📚 AS WJEC Engineering: Formula & Theorem Quick Reference Handbook | AS WJEC 工程：公式定理速查手册
This quick-reference handbook summarises all the essential formulas and theorems needed for the WJEC AS Engineering specification. It covers mechanics, materials, thermodynamics, fluid mechanics, electrical principles, electronics, and engineering mathematics, providing a concise revision aid for students who need to select, rearrange and apply equations accurately under exam conditions.
本速查手册汇总了 WJEC AS 工程课程所需的所有核心公式和定理，涵盖力学、材料、热力学、流体力学、电学原理、电子学和工程数学等内容，为学生提供简洁的复习参考。手册的目标不仅是"记住公式"，更是在考试中快速判断该用哪一条、如何变形、单位是否统一。
Each section below lists the formula, states what every symbol means, and explains the condition under which the relationship is valid. Use the section headings as a checklist: if you can explain the meaning and the limits of each equation without looking, you are ready for the calculation questions.
下文每一节都给出公式本身、各符号含义，以及该关系式成立的条件。请把小节标题当作自测清单：如果你能不看笔记就说出每条公式的含义与适用前提，那么计算题部分就已经准备到位。
1. Mechanics and Kinematics | 力学与运动学
v = u + a t
Relates final velocity v, initial velocity u, constant acceleration a, and time t. It is valid only when acceleration is uniform and motion is along a straight line.
联系末速度 v、初速度 u、恒定加速度 a 和时间 t。仅当加速度恒定且物体沿直线运动时成立。
s = u t + ½ a t²
Gives displacement s in terms of initial velocity, acceleration, and time. The sign of s tells you the direction of displacement relative to the chosen positive axis.
利用初速度、加速度和时间求出位移 s。s 的正负号表示位移相对于所选正方向的方向。
v² = u² + 2 a s
Links final velocity, initial velocity, acceleration, and displacement, without involving time. Use it whenever a question mentions distance but gives no time data.
将末速度、初速度、加速度和位移联系起来，不涉及时间。当题目给出距离却没有给出时间时优先使用。
s = ½ (u + v) t
Displacement equals average velocity multiplied by time, which is only true for uniform acceleration because the velocity changes linearly.
位移等于平均速度乘以时间。仅在匀加速条件下成立，因为此时速度随时间线性变化。
Projectile: vx = u cos θ, vy = u sin θ − g t
orizontal velocity stays constant; vertical velocity changes under gravity g = 9.81 m s⁻².
以初速度 u 和角度 θ 抛出的物体，其水平和垂直速度分量。水平速度保持不变，垂直速度在重力 g = 9.81 m s⁻² 作用下变化。
x = u cos θ · t, y = u sin θ · t − ½ g t²
Displacement components in projectile motion under uniform gravity g. At the highest point vy = 0, which is the standard shortcut for finding the time to the apex.
在匀重力 g 作用下的抛体位移分量。在最高点处 vy = 0，这是求到达最高点所需时间的标准捷径。
2. Newton's Laws and Forces | 牛顿定律与力
F = m a
Newton's second law: the resultant force F on a body equals its mass m multiplied by its acceleration a. F must be the vector sum of all forces, not a single applied force.
牛顿第二定律：物体所受合力 F 等于其质量 m 乘以加速度 a。这里的 F 必须是所有力的矢量和，而不是某一个施加的力。
W = m g
Weight is the gravitational force acting on a mass, where g is the local gravitational field strength, approximately 9.81 N kg⁻¹ at the Earth's surface.
重力是作用在质量上的引力，其中 g 为当地重力场强度，地球表面约为 9.81 N kg⁻¹。
F = μ N
Limiting friction equals the coefficient of friction μ multiplied by the normal reaction N. Static friction is a variable up to this maximum value, while kinetic friction is roughly constant.
极限摩擦力等于摩擦因数 μ 乘以法向反力 N。静摩擦在达到该最大值之前是可变的，而动摩擦基本保持恒定。
Inclined plane: W sin θ (along slope), W cos θ (normal)
On a smooth slope of angle θ, the component of weight pulling the object down the slope is W sin θ and the normal reaction is W cos θ.
在倾角为 θ 的光滑斜面上，使物体下滑的重力分量为 W sin θ，法向反力为 W cos θ。
Newton's first law states that a body remains at rest or in uniform motion unless acted on by a resultant force; Newton's third law states that action and reaction are equal, opposite and act on different bodies.
牛顿第一定律指出：物体在不受合力作用时保持静止或匀速直线运动；牛顿第三定律指出：作用力与反作用力大小相等、方向相反，且作用在不同物体上。
3. Work, Energy, Power and Momentum | 功、能、功率与动量
W = F s cos θ
Work done equals force times displacement times the cosine of the angle between them. If the force is perpendicular to the displacement, no work is done.
功等于力乘以位移再乘以两者夹角的余弦。若力与位移垂直，则做功为零。
Ek = ½ m v², Ep = m g h
Kinetic energy depends on the square of speed, so doubling speed quadruples kinetic energy. Gravitational potential energy is measured relative to a stated datum level.
动能与速度的平方成正比，因此速度加倍则动能变为四倍。重力势能需相对于指定的参考基准面来计量。
P = W ÷ t = F v
Power is the rate of doing work or transferring energy. The form F v applies when the force and velocity are in the same direction.
功率是做功或传递能量的速率。F v 形式适用于力与速度方向相同的情形。
Efficiency = (useful output ÷ total input) × 100%
Efficiency is always less than 100% in real machines because energy is dissipated, mainly as heat through friction.
由于能量会耗散（主要通过摩擦转化为热），实际机械的效率永远低于 100%。
p = m v, F t = Δp
Momentum is mass times velocity; impulse is force times time and equals the change in momentum. This is why crumple zones and seat belts reduce the force in a collision by extending the collision time.
动量等于质量乘以速度；冲量等于力乘以时间，并等于动量的变化量。这正是溃缩区和安全带通过延长碰撞时间来减小碰撞力的原因。
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
The principle of conservation of momentum applies to any collision or explosion in a closed system where no external resultant force acts.
动量守恒原理适用于封闭系统中无外力合力的任何碰撞或爆炸过程。
4. Materials and Stress Analysis | 材料与应力分析
σ = F ÷ A, ε = ΔL ÷ L₀
Direct stress is the applied force divided by the original cross-sectional area, and direct strain is the extension divided by the original length. Strain has no units.
正应力等于施加的力除以原始横截面积，正应变等于伸长量除以原始长度。应变是无量纲量。
E = σ ÷ ε
Young's modulus is the gradient of the straight-line region of a stress-strain graph. It is a measure of stiffness and applies only up to the limit of proportionality.
杨氏模量是应力—应变图中直线段的斜率，用来衡量材料的刚度，且只在比例极限以内适用。
Strain energy = ½ F ΔL
The energy stored elastically in a stretched specimen equals half the average force multiplied by the extension, equal to the area under a force-extension graph within the elastic region.
拉伸试样中储存的弹性应变能等于平均力乘以伸长量的一半，也等于弹性范围内力—伸长曲线下的面积。
Factor of safety = failure stress ÷ working stress
A design factor of safety greater than one allows for uncertainty in loading, material defects, fatigue and environmental degradation.
设计安全系数大于 1，用以考虑载荷不确定性、材料缺陷、疲劳以及环境老化等因素。
ΔL = α L₀ ΔT, ρ = m ÷ V
Linear thermal expansion depends on the coefficient of linear expansion α, and density is mass per unit volume. Both are frequently combined with stress formulas in thermal stress problems.
线热膨胀取决于线膨胀系数 α，密度是单位体积的质量。在热应力问题中两者常与应力公式配合使用。
5. Thermodynamics and Heat Transfer | 热力学与传热
Q = m c ΔT
Sensible heat transfer equals mass times specific heat capacity times temperature change. No phase change occurs during this process.
显热传递等于质量乘以比热容再乘以温度变化。此过程中不发生相变。
Q = m L
Latent heat is the energy absorbed or released during a change of state at constant temperature, where L is the specific latent heat of fusion or vaporisation.
潜热是物态变化过程中在温度不变时吸收或释放的能量，其中 L 为比熔化潜热或比汽化潜热。
Thermal efficiency = (useful work output ÷ heat energy input) × 100%
For any heat engine, the difference between heat input and work output is rejected to a cold sink and cannot be fully recovered.
对任何热机而言，输入热量与输出功之间的差值会被排放到低温热汇，无法完全回收。
Conduction: Q ÷ t = k A ΔT ÷ L
The rate of heat conduction through a solid depends on thermal conductivity k, cross-sectional area A, temperature difference ΔT and thickness L. Doubling thickness halves the rate.
固体导热速率取决于导热系数 k、横截面积 A、温差 ΔT 和厚度 L。厚度加倍则导热速率减半。
p V = n R T, p₁V₁ ÷ T₁ = p₂V₂ ÷ T₂
The ideal gas equation uses absolute temperature in kelvin; the combined gas law applies to a fixed mass of gas between two states.
理想气体方程使用开尔文绝对温度；联合气体定律适用于一定质量气体在两个状态之间的变化。
6. Fluid Mechanics | 流体力学
p = F ÷ A, p = ρ g h
Pressure is force per unit area, and hydrostatic pressure increases linearly with depth h in a fluid of density ρ. It acts equally in all directions.
压强是单位面积上的力，静水压强在密度为 ρ 的流体中随深度 h 线性增大，且向各个方向大小相等。
Flow rate: Q = A v
Volume flow rate equals cross-sectional area times mean flow velocity. This is the basis of the continuity equation.
体积流量等于横截面积乘以平均流速，这是连续性方程的基础。
Continuity: A₁v₁ = A₂v₂
For an incompressible fluid in a closed pipe, the mass entering per second equals the mass leaving, so a narrower section gives a higher velocity.
对封闭管道中的不可压缩流体，每秒流入的质量等于流出的质量，因此截面变窄处流速升高。
Bernoulli: p + ½ ρ v² + ρ g h = constant
Along a streamline in steady, non-viscous, incompressible flow, the sum of pressure energy, kinetic energy and potential energy per unit volume is constant.
在定常、无粘、不可压缩流动的同一流线上，单位体积的压能、动能与势能之和保持不变。
Upthrust = ρ g V
Archimedes' principle: the upthrust on a submerged body equals the weight of fluid displaced, where V is the submerged volume.
阿基米德原理：浸没物体所受浮力等于所排开流体的重力，其中 V 为浸没体积。
7. Electrical Principles | 电学原理
V = I R
Ohm's law applies to a conductor at constant temperature where resistance R is constant. It does not apply to non-ohmic devices such as filament lamps.
欧姆定律适用于温度不变、电阻 R 恒定的导体，不适用于灯丝等非欧姆器件。
P = V I = I² R = V² ÷ R
Electrical power can be expressed in three equivalent forms. Use I² R when current is known and V² ÷ R when voltage is known.
电功率有三种等价表达形式。已知电流时用 I² R，已知电压时用 V² ÷ R。
Series: R = R₁ + R₂; Parallel: 1 ÷ R = 1 ÷ R₁ + 1 ÷ R₂
Resistors in series share the same current; resistors in parallel share the same voltage and the combined resistance is always smaller than the smallest individual resistance.
串联电阻电流相同；并联电阻电压相同，且总电阻始终小于其中最小的单个电阻。
V = E − I r, EMF: E = I (R + r)
e of the internal resistance r of the source. Short-circuit current is maximum when R = 0.
由于电源存在内阻 r，端电压低于电动势。当 R = 0 时短路电流达到最大。
Kirchhoff: ΣI = 0 at a junction; ΣV = 0 around a loop
Kirchhoff's current law is a statement of conservation of charge; his voltage law is a statement of conservation of energy.
基尔霍夫电流定律是电荷守恒的表述；电压定律是能量守恒的表述。
Q = C V, E = ½ C V², τ = R C
tor is proportional to voltage; the time constant τ = R C gives the time to reach about 63% of the final voltage.
电容器储存的电荷与电压成正比；时间常数 τ = R C 表示达到最终电压约 63% 所需的时间。
Φ = B A, EMF = −N ΔΦ ÷ Δt, Vrms = Vpeak ÷ √2
Faraday's law states that a changing magnetic flux induces an EMF; Lenz's law explains the negative sign as opposition to the change producing it.
法拉第定律指出磁通量变化会产生感应电动势；楞次定律解释了负号表示感应效应反抗产生它的变化。
8. Electronics | 电子学
Potential divider: Vout = Vin × R₂ ÷ (R₁ + R₂)
A potential divider produces an output voltage proportional to the ratio of resistances. It is the basis of sensor circuits using thermistors and LDRs.
分压器产生的输出电压与电阻之比成正比，是使用热敏电阻和光敏电阻的传感器电路的基础。
Op-amp gain = −Rf ÷ Rin
For an inverting amplifier with negative feedback, the gain depends only on the two resistors, not on the open-loop gain of the op-amp. The negative sign indicates phase inversion.
带负反馈的反相放大器，其增益只取决于两个电阻，而与运放的开环增益无关。负号表示相位反转。
Transistor gain: hFE = Ic ÷ Ib
Current gain relates collector current to base current. In saturation the transistor behaves as a closed switch; in cut-off it behaves as an open switch.
电流增益表示集电极电流与基极电流之比。饱和状态下晶体管相当于闭合开关，截止状态下相当于断开开关。
Voltage gain (dB) = 20 log₁₀ (Vout ÷ Vin)
Decibel notation compresses a wide range of gains into a manageable scale. A gain of 10 corresponds to 20 dB, and a halving of output corresponds to about −6 dB.
分贝表示法把很宽的增益范围压缩到便于使用的刻度上。增益为 10 时对应 20 dB，输出减半约对应 −6 dB。
Logic gates: NOT inverts, AND requires all inputs high, OR requires any input high, NAND and NOR are the inverted forms of AND and OR, and XOR is high only when inputs differ.
逻辑门：非门取反，与门要求所有输入为高，或门要求任一输入为高，与非门和或非门分别是与门和或门的取反形式，异或门仅在输入不同时为高。
9. Engineering Mathematics | 工程数学
x = (−b ± √b² − 4 a c) ÷ (2 a)
criminant b² − 4 a c tells you whether the roots are real and distinct, repeated, or complex.
二次公式用于求解 a x² + b x + c = 0 形式的方程。判别式 b² − 4 a c 决定根是两实根、重根还是复根。
Pythagoras: c² = a² + b²; sin θ = opp ÷ hyp, cos θ = adj ÷ hyp, tan θ = opp ÷ adj
Trigonometry underpins the resolution of forces, vector addition and the analysis of alternating quantities. Always confirm whether your calculator is in degrees or radians.
三角学是力的分解、矢量合成以及交流量分析的基础。务必确认计算器处于角度制还是弧度制。
Sine rule: a ÷ sin A = b ÷ sin B; Cosine rule: a² = b² + c² − 2 b c cos A
Use the sine rule when you know a matching side-angle pair; use the cosine rule for two sides and the included angle, or for all three sides.
已知一组对应边角时用正弦定理；已知两边及夹角，或已知三边时用余弦定理。
Area and volume: A = π r², V = π r² h, V = 4/3 π r³
Standard mensuration formulas appear in material costing, mass estimation and flow calculations, so keep units consistent throughout.
标准求积公式常见于材料成本、质量估算与流量计算中，使用时要保证全程单位一致。
og (aⁿ) = n log a. These are essential for decibel, exponential growth and decay problems.
对数运算法则：log (a b) = log a + log b，log (a ÷ b) = log a − log b，log (aⁿ) = n log a。这些法则在分贝、指数增长与衰减问题中必不可少。
Gradient = Δy ÷ Δx; area under curve ≈ ½ (y₁ + y₂) Δx
Differentiation gives rates of change such as velocity from displacement; integration gives accumulated quantities such as work from a force-displacement graph.
微分给出位移求速度等变化率；积分给出由力—位移图求功等累积量。
10. Statics, Moments and Structures | 静力学、力矩与结构
M = F d, T = F r
The moment of a force about a point equals the force multiplied by the perpendicular distance from the pivot. Torque on a shaft uses the radius r as the perpendicular distance.
力对某点的力矩等于力乘以到支点的垂直距离。轴上的转矩则以半径 r 作为垂直距离。
Equilibrium: ΣF = 0 and ΣM = 0
A body in equilibrium has no resultant force and no resultant moment. Taking moments about a point where an unknown force acts eliminates that unknown from the equation.
处于平衡状态的物体既无合力也无合力矩。对某个未知力作用点取矩，可在方程中消去该未知量。
Couple: M = F × d
A couple consists of two equal, opposite and parallel forces separated by a perpendicular distance d. It produces rotation without any resultant translational force.
力偶由两个大小相等、方向相反且平行的力组成，二者相隔垂直距离 d，只产生转动效应而无平移合力。
For simply supported beams, calculate the reactions first using moments, then draw the shear force diagram, and finally the bending moment diagram. Maximum bending moment usually occurs where shear force is zero.
对简支梁，先用取矩法求支座反力，再画剪力图，最后画弯矩图。最大弯矩通常出现在剪力为零的位置。
11. Units
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