📚 Pre-U Cambridge Engineering: Quick Reference Handbook of Formulas and Theorems | Pre-U Cambridge 工程：公式定理速查手册
This quick reference handbook provides a concise collection of essential formulas, definitions and theorems for the Cambridge Pre-U Engineering course. It covers mechanics, materials, thermodynamics, fluid mechanics, electronics and digital systems, allowing you to rapidly review the key equations required for both examinations and practical applications. Each formula is presented with brief explanatory notes to reinforce understanding.
本速查手册为 Cambridge Pre-U 工程课程提供了精选的基本公式、定义与定理的集合。内容涵盖力学、材料、热力学、流体力学、电子学与数字系统，帮助您快速复习考试和实际应用所需的关键方程。每个公式都配有简要说明以加深理解。
1. SI Units and Prefixes | 国际单位制与词头
The International System of Units (SI) forms the foundation of all engineering measurements. The base units are: metre (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for temperature, mole (mol) for amount of substance, and candela (cd) for luminous intensity. Common prefixes include giga (G, 109), mega (M, 106), kilo (k, 103), centi (c, 10−2), milli (m, 10−3), micro (μ, 10−6) and nano (n, 10−9).
国际单位制 (SI) 是所有工程测量的基础。基本单位包括：米 (m) 表示长度，千克 (kg) 表示质量，秒 (s) 表示时间，安培 (A) 表示电流，开尔文 (K) 表示温度，摩尔 (mol) 表示物质的量，坎德拉 (cd) 表示发光强度。常用词头有：吉咖 (G, 109)、兆 (M, 106)、千 (k, 103)、厘 (c, 10−2)、毫 (m, 10−3)、微 (μ, 10−6) 和纳诺 (n, 10−9)。
2. Static Equilibrium | 静力平衡
A body is in static equilibrium when the resultant force and the resultant moment about any point are both zero. This is expressed by the two vector equations. For coplanar forces, we often resolve into perpendicular directions.
当物体所受的合外力和对任一点的合力矩均为零时，该物体处于静力平衡状态。这由两个矢量方程表达。对于共面力系，通常分解到两个垂直方向进行分析。
ΣF = 0 and ΣM = 0
ΣFx = 0, ΣFy = 0, ΣMO = 0
ΣF = 0 表示合力为零，ΣM = 0 表示合力矩为零。在平面问题中，通常列出水平方向合力 ΣFx = 0，垂直方向合力 ΣFy = 0，以及对某点 O 的力矩 ΣMO = 0。
3. Stress, Strain and Elasticity | 应力、应变与弹性
Direct normal stress is the force applied perpendicular to a surface divided by the cross-sectional area. Linear strain is the ratio of change in length to original length. For many materials within the elastic limit, stress is proportional to strain (Hooke’s law).
正应力是垂直于表面的力除以横截面积。线应变是长度变化量与原始长度的比值。对许多材料而言，在弹性极限内应力与应变成正比（胡克定律）。
σ = F / A
ε = ΔL / L0
E = σ / ε (Young’s modulus)
σ = F / A 为正应力，ε = ΔL / L0 为纵向应变。杨氏模量 E 定义为 σ / ε，表征材料刚度。泊松比 ν 描述横向与纵向应变的关系：ν = −εlateral / εaxial。剪切应力 τ = Fshear / A 与剪切应变 γ 通过剪切模量 G = τ / γ 关联。
4. Linear Kinematics | 直线运动学
For constant linear acceleration, the relationships between displacement s, initial velocity u, final velocity v, acceleration a and time t are given by the SUVAT equations. These assume motion in a straight line.
对于匀加速直线运动，位移 s、初速度 u、末速度 v、加速度 a 和时间 t 之间的关系由 SUVAT 方程给出。这些方程适用于直线运动。
v = u + a t
s = u t + ½ a t2
v2 = u2 + 2 a s
s = ½ (u + v) t
v = u + a t 给出末速度；s = u t + ½ a t2 给出位移；v2 = u2 + 2 a s 消去时间；s = ½ (u + v) t 用平均速度求位移。在自由下落运动中，a 等于重力加速度 g (约 9.81 m s−2)。抛体运动可分解为水平匀速和垂直匀加速独立处理。
5. Newton’s Laws and Dynamics | 牛顿定律与动力学
Newton’s second law states that the net force acting on a body equals the rate of change of its momentum. For constant mass, this reduces to the familiar F = m a. Weight is the gravitational force on a mass.
牛顿第二定律指出，作用在物体上的净力等于其动量的变化率。对于质量不变的物体，这简化为常见的 F = m a。重量是质量在重力场中受到的力。
F = m a
W = m g
p = m v (momentum)
Ffriction ≤ μ N
F = m a 用于平动动力学分析。动量 p = m v 是矢量。冲量-动量定理：F Δt = Δp。摩擦力 Ffriction 的最大值为 μ N，其中 μ 是摩擦系数，N 为正压力，静摩擦系数 μs 通常大于动摩擦系数 μk。
6. Work, Energy and Power | 功、能与功率
Work is done when a force moves its point of application along a displacement. Energy is the capacity to do work, and power is the rate of doing work. The principle of conservation of mechanical energy often applies when non-conservative forces do no net work.
当力使其作用点沿位移方向移动时做功。能量是做功的本领，功率是做功的速率。当非保守力不做净功时，机械能守恒原理通常适用。
W = F d cos θ
Ek = ½ m v2
Ep = m g h (gravitational)
Ep = ½ k x2 (elastic)
P = W / t = F v
W = F d cos θ 计算恒力做功，θ 为力与位移的夹角。动能 Ek = ½ m v2，重力势能 Ep = m g h，弹性势能 Ep = ½ k x2。功率 P 是做功率，运动物体瞬时功率可表示为 P = F v。效率 η = (有用输出功/输入能量) × 100%.
7. Torque and Rotational Motion | 扭矩与转动
Torque is the rotational analogue of force and is defined as the product of force and the perpendicular distance from the pivot. Moment of inertia I quantifies an object’s resistance to angular acceleration.
扭矩是力的转动类比，定义为力与力臂的乘积。转动惯量 I 量化物体对角加速度的抵抗能力。
τ = r F sin θ
Στ = I α
I = Σ mi ri2
Ek,rot = ½ I ω2
L = I ω (angular momentum)
τ = r F sin θ 其中 r 是力臂长度。牛顿第二定律的转动形式为 Στ = I α，α 是角加速度。转动惯量 I 取决于质量分布，常见形状：圆盘 I = ½ M R2，细杆绕端点 I = ⅓ M L2。转动动能 ½ I ω2 与角动量 L = I ω 在无外力矩时守恒。
8. Bending of Beams | 梁的弯曲
When a beam is subjected to bending moments, longitudinal stresses are induced. The simple bending theory relates stress to the applied moment through the section’s second moment of area I.
当梁承受弯矩时，会产生纵向应力。简单弯曲理论通过截面二次矩 I 将应力与作用弯矩联系起来。
σ / y = M / I = E / R
Irect = b h3 / 12 (about centroidal axis)
Icircle = π d4 / 64
弯曲公式中 σ 为距中性轴 y 处的弯曲应力，M 为弯矩，I 为截面二次矩，E 为杨氏模量，R 为曲率半径。对于矩形截面 (宽 b，高 h)，I = b h3/12；实心圆截面 I = π d4/64。最大应力出现在 y 最大处，设计时需确保 σmax 小于许用应力。
9. Fluid Mechanics | 流体力学
Fluid statics deals with pressure variation in a stationary fluid. Fluid dynamics involves conservation of mass (continuity) and energy (Bernoulli’s equation) along a streamline for steady, incompressible, inviscid flow.
流体静力学研究静止流体中的压力变化。流体动力学则涉及沿流线的质量守恒（连续性方程）和能量守恒（伯努利方程），适用于稳定、不可压缩、无粘性流动。
p = ρ g h (hydrostatic pressure)
FB = ρfluid g Vdisplaced (buoyancy)
A1 v1 = A2 v2 (continuity)
p + ½ ρ v2 + ρ g h = constant (Bernoulli)
静水压力 p = ρ g h，其中 ρ 为流体密度，h 为液面下的深度。阿基米德浮力等于排开流体的重量。连续性方程表明体积流量 A v 为常数。伯努利方程中三项分别为静压、动压和位压，适用于无摩擦稳态流动。实际有粘性时需引入水头损失。
10. Thermodynamics | 热力学
The First Law of Thermodynamics is a statement of energy conservation for a system, relating internal energy change, heat transfer and work. Ideal gas behaviour is modelled by the equation of state.
热力学第一定律是系统能量守恒的表述，将内能变化、热量传递和做功联系起来。理想气体行为由状态方程建模。
ΔU = Q − W (First Law, engineering sign convention)
P V = n R T (ideal gas law)
Q = m c ΔT (sensible heat)
ηthermal = Wnet / Qin
工程惯例中，系统对外做功 W 为正，吸热 Q 为正，ΔU = Q − W。理想气体定律 P V = n R T，R = 8.31 J mol−1 K−1。比热容公式 Q = m c ΔT，c 为比热。热机效率 η = 净输出功 / 吸热量。热传导：Q/t = k A ΔT / L，k 为导热系数。
11. DC Circuits | 直流电路
Ohm’s law and Kirchhoff’s rules are fundamental to the analysis of direct current circuits. The total resistance of series and parallel combinations and the power dissipated by components must be mastered.
欧姆定律和基尔霍夫定则是直流电路分析的基础。必须掌握串联和并联组合的总电阻以及元件耗散的功率。
V = I R (Ohm’s law)
P = I V = I2 R = V2 / R
Rseries = R1 + R2 + …
1 / Rparallel = 1 / R1 + 1 / R2 + …
欧姆定律 V = I R 定义电阻。功率可表示为多种形式。基尔霍夫电流定律 (KCL)：ΣI进入 = ΣI离开。基尔霍夫电压定律 (KVL)：闭合回路中各段电压代数和为零。分压公式：VR2 = Vtotal × R2 / (R1 + R2)。分流公式：IR2 = Itotal × R1 / (R1 + R2)。
12. Digital Logic | 数字逻辑
Digital systems rely on Boolean algebra and logic gates. The basic gates are AND, OR, NOT, NAND, NOR, XOR and XNOR. De Morgan’s theorems allow conversion between AND/OR forms.
数字系统依赖布尔代数和逻辑门。基本门包括与、或、非、与非、或非、异或和同或门。德摩根定理允许在与/或形式之间进行转换。
A + A = A, A ⋅ A = A (idempotent)
A + 0 = A, A ⋅ 1 = A
A + A’ = 1, A ⋅ A’ = 0
(A + B)’ = A’ ⋅ B’, (A ⋅ B)’ = A’ + B’ (De Morgan)
布尔代数中 ‘+’ 表示 OR，’⋅’ 表示 AND，A’ 表示 NOT A。德摩根定理常用于化简逻辑表达式。基本门真值表：AND 输出为 1 当所有输入为 1；OR 输出为 1 当任一输入为 1；NAND 和 NOR 分别是 AND 和 OR 的反相输出。
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