📚 A-Level AQA Engineering: Formula & Theorem Quick Reference Handbook | A-Level AQA 工程：公式定理速查手册
This quick reference handbook compiles all the essential formulas, laws and theorems required for the AQA A-Level Engineering examination. Whether you are revising forces and moments, analysing electrical circuits, or solving thermodynamics problems, this guide helps you locate key equations and understand their applications through paired English and Chinese explanations. Use it alongside your notes and past papers to strengthen your command of engineering principles.
本速查手册汇总了 AQA A-Level 工程考试所需的所有关键公式、定律与定理。无论你是在复习力与力矩、分析电路，还是求解热力学问题，这份指南都能帮助你快速定位核心方程，并通过中英对照的解释加深理解。请结合笔记和历年真题使用，以巩固对工程原理的掌握。
1. Mechanics: Forces and Moments | 力学：力与力矩
A force is a vector quantity measured in newtons (N). When multiple forces act on a body, the resultant force can be found by vector addition. A moment is the turning effect of a force about a pivot, calculated as the product of the force and the perpendicular distance from the pivot to the line of action of the force.
力是一个矢量，单位为牛顿 (N)。当多个力作用于一个物体时，合力可通过矢量加法求得。力矩是力对于支点的转动效应，等于力与支点到力作用线的垂直距离的乘积。
M = F × d
where M = moment (N m), F = force (N), d = perpendicular distance from pivot (m).
其中 M = 力矩 (N m)，F = 力 (N)，d = 支点到力作用线的垂直距离 (m)。
For a body in static equilibrium, the sum of clockwise moments about any pivot equals the sum of anticlockwise moments about that same pivot. This is known as the principle of moments.
对于处于静力平衡的物体，对任意支点，所有顺时针力矩之和等于所有逆时针力矩之和。这就是力矩原理。
ΣMclockwise = ΣManticlockwise
In vector addition, forces can be resolved into horizontal and vertical components. For a force F at an angle θ to the horizontal:
在矢量加法中，力可分解为水平和垂直分量。对于一个与水平方向成 θ 角的力 F：
Fx = F cos θ, Fy = F sin θ
Couples consist of two equal and opposite parallel forces whose lines of action do not coincide. The moment of a couple is F × d, where d is the perpendicular distance between the forces. This creates pure rotation without resultant linear force.
力偶由两个大小相等、方向相反且作用线不重合的平行力组成。力偶的力矩为 F × d，其中 d 是两力之间的垂直距离。力偶产生无合力的纯转动效应。
2. Mechanical Energy and Power | 机械能与功率
Work done by a constant force is the product of the force and the distance moved in the direction of the force. Energy is the capacity to do work, and both are measured in joules (J).
恒力所做的功等于力与在力方向上移动的距离的乘积。能量是做功的能力，功和能量的单位都是焦耳 (J)。
W = F × s or W = F s cos θ
where F is the force, s is the displacement, and θ is the angle between the force and displacement vectors.
其中 F 是力，s 是位移，θ 是力与位移矢量之间的夹角。
Gravitational potential energy (GPE) and kinetic energy (KE) are two fundamental mechanical energy forms.
重力势能 (GPE) 和动能 (KE) 是两种基本的机械能形式。
GPE = m g h
KE = ½ m v²
where m = mass (kg), g = acceleration of free fall (9.81 m s⁻²), h = height (m), v = velocity (m s⁻¹).
其中 m = 质量 (kg)，g = 自由落体加速度 (9.81 m s⁻²)，h = 高度 (m)，v = 速度 (m s⁻¹)。
Power is the rate of doing work or transferring energy. The efficiency of a system is the ratio of useful output power to total input power.
功率是做功或转换能量的速率。系统的效率是有用输出功率与总输入功率之比。
P = W / t or P = F v
η = (useful power output / total power input) × 100%
The principle of conservation of energy states that energy can neither be created nor destroyed, only transferred or converted from one form to another.
能量守恒定律表明，能量既不会凭空产生也不会凭空消失，只能从一种形式转移或转化为另一种形式。
3. Kinematics: Linear and Angular Motion | 运动学：线性和角运动
Linear motion with constant acceleration is described by the SUVAT equations. These link displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t).
匀加速直线运动由 SUVAT 方程描述。这些方程联系了位移 (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
v² = u² + 2 a s
s = ½ (u + v) t
For angular motion, analogous quantities are angular displacement (θ), angular velocity (ω), angular acceleration (α) and time (t). The equations have the same form, with angular quantities replacing linear ones.
对于角运动，类似量有角位移 (θ)、角速度 (ω)、角加速度 (α) 和时间 (t)。方程形式相同，只需用角量替换线量即可。
ω = ω₀ + α t, θ = ω₀ t + ½ α t², ω² = ω₀² + 2 α θ
The relationship between linear and angular velocity for a point at radius r is v = r ω. Similarly, tangential acceleration a = r α.
半径为 r 的点，其线速度与角速度的关系为 v = r ω。类似地，切向加速度 a = r α。
4. Dynamics and Newton’s Laws | 动力学与牛顿定律
Newton’s three laws of motion are the foundation of dynamics. The first law defines inertia: a body remains at rest or in uniform motion unless acted upon by a resultant force. The second law quantifies the relationship between force, mass and acceleration. The third law states that every action has an equal and opposite reaction.
牛顿运动三定律是动力学的基础。第一定律定义了惯性：物体将保持静止或匀速直线运动状态，除非有合力作用于它。第二定律量化了力、质量与加速度的关系。第三定律指出，每个作用力都有一个大小相等、方向相反的反作用力。
F = m a
where F = resultant force (N), m = mass (kg), a = acceleration (m s⁻²).
其中 F = 合力 (N)，m = 质量 (kg)，a = 加速度 (m s⁻²)。
Momentum is the product of mass and velocity. The principle of conservation of momentum is crucial for analysing collisions and explosions.
动量是质量与速度的乘积。动量守恒定律在分析碰撞和爆炸时至关重要。
p = m v
Σ pbefore = Σ pafter
Impulse is the change in momentum, and it equals the average force multiplied by the time for which it acts.
冲量是动量的变化量，等于平均力乘以力作用的时间。
Impulse = F Δ t = Δ p = m v – m u
Frictional forces oppose relative motion. The maximum static friction is proportional to the normal reaction force, Ff ≤ μ R, where μ is the coefficient of friction.
摩擦力阻碍相对运动。最大静摩擦力与法向反力成正比，Ff ≤ μ R，其中 μ 是摩擦系数。
5. Stress, Strain and Young’s Modulus | 应力、应变与杨氏模量
Materials respond to applied forces with deformation. Stress is the internal force per unit cross-sectional area, while strain is the ratio of extension to original length.
材料在外力作用下会发生变形。应力是单位横截面积上的内力，应变则是伸长量与原长之比。
Stress σ = F / A
Strain ε = ΔL / L
where F = applied force (N), A = cross-sectional area (m²), ΔL = change in length (m), L = original length (m).
其中 F = 施加的力 (N)，A = 横截面积 (m²)，ΔL = 长度变化量 (m)，L = 原始长度 (m)。
Young’s modulus (E) measures the stiffness of a material in the elastic region. It is the gradient of the stress-strain graph up to the limit of proportionality.
杨氏模量 (E) 衡量材料在弹性范围内的刚度。它是应力-应变图在比例极限内的斜率。
E = σ / ε = (F L) / (A ΔL)
Units are pascals (Pa) or N m⁻². Toughness is the energy absorbed up to fracture, indicated by the area under the stress-strain curve. Factor of safety = ultimate tensile stress / allowable working stress.
单位是帕斯卡 (Pa) 或 N m⁻²。韧性是材料断裂前吸收的能量，由应力-应变曲线下的面积表示。安全系数 = 极限抗拉应力 / 许用工作应力。
6. Fluid Mechanics | 流体力学
Pressure in a fluid at rest is given by the hydrostatic equation. It depends on the fluid density, depth and gravitational field strength. Pressure acts equally in all directions.
静止流体中的压力由流体静力学方程给出。它取决于流体密度、深度和重力场强度。压力在各个方向上均等作用。
p = ρ g h
where p = pressure (Pa), ρ = fluid density (kg m⁻³), g = 9.81 m s⁻², h = height/depth (m).
其中 p = 压力 (Pa)，ρ = 流体密度 (kg m⁻³)，g = 9.81 m s⁻²，h = 高度/深度 (m)。
For incompressible fluids flowing along a pipe, the volume flow rate Q = A v remains constant (continuity equation). A is the cross-sectional area and v is the fluid velocity.
对于沿管道流动的不可压缩流体，体积流量 Q = A v 保持恒定（连续性方程）。A 为横截面积，v 为流体速度。
A₁ v₁ = A₂ v₂
Bernoulli’s principle relates pressure, kinetic energy per unit volume and potential energy per unit volume for steady, inviscid, incompressible flow along a streamline.
伯努利原理联系了沿流线的稳定、无黏、不可压缩流动中的压力、单位体积动能和单位体积势能。
p + ½ ρ v² + ρ g h = constant
Hydraulic systems use Pascal’s principle: pressure applied to an enclosed fluid is transmitted undiminished to all parts. Force multiplication is based on area ratios: F₁/A₁ = F₂/A₂.
液压系统利用帕斯卡原理：施加在密闭流体上的压力会无衰减地传递到各处。力的放大基于面积比：F₁/A₁ = F₂/A₂。
7. Thermodynamics and Heat Transfer | 热力学与热传递
Temperature is a measure of the average kinetic energy of particles. Heat is energy transferred due to a temperature difference. The specific heat capacity quantifies the energy required to raise the temperature of 1 kg of a substance by 1 K.
温度是粒子平均动能的量度。热量是由于温差而传递的能量。比热容量表示将 1 kg 物质升高 1 K 所需的能量。
Q = m c Δθ
where Q = heat energy (J), m = mass (kg), c = specific heat capacity (J kg⁻¹ K⁻¹), Δθ = temperature change (K or °C).
其中 Q = 热能 (J)，m = 质量 (kg)，c = 比热容量 (J kg⁻¹ K⁻¹)，Δθ = 温度变化 (K 或 °C)。
The ideal gas law connects pressure, volume, temperature and the amount of gas. It is often used when analysing thermodynamic cycles in engines.
理想气体定律将压力、体积、温度和气体量联系起来。在分析发动机热力学循环时经常使用。
p V = n R T
where p = pressure (Pa), V = volume (m³), n = number of moles, R = 8.31 J mol⁻¹ K⁻¹, T = absolute temperature (K).
其中 p = 压力 (Pa)，V = 体积 (m³)，n = 摩尔数，R = 8.31 J mol⁻¹ K⁻¹，T = 绝对温度 (K)。
Heat transfer occurs by conduction, convection and radiation. The rate of conductive heat transfer through a material is given by Fourier’s law in simplified form:
热传递通过传导、对流和辐射发生。通过材料的导热速率可由傅里叶定律的简化形式给出：
P = (k A ΔT) / d
where k = thermal conductivity (W m⁻¹ K⁻¹), A = area, ΔT = temperature difference, d = thickness.
其中 k = 导热系数 (W m⁻¹ K⁻¹)，A = 面积，ΔT = 温差，d = 厚度。
8. Electrical Circuits: Ohm’s Law and Kirchhoff’s Laws | 电路：欧姆定律与基尔霍夫定律
The relationship between voltage, current and resistance in a metallic conductor at constant temperature is given by Ohm’s law. Components that obey this linear relationship are called ohmic.
在恒定温度下，金属导体中电压、电流与电阻之间的关系由欧姆定律给出。服从这种线性关系的元件称为欧姆元件。
V = I R
where V = potential difference (V), I = current (A), R = resistance (Ω).
其中 V = 电势差 (V)，I = 电流 (A)，R = 电阻 (Ω)。
Resistors in series and parallel can be combined using these rules:
串联和并联电阻可按照以下规则合并：
Series: Rtotal = R₁ + R₂ + …
Parallel: 1/Rtotal = 1/R₁ + 1/R₂ + …
Kirchhoff’s current law (KCL) states that the total current entering a junction equals the total current leaving. Kirchhoff’s voltage law (KVL) states that the sum of the emfs around any closed loop equals the sum of the pds across the components.
基尔霍夫电流定律 (KCL) 指出，流入节点的电流总和等于流出节点的电流总和。基尔霍夫电压定律 (KVL) 指出，任一闭合回路中电动势的代数和等于各元件上电势差的代数和。
ΣIin = ΣIout
Σε = Σ I R
Electrical power and energy are essential in engineering applications:
电功率和电能是工程应用中的关键：
P = I V = I² R = V² / R, E = P t
The efficiency of electrical systems can be calculated as output power divided by input power.
电气系统的效率可由输出功率除以输入功率计算得出。
9. Electronics: Logic Gates and Boolean Algebra | 电子学：逻辑门与布尔代数
Digital electronic systems process binary signals (0 and 1) using logic gates. The fundamental gate types and their Boolean expressions are tabulated below.
数字电子系统使用逻辑门处理二进制信号（0 和 1）。基本门类型及其布尔表达式见下表。
| Gate | Boolean Expression | Symbol (standard) |
|---|---|---|
| AND | Y = A · B | & |
| OR | Y = A + B | ≥1 |
| NOT | Y = Ā | 1 |
| NAND | Y = A · B (overbar) | & with circle |
| NOR | Y = A + B (overbar) | ≥1 with circle |
| XOR | Y = A ⊕ B | =1 |
Truth tables define the output for every combination of inputs. Boolean algebra allows simplification of logic circuits. Key identities include:
真值表定义了每种输入组合下的输出。布尔代数可用于简化逻辑电路。重要恒等式包括：
A + A = A, A · A = A, A + 0 = A, A + 1 = 1, A · 0 = 0, A + Ā = 1, A · Ā = 0
De Morgan’s theorems are vital for transforming expressions:
德摩根定理对于表达式变换至关重要：
A · B = A + B (overbar), A + B = A · B (overbar)
Combinational logic circuits combine gates to perform functions like addition, multiplexing and decoding. Sequential logic, such as flip-flops, uses feedback and a clock signal to store state, but is explored further in specific design tasks.
组合逻辑电路将门组合以执行加法、多路复用和解码等功能。时序逻辑（如触发器）利用反馈和时钟信号来存储状态，但将在具体设计任务中进一步探讨。
10. Engineering Mathematics and Units | 工程数学与单位
Engineers regularly use trigonometry, vectors and calculus. Sine, cosine and tangent rules help solve non‑right‑angled triangles. In any triangle ABC with sides a, b, c opposite angles A, B, C:
工程师经常使用三角学、矢量与微积分。正弦定理、余弦定理和正切定理有助于求解非直角三角形。对于任意三角形 ABC，边 a、b、c 分别对应角 A、B、C：
Sine rule: a / sin A = b / sin B = c / sin C
Cosine rule: a² = b² + c² – 2 b c cos A
Vectors can be added using the head-to-tail method or by resolving into components. The magnitude and direction of a resultant vector are found from its components.
矢量可通过头尾相接法或分解为分量来相加。合矢量的模和方向可由其分量求得。
R = √(Rx² + Ry²), θ = tan⁻¹(Ry / Rx)
SI base units in engineering are: metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), candela (cd). Derived units such as newton (N = kg m s⁻²), joule (J = N m), watt (W = J s⁻¹), pascal (Pa = N m⁻²) and volt (V = J C⁻¹) must be understood. Prefixes from pico (p, 10⁻¹²) to tera (T, 10¹²) often appear in exam data.
工程中的国际单位制基本单位有：米 (m)、千克 (kg)、秒 (s)、安培 (A)、开尔文 (K)、摩尔 (mol)、坎德拉 (cd)。导出单位如牛顿 (N = kg m s⁻²)、焦耳 (J = N m)、瓦特 (W = J s⁻¹)、帕斯卡 (Pa = N m⁻²) 和伏特 (V = J C⁻¹) 必须掌握。从皮 (p, 10⁻¹²) 到太 (T, 10¹²) 的词头常出现在考试数据中。
Calculus applications include differentiation for rates of change (e.g., velocity as ds/dt, acceleration as dv/dt) and integration for accumulation (e.g., area under a velocity-time graph gives displacement). In simple electric circuits, V = L dI/dt for an inductor and I = C dV/dt for a capacitor.
微积分的应用包括用微分求变化率（如速度 v = ds/dt、加速度 a = dv/dt）以及用积分求累积量（如速度-时间图下的面积给出位移）。在简单电路中，电感 V = L dI/dt，电容 I = C dV/dt。
Dimensional analysis can verify the consistency of equations. For instance, checking that the dimensions of work (M L² T⁻²) match those of force times displacement.
量纲分析可验证方程的一致性。例如，检查功的量纲 (M L² T⁻²) 是否与力乘以位移的量纲匹配。
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