📚 Year 9 Edexcel Engineering: Formula & Theorem Quick Reference Handbook | Year 9 Edexcel 工程：公式定理速查手册
This quick reference handbook brings together the essential formulas and theorems you will encounter in the Year 9 Edexcel Engineering curriculum. From basic mechanics and electrical principles to simple machines and material behaviour, every equation is presented side‑by‑side in English and Chinese, with clear explanations and worked examples. Keep this guide handy for revision, homework and putting engineering theory into practice.
本速查手册汇集了 Year 9 Edexcel 工程课程中的核心公式与定理。从基础力学、电学原理到简单机械与材料性能，每一个公式均以中英对照呈现，并配有清晰的说明与计算示例。复习、作业或理论联系实际时，可随时查阅。
1. Force and Acceleration (Newton’s Second Law) | 力与加速度（牛顿第二定律）
s (N), m is mass in kilograms (kg) and a is acceleration in metres per second squared (m/s2). It tells us that a larger force produces a larger acceleration, and a more massive object needs a bigger force to achieve the same acceleration.
作用在物体上的合外力等于物体的质量乘以加速度，常用公式为 F = m × a。其中 F 的单位是牛顿 (N)，m 是千克 (kg)，a 是米每二次方秒 (m/s2)。这一定律说明：力越大加速度越大；在相同加速度下，质量越大的物体所需的力也越大。
F = m × a
s pushed with a resultant force of 200 N. Its acceleration is a = F / m = 200 / 50 = 4 m/s2.
示例：一辆质量为 50 kg 的小车受到 200 N 的合外力，其加速度为 a = F / m = 200 / 50 = 4 m/s2。
2. Pressure | 压力
re P is pressure in pascals (Pa), F is force in newtons (N) and A is area in square metres (m2). A sharp knife cuts easily because the force is concentrated on a very small area, resulting in high pressure.
压力表示力在单位面积上的集中程度，即单位面积所受的力：P = F / A，P 的单位为帕斯卡 (Pa)，F 为力 (N)，A 为面积 (m2)。锋利的小刀容易切割，是因为力集中在极小的面积上，产生很大的压力。
P = F ÷ A
Example: A block weighing 600 N rests on a base of area 0.5 m2. The pressure on the floor is P = 600 / 0.5 = 1200 Pa.
示例：一个重 600 N 的物体放置在底面积为 0.5 m2 的基座上，对地面的压力为 P = 600 / 0.5 = 1200 Pa。
3. Moments and Equilibrium | 力矩与平衡
nt. Moment (M) = force (F) × perpendicular distance from the pivot (d). The unit is newton‑ ional equilibrium, the total clockwise moments must equal the total anticlockwise moments.
力对某一支点的转动效应称为力矩。力矩 M = 力 F × 支点到力作用线的垂直距离 d，单位是牛顿·米 (N m)。若物体处于转动平衡状态，则顺时针方向的总力矩等于逆时针方向的总力矩。
M = F × d
nd, perpendicular to the spanner. The moment applied to the nut is M = 40 × 0.25 = 10 N m.
示例：一把长 0.25 m 的扳手，在末端施加垂直于扳手的 40 N 力，对螺母产生的力矩为 M = 40 × 0.25 = 10 N m。
Principle of moments: For a seesaw in balance, F1 × d1 = F2 × d2.
力矩原理：跷跷板平衡时，F1 × d1 = F2 × d2。
4. Work and Mechanical Energy | 功与机械能
n of the force (d). The unit of work is the joule (J). If you lift a load vertically, you increase its gravitational potential energy (Ep): Ep = m × g × h, where g = 10 m/s2 (on Earth) and h is the height gained.
当力使物体沿力的方向移动一段距离时，便做了功。功 W = 力 F × 沿力方向移动的距离 d。功的单位是焦耳 (J)。将重物竖直提起，会使其重力势能增加：Ep = m × g × h，其中 g 取 10 m/s2，h 为上升高度。
W = F × d
Ep = m g h
Kinetic energy is the energy of motion. Ek = ½ m v2, where m is mass and v is velocity.
动能是物体由于运动而具有的能量：Ek = ½ m v2，m 为质量，v 为速度。
Ek = ½ m v2
Example: A 2 kg ball moves at 3 m/s. Its kinetic energy is Ek = ½ × 2 × 32 = 9 J.
示例：一个 2 kg 的球以 3 m/s 运动，其动能为 Ek = ½ × 2 × 32 = 9 J。
5. Power and Efficiency | 功率与效率
ts (W), where 1 W = 1 J/s. A more powerful engine can do the same amount of work in less time.
功率是做功或传递能量的快慢程度。功率 P = 做功 W ÷ 时间 t，也可表示为 P = 能量转换量 ÷ 时间，单位是瓦特 (W)，1 W = 1 J/s。功率越大的发动机，完成同样功所需的时间越短。
P = W ÷ t
al input. Efficiency = (useful output energy or power ÷ total input energy or power) × 100%. No real machine is 100% efficient because some energy is always lost as heat or sound.
效率反映有用输出占总输入的比值。效率 = (有用输出能量或功率 ÷ 总输入能量或功率) × 100%。实际机器的效率不可能达到 100%，因为总有部分能量以热或声音等形式散失。
η = (Useful output ÷ Total input) × 100%
ectrical energy and delivers 350 J of mechanical work. Its efficiency is (350 ÷ 500) × 100% = 70%.
示例：一台电动机消耗 500 J 的电能，输出 350 J 的机械功，其效率为 (350 ÷ 500) × 100% = 70%。
6. Ohm’s Law and Resistance | 欧姆定律与电阻
e V is voltage in volts (V), I is current in amperes (A) and R is resistance in ohms (Ω). For a given resistance, the current increases proportionally with the voltage.
欧姆定律给出了电路中电压、电流与电阻的关系：V = I × R，V 的单位为伏特 (V)，I 为安培 (A)，R 为欧姆 (Ω)。在电阻值固定时，电流随电压成比例增加。
V = I × R
istance of 12 Ω and draws a current of 0.5 A. The voltage across it is V = 0.5 × 12 = 6 V.
示例：一盏灯的电阻为 12 Ω，通过的电流为 0.5 A，则其两端电压为 V = 0.5 × 12 = 6 V。
Resistance can also be calculated from power: P = I × V or P = I2R = V2/R.
由功率也可计算电阻：P = I × V，或 P = I2R = V2/R。
7. Series and Parallel Circuits | 串联与并联电路
In a series circuit, the total resistance is the sum of all individual resistances: Rtotal = R1 + R2 + R3 + ... The current is the same everywhere, but the voltage divides across each resistor.
在串联电路中，总电阻等于各个电阻之和：Rtotal = R1 + R2 + R3 + ... 电流处处相等，电压则分配在各电阻上。
Rtotal = R1 + R2 + R3 + …
In a parallel circuit, the total resistance is found using the reciprocal formula: 1/Rtotal = 1/R1 + 1/R2 + 1/R3 + ... The voltage across each branch is the same, while the total current splits among the branches.
在并联电路中，总电阻通过倒数公式求得：1/Rtotal = 1/R1 + 1/R2 + 1/R3 + ... 各支路两端电压相同，总电流则分配到各支路中。
1/Rtotal = 1/R1 + 1/R2 + 1/R3 + …
Example: Two resistors of 6 Ω and 3 Ω in parallel. 1/Rtotal = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2, so Rtotal = 2 Ω.
示例：两个电阻 6 Ω 与 3 Ω 并联。1/Rtotal = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2，故 Rtotal = 2 Ω。
8. Gear Ratios and Velocity Ratio | 齿轮比与速度比
Gears transmit rotary motion and torque. The gear ratio is the ratio of the number of teeth on the driven gear (Tdriven) to the number of teeth on the driver gear (Tdriver): Gear Ratio = Tdriven / Tdriver. A ratio greater than 1 reduces speed but increases torque; a ratio less than 1 increases speed but reduces torque.
齿轮用来传递旋转运动和扭矩。齿轮比等于从动齿轮的齿数 Tdriven 除以主动齿轮的齿数 Tdriver：齿轮比 = Tdriven / Tdriver。比值大于 1 时降速增扭，小于 1 时增速减扭。
Gear Ratio = Tdriven ÷ Tdriver
e output (driven): VR = Speed of driver ÷ Speed of driven. For a simple gear train, VR = Tdriven / Tdriver as well.
速度比 VR 表示输入转速与输出转速之比：VR = 主动轮转速 ÷ 从动轮转速。对于简单的齿轮系，VR 同样等于 Tdriven / Tdriver。
VR = Input speed ÷ Output speed
with a driven gear of 60 teeth. Gear ratio = 60/20 = 3. Output speed = 100 / 3 ≈ 33.3 rpm.
示例：主动轮 20 齿，转速 100 rpm，与 60 齿的从动轮啮合。齿轮比 = 60/20 = 3，输出转速 = 100 / 3 ≈ 33.3 rpm。
9. Levers and Mechanical Advantage | 杠杆与机械效益
actor by which the machine multiplies the effort: MA = Load (output force) ÷ Effort (input force). An ideal lever has MA = VR, where VR = distance moved by effort ÷ distance moved by load.
杠杆是一种能放大输入力的简单机械，由支点、刚杆组成。机械效益 MA 表示机器将作用力放大的倍数：MA = 负载（输出力）÷ 作用力（输入力）。理想杠杆中 MA = VR，而 VR = 作用力移动距离 ÷ 负载移动距离。
MA = Load ÷ Effort
For a first‑ states: Effort × Effort arm = Load × Load arm. By rearranging, MA = Effort arm ÷ Load arm.
对于第一类杠杆（支点在作用力与负载之间），杠杆定律为：作用力 × 作用力臂 = 负载 × 负载臂。整理后，MA = 作用力臂 ÷ 负载臂。
Effort × deffort = Load × dload
The effort arm is 1.2 m, the load arm is 0.3 m. The required effort is Effort = (Load × dload) / deffort = (800 × 0.3) / 1.2 = 200 N. MA = 800 / 200 = 4.
示例：用撬棍抬起 800 N 的岩石。作用力臂 1.2 m，负载臂 0.3 m。所需作用力 = (800 × 0.3) / 1.2 = 200 N，MA = 800 / 200 = 4。
10. Hooke’s Law and Spring Extension | 胡克定律与弹簧伸长
tly proportional to its extension (x). F = k × x, where k is the spring constant (stiffness) in N/m. The graph of force against extension is a straight line through the origin up to the limit of proportionality.
胡克定律描述了许多弹性材料在弹性限度内拉伸时的行为：施加在弹簧上的力 F 与其伸长量 x 成正比。F = k × x，其中 k 为弹簧常数（劲度系数），单位为 N/m。在比例极限内，力‑伸长图是一条通过原点的直线。
F = k x
/ 0.04 = 300 N/m. The extension under a 24 N load would be x = F / k = 24 / 300 = 0.08 m.
示例：一弹簧在 12 N 力作用下伸长 0.04 m。其劲度系数 k = F / x = 12 / 0.04 = 300 N/m。若施加 24 N 的力，伸长量 x = F / k = 24 / 300 = 0.08 m。
11. Density and Material Properties | 密度与材料性质
mass is packed into a given volume. ρ = m / V, where m is mass in kg and V is volume in m3. It helps engineers select lightweight or heavy materials for different applications. Knowing the density also allows you to calculate the mass of a component from its volume.
密度 ρ 表示单位体积内所含的质量：ρ = m / V，m 的单位为 kg，V 为 m3。密度帮助工程师选择适合不同用途的轻质或重质材料，也可根据体积反算零件的质量。
ρ = m ÷ V
Stress and strain describe how a material responds to forces. Stress σ = Force ÷ Cross‑sectional area (N/m2 l length (no units). These introduce the stiffness of materials but detailed Young’s modulus calculations are covered in later years.
应力和应变描述材料受力后的反应。应力 σ = 力 ÷ 横截面积 (N/m2 或 Pa)；应变 ε = 伸长量 ÷ 原长（无单位）。这些概念初步引入了材料的刚度概念，详细的杨氏模量计算将在后续年级学习。
Stress σ = F ÷ A
Strain ε = extension ÷ original length
12. Summary of Key Units and Symbols | 关键单位与符号汇总
Being comfortable with units is essential for solving engineering problems. Below is a quick reference table:
熟练掌握单位是解决工程问题的基础。下表为一览：
| Quantity 物理量 | Symbol 符号 | Unit 单位 |
|---|---|---|
| Force | F | newton (N) |
| Mass | m | kilogram (kg) |
| Distance / length | d, x, L | metre (m) |
| Area | A | m2 |
| Volume | V | m3 |
| Velocity / speed | v | m/s |
| Acceleration | a | m/s2 |
| Work / Energy | W, E | joule (J) |
| Power | P | watt (W) |
| Voltage | V | volt (V) |
| Current | I | ampere (A) |
| Resistance | R | ohm (Ω) |
| Density | ρ (rho) | kg/m3 |
| Moment | M | N m |
| Pressure | P | pascal (Pa) |
| Spring constant | k | N/m |
Always convert values to these standard units before substituting into formulas to avoid errors.
在代入公式之前，始终将数值转换为以上标准单位，可避免计算错误。
或截图后用「扫一扫 → 相册」选择本图