Year 9 CCEA Engineering: Formula & Theorem Quick Reference Handbook | 九年级 CCEA 工程：公式定理速查手册

📚 Year 9 CCEA Engineering: Formula & Theorem Quick Reference Handbook | 九年级 CCEA 工程：公式定理速查手册

Engineering at Year 9 level introduces the fundamental principles that underpin mechanical and electrical systems. This handbook gathers the key formulas, theorems, and constants that students need for quick revision, solving problems, and understanding how forces, energy, and circuits behave in real-world applications.

九年级工程课程介绍了支撑机械与电气系统的基本原理。本手册汇总了学生在快速复习、解题和理解力、能量与电路在实际应用中的行为时所需的关键公式、定理和常数。

1. Basic Quantities and Units | 基础物理量及单位

Before using any formula, it is essential to know the standard quantities, their symbols, and the SI units used in engineering calculations.

在使用任何公式之前，必须了解工程计算中的标准物理量、符号及其国际单位制（SI）单位。

Quantity 物理量	Symbol 符号	SI Unit 单位	Unit Symbol 单位符号
Length 长度	l, s, d	metre 米	m
Mass 质量	m	kilogram 千克	kg
Time 时间	t	second 秒	s
Force 力	F	newton 牛顿	N
Energy / Work 能量/功	E, W	joule 焦耳	J
Power 功率	P	watt 瓦特	W
Voltage 电压	V	volt 伏特	V
Current 电流	I	ampere 安培	A
Resistance 电阻	R	ohm 欧姆	Ω
Area 面积	A	square metre 平方米	m²

2. Force and Newton’s Second Law | 力与牛顿第二定律

Newton’s Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This is the foundation of dynamics.

牛顿第二定律指出，物体的加速度与作用在其上的净外力成正比，与其质量成反比。这是动力学的基础。

F = m × a

where F is the net force in newtons (N), m is the mass in kilograms (kg), and a is the acceleration in metres per second squared (m/s²).

其中 F 为净外力，单位牛顿（N）；m 为质量，单位千克（kg）；a 为加速度，单位米每二次方秒（m/s²）。

Weight is a specific force caused by gravity: W = m × g, where g = 9.81 m/s² on Earth (often approximated as 10 m/s²).

重力是由引力引起的一种特定力：W = m × g，地球上的 g = 9.81 m/s²（常近似取 10 m/s²）。

3. Moments and Levers | 力矩与杠杆

A moment is the turning effect of a force. It depends on the size of the force and the perpendicular distance from the pivot to the line of action of the force.

力矩是力的转动效应，它取决于力的大小以及从支点到力作用线的垂直距离。

M = F × d

where M is the moment in newton-metres (N m), F is the force in newtons (N), and d is the perpendicular distance from the pivot in metres (m).

其中 M 为力矩，单位牛顿·米（N m）；F 为力，单位牛顿（N）；d 为从支点到力作用线的垂直距离，单位米（m）。

The principle of moments states that for a system in equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments.

力矩原理指出，对于处于平衡状态的系统，顺时针力矩之和等于逆时针力矩之和。

Levers are simple machines that use the principle of moments to amplify an effort force. There are three classes of levers, categorised by the relative positions of load, effort and pivot.

杠杆是利用力矩原理来放大施力力的简单机械。根据负载、施力点和支点的相对位置，杠杆分为三类。

4. Stress, Strain and Young’s Modulus | 应力、应变与杨氏模量

When a material is subjected to a force, it experiences stress and strain. These quantities help engineers select suitable materials for structures.

当材料受力时，会产生应力和应变。这些物理量帮助工程师为结构选择合适的材料。

Stress, σ = F / A

Stress (σ) is the force applied per unit cross-sectional area. Its unit is the pascal (Pa) or N/m².

应力（σ）是单位横截面积上所承受的力，单位为帕斯卡（Pa）或 N/m²。

Strain, ε = ΔL / L₀

Strain (ε) is the extension per unit original length. It has no units because it is a ratio of two lengths.

应变（ε）是单位原始长度的伸长量，由于是两个长度的比值，因此没有单位。

Young’s Modulus, E = σ / ε

Young’s modulus (E) measures the stiffness of a material within the elastic limit. Its unit is the pascal (Pa). A higher E means a stiffer material.

杨氏模量（E）衡量材料在弹性极限内的刚度，单位为帕斯卡（Pa）。E 值越高表示材料越坚硬。

5. Simple Machines: Mechanical Advantage (MA) | 简单机械：机械效益（MA）

Simple machines such as levers, pulleys, and inclined planes can reduce the effort needed to perform a task. Mechanical advantage (MA) quantifies this reduction.

杠杆、滑轮和斜面等简单机械可以减少完成任务所需的作用力。机械效益（MA）用来量化这一减省程度。

MA = Load (L) / Effort (E)

where Load is the force to be overcome and Effort is the force applied by the user. MA is a dimensionless ratio.

其中 Load 是需要克服的力，Effort 是用户施加的力。MA 是一个无量纲的比值。

Velocity Ratio (VR) is another important concept: VR = distance moved by effort / distance moved by load in the same time.

速比（VR）是另一个重要概念：VR = 作用力移动的距离 / 负载移动的距离（在同一时间内）。

Efficiency of a machine can also be expressed as: Efficiency = (MA / VR) × 100%.

机器的效率也可以表示为：效率 = (MA / VR) × 100%。

6. Work, Energy and Power | 功、能与功率

Work is done when a force moves an object through a distance in the direction of the force. Energy is the capacity to do work, and power is the rate of doing work.

当力使物体沿力的方向移动一段距离时，我们就说力做了功。能量是做功的能力，功率是做功的快慢。

Work Done, W = F × d

where W is work in joules (J), F is force in newtons (N), and d is distance in metres (m).

其中 W 为功，单位焦耳（J）；F 为力，单位牛顿（N）；d 为距离，单位米（m）。

Gravitational potential energy (GPE) gained by raising an object: ΔEₚ = m × g × h, where h is the height raised.

物体被抬高时获得的引力势能变化：ΔEₚ = m × g × h，其中 h 为上升高度。

Kinetic energy (KE) of a moving object: Eₖ = ½ × m × v², where v is velocity in m/s.

运动物体的动能：Eₖ = ½ × m × v²，v 为速度，单位 m/s。

Power is the rate of energy transfer: P = W / t or P = E / t, with power in watts (W).

功率是能量转移的速率：P = W / t 或 P = E / t，功率单位为瓦特（W）。

7. Ohm’s Law and Electric Circuits | 欧姆定律与电路

Ohm’s Law is the most fundamental rule in electrical engineering. It relates voltage, current, and resistance in a conductor at constant temperature.

欧姆定律是电气工程中最基本的定律，它将恒定温度下导体中的电压、电流和电阻联系起来。

V = I × R

where V is voltage in volts (V), I is current in amperes (A), and R is resistance in ohms (Ω).

其中 V 为电压，单位伏特（V）；I 为电流，单位安培（A）；R 为电阻，单位欧姆（Ω）。

Electrical power can be calculated using two useful forms:

电功率可以用两个有用的公式计算：

P = V × I and P = I² × R

If a circuit has a known resistance and current, the second formula avoids measuring voltage.

如果电路电阻和电流已知，第二个公式可以避免测量电压。

8. Resistors in Series and Parallel | 电阻的串联与并联

Resistors can be combined to control current and voltage in circuits. The total resistance depends on whether they are arranged in series or parallel.

电阻可以组合使用以控制电路中的电流和电压。总电阻取决于它们是串联还是并联。

For resistors in series (connected end-to-end):

对于串联电阻（首尾相连）：

Rₜₒₜₐₗ = R₁ + R₂ + R₃ + …

The same current flows through each resistor, but the voltage is divided.

相同电流流过每个电阻，但电压被分压。

For resistors in parallel (connected side-by-side):

对于并联电阻（并排连接）：

1 / Rₜₒₜₐₗ = 1 / R₁ + 1 / R₂ + 1 / R₃ + …

The voltage across each branch is the same, but the currents add up. The total resistance is always less than the smallest individual resistance.

每个支路两端的电压相同，但电流相加。总电阻始终小于最小的单个电阻。

9. Efficiency | 效率

No machine converts all input energy into useful output. Efficiency measures how well a system transfers energy into the desired form.

没有机器能将所有输入能量转化为有用输出。效率用于衡量系统将能量转化为目标形式的程度。

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

It can also be calculated from power: Efficiency = (Useful power output / Total power input) × 100%.

也可以根据功率计算：效率 = (有用输出功率 / 总输入功率) × 100%。

In an ideal machine, efficiency would be 100%, but friction and heat losses always reduce real efficiency. Understanding efficiency helps engineers improve designs and reduce energy waste.

在理想机器中，效率将达到 100%，但摩擦和热损失总会降低实际效率。了解效率有助于工程师改进设计并减少能源浪费。

10. Key Engineering Constants and Approximations | 工程常用常数与近似值

Engineering calculations often rely on standard values that simplify problem-solving. Here are some constants and safe approximations used at Year 9 level.

工程计算通常会依赖标准值来简化问题求解。以下是一些在九年级阶段使用的常数和安全近似值。

Acceleration due to gravity, g: 9.81 m/s² (often 10 m/s² for estimates)

重力加速度 g：9.81 m/s²（估算时常取 10 m/s²）
Density of water: 1000 kg/m³ at 4 °C

水的密度：1000 kg/m³（4°C 时）
Speed of light in vacuum: 3.00 × 10⁸ m/s (used in electronics and communication)

真空中的光速：3.00 × 10⁸ m/s（用于电子学与通信）
1 kilowatt-hour (kWh) = 3.6 × 10⁶ J (energy unit in electricity)

1 千瓦时（kWh）= 3.6 × 10⁶ J（电能的能量单位）
Young’s modulus of steel: approx. 200 GPa (2.0 × 10¹¹ Pa)

钢的杨氏模量：约 200 GPa（2.0 × 10¹¹ Pa）

These numbers provide a quick reference and reduce the need for looking up values during exercises.

这些数据提供了快速参考，减少了在练习中查找数值的需要。

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Year 9 CCEA Engineering: Formula & Theorem Quick Reference Handbook | 九年级 CCEA 工程：公式定理速查手册