📚 Year 7 CIE Engineering: Quick-Reference Formula & Theorem Handbook | Year 7 CIE 工程：公式定理速查手册
This handbook brings together the essential formulas, principles, and theorems that every Year 7 CIE Engineering student needs to know. You will find clear explanations grouped by topic, with worked examples and side-by-side English-Chinese translations to support bilingual learning. Use it to revise for tests, complete design challenges, or simply reinforce your understanding of how things work.
本手册汇集了每一位 Year 7 CIE 工程学生必须掌握的核心公式、原理和定理。内容按主题分组，配有清晰的解释和示例，并采用中英文对照形式，助力双语学习。你可以用它来备考、完成设计挑战，或巩固自己对事物运作原理的理解。
1. Basic Engineering Quantities and Units | 基本工程量和单位
All engineering calculations start from a small set of fundamental quantities. In Year 7, you will most often work with length, mass, time, temperature, electric current, force, and energy. Each quantity has a standard SI unit and a symbol used in formulas.
所有工程计算都源于一小部分基本量。在 Year 7 课程中，你最常接触的是长度、质量、时间、温度、电流、力和能量。每个量都有标准国际单位（SI）和公式中使用的符号。
- Length (l, d, h) – metre (m). Sometimes centimetre (cm) or millimetre (mm) are used; 1 m = 100 cm = 1000 mm.
- 长度 (l, d, h) – 米 (m)。有时也使用厘米 (cm) 或毫米 (mm)；1 m = 100 cm = 1000 mm。
- Mass (m) – kilogram (kg). Grams (g) are also common; 1 kg = 1000 g.
- 质量 (m) – 千克 (kg)。克 (g) 也很常用；1 kg = 1000 g。
- Time (t) – second (s). Minutes and hours are used for longer intervals.
- 时间 (t) – 秒 (s)。较长的时间间隔用分钟和小时表示。
- Temperature (T or θ) – gree Celsius (°C) or kelvin (K). The size of one degree is the same in both; 0 °C = 273 K.
- 温度 (T 或 θ) – 摄氏度 (°C) 或开尔文 (K)。两者的温度间隔大小相同；0 °C = 273 K。
- Electric current (I) – ampere (A).
- 电流 (I) – 安培 (A)。
- Force (F) – newton (N). 1 N is roughly the weight of a small apple.
- 力 (F) – 牛顿 (N)。1 N 大约是一个小苹果的重量。
- Energy, work, heat (E, W, Q) – joule (J).
- 能量、功、热量 (E, W, Q) – 焦耳 (J)。
2. Newton's Laws of Motion | 牛顿运动定律
Newton's three laws explain how forces change the motion of objects. They are the foundation of mechanical engineering and appear again and again in design problems.
牛顿三大定律解释了力如何改变物体的运动。它们是机械工程的基础，并在设计问题中反复出现。
First law (law of inertia): An object stays at rest, or keeps moving at a steady speed in a straight line, unless a resultant force acts on it.
第一定律（惯性定律）： 除非受到合力作用，否则物体将保持静止或匀速直线运动状态。
Second law: The acceleration (a) of an object is directly proportional to the resultant force (F) and inversely proportional to its mass (m). This is written as:
第二定律： 物体的加速度 (a) 与所受合力 (F) 成正比，与自身质量 (m) 成反比。公式为：
F = m × a
Where F is in newtons (N), m in kilograms (kg), and a in metres per second squared (m/s²).
其中 F 的单位是牛顿 (N)，m 的单位是千克 (kg)，a 的单位是米每二次方秒 (m/s²)。
Third law: For every action force, there is an equal and opposite reaction force. The two forces act on different objects.
第三定律： 每一个作用力都有一个大小相等、方向相反的反作用力。这两个力作用在不同的物体上。
3. Forces, Mass, and Weight | 力、质量和重量
In everyday language, mass and weight are often mixed up. In engineering, the distinction is crucial. Mass is the amount of matter; weight is the gravitational force on that mass.
在日常用语中，质量和重量常被混淆。在工程学里，区分它们至关重要。质量是物质的多少；重量是作用在该质量上的万有引力。
W = m × g
d to 10 N/kg). Weight is measured in newtons (N). Do not confuse mass (kilograms) with weight (newtons).
重量 (W) 等于质量 (m) 乘以引力场强度 (g)。在地球表面，g ≈ 9.8 N/kg（常取 10 N/kg）。重量单位是牛顿 (N)。不要把质量（千克）和重量（牛顿）混为一谈。
Example: A 5 kg toolbox has a weight of 5 × 10 = 50 N.
示例：一个 5 kg 的工具箱，重量为 5 × 10 = 50 N。
4. Work, Energy, and Power | 功、能量和功率
Whenever a force moves an object, work is done. Energy is the ability to do work. Power tells you how quickly that work is done or energy is transferred.
只要力使物体移动，就做了功。能量是做功的本领。功率则表示做功或能量传递的快慢。
W = F × d
nce moved in the direction of the force (d). Both W and energy are measured in joules (J).
做功 (W) = 力 (F) × 沿力方向移动的距离 (d)。功和能量的单位都是焦耳 (J)。
P = E ÷ t or P = W ÷ t
transferred or work done ÷ time taken. Power is measured in watts (W), where 1 W = 1 J/s.
功率 (P) = 传递的能量或做的功 ÷ 所用时间。功率单位是瓦特 (W)，1 W = 1 J/s。
200 × 3 = 600 J of work. If the lift takes 4 seconds, the power output is 600 ÷ 4 = 150 W.
示例：将一个 200 N 的负载提升 3 m 需做功 200 × 3 = 600 J。若提升耗时 4 秒，则输出功率为 600 ÷ 4 = 150 W。
5. Simple Machines and Mechanical Advantage | 简单机械与机械利益
Simple machines like levers, pulleys, and inclined planes make work easier by reducing the effort force needed. The mechanical advantage (MA) tells you how many times the machine multiplies your effort.
杠杆、滑轮、斜面等简单机械能减小所需的动力，从而让工作更省力。机械利益 (MA) 告诉你机械将你的动力放大了多少倍。
MA = Load ÷ Effort
ffort force. If MA is greater than 1, the machine is a force multiplier. If MA is less than 1, it is a distance (or speed) multiplier.
机械利益 = 负载力 ÷ 动力。若 MA 大于 1，该机械是一个省力装置；若 MA 小于 1，则是一个省距离（或增速）装置。
For an ideal machine with no friction: Work input = Work output, so effort × distance moved by effort = load × distance moved by load.
对于无摩擦的理想机械：输入功 = 输出功，即 动力 × 动力移动距离 = 负载 × 负载移动距离。
ulcrum equals the product of load and its distance: Effort × effort arm = Load × load arm.
在杠杆中，动力 × 动力臂 = 负载 × 负载臂。
6. Ohm's Law and Electrical Circuits | 欧姆定律与电路
Electricity is central to modern engineering. Ohm’s law links voltage, current, and resistance in a single, powerful equation that is used almost every day when designing or testing circuits.
电是现代工程的核心。欧姆定律用一个简洁有力的方程将电压、电流和电阻联系起来，在设计或测试电路时几乎每天都会用到。
V = I × R
Voltage (V) across a resistor equals current (I) through it multiplied by its resistance (R). Voltage is in volts (V), current in amperes (A), resistance in ohms (Ω).
电阻两端的电压 (V) 等于通过它的电流 (I) 乘以其电阻 (R)。电压单位为伏特 (V)，电流单位为安培 (A)，电阻单位为欧姆 (Ω)。
In a series circuit: current is the same at all points; total resistance is the sum of individual resistances; the supply voltage is shared across components.
在串联电路中：电流处处相等；总电阻等于各个电阻之和；电源电压被分配在各个元件上。
In a parallel circuit: the voltage across each branch is the same; total current is the sum of branch currents; total resistance is less than the smallest individual resistance.
在并联电路中：各支路两端电压相同；总电流等于各支路电流之和；总电阻小于任何一个单独的电阻。
7. Density and Material Properties | 密度与材料性质
Choosing the right material for a design means looking at properties such as density, strength, stiffness, and conductivity. Density tells you how much mass is packed into a certain volume.
为设计选择合适的材料需要考虑密度、强度、刚度和导电性等性质。密度告诉你一定体积内聚集了多少质量。
ρ = m ÷ V
Density (ρ, Greek letter rho) = mass (m) ÷ volume (V). Common units are g/cm³ or kg/m³.
密度 (ρ，希腊字母 rho) = 质量 (m) ÷ 体积 (V)。常用单位为 g/cm³ 或 kg/m³。
| Material | Density (g/cm³) | Typical use in engineering |
|---|---|---|
| Aluminium | 2.7 | Aeroplane frames, drink cans |
| Steel (mild) | 7.85 | Bridges, car bodies, tools |
| Copper | 8.96 | Electrical wires |
| Pine wood | 0.4–0.7 | Furniture, model structures |
Other important properties: tensile strength, compressive strength, stiffness (Young's modulus), thermal conductivity, and electrical conductivity. These are often found in data tables and must be considered together when selecting a material.
其他重要性质：抗拉强度、抗压强度、刚度（杨氏模量）、导热性和导电性。这些数据通常可在表格中查到，选材时必须综合考虑。
8. Hooke's Law and Springs | 胡克定律与弹簧
Many engineering materials and components, such as springs and rubber mounts, behave elastically. Hooke’s law describes the relationship between the force applied and the extension produced, as long as the material is not stretched beyond its elastic limit.
许多工程材料和构件（如弹簧和橡胶座）都具有弹性行为。只要材料未超出其弹性极限，胡克定律就能描述施加的力与产生的伸长量之间的关系。
F = k × x
nsion or compression (x). The spring constant k is in N/m and describes the stiffness of the spring.
拉伸或压缩弹簧所需的力 (F) = 劲度系数 (k) × 伸长量或压缩量 (x)。劲度系数 k 的单位是 N/m，它描述了弹簧的刚度。
ple: A spring with k = 50 N/m extends by 0.2 m when a force of 50 × 0.2 = 10 N is applied.
示例：一根 k = 50 N/m 的弹簧，在施加 10 N 的力时会伸长 0.2 m（因为 50 × 0.2 = 10 N）。
Always stop using Hooke’s law if the material permanently deforms or the force and extension stop being proportional.
如果材料发生永久变形，或者力与伸长量不再成正比，就不能继续使用胡克定律。
9. Pressure, Area, and Hydraulics | 压强、面积与液压
Pressure is a measure of how concentrated a force is. It plays a key role in hydraulics, pneumatics, and structural design. The same force can produce a very different effect depending on the area over which it acts.
压强衡量的是力的集中程度。它在液压、气动以及结构设计中扮演着关键角色。同样的力，作用面积不同，效果可能大相径庭。
P = F ÷ A
Pressure (P) = force (F) ÷ area (A). The SI unit is the pascal (Pa), where 1 Pa = 1 N/m².
压强 (P) = 力 (F) ÷ 面积 (A)。国际单位是帕斯卡 (Pa)，1 Pa = 1 N/m²。
In a hydraulic system, a liquid transmits pressure equally in all directions (Pascal's principle). A small input force on a small piston creates a large output force on a bigger piston, because pressure is constant while area increases.
在液压系统中，液体朝各个方向等大地传递压强（帕斯卡原理）。在小活塞上施加一个小的输入力，可以在大活塞上产生一个大的输出力，因为当压强恒定时，面积增大会使力增大。
F₁ ÷ A₁ = F₂ ÷ A₂
Where F₁, A₁ are the force and area of the small piston, and F₂, A₂ belong to the large piston.
其中 F₁、A₁ 是小活塞的力和面积，F₂、A₂ 是大活塞的力和面积。
10. Moments and Turning Effect | 力矩与转动效应
A moment is the turning effect of a force. It is extremely important when designing levers, beams, cranes, and any structure that can rotate about a pivot.
力矩指的是力的转动效应。在设计杠杆、梁、起重机以及任何可能绕支点旋转的结构时，力矩都极为重要。
M = F × d
r distance from the pivot to the line of action of the force (d). The unit is newton metre (N m).
力矩 (M) = 力 (F) × 从支点到力作用线的垂直距离 (d)。单位是牛顿·米 (N m)。
For an object to be in rotational equilibrium, the total clockwise moment must equal the total anticlockwise moment about the pivot:
物体要处于转动平衡，关于支点的顺时针力矩总和必须等于逆时针力矩总和：
Σ clockwise moments = Σ anticlockwise moments
This is the principle of moments. It is used to calculate unknown forces or distances in balanced systems such as see-saws, levers, and bridges.
这就是力矩原理，用于计算平衡系统（如跷跷板、杠杆和桥梁）中未知的力或距离。
11. Thermal Expansion and Heat Transfer | 热膨胀与传热
Almost all materials expand when heated and contract when cooled. Engineers must account for this in bridges, railways, pipelines, and electronic devices.
几乎所有材料都会热胀冷缩。工程师在桥梁、铁路、管道和电子设备中都必须考虑这一点。
Linear expansion: gth, α the coefficient of linear expansion, l₀ the original length, and ΔT the temperature change. The value of α is a property of the material.
线膨胀： Δl = α × l₀ × ΔT，其中 Δl 为长度变化量，α 为线膨胀系数，l₀ 为原长，ΔT 为温度变化量。α 的数值取决于材料。
Heat transfer attern: P = k A (ΔT)/d, where k is thermal conductivity, A is cross-sectional area, and d is thickness.
热传递通过传导（固体）、对流（流体）和辐射（红外波）进行。通过材料的热流率 P 通常遵循：P = k A (ΔT)/d，其中 k 为导热系数，A 为横截面积，d 为厚度。
Year 7 students are expected to understand these concepts qualitatively and perform simple calculations using given formulas.
Year 7 学生需要定性地理解这些概念，并能用给出的公式进行简单计算。
12. Engineering Design Process and Efficiency | 工程设计程序与效率
Engineering is not just about equations; it is a systematic process of solving problems. A typical design cycle includes: identify the need, research, brainstorm solutions, select and develop a design, create a prototype, test and evaluate, and finally refine and communicate.
工程不仅是套用公式，更是一套系统性的问题解决流程。典型的设计循环包括：识别需求、研究、头脑风暴、选择并深化设计方案、制作原型、测试与评估，最后改进并沟通。
No machine is 100% efficient. Efficiency compares the useful output with the input, usually expressed as a percentage:
任何机器的效率都无法达到 100%。效率将有用输出与输入进行比较，通常以百分数表示：
Efficiency = (Useful output energy ÷ Total input energy) × 100%
or Efficiency = (Useful power output ÷ Total power input) × 100%
或 效率 = (有用输出功率 ÷ 总输入功率) × 100%
Friction, heat loss, air resistance, and electrical heating are common reasons why efficiency is less than 100%. Understanding losses helps engineers improve designs to save energy and reduce waste.
摩擦、散热、空气阻力和电发热是效率低于 100% 的常见原因。了解这些损耗有助于工程师改进设计，节约能源并减少浪费。
Always remember that real-world measurements involve some uncertainty. Year 7 students should record data with the correct number of decimal places based on the instrument used and note any possible sources of error.
请始终记住，真实世界的测量存在一定的不确定性。Year 7 学生应根据所用仪器记录正确的小数位数，并注明可能的误差来源。
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