📚 Year 9 OCR Engineering: Formula & Theorem Quick Reference Guide | 9年级OCR工程：公式定理速查手册
This quick reference handbook brings together the essential formulae and theorems that form the backbone of Year 9 OCR Engineering. Mastery of these principles enables you to apply mathematics confidently to real-world engineering problems, from calculating forces in structures to analysing electrical circuits.
本速查手册汇集了支撑9年级OCR工程课程的核心公式与定理。掌握这些原理能使你自信地将数学应用于实际工程问题，从计算结构受力到分析电路。
1. Speed and Velocity | 速度与速率
Speed is a scalar quantity measuring how fast an object moves, defined as the distance travelled divided by the time taken. The average speed v is given by v = d ÷ t, where d is distance in metres (m) and t is time in seconds (s).
速度是一个标量，衡量物体运动的快慢，定义为移动距离除以所用时间。平均速度 v 的计算公式为 v = d ÷ t，其中 d 为距离（单位米 m），t 为时间（单位秒 s）。
v = d ÷ t
Velocity is the vector equivalent, specifying both magnitude and direction. In many engineering calculations, when direction is constant, the same formula applies but displacement s is used instead of distance d.
速率是速度的矢量对应量，同时说明大小和方向。在许多工程计算中，当方向不变时，可使用同样公式，但用位移 s 代替距离 d。
2. Acceleration | 加速度
Acceleration is the rate of change of velocity. For uniform acceleration, it is calculated as the change in velocity divided by the time taken: a = (v – u) ÷ t, where v is final velocity (m/s), u is initial velocity (m/s) and t is time (s).
加速度是速度变化的快慢。对于匀加速运动，计算公式为速度变化量除以所用时间：a = (v – u) ÷ t，其中 v 为末速度（m/s），u 为初速度（m/s），t 为时间（s）。
a = (v – u) ÷ t
The SI unit of acceleration is metres per second squared (m/s²). A negative value indicates deceleration or retardation.
加速度的国际单位是米每二次方秒（m/s²）。负值表示减速或负加速度。
3. Newton’s Second Law of Motion | 牛顿第二运动定律
Newton’s second law describes the relationship between the resultant force acting on an object, its mass, and the acceleration produced. It states that resultant force F (N) = mass m (kg) × acceleration a (m/s²).
牛顿第二定律描述了作用在物体上的合力、物体质量以及产生的加速度之间的关系。定律指出，合力 F（N）= 质量 m（kg）× 加速度 a（m/s²）。
F = m × a
This law is fundamental for analysing dynamic systems, vehicle motion and structural loads. It also explains why heavier objects require greater force to achieve the same acceleration.
这条定律是分析动态系统、车辆运动和结构载荷的基础。它也解释了为何较重的物体需要更大的力才能达到相同的加速度。
4. Moments | 力矩
A moment is the turning effect of a force about a pivot. The size of the moment is calculated by multiplying the force by the perpendicular distance from the pivot to the line of action of the force: M = F × d.
力矩是力绕支点产生的转动效应。力矩的大小等于力乘以从支点到力作用线的垂直距离：M = F × d。
M = F × d
Moment is measured in newton metres (N m). In equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments – a principle used extensively in lever and gear design.
力矩的单位是牛顿米（N m）。在平衡状态下，顺时针力矩之和等于逆时针力矩之和——这一原理广泛应用于杠杆和齿轮设计中。
5. Work and Energy | 功与能
Work is done when a force moves an object through a distance in the direction of the force. Work done W (J) = force F (N) × displacement d (m).
当力使物体沿力的方向移动一段距离时，力做了功。做功 W（J）= 力 F（N）× 位移 d（m）。
W = F × d
Moving objects possess kinetic energy (KE). The formula for kinetic energy is KE = ½ × m × v², where m is mass (kg) and v is velocity (m/s).
运动的物体具有动能（KE）。动能的计算公式为 KE = ½ × m × v²，其中 m 为质量（kg），v 为速度（m/s）。
KE = ½ × m × v²
Objects lifted against gravity gain gravitational potential energy (GPE). The change in GPE is calculated as GPE = m × g × h, with g being gravitational field strength (9.8 N/kg on Earth) and h the height (m).
被举高的物体反抗重力做功，获得了重力势能（GPE）。重力势能的变化量公式为 GPE = m × g × h，其中 g 为重力场强度（地球上约为 9.8 N/kg），h 为高度（m）。
GPE = m × g × h
6. Power and Efficiency | 功率与效率
Power is the rate of doing work or transferring energy. It is defined as power P (W) = work done W (J) ÷ time t (s), or equally P = E ÷ t where E is energy transferred in joules.
功率是做功或传递能量的快慢。定义为功率 P（W）= 做功 W（J）÷ 时间 t（s），或等效为 P = E ÷ t，其中 E 为以焦耳为单位的转换能量。
P = W ÷ t
Efficiency measures how well a system converts input energy into useful output. It is expressed as a percentage: efficiency = (useful output energy ÷ total input energy) × 100%.
效率衡量系统将输入能量转化为有用输出能量的有效程度。以百分比表示：效率 =（有用输出能量 ÷ 总输入能量）× 100%。
η = (Useful output ÷ Total input) × 100%
7. Ohm’s Law | 欧姆定律
Ohm’s law is a fundamental principle in electrical engineering, linking voltage, current and resistance in a conductor at constant temperature. It states that voltage V (V) = current I (A) × resistance R (Ω).
欧姆定律是电气工程中的基本原则，描述了恒温条件下导体中电压、电流与电阻的关系。公式为：电压 V（V）= 电流 I（A）× 电阻 R（Ω）。
V = I × R
This relationship allows engineers to design circuits by selecting appropriate resistors and calculating current flow. The resistance of a component remains constant only if the temperature is stable.
这一关系使工程师能够通过选择合适的电阻和计算电流来设计电路。只有在温度稳定的情况下，元件的电阻才会保持恒定。
8. Electrical Power | 电功率
Electrical power is the rate at which electrical energy is transferred in a circuit. It can be calculated in two useful ways: P = V × I, and by combining with Ohm’s law, P = I² × R.
电功率是电路传输电能的快慢。可通过两种实用方式计算：P = V × I，以及结合欧姆定律得到的 P = I² × R。
P = V × I
P = I² × R
Power is measured in watts (W). These formulae are essential when selecting components such as resistors, cables and power supplies to prevent overheating and ensure safe operation.
功率以瓦特（W）为单位。这些公式在选择电阻、电缆和电源等元件时至关重要，可防止过热并确保安全运行。
9. Stress and Strain | 应力与应变
Stress quantifies the internal forces that particles of a material exert on each other when an external load is applied. Tensile or compressive stress σ (sigma) is force F divided by cross-sectional area A: σ = F ÷ A, measured in pascals (Pa).
应力量化了施加外部载荷时，材料内部粒子间相互作用的内力。拉伸或压缩应力 σ 等于力 F 除以截面积 A：σ = F ÷ A，单位为帕斯卡（Pa）。
σ = F ÷ A
Strain describes the deformation caused by stress. It is defined as the change in length ΔL divided by the original length L: ε = ΔL ÷ L. Strain is dimensionless as it is a ratio of lengths.
应变描述了应力引起的变形。其定义为长度变化量 ΔL 除以原始长度 L：ε = ΔL ÷ L。应变是无量纲的，因为它是长度的比值。
ε = ΔL ÷ L
10. Hooke’s Law | 胡克定律
Hooke’s law applies to materials that deform elastically, meaning they return to their original shape when the load is removed. It states that the force F applied to a spring is directly proportional to its extension e, as long as the elastic limit is not exceeded: F = k × e, where k is the spring constant (N/m).
胡克定律适用于发生弹性变形的材料，即卸载后能恢复原状的材料。定律指出，只要不超过弹性极限，施加在弹簧上的力 F 与弹簧的伸长量 e 成正比：F = k × e，其中 k 为弹簧刚度（N/m）。
F = k × e
The spring constant k represents the stiffness of the spring; a stiffer spring has a higher k value and requires more force to produce the same extension.
弹簧刚度 k 代表了弹簧的硬度；越硬的弹簧 k 值越大，需要更大的力才能产生相同的伸长量。
11. Density | 密度
Density is a material property that links mass and volume. It is defined as density ρ (rho) = mass m (kg) ÷ volume V (m³). The SI unit is kilograms per cubic metre (kg/m³).
密度是将质量与体积联系起来的材料属性。定义为密度 ρ = 质量 m（kg）÷ 体积 V（m³）。国际单位为千克每立方米（kg/m³）。
ρ = m ÷ V
Knowledge of density is crucial for selecting materials in engineering, calculating buoyancy, and determining whether a structural component is light enough for its intended use.
了解密度对于工程中的材料选择、计算浮力以及判断结构部件是否足够轻以满足其预期用途至关重要。
12. Pressure | 压力
Pressure measures how concentrated a force is over a surface area. It is calculated as pressure p (Pa) = force F (N) ÷ area A (m²). A sharp tool exerts high pressure because the same force is applied to a small area.
压力衡量力在表面积上的集中程度。计算公式为压力 p（Pa）= 力 F（N）÷ 面积 A（m²）。锋利工具因相同力作用于小面积而产生高压力。
p = F ÷ A
In hydraulic systems, pressure is transmitted equally throughout a fluid, enabling force multiplication – Pascal’s principle, which directly stems from this pressure definition.
在液压系统中，压力在液体中均匀传递，可实现力的放大——帕斯卡原理，这直接源自此压力定义。
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