AS CCEA Engineering: Formula & Theorem Quick Reference Guide | AS CCEA 工程：公式定理速查手册

📚 AS CCEA Engineering: Formula & Theorem Quick Reference Guide | AS CCEA 工程：公式定理速查手册

This Quick Reference Guide summarises the essential formulas, theorems and key relationships required for the AS CCEA Engineering specification. It covers mechanics, materials, electrical principles, fluids and thermodynamics. Each entry includes the formula, variable definitions, SI units and a brief contextual explanation, helping you to revise efficiently and apply concepts accurately in problem-solving.

本速查手册汇总了 AS CCEA 工程课程中必备的公式、定理和关键关系，涵盖力学、材料、电学原理、流体和热力学。每一项条目均包含公式、变量定义、SI 单位及简要背景说明，助您高效复习，在解题时准确应用。

1. Linear Motion & Kinematics | 直线运动与运动学

The SUVAT equations describe uniformly accelerated motion along a straight line. They link displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t).

SUVAT 方程描述匀加速直线运动，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

v² = u² + 2 a s

s = ½ (u + v) t

• All quantities are scalar for motion in one dimension. Take direction as positive for sign convention.

  所有量在单维运动中是标量，使用正方向符号约定。

• Free-fall acceleration g = 9.81 m/s² downwards is commonly used.

  自由落体重力加速度 g 通常取 9.81 m/s²，方向向下。

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2. Forces and Newton’s Laws | 力与牛顿定律

Newton’s laws form the foundation of classical mechanics. The second law relates net force, mass and acceleration.

牛顿定律是经典力学的基础。第二定律关联了合外力、质量和加速度。

ΣF = m a

Weight is a gravitational force acting on a mass.

重力是作用在质量上的引力。

W = m g

• ΣF is the vector sum of all forces (N), m is mass (kg), a is acceleration (m/s²).

  ΣF 是所有力的矢量和 (N)，m 是质量 (kg)，a 是加速度 (m/s²)。

• Friction force F_f ≤ μ R, where μ is the coefficient of friction and R is the normal reaction.

  摩擦力 F_f ≤ μ R，μ 为摩擦系数，R 为法向反作用力。

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3. Moments and Equilibrium | 力矩与平衡

The moment of a force about a point measures its turning effect. Equilibrium requires both resultant force and resultant moment to be zero.

力矩衡量力对某点的转动效应。平衡要求合力与合力矩均为零。

M = F × d

where d is the perpendicular distance from the pivot to the line of action of the force.

其中 d 是从支点到力作用线的垂直距离。

ΣM = 0 and ΣF = 0

• Principle of moments: for a body in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any pivot.

  力矩原理：对处于转动平衡的物体，绕任一支点，顺时针力矩之和等于逆时针力矩之和。

• Use these conditions to solve for unknown support reactions in beams and structures.

  利用这些条件求解梁和结构中的未知支座反力。

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4. Work, Energy and Power | 功、能与功率

Mechanical work is done when a force moves its point of application. The work–energy principle links net work to the change in kinetic energy.

当力的作用点发生位移时便做了机械功。功–能原理将净功与动能变化联系起来。

W = F s cosθ

Kinetic energy and gravitational potential energy:

动能与重力势能：

Eₖ = ½ m v²

Eₚ = m g h

Power is the rate of doing work.

功率是做功的速率。

P = W / t = F v

• Efficiency η = (useful output power / input power) × 100%.

  效率 η = (有用输出功率 / 输入功率) × 100%.

• Conservation of energy: total energy in an isolated system remains constant, transferred between forms.

  能量守恒：孤立系统的总能量保持恒定，在不同形式间转化。

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5. Stress and Strain | 应力与应变

These quantities describe how materials respond to applied forces. Stress is a measure of internal load intensity, and strain measures deformation.

这些量描述材料在受力时的响应。应力是内部载荷强度的量度，应变是形变的量度。

σ = F / A

ε = ΔL / L₀

• σ is direct stress (Pa or N/m²), F is force (N), A is cross-sectional area (m²).

  σ 为正应力 (Pa 或 N/m²)，F 为力 (N)，A 为截面积 (m²)。

• ε is strain (dimensionless), ΔL is change in length (m), L₀ is original length (m).

  ε 为应变 (无量纲)，ΔL 为长度变化 (m)，L₀ 为原始长度 (m)。

• Tensile stress and compressive stress follow the same formula; shear stress uses τ = F/A for parallel forces.

  拉应力和压应力使用同一公式；剪应力用 τ = F/A 表示平行力情况。

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6. Young’s Modulus | 杨氏模量

Young’s modulus characterises the stiffness of a material within the linear elastic region of the stress–strain curve.

杨氏模量表征材料在应力–应变曲线线弹性区域内的刚度。

E = σ / ε

It is the gradient of the initial straight-line portion of the curve.

它是曲线初始直线段的斜率。

• E has units of Pa (N/m²). A higher E indicates a stiffer material.

  E 的单位为 Pa (N/m²)。E 值越高表示材料越刚硬。

• The relationship holds only up to the limit of proportionality, beyond which Hooke’s law ceases to apply.

  此关系仅在比例极限以下成立，超出后胡克定律不再适用。

• Factor of safety = ultimate stress / working stress, used in design to avoid failure.

  安全系数 = 极限应力 / 许用应力，用于设计中避免失效。

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7. Ohm’s Law and Resistance | 欧姆定律与电阻

Ohm’s law relates the potential difference across a conductor to the current flowing through it, provided temperature is constant.

欧姆定律给出了导体两端电势差与流过导体的电流之间的关系，前提是温度恒定。

V = I R

The resistance of a uniform wire depends on its resistivity, length and cross-sectional area.

均匀导线的电阻值取决于其电阻率、长度和截面积。

R = ρ L / A

• V is voltage (V), I is current (A), R is resistance (Ω). ρ is resistivity (Ω·m).

  V 为电压 (V)，I 为电流 (A)，R 为电阻 (Ω)。ρ 为电阻率 (Ω·m)。

• Resistance increases with length and decreases with larger cross-sectional area; temperature effects alter ρ.

  电阻随长度增加而增大，随截面积增大而减小；温度变化会影响 ρ。

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8. Series and Parallel Circuits | 串联与并联电路

Resistors can be combined to simplify circuit analysis. The total resistance depends on the configuration.

电阻器可加以组合以简化电路分析，总电阻取决于连接方式。

Series:

串联：

R_total = R₁ + R₂ + R₃ + …

Parallel:

并联：

1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …

• In series, current is the same through all resistors; in parallel, voltage is the same across all branches.

  串联时通过各电阻的电流相同；并联时各支路两端的电压相同。

• For two parallel resistors: R_total = (R₁ × R₂) / (R₁ + R₂).

  两个电阻并联时：R_total = (R₁ × R₂) / (R₁ + R₂)。

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9. Kirchhoff’s Laws | 基尔霍夫定律

Kirchhoff’s current and voltage laws are used to analyse complex circuits where Ohm’s law alone is insufficient.

基尔霍夫电流定律和电压定律用于分析仅靠欧姆定律无法解决的复杂电路。

Kirchhoff’s Current Law (KCL): The sum of currents entering a junction equals the sum leaving.

基尔霍夫电流定律 (KCL)：流入节点的电流之和等于流出之和。

Σ I_in = Σ I_out

Kirchhoff’s Voltage Law (KVL): The algebraic sum of potential differences around any closed loop is zero.

基尔霍夫电压定律 (KVL)：沿任一闭合回路，各段电势差的代数和为零。

Σ V = 0 (closed loop)

• Choose loop direction and assign consistent sign conventions for voltage rises and drops.

  选择回路方向并为电压升和电压降指定一致的符号约定。

• These laws underpin mesh and nodal analysis widely used in engineering.

  这些定律构成了工程中广泛使用的网孔分析和节点分析的基础。

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10. Capacitance in DC Circuits | 直流电路中的电容

A capacitor stores charge and energy in an electric field. Its capacitance defines the charge stored per unit potential difference.

电容器在电场中储存电荷和能量。电容定义了单位电势差下储存的电荷量。

C = Q / V

For parallel plate capacitors, capacitance depends on plate area A, separation d and permittivity ε.

对于平行板电容器，电容取决于极板面积 A、间距 d 和介电常数 ε。

C = ε A / d

Time constant for an RC circuit indicates charge/discharge speed.

RC 电路的时间常数表示充放电的快慢。

τ = R C

• In series: 1/C_total = 1/C₁ + 1/C₂ + … . In parallel: C_total = C₁ + C₂ + … .

  串联时：1/C_total = 1/C₁ + 1/C₂ + … 。并联时：C_total = C₁ + C₂ + … 。

• Stored energy: E = ½ C V² = ½ Q V.

  储存的能量：E = ½ C V² = ½ Q V。

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11. Fluid Pressure and Hydrostatics | 流体压力与静力学

Pressure in a static fluid increases with depth and is the same in all directions at a given point.

静止流体中的压力随深度增加而增大，且在同一点向各个方向均相等。

p = ρ g h

Pascal’s principle states that pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid.

帕斯卡原理指出，施加于密闭流体的压强会大小不变地向流体各处传递。

F₁ / A₁ = F₂ / A₂ (hydraulic systems)

• p is pressure (Pa), ρ is fluid density (kg/m³), h is depth or height difference (m). Atmospheric pressure is often added for absolute pressure.

  p 为压强 (Pa)，ρ 为流体密度 (kg/m³)，h 为深度或高度差 (m)。绝对压强常需加上大气压。

• Upthrust = weight of fluid displaced (Archimedes’ principle).

  浮力 = 排开流体的重量（阿基米德原理）。

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12. Thermodynamics – First Law | 热力学第一定律

The first law of thermodynamics expresses the conservation of energy for a system, linking internal energy change, heat and work.

热力学第一定律表达了系统的能量守恒，关联了内能变化、热量与功。

ΔU = Q − W

Here ΔU is the change in internal energy (J), Q is heat added to the system (J), and W is work done by the system (J). The sign convention may vary but is consistent in this formulation.

这里 ΔU 是内能的变化 (J)，Q 是系统吸收的热量 (J)，W 是系统对外做的功 (J)。符号约定可能不同，但在此公式中保持一致。

• For an ideal gas, ΔU depends only on temperature change: ΔU = n C_v ΔT.

  对于理想气体，ΔU 仅取决于温度变化：ΔU = n C_v ΔT。

• In engineering contexts, the steady flow energy equation extends this to open systems including enthalpy and kinetic energy terms.

  在工程情境中，稳态流动能量方程将此推广到开系，包括焓和动能项。

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