📚 Engineering Formula & Theorem Quick Reference Guide | 工程公式定理速查手册
This quick reference guide is designed for Year 8 WJEC Engineering students. It brings together the essential formulas, principles and theorems you will meet in your lessons and projects, from mechanics and structures to electronics and energy. Keep it handy for homework, revision and practical work.
本速查手册专为 Year 8 WJEC 工程课程学生设计。它汇集了你在课堂和项目中将遇到的基本公式、原理和定理，涵盖力学、结构、电子和能源等领域。无论是作业、复习还是实际操作，请随身携带。
1. Force and Pressure | 力与压强
In engineering, force is a push or a pull that can cause an object to move, stop or change shape. Pressure measures how concentrated a force is over a given area — a smaller area means higher pressure for the same force.
在工程中，力是能使物体移动、停止或改变形状的推或拉。压强衡量力在给定面积上的集中程度 — 相同的力，面积越小，压强越大。
The pressure formula linking force and area is:
压强与力、面积的关系式为：
P = F / A
where P is pressure in pascals (Pa), F is force in newtons (N) and A is the area in square metres (m²). 1 Pa equals 1 N/m².
其中 P 为压强，单位帕斯卡 (Pa)；F 为力，单位牛顿 (N)；A 为面积，单位平方米 (m²)。1 Pa = 1 N/m²。
For example, if a force of 500 N is applied to an area of 2 m², the pressure is 250 Pa. In hydraulic systems, engineers use this principle to multiply force.
例如，若 500 N 的力作用在 2 m² 的面积上，压强为 250 Pa。在液压系统中，工程师利用这一原理来增大力。
2. Speed, Distance and Time | 速度、距离与时间
Speed describes how fast an object moves. It is the distance travelled per unit of time. The average speed of any moving part in a mechanism can be found using this relationship.
速度描述物体运动的快慢，即单位时间内通过的距离。任何机构中运动部件的平均速度都可用此关系求得。
speed = distance / time
Common units are metres per second (m/s) or kilometres per hour (km/h). In engineering design, knowing the speed of a conveyor belt or a robot arm is essential for timing and safety.
常用单位为米每秒 (m/s) 或千米每小时 (km/h)。在工程设计中，了解传送带或机械臂的速度对于时序安排和安全至关重要。
and time = distance ÷ speed. These three forms help you solve motion problems in projects.
公式变形可得 距离 = 速度 × 时间 以及 时间 = 距离 ÷ 速度。这三种形式可帮助你解决项目中的运动问题。
3. Acceleration | 加速度
Acceleration is the rate at which an object's speed changes. If an object speeds up, slows down or changes direction, it is accelerating. Uniform acceleration can be calculated with:
加速度是物体速度变化的速率。物体加速、减速或改变方向时均有加速度。匀加速可用以下公式计算：
a = (v – u) / t
where a is acceleration (m/s²), v is final velocity (m/s), u is initial velocity (m/s) and t is time (s).
其中 a 为加速度（单位 m/s²），v 为末速度（m/s），u 为初速度（m/s），t 为时间（s）。
This formula is particularly useful when analysing launch mechanisms, braking systems or lifts. A negative acceleration indicates deceleration.
分析发射机构、制动系统或电梯时，该公式特别有用。加速度为负表示减速（减速度）。
4. Newton's Second Law | 牛顿第二定律
Newton's Second Law connects force, mass and acceleration. It is one of the most important relationships in mechanical engineering.
牛顿第二定律将力、质量和加速度联系起来，是机械工程中最重要的关系之一。
F = m × a
where F is the resultant force in newtons (N), m is mass in kilograms (kg) and a is acceleration in metres per second squared (m/s²).
其中 F 为合力（牛顿 N），m 为质量（千克 kg），a 为加速度（米每二次方秒 m/s²）。
This means a larger force is needed to accelerate a heavier object at the same rate. Engineers use this law to calculate motor forces, rocket thrust and structural loads.
这表明要使较重的物体获得相同的加速度，需要更大的力。工程师利用该定律计算机械动力、火箭推力和结构载荷。
5. Moments and the Principle of Moments | 力矩与力矩原理
A moment is the turning effect of a force about a pivot. It depends on the size of the force and its perpendicular distance from the pivot.
力矩是力绕支点产生的转动效应，取决于力的大小及其到支点的垂直距离。
M = F × d
where M is the moment in newton metres (N m), F is force (N) and d is the perpendicular distance from the pivot (m).
其中 M 为力矩，单位牛顿米 (N m)；F 为力 (N)；d 为支点到力作用线的垂直距离 (m)。
The principle of moments states that for a balanced lever or beam, the total clockwise moments equal the total anticlockwise moments:
力矩原理指出，杠杆或横梁平衡时，顺时针力矩总和等于逆时针力矩总和：
∑ clockwise moments = ∑ anticlockwise moments
This principle is used to design levers, seesaws, cranes and balance systems. It helps engineers calculate exactly where to place a counterweight.
此原理用于设计杠杆、跷跷板、起重机和平衡系统，帮助工程师精确计算配重的位置。
6. Work and Power | 功与功率
Work is done when a force moves an object in the direction of the force. The amount of work measures the energy transferred.
力使物体沿力的方向移动时做了功。功的大小衡量了转移的能量。
W = F × d
J), F is force (N) and d is distance moved in the direction of the force (m). 1 J = 1 N m.
其中 W 为功，单位焦耳 (J)；F 为力 (N)；d 为沿力方向移动的距离 (m)。1 J = 1 N m。
Power is the rate of doing work, or how quickly energy is transferred:
功率是做功的快慢，即能量转换的速率：
P = W / t
machines, a higher power rating means the machine can complete the same task in less time.
其中 P 为功率，单位瓦特 (W)；W 为功 (J)；t 为时间 (s)。1 W = 1 J/s。在机器中，额定功率越高，表示完成相同任务所需时间越短。
7. Mechanical Advantage | 机械利益
Mechanical advantage (MA) tells us how much a simple machine multiplies the input force. It is a ratio without units.
机械利益 (MA) 反映简单机械将输入力放大的倍数，是一个无单位的比值。
MA = load / effort
where load is the output force (often the weight being lifted) and effort is the input force you apply. If MA > 1, the machine makes the task easier by requiring less effort force.
其中 load 为输出力（通常是需抬起的重量），effort 为你施加的输入力。若 MA > 1，机器则因所需施力更小而省力。
Levers, pulleys and gears all have mechanical advantage. An engineer selects the arrangement that gives the best MA for the task while considering speed and distance trade‑offs.
杠杆、滑轮和齿轮都有机械利益。工程师在权衡速度与距离后，选择对任务最有利的 MA 配置。
8. Efficiency | 效率
No machine is perfect — some energy is always lost, usually as heat or sound. Efficiency measures how much of the input energy is converted into useful output.
没有完美的机器，总会有能量损耗（通常以热或声的形式）。效率衡量输入能量转化为有用输出的比例。
Efficiency = (useful output energy / total input energy) × 100%
so be calculated using power: Efficiency = (useful power output / total power input) × 100%. It is always less than 100% for real machines.
效率也可用功率计算：效率 = (有用输出功率 / 总输入功率) × 100%。对于实际机器，效率总是低于 100%。
Engineers aim to improve efficiency by reducing friction, using lighter materials and improving designs. High efficiency means lower running costs and less wasted energy.
工程师通过减小摩擦、采用轻质材料和优化设计来提高效率。高效率意味着运行成本更低，能量浪费更少。
9. Ohm's Law | 欧姆定律
Ohm's Law is fundamental in electronic engineering. It links voltage, current and resistance in an electrical circuit.
欧姆定律是电子工程的基础，描述了电路中电压、电流和电阻之间的关系。
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（伏特 V），I 为电流（安培 A），R 为电阻（欧姆 Ω）。
The three forms of Ohm's Law are shown in the table below.
下表列出了欧姆定律的三种形式。
| Quantity to find | Formula |
|---|---|
| Voltage (V) | V = I × R |
| Current (I) | I = V / R |
| Resistance (R) | R = V / I |
Using Ohm's Law, an engineer can select the correct resistor to protect an LED or calculate the current in a motor circuit. It is essential for safe circuit design.
利用欧姆定律，工程师可选择合适的电阻保护 LED，或计算电机电路中的电流。这对安全设计电路至关重要。
10. Electrical Power | 电功率
Electrical power measures the rate at which an electrical device converts energy. It helps determine how much energy a component or system uses.
电功率衡量电气设备转换能量的速率，可用于确定元件或系统的能耗。
P = V × I
where P is power in watts (W), V is voltage (V) and I is current (A).
其中 P 为功率（瓦特 W），V 为电压 (V)，I 为电流 (A)。
The following table shows useful variations of the power formula.
下表列出了电功率公式的实用变形。
| Known quantities | Formula |
|---|---|
| Voltage and current | P = V × I |
| Voltage and resistance | P = V² / R |
| Current and resistance | P = I² × R |
These forms are used when designing power supplies, choosing wire thickness, and ensuring components do not overheat. The total power drawn by a circuit must not exceed the supply rating.
在设计电源、选择导线粗细以及确保元件不发生过热时，就会用到这些公式。电路的总功率不能超过电源额定值。
11. Density | 密度
Density describes how much mass is packed into a given volume. It is a key property of materials in engineering, affecting weight, strength and buoyancy.
密度描述单位体积内所含质量的大小，是工程材料的关键属性，影响重量、强度和浮力。
ρ = m / V
where ρ (rho) is density (kg/m³ or g/cm³), m is mass (kg or g) and V is volume (m³ or cm³). In practice, engineers often use kg/m³ for large structures and g/cm³ for small components.
其中 ρ 为密度（kg/m³ 或 g/cm³），m 为质量（kg 或 g），V 为体积（m³ 或 cm³）。实际中，大型结构常用 kg/m³，小型部件常用 g/cm³。
Choosing a material with the right density is vital — a bicycle frame needs low density for lightness, whereas a counterweight needs high density to be compact. Density also helps identify unknown materials.
选择密度合适的材料至关重要 — 自行车架需要低密度以实现轻便，而配重则需要高密度以保持紧凑。密度还可用于鉴别未知材料。
12. Gear Ratios | 齿轮比
Gears transmit motion and force between rotating shafts. The gear ratio determines how speed and torque change from the driver gear to the driven gear.
齿轮在旋转轴之间传递运动和力。齿轮比决定了从主动轮到从动轮的速度和扭矩如何变化。
Gear Ratio = Number of teeth on driven gear / Number of teeth on driver gear
A gear ratio greater than 1 reduces speed but increases torque (low gear), while a ratio less than 1 increases speed but reduces torque (high gear).
齿轮比大于 1 时减速增扭（低速档），小于 1 时增速减扭（高速档）。
The relationship between speed and teeth is:
转速与齿数的关系为：
Output speed = Input speed × (Teeth on driver / Teeth on driven)
Engineers use gear ratios to match motor RPM to the required output speed in robots, vehicles and machinery. Compound gear trains combine several pairs to achieve precise ratios.
工程师利用齿轮比，将电机转速与机器人、车辆及机械所需的输出速度匹配。复合齿轮系将多个齿轮副组合，以实现精确的传动比。
Published by TutorHao | Engineering Revision Series | aleveler.com
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