout.
基尔霍夫电流定律 (KCL)：流进一个节点的电流之和等于流出该节点的电流之和。ΣIin = ΣIout。
ny closed loop, the sum of the e.m.f.s equals the sum of the p.d.s (IR drops). Σε = Σ(IR).
基尔霍夫电压定律 (KVL)：在任一闭合回路中，电动势的代数和等于各元件上的电压降之和。Σε = Σ(IR)。
These laws are fundamental for analysing multi-loop and multi-branch circuits.
这两条定律是分析多回路和多支路电路的基础。
10. Thermodynamics – First Law | 热力学第一定律
of a system equals the heat added to the system minus the work done by the system. ΔU = Q – W.
系统内能的增加等于系统吸收的热量减去系统对外做的功。ΔU = Q – W。
a gas: W = p ΔV for a constant-pressure process, and for any reversible process W = ∫p dV.
对气体：等压过程中 W = p ΔV；对于任何可逆过程，W = ∫p dV。
In an adiabatic process (Q = 0), ΔU = –W; in an isothermal process for an ideal gas ΔU = 0, therefore Q = W.
绝热过程中 (Q = 0)，ΔU = –W；理想气体等温过程中 ΔU = 0，因此 Q = W。
11. Bernoulli’s Principle for Inviscid Fluids | 理想流体伯努利原理
For steady, incompressible, inviscid flow along a streamline, the total mechanical energy per unit volume is constant:
对于沿流线的定常、不可压缩、无黏流动，单位体积流体的总机械能守恒：
p + ½ ρ v² + ρ g h = constant (常数)
Here p is static pressure, ρ is fluid density, v is flow speed, and h is height above a reference plane.
式中 p 为静压，ρ 为流体密度，v 为流速，h 为距参考面的高度。
Volumetric flow rate: Q = A v, where A is the cross-sectional area.
体积流量：Q = A v，A 为横截面积。
12. Engineering Mathematics Quick References | 工程数学速查
Cosine rule for a triangle: a² = b² + c² – 2bc cos A.
三角形的余弦定理：a² = b² + c² – 2bc cos A。
Sine rule: a / sin A = b / sin B = c / sin C.
正弦定理：a / sin A = b / sin B = c / sin C。
Vector resolution: a vector of magnitude V at angle θ has components Vx = V cos θ, Vy = V sin θ.
矢量分解：大小为 V、与 x 轴夹角为 θ 的矢量的分量为 Vx = V cos θ，Vy = V sin θ。
Scalar (dot) product: A·B = |A||B| cos θ. Useful for work (W = F·s) and component projections.
标量积（点积）：A·B = |A||B| cos θ。常用于功 (W = F·s) 和分量投影。
Trigonometric identities: sin²θ + cos²θ = 1; tan θ = sin θ / cos θ.
三角恒等式：sin²θ + cos²θ = 1；tan θ = sin θ / cos θ。
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