📚 Year 9 AQA Engineering: Formula & Theorem Quick Reference Handbook | 公式定理速查手册
This quick reference handbook covers the essential formulas and theorems needed for Year 9 AQA Engineering. Understanding these relationships is fundamental to solving engineering problems and building a strong foundation for future studies. All symbols and units follow standard conventions, and every concept is explained with clear examples, worked substitutions and exam-focused notes.
本速查手册涵盖了 Year 9 AQA 工程所需的关键公式和定理。理解这些关系是解决工程问题的基础，也为将来的学习打下坚实的根基。所有符号与单位均遵循标准惯例，每个概念都配有清晰的示例、代入过程以及紧扣考试的要点说明。
Use the handbook in three ways: learn the formula, practise rearranging it, and always write the unit next to the number. Marks in engineering questions are usually split between the correct formula, the correct substitution and the correct unit.
建议按三种方式使用本手册：记住公式、练习变形公式、并始终在数字后写出单位。工程题目的分值通常分为三部分：正确的公式、正确的代入、以及正确的单位。
1. SI Units, Prefixes and Dimensional Consistency | 国际单位、词头与量纲一致性
Every engineering calculation depends on consistent units. The seven SI base units are the metre (m) for length, the kilogram (kg) for mass, the second (s) for time, the ampere (A) for electric current, the kelvin (K) for temperature, the mole (mol) for amount of substance and the candela (cd) for luminous intensity. In Year 9 you will mainly use the first five.
每一个工程计算都依赖统一的单位。国际单位制的七个基本单位是：长度的米(m)、质量的千克(kg)、时间的秒(s)、电流的安培(A)、热力学温度的开尔文(K)、物质的量的摩尔(mol)以及发光强度的坎德拉(cd)。在 Year 9 阶段，你主要会用到前五个。
watt (W), where 1 W = 1 J/s. Pressure and stress are the pascal (Pa), where 1 Pa = 1 N/m².
导出单位由基本单位组合而成。力的单位是牛顿(N)，1 N = 1 kg·m/s²；能量的单位是焦耳(J)，1 J = 1 N·m；功率的单位是瓦特(W)，1 W = 1 J/s；压强与应力的单位是帕斯卡(Pa)，1 Pa = 1 N/m²。
| Prefix 词头 | Symbol 符号 | Factor 倍数 | Example 例子 |
| giga 吉 | G | × 10⁹ | 1 GPa = 10⁹ Pa |
| mega 兆 | M | × 10⁶ | 1 MPa = 10⁶ Pa |
| kilo 千 | k | × 10³ | 1 kN = 1000 N |
| centi 厘 | c | × 10⁻² | 1 cm = 0.01 m |
| milli 毫 | m | × 10⁻³ | 1 mm = 0.001 m |
| micro 微 | μ | × 10⁻⁶ | 1 μm = 10⁻⁶ m |
Always convert before substituting. A length given in mm must become metres, and a load given in kN must become newtons, otherwise the answer will be wrong by a factor of 1000.
务必先换算再代入。若长度以 mm 给出，必须换算为米；若载荷以 kN 给出，必须换算为牛顿，否则答案会差 1000 倍。
2. Motion: Speed, Velocity and Acceleration | 运动：速度、速率与加速度
The average speed (v) of an object is is defined as the distance moved (d) divided by the time taken (t). Speed is a scalar quantity measured in metres per second (m/s).
物体的平均速率(v)定义为移动的距离(d)除以所用时间(t)。速率是标量，单位为米每秒(m/s)。
v = d / t d = v × t t = d / v
Velocity is the same calculation but with direction included, so velocity is a vector. A car travelling at 20 m/s north and a car travelling at 20 m/s south have the same speed but different velocities.
速度矢量(velocity)使用同样的计算，但必须包含方向，因此它是矢量。向北以 20 m/s 行驶的车与向南以 20 m/s 行驶的车速率相同，但速度矢量不同。
eleration is a = (v − u) / t. Acceleration is measured in metres per second squared (m/s²).
加速度(a)描述速度变化的快慢。若物体从初速度(u)变为末速度(v)，经过时间(t)，则加速度为 a = (v − u) / t。加速度的单位是米每二次方秒(m/s²)。
a = (v − u) / t v = u + a t
/s to 5 m/s in 4 s, so a = (5 − 15) / 4 = −2.5 m/s². The negative sign means deceleration.
示例：一名骑行者 60 s 内前进 900 m，则 v = 900 / 60 = 15 m/s。随后刹车，在 4 s 内由 15 m/s 减到 5 m/s，故 a = (5 − 15) / 4 = −2.5 m/s²。负号表示减速。
On a distance-time graph the gradient is the speed. On a velocity-time graph the gradient is the acceleration and the area under the line is the distance travelled.
在位移-时间图像上，斜率就是速率；在速度-时间图像上，斜率是加速度，而图线下的面积就是移动的距离。
3. Force and Newton's Second Law | 力与牛顿第二定律
Newton' s acceleration (a): F = m × a. The unit of force is the newton (N), where 1 N = 1 kg·m/s².
牛顿第二定律指出，物体所受的合力(F)等于其质量(m)与加速度(a)的乘积：F = m × a。力的单位是牛顿(N)，1 N = 1 kg·m/s²。
F = m × a m = F / a a = F / m
imately 9.8 N/kg: W = m × g. Mass is measured in kilograms and does not change with location, but weight is a force and does change on the Moon or on Mars.
重量(W)是重力施加在质量上的力。它等于质量(m)乘以重力场强度(g)，地球表面 g ≈ 9.8 N/kg：W = m × g。质量以千克计量，不随位置改变；但重量是一种力，在月球或火星上会改变。
W = m × g g ≈ 9.8 N/kg (Earth)
needs a resultant force of F = 1200 × 3 = 3600 N. Its weight is W = 1200 × 9.8 = 11 760 N.
当多个力作用于同一物体时，需用矢量方法求出合力。一辆 1200 kg 的汽车以 3 m/s² 加速，所需合力为 F = 1200 × 3 = 3600 N；其重量为 W = 1200 × 9.8 = 11 760 N。
Balanced forces give zero resultant force, so the object stays at rest or moves at constant velocity. Unbalanced forces always produce acceleration.
平衡力的合力为零，物体保持静止或匀速直线运动；不平衡的力一定会产生加速度。
4. Moments and Equilibrium | 力矩与平衡
A moment (M) is the turning effect of a force. It equals the force (F) multiplied by the perpendicular distance (d) from the pivot to the line of action of the force. The unit of a moment is the newton metre (N·m).
力矩(M)是力产生的转动效应。它等于力(F)乘以从支点到力作用线的垂直距离(d)。力矩的单位是牛·米(N·m)。
M = F × d F = M / d d = M / F
bling the distance doubles the moment, which is why a longer spanner loosens a tight bolt more easily.
在距支点 0.35 m 处施加 200 N 的力，产生的力矩为 M = 200 × 0.35 = 70 N·m。距离加倍，力矩也加倍，这就是长扳手更容易拧松紧固螺栓的原因。
The principle of moments states that for an object in equilibrium, the sum of clockwise moments about a point equals the sum of anticlockwise moments about the same point.
力矩平衡原理指出：当物体处于平衡状态时，绕某点的顺时针力矩之和等于绕同一点的逆时针力矩之和。
Σ M clockwise = Σ M anticlockwise
he pivot on the other side. For balance, F × 0.80 = 300 × 1.20, so F = 360 / 0.80 = 450 N.
示例：300 N 的载荷位于杠杆支点一侧 1.20 m 处，另一侧 0.80 m 处施加动力。平衡时 F × 0.80 = 300 × 1.20，故 F = 360 / 0.80 = 450 N。
ction. If the force acts at an angle θ, the effective perpendicular distance is d × sin θ.
切记：距离必须沿垂直于力作用线的方向测量。若力与杠杆成 θ 角作用，则有效的垂直距离为 d × sin θ。
5. Work, Energy and Power | 功、能量与功率
tion of the force (d). Work and energy are both measured in joules (J), where 1 J = 1 N·m.
当力使物体沿力的方向移动一段距离时，就做了功。功(W)等于力(F)乘以沿力方向移动的距离(d)。功与能量的单位都是焦耳(J)，1 J = 1 N·m。
W = F × d (1 J = 1 N·m)
Kinetic energy is the energy stored in a moving object, while gravitational potential energy is the energy stored in an object raised above the ground. These two relationships explain most energy transfers in machine problems.
动能是运动物体储存的能量，重力势能是被举高的物体储存的能量。这两个关系式可以解释机械题目中的大多数能量转化过程。
Eₖ = ½ × m × v² Eₚ = m × g × h
(W), where 1 W = 1 J/s. A motor that transfers 1500 J in 5 s has a power output of 300 W.
功率是能量转化的速率，所以 P = E / t 或 P = W / t，单位为瓦特(W)，1 W = 1 J/s。若电动机在 5 s 内输出 1500 J 能量，则其功率为 300 W。
P = E / t = W / t E = P × t
climbs to a height of 3 m, the gain in potential energy is Eₚ = 1200 × 9.8 × 3 = 35 280 J.
示例：一辆 1200 kg 的汽车以 15 m/s 行驶，其动能为 Eₖ = ½ × 1200 × 15² = 135 000 J。若它再爬升 3 m，则增加的势能为 Eₚ = 1200 × 9.8 × 3 = 35 280 J。
6. Efficiency and Energy Losses | 效率与能量损耗
No machine is perfectly efficient. Friction, air resistance, sound and heat all remove some of the input energy, so the useful output energy is always less than the total input energy.
没有机器是百分之百高效的。摩擦力、空气阻力、声音和热量都会带走一部分输入能量，因此有用的输出能量总是小于总输入能量。
efficiency = (useful output energy / total input energy) × 100%
calculated with power, using efficiency = (useful output power / total input power) × 100%. The value is a percentage and can never exceed 100% in a real machine.
效率也可以用功率来计算：效率 = (有用输出功率 / 总输入功率) × 100%。结果是一个百分数，在真实机械中永远不可能超过 100%。
energy and delivers 350 J of useful mechanical energy. Its efficiency is (350 / 500) × 100% = 70%. The remaining 150 J is mostly lost as heat and sound.
示例：一台电动机输入 500 J 电能，输出 350 J 有用机械能，其效率为 (350 / 500) × 100% = 70%。其余 150 J 主要以热量和声音的形式损失。
A Sankey diagram shows the same information graphically: the width of each arrow is proportional to the energy it represents, and the total width never increases from left to right.
桑基图(Sankey diagram)以图形方式表达同样的信息：每条箭头的宽度与它代表的能量成正比，且从左到右总宽度不会增加。
7. Stress, Strain and Young's Modulus | 应力、应变与杨氏模量
divided by cross-sectional area (A) and is measured in pascals (Pa), where 1 Pa = 1 N/m².
应力(σ)是材料单位面积所承受的内部力。它等于力(F)除以横截面积(A)，单位为帕斯卡(Pa)，1 Pa = 1 N/m²。
σ = F / A (1 Pa = 1 N/m² 1 MPa = 10⁶ Pa)
Strain (ε) is the fractional change in length. It is the extension (ΔL) divided by the original length (L₀), and because it is a ratio of two lengths it has no units.
应变(ε)是长度的相对变化量。它等于伸长量(ΔL)除以原长(L₀)。由于它是两个长度的比值，因此没有单位。
ε = ΔL / L₀ (no unit 无单位)
Young's modulus (E) describes the stiffness of a material and is the ratio of stress to strain within the elastic region. Its unit is the same as stress, the pascal.
杨氏模量(E)描述材料的刚度，在弹性范围内等于应力与应变之比。它的单位与应力相同，即帕斯卡。
E = σ / ε = (F × L₀) / (A × ΔL)
rain is 0.001, then E = 200 / 0.001 = 200 000 N/mm² = 200 GPa, which is typical for steel.
示例：一根横截面积为 2 mm² 的拉杆承受 400 N 载荷，应力 σ = 400 / 2 = 200 N/mm² = 200 MPa。若测得应变为 0.001，则 E = 200 / 0.001 = 200 000 N/mm² = 200 GPa，这与钢材的典型值相符。
8. Pressure, Density and Fluids | 压强、密度与流体
Density (ρ) is mass per unit volume. It is calculated as mass (m) divided by volume (V) and is measured in kilograms per cubic metre (kg/m³).
密度(ρ)是单位体积的质量。它等于质量(m)除以体积(V)，单位为千克每立方米(kg/m³)。
ρ = m / V m = ρ × V V = m / ρ
Pressure (p) is the force acting normally per unit
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