📚 Year 8 CIE Engineering: Quick Reference Handbook of Formulas and Theorems | Year 8 CIE 工程：公式定理速查手册
This handbook brings together the key formulas, theorems, and relationships you will meet in Year 8 CIE Engineering. Keep it handy for quick revision, problem-solving, and practical project work.
本手册汇集了 Year 8 CIE 工程课程中将会遇到的关键公式、定理与关系式。你可以在快速复习、解题和完成实践项目时随时查阅。
1. Forces and Motion | 力与运动
Average speed is defined as the total distance travelled divided by the total time taken.
平均速率等于总路程除以总时间。
v = d / t
Acceleration is the rate of change of velocity. If an object changes from an initial velocity u to a final velocity v over a time t, its acceleration a is given by:
加速度是速度变化的快慢。若物体在时间 t 内从初始速度 u 变为最终速度 v，其加速度 a 为：
a = (v – u) / t
Newton's Second Law states that the resultant force acting on an object is equal to the product of its mass and its acceleration.
牛顿第二定律指出，作用在物体上的合力等于物体质量与加速度的乘积。
F = m a
Weight is the force of gravity on an object. On Earth, the gravitational field strength g is approximately 9.8 N/kg (often taken as 10 N/kg).
重量是地球对物体的引力。地球表面的重力场强度 g 约为 9.8 N/kg（计算中常取 10 N/kg）。
W = m g
2. Work, Energy and Power | 功、能与功率
Work is done when a force moves an object in the direction of the force. Work W equals the force F multiplied by the distance d moved along the line of the force.
当力使物体沿力的方向移动时，力就对物体做了功。功 W 等于力 F 乘以沿力方向移动的距离 d。
W = F d
Kinetic energy is the energy an object possesses due to its motion. It depends on the mass m and the speed v of the object.
动能是物体由于运动而具有的能量，它取决于物体的质量 m 和速度 v。
Eₖ = ½ m v²
Gravitational potential energy is the energy stored in an object because of its height above the ground. It depends on the mass m, gravitational field strength g, and height h.
重力势能是物体因离地高度而储存的能量，它取决于质量 m、重力场强度 g 和高度 h。
Eₚ = m g h
Power is the rate at which work is done or energy is transferred. It is measured in watts (W).
功率是做功或能量转移的快慢，单位是瓦特 (W)。
P = W / t or P = E / t
Efficiency compares the useful output energy or power to the total input energy or power. It is often expressed as a percentage.
效率将有用的输出能量或功率与总输入能量或功率进行对比，通常以百分数表示。
η = (useful output / total input) × 100%
3. Levers and Moments | 杠杆与力矩
The moment of a force about a pivot is the turning effect. It is calculated by multiplying the force by the perpendicular distance from the pivot to the line of action of the force.
力对某个支点的力矩是其转动效应，大小等于力乘以支点到力作用线的垂直距离。
M = F × d
For a lever in equilibrium, the sum of the clockwise moments equals the sum of the anticlockwise moments about the pivot.
对于处于平衡状态的杠杆，相对于支点的顺时针力矩之和等于逆时针力矩之和。
F₁ × d₁ = F₂ × d₂
Mechanical advantage (MA) tells you how much a machine multiplies the effort force. It is the ratio of load to effort.
机械效益 (MA) 表示机器将动力放大的倍数，它是载荷与动力的比值。
MA = load / effort
Velocity ratio (VR) is the ratio of the distance moved by the effort to the distance moved by the load in the same time.
速度比 (VR) 是同一时间内动力移动的距离与载荷移动的距离之比。
VR = distance moved by effort / distance moved by load
The efficiency of a machine can also be found from the mechanical advantage and the velocity ratio.
机器的效率也可以由机械效益和速度比求得。
η = (MA / VR) × 100%
4. Pressure | 压强
Pressure is the force acting per unit area. A larger force or a smaller area gives a higher pressure.
压强是单位面积上所受的力。力越大或面积越小，压强就越大。
p = F / A
In a hydraulic system, the pressure is transmitted equally throughout the enclosed liquid. This leads to a force multiplication relationship between two connected pistons.
在液压系统中，压强在整个密闭液体中大小不变地传递，这使得两个相连活塞之间产生力的放大关系。
p₁ = p₂ → F₁ / A₁ = F₂ / A₂
5. Density | 密度
Density measures how much mass is packed into a given volume. It helps engineers choose the right material for a job.
密度用来衡量给定体积内所含质量的多少，它帮助工程师选择合适的材料。
ρ = m / V
An object will float if its density is less than the density of the fluid it is placed in; it will sink if its density is greater.
如果物体的密度小于它所在流体的密度，物体会漂浮；如果物体的密度更大，它则会下沉。
6. Electricity: Charge, Current and Voltage | 电学：电荷、电流与电压
Electric charge (Q) is measured in coulombs (C). The amount of charge that passes a point in a circuit is related to the current (I) and the time (t).
电荷 (Q) 的单位是库仑 (C)。流过电路中某一点的电荷量取决于电流 (I) 和时间 (t)。
Q = I t
Voltage (potential difference) is the energy transferred per unit charge. It shows how much energy is given to or taken from each coulomb of charge.
电压（电势差）是每单位电荷转移的能量，表明每库仑电荷获得或消耗了多少能量。
V = E / Q
7. Ohm's Law and Resistance | 欧姆定律与电阻
Ohm's Law states that, at constant temperature, the current through a conductor is directly proportional to the potential difference across it. The constant of proportionality is the resistance.
欧姆定律指出，在温度不变的条件下，通过导体的电流与导体两端的电势差成正比，这个比例常数就是电阻。
V = I R
Resistance measures how much a component opposes the flow of current. It can be calculated from the voltage and current.
电阻衡量一个元件对电流阻碍作用的大小，可以根据电压和电流计算得出。
R = V / I
8. Series and Parallel Circuits | 串联与并联电路
In a series circuit, the current is the same at all points, and the total voltage is the sum of the individual voltages across each component. The total resistance is the sum of the individual resistances.
在串联电路中，各处电流相等，总电压等于各元件两端电压之和。总电阻等于各个电阻之和。
I = I₁ = I₂ V = V₁ + V₂ Rtotal = R₁ + R₂ + ...
In a parallel circuit, the voltage across each branch is the same as the source voltage. The total current is the sum of the currents in the branches. The total resistance can be found from the reciprocal formula.
在并联电路中，各支路两端的电压相等，都等于电源电压。总电流等于各支路电流之和。总电阻可由倒数公式求得。
V = V₁ = V₂ I = I₁ + I₂ 1/Rtotal = 1/R₁ + 1/R₂ + ...
9. Electrical Power | 电功率
Electrical power is the rate at which electrical energy is transferred in a circuit. It is the product of current and voltage.
电功率是电路中电能转移的快慢，它等于电流与电压的乘积。
P = I V
Using Ohm's Law, two other useful forms of the power equation can be derived. These are especially helpful when you know the resistance and either the current or the voltage.
结合欧姆定律，可以推导出电功率的另外两个实用形式。当已知电阻和电流或电压时，这些公式非常有用。
P = I² R and P = V² / R
10. Material Properties and Hooke's Law | 材料特性与胡克定律
Hooke's Law describes the behaviour of many elastic materials, such as springs, when they are stretched or compressed. The extension x is directly proportional to the applied force F, up to the limit of proportionality.
胡克定律描述了许多弹性材料（如弹簧）被拉伸或压缩时的特性。在比例限度内，伸长量 x 与施加的力 F 成正比。
F = k x
Stress is a measure of the internal force per unit area within a material. It helps engineers assess whether a part can withstand the applied load.
应力是材料内部单位面积上所承受的内力，它帮助工程师判断零件能否承受外加的载荷。
σ = F / A
Strain is a measure of deformation representing the change in length relative to the original length. It has no units.
应变是材料变形程度的量度，表示长度变化与原长之比，没有单位。
ε = ΔL / L₀
Young's modulus (or the elastic modulus) is the ratio of stress to strain in the linear elastic region. It is a measure of the stiffness of a material.
杨氏模量（弹性模量）是材料在弹性范围内应力与应变的比值，它衡量材料的刚度。
E = σ / ε
11. Engineering Units and Prefixes | 工程单位与前缀
Engineering calculations often deal with very large or very small quantities. Standard prefixes make these numbers easier to write and understand. The table below shows the most common SI prefixes you need at this level.
工程计算中经常碰到非常大或非常小的数值。标准前缀使得这些数字更易于书写和理解。下表展示了本阶段需要掌握的最常见SI前缀。
| Prefix | Symbol | Multiplier | 中文 |
|---|---|---|---|
| kilo | k | 10³ = 1000 | 千 |
| mega | M | 10⁶ = 1,000,000 | 兆 |
| centi | c | 10⁻² = 0.01 | 厘 |
| milli | m | 10⁻³ = 0.001 | 毫 |
| micro | µ | 10⁻⁶ = 0.000 001 | 微 |
Always check the units in your final answer and convert them to a standard form, for example metres instead of centimetres, unless the question asks otherwise.
始终检查最终答案的单位并将其转换为标准形式，例如除非题目另有要求，否则应使用米而不是厘米。
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