📚 Year 8 CAIE Engineering: Formula & Theorem Quick Reference | Year 8 CAIE 工程：公式定理速查手册
This quick-reference handbook compiles all the essential formulas and theorems you need for the CAIE Year 8 Engineering course. From Ohm’s Law to mechanical advantage, each entry is presented with a clear explanation, units, and practical tips to help you solve problems confidently. Keep this guide handy for homework, revision, and exam preparation.
本速查手册汇编了 CAIE 8 年级工程课程所需的所有基本公式和定理。从欧姆定律到机械优势，每个条目都附有清晰的解释、单位以及实用提示，帮助你自信地解决问题。在完成作业、复习和备考时，请随时参考本指南。
1. Ohm’s Law & Electrical Resistance | 欧姆定律与电阻
Ohm’s Law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points, provided the temperature remains constant. The constant of proportionality is the resistance of the conductor.
欧姆定律指出，在温度保持不变的条件下，通过导体两点的电流与这两点之间的电压成正比。比例常数就是该导体的电阻。
V = I R
Here, V is the voltage in volts (V), I is the current in amperes (A), and R is the resistance in ohms (Ω). 1 Ω equals 1 V / A. You can rearrange the formula to find current: I = V / R, or to find resistance: R = V / I.
公式中，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)。1 Ω 等于 1 V/A。可将公式变形为求电流的 I = V / R，或求电阻的 R = V / I。
In a series circuit, the total resistance is the sum of individual resistances: Rₜ = R₁ + R₂ + ... In a parallel circuit, the reciprocal of total resistance equals the sum of the reciprocals: 1/Rₜ = 1/R₁ + 1/R₂ + ...
在串联电路中，总电阻等于各个电阻之和：Rₜ = R₁ + R₂ + ... 在并联电路中，总电阻的倒数等于各支路电阻的倒数之和：1/Rₜ = 1/R₁ + 1/R₂ + ...
2. Electrical Power | 电功率
Power in an electrical circuit measures how quickly electrical energy is converted into other forms, such as heat or light. The basic power formula relates voltage and current.
电路中的电功率衡量电能转化为其他形式（如热或光）的快慢。基本功率公式联系了电压与电流。
P = I V
per second (1 W = 1 J/s). By substituting Ohm’s Law, you can derive two alternative forms: P = I² R and P = V² / R. These are useful when you know the resistance and either current or voltage.
P 为功率，单位瓦特 (W)；I 为电流 (A)；V 为电压 (V)。1 瓦特等于每秒 1 焦耳 (1 W = 1 J/s)。结合欧姆定律可导出另外两种形式：P = I² R 和 P = V² / R。当已知电阻和电流或电压时，这两个公式非常实用。
3. Mechanical Advantage (MA) | 机械优势
Mechanical advantage tells you how much a machine multiplies the effort force you apply. It is defined as the ratio of the load (output force) to the effort (input force).
机械优势表示机器将你施加的动力放大了多少倍。它定义为负载（输出力）与动力（输入力）的比值。
MA = Load / Effort = L / E
MA has no units because it is a ratio of two forces. If MA > 1, the machine amplifies the effort; if MA < 1, the machine allows faster movement but less force output, as in certain gear or lever arrangements. In an ideal machine with no friction, MA equals the velocity ratio (VR), but in reality MA is always less than VR due to friction and other losses.
MA 没有单位，因为它是两个力的比值。若 MA > 1，机器放大了动力；若 MA < 1，机器可提供更快的移动但输出力较小，如某些齿轮或杠杆设计。在理想无摩擦的机器中，MA 等于速度比 (VR)；但由于摩擦和损耗，实际 MA 总是小于 VR。
4. Velocity Ratio (VR) | 速度比
The velocity ratio compares the distance moved by the effort to the distance moved by the load in the same time. It is a purely geometric property of a machine and does not depend on friction.
速度比比较了在相同时间内动力移动的距离与负载移动的距离。它完全是机器的几何特性，与摩擦无关。
VR = distance moved by effort / distance moved by load = dₑ / dₗ
hed gears, VR = number of teeth on driven gear / number of teeth on driver gear. For a belt drive, VR = diameter of driven pulley / diameter of driver pulley. Knowing VR helps you predict how fast or far the load moves relative to the effort.
对于简单杠杆，VR = 动力臂长度 / 负载臂长度。对于滑轮组，VR 等于支撑负载的绳段数。对于一对啮合齿轮，VR = 从动轮齿数 / 主动轮齿数。对于带传动，VR = 从动轮直径 / 主动轮直径。了解 VR 有助于预测负载相对于动力的运动速度和距离。
5. Efficiency | 效率
Efficiency measures how well a machine converts the input work into useful output work. It is expressed as a percentage and accounts for energy lost due to friction, heat, and sound.
效率衡量机器将输入功转化为有用输出功的程度。它以百分比表示，并考虑了摩擦、热量和声音造成的能量损失。
Efficiency η = (useful work output / total work input) × 100%
Since work = force × distance, you can also express efficiency using MA and VR:
η = (MA / VR) × 100%
因为功 = 力 × 距离，所以也可用 MA 和 VR 表示效率：η = (MA / VR) × 100%。
Efficiency is always less than 100% for real-world machines. A higher efficiency means less energy is wasted; ideal machines would have 100% efficiency but do not exist in practice.
实际机器的效率始终低于 100%。效率越高，意味着浪费的能量越少；理想机器的效率为 100%，但现实中并不存在。
| Formula | Description |
|---|---|
| η = (MA / VR) × 100% | Efficiency from mechanical advantage and velocity ratio |
| η = (Woutput / Winput) × 100% | Efficiency from work input and output |
6. Principle of Moments | 力矩原理
When an object is in rotational equilibrium, the total clockwise moment about any pivot equals the total anticlockwise moment. This principle applies to levers, seesaws, and many mechanical systems.
当物体处于转动平衡时，对于任何一个支点，所有顺时针力矩的总和等于所有逆时针力矩的总和。这一原理适用于杠杆、跷跷板和许多机械系统。
Sum of clockwise moments = Sum of anticlockwise moments
One moment is calculated as force multiplied by the perpendicular distance from the pivot:
Moment = F × d
Where F is the force in newtons (N) and d is the perpendicular distance in metres (m). The SI unit of moment is the newton-metre (N m). Remember always to use the perpendicular distance from the line of action of the force to the pivot.
力矩的计算方式为力乘以从支点到力作用线的垂直距离：力矩 = F × d，其中 F 是力 (N)，d 是垂直距离 (m)。力矩的国际单位是牛·米 (N m)。请务必使用支点到力作用线的垂直距离。
7. Pressure | 压力
Pressure quantifies how the effect of a force is spread over an area. A small force applied to a tiny area can generate a very large pressure, which is why sharp tools are effective.
压力衡量力的作用在面积上的分布情况。作用于极小面积上的小力也能产生非常大的压力，这就是锋利工具作用明显的原理。
P = F / A
Pa = 1 N/m². For fluids at rest, you may also encounter the hydrostatic pressure equation P = h ρ g, where h is depth, ρ is fluid density, and g is gravitational field strength. This explains why pressure increases deeper in a liquid.
P 为压力 (Pa)，F 为力 (N)，A 为面积 (m²)。1 Pa = 1 N/m²。对于静止流体，你可能还会遇到静水压力公式 P = h ρ g，其中 h 为深度，ρ 为流体密度，g 为重力场强度。这个公式解释了液体内越深处压力越大的原因。
8. Hooke’s Law | 胡克定律
Hooke’s Law describes the behaviour of many elastic materials, such as springs and wires, when they are stretched or compressed within their elastic limit.
胡克定律描述了许多弹性材料（如弹簧和金属丝）在弹性极限内被拉伸或压缩时的行为。
F = k x
F is the applied force (N), x is the extension or compression from the original length (m), and k is the spring constant (N/m). A stiffer spring has a larger k. As long as the material does not exceed its elastic limit, the extension is directly proportional to the force. Beyond this limit, permanent deformation occurs and Hooke’s Law no longer holds.
F 为施加的力 (N)，x 为从原始长度算起的伸长或压缩量 (m)，k 是弹簧常数 (N/m)。弹簧越硬，k 值越大。只要材料不超出弹性极限，伸长量与力成正比。超出此极限，将发生永久变形，胡克定律不再适用。
rain: stress = E × strain, where E is the Young modulus. Stress = F / A, strain = ΔL / L₀.
在工程中，胡克定律常写作应力-应变形式：应力 = E × 应变，其中 E 为杨氏模量。应力 σ = F / A，应变 ε = ΔL / L₀。
9. Work | 功
In physics and engineering, work is done when a force moves an object. The amount of work tells you how much energy has been transferred.
在物理和工程中，当一个力使物体产生位移时就做了功。功的大小表示转移了多少能量。
W = F × d
W is work in joules (J), F is the constant force applied (N), and d is the distance moved in the direction of the force (m). One joule equals one newton-metre. If force and displacement are at an angle, only the component of force in the direction of displacement does work.
W 为功 (J)，F 为施加的恒力 (N)，d 为物体沿力的方向移动的距离 (m)。1 焦耳等于 1 牛·米。如果力与位移方向成角度，则只有沿位移方向的分力做功。
It is important to distinguish between work input and work output when studying machines. Some work is always wasted overcoming friction, so output work is always less than input work.
在研究机器时，区分输入功和输出功很重要。总会有一部分功在克服摩擦时消耗掉，因此输出功总是小于输入功。
10. Gravitational Potential Energy & Kinetic Energy | 重力势能与动能
Energy can be stored in an object due to its position (gravitational potential energy, GPE) or its motion (kinetic energy, KE). These two forms are frequently converted one into another in engineering systems.
物体可因其位置（重力势能 GPE）或运动（动能 KE）而储存能量。在工程系统中，这两种能量形式经常相互转换。
GPE = m g h
/s² on Earth), and h is height above a reference level (m). GPE is measured in joules (J).
m 为质量 (kg)，g 为重力加速度（地球上约为 9.8 m/s²），h 为相对参考点的高度 (m)。GPE 的单位是焦耳 (J)。
KE = ½ m v²
Here v is speed in metres per second (m/s). Notice that KE depends on the square of the speed, so doubling the speed makes the kinetic energy four times as great. In an ideal energy conversion (no friction), the loss in GPE equals the gain in KE.
v 为速度，单位米/秒 (m/s)。注意 KE 与速度的平方成正比，因此速度加倍可使动能变为原来的四倍。在理想能量转换中（无摩擦），减少的重力势能等于增加的动能。
11. Power in Mechanical Systems | 机械系统中的功率
Power measures the rate at which work is done or energy is transferred. In mechanical systems, it can be expressed in terms of work and time, or force and velocity.
功率衡量做功或能量转移的速率。在机械系统中，它可用功与时间的关系，或力与速度的关系来表示。
P = W / t
P is power in watts (W), W is work done (J), and t is time (s). When a constant force moves an object at a steady speed, power can also be calculated as:
P 为功率 (W)，W 为做的功 (J)，t 为时间 (s)。当恒力使物体匀速运动时，功率也可计算为：
P = F v
rticularly handy for analysing vehicles and lifting mechanisms. Remember that 1 W = 1 J/s.
F 为力 (N)，v 为速度 (m/s)。这个公式在分析车辆和起重机构时特别方便。请记住 1 W = 1 J/s。
12. Density | 密度
Density describes how much mass is packed into a given volume of a material. It is a key property when selecting materials for lightweight or heavyweight engineering applications.
密度描述单位体积所包含的质量。在选择用于轻质或重载工程应用的材料时，密度是一项关键属性。
ρ = m / V
ρ is density, m is mass (kg or g), and V is volume (m³ or cm³). The SI unit is kg/m³, but g/cm³ is often used for practical measurements. For example, the density of water is 1.0 g/cm³ or 1000 kg/m³. Relative density compares the density of a substance to that of water.
ρ 为密度，m 为质量（kg 或 g），V 为体积（m³ 或 cm³）。SI 单位是 kg/m³，但实际中常用 g/cm³。例如，水的密度为 1.0 g/cm³ 或 1000 kg/m³。相对密度是物质密度与水的密度的比值。
Understanding density helps you calculate buoyancy, estimate whether objects will float, and choose the appropriate material when weight is critical, such as in bicycle frames or aircraft parts.
了解密度有助于计算浮力、估计物体是否会漂浮，并在重量至关重要的场合（如自行车车架或飞机零件）选择合适的材料。
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