Year 9 AQA Engineering: Formula & Theorem Quick Reference Handbook | Year 9 AQA 工程：公式定理速查手册

📚 Year 9 AQA Engineering: Formula & Theorem Quick Reference Handbook | Year 9 AQA 工程：公式定理速查手册

This quick reference handbook is designed for Year 9 students following the AQA Engineering pathway. It brings together the essential formulas, theorems and key relationships you will need when analysing forces, materials, mechanisms, electrical systems and thermodynamics. Each section presents the core principle in a clear, paired English–Chinese format, with every formula highlighted for easy revision.

本速查手册专为修读 AQA 工程课程的 Year 9 学生编写。手册汇集了分析力、材料、机构、电气系统和热力学时所需的必备公式、定理和关键关系。每个小节均以清晰的中英对照形式呈现核心原理，所有公式均突出显示，便于复习。

1. Basic Mechanics: Force, Mass, Acceleration, Weight & Density | 基础力学：力、质量、加速度、重量与密度

Newton’s Second Law relates the net force acting on an object to its mass and acceleration. It is the foundation of dynamic analysis in engineering.

F = m a

牛顿第二定律将作用在物体上的合外力与其质量和加速度联系起来。它是工程动力学分析的基础。

Weight is the gravitational force on a mass. It differs from mass and varies with gravitational field strength g (on Earth approximately 9.8 m/s²).

W = m g

重量是作用在质量上的引力。它不同于质量，并随引力场强度 g 变化（地球表面约 9.8 m/s²）。

Density describes how much mass is contained in a given volume. It is a key material property used for selection and fluid calculations.

ρ = m / V

密度描述单位体积内所含质量的大小，是材料选择和流体计算中一项关键的材料属性。

F = force (N), m = mass (kg), a = acceleration (m/s²)

F = 力（牛顿），m = 质量（千克），a = 加速度（米/秒²）
W = weight (N), g = gravitational field strength (N/kg or m/s²)

W = 重量（牛顿），g = 引力场强度（牛/千克或米/秒²）
ρ = density (kg/m³), V = volume (m³)

ρ = 密度（千克/米³），V = 体积（米³）

2. Stress and Strain | 应力与应变

Engineering stress (σ) is the internal force per unit area within a material when an external load is applied. It is measured in pascals (Pa).

σ = F / A

工程应力（σ）是材料承受外加载荷时单位面积上的内力，单位为帕斯卡（Pa）。

Strain (ε) is the dimensionless ratio of the change in length to the original length. It indicates how much a material deforms.

ε = ΔL / L₀

应变（ε）是长度变化量与原长之比，是一个无量纲量，表示材料的变形程度。

σ = stress (Pa or N/m²), F = applied force (N), A = cross-sectional area (m²)

σ = 应力（Pa 或 N/m²），F = 施加的力（N），A = 横截面积（m²）
ε = strain, ΔL = change in length (m), L₀ = original length (m)

ε = 应变，ΔL = 长度变化量（m），L₀ = 原始长度（m）

3. Young’s Modulus (Elastic Modulus) | 杨氏模量（弹性模量）

Young’s modulus (E) measures the stiffness of a solid material. It is the constant of proportionality between stress and strain in the linear elastic region.

E = σ / ε

杨氏模量（E）衡量固体材料的刚度，是线弹性范围内应力与应变的比例常数。

A high Young’s modulus means the material is very stiff (e.g. steel); a low value indicates a more flexible material (e.g. rubber). This relationship is fundamental to structural engineering.

高杨氏模量意味着材料非常坚硬（如钢）；低值则表示材料更具弹性（如橡胶）。这一关系是结构工程的基础。

E = Young’s modulus (Pa), σ = stress (Pa), ε = strain (no units)

E = 杨氏模量（Pa），σ = 应力（Pa），ε = 应变（无单位）

4. Moments and Levers | 力矩与杠杆

The moment of a force is its turning effect about a pivot. It depends on the force size and the perpendicular distance from the pivot.

M = F × d

力矩是力对支点的转动效应，取决于力的大小和力到支点的垂直距离。

For a lever in equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments. This principle is used in countless mechanisms, from seesaws to hydraulic arms.

杠杆平衡时，顺时针力矩之和等于逆时针力矩之和。从跷跷板到液压臂，这一原理被广泛应用于无数机构中。

M = moment (N m), F = force (N), d = perpendicular distance from pivot (m)

M = 力矩（牛·米），F = 力（N），d = 支点的垂直距离（m）

5. Gears, Velocity Ratio and Mechanical Advantage | 齿轮、速度比与机械效益

The velocity ratio (VR) of a simple gear train is determined by the number of teeth on the driven gear compared to the driver gear. It describes the ideal speed change.

VR = Number of teeth on driven gear / Number of teeth on driver gear

简单齿轮装置的速度比（VR）由从动轮齿数与主动轮齿数之比决定，描述理想的速度变化。

Mechanical advantage (MA) is the ratio of the load force to the effort force. In an ideal (frictionless) system, MA equals VR.

MA = Load / Effort

机械效益（MA）是负载力与作用力之比。在理想（无摩擦）系统中，MA 等于 VR。

VR has no units; MA is also dimensionless.

VR 无单位；MA 无量纲。
When friction is present, MA < VR.

当存在摩擦时，MA < VR。

6. Work, Gravitational Potential Energy and Power | 功、重力势能与功率

Work is done when a force moves an object in the direction of the force. It is a scalar quantity measured in joules (J).

W = F × d

力沿其方向推动物体移动时即做功，功是标量，单位为焦耳（J）。

Gravitational potential energy (GPE) is the energy stored due to an object’s position in a gravitational field.

Eₚ = m g h

重力势能（GPE）是因物体在引力场中的位置而储存的能量。

Power is the rate of doing work or transferring energy. The standard unit is the watt (W).

P = W / t

功率是做功或传递能量的速率，标准单位为瓦特（W）。

W = work (J), F = force (N), d = displacement in direction of force (m)

W = 功（J），F = 力（N），d = 沿力方向的位移（m）
Eₚ = gravitational potential energy (J), h = height (m)

Eₚ = 重力势能（J），h = 高度（m）
P = power (W), t = time (s)

P = 功率（W），t = 时间（s）

For energy transfer systems, efficiency indicates how much input energy is converted to useful output:

Efficiency = (Useful output energy / Total input energy) × 100%

在能量传递系统中，效率表示输入能量中有多少被转化为有用输出：效率 = （有用输出能量 / 总输入能量）× 100%。

7. Ohm’s Law and Electrical Power | 欧姆定律与电功率

Ohm’s Law states that the current through a conductor is directly proportional to the potential difference across it, provided temperature remains constant.

V = I R

欧姆定律指出，在温度保持不变的条件下，通过导体的电流与导体两端的电势差成正比。

Electrical power is the product of voltage and current. It indicates how much electrical energy is transferred per unit time.

P = I V

电功率是电压与电流的乘积，表示单位时间内传递的电能。

V = potential difference (V), I = current (A), R = resistance (Ω)

V = 电势差（伏特），I = 电流（安培），R = 电阻（欧姆）
P = electrical power (W), also can be expressed as P = I² R or P = V² / R

P = 电功率（瓦特），也可表示为 P = I² R 或 P = V² / R

8. Pressure in Fluids | 流体压强

Pressure measures how concentrated a force is over an area. For fluids, pressure acts equally in all directions.

P = F / A

压强衡量力在面积上的集中程度。对于流体，压强在各个方向上均等作用。

In a liquid column, pressure increases with depth due to the weight of the fluid above.

p = ρ g h

在液柱中，由于上方液体的重量，压强随深度增加而增大。

P (or p) = pressure (Pa), F = force (N), A = area (m²)

P (或 p) = 压强（Pa），F = 力（N），A = 面积（m²）
h = depth/height of the liquid column (m)

h = 液柱深度/高度（m）

9. Thermal Expansion | 热膨胀

Most materials expand when heated and contract when cooled. The linear expansion formula predicts the change in length for a solid.

ΔL = α L₀ ΔT

大多数材料受热膨胀、冷却收缩。线膨胀公式可预测固体的长度变化。

This effect must be allowed for in engineering structures such as bridges, railways and pipelines, where expansion gaps are provided.

在桥梁、铁轨和管道等工程结构中必须考虑这一效应，因此需要设置伸缩缝。

ΔL = change in length (m), α = coefficient of linear expansion (/°C or /K), L₀ = original length (m), ΔT = temperature change (°C or K)

ΔL = 长度变化量（m），α = 线膨胀系数（/°C 或 /K），L₀ = 原始长度（m），ΔT = 温度变化量（°C 或 K）

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Year 9 AQA Engineering: Formula & Theorem Quick Reference Handbook | Year 9 AQA 工程：公式定理速查手册