📚 Year 8 OCR Engineering: Formula & Theorem Quick Reference Handbook | 八年级OCR工程：公式定理速查手册
This quick reference handbook provides a concise collection of essential formulas, theorems, and key engineering concepts for Year 8 OCR Engineering students. Use it to review and reinforce your understanding of physical principles applied in mechanical, electrical, and structural systems.
本速查手册为八年级OCR工程学生精简整理了基本公式、定理和关键工程概念，帮助复习和巩固在机械、电气和结构系统中应用的物理原理。
1. Speed and Motion | 速度与运动
Speed describes how fast an object moves, defined as the distance travelled per unit time.
速度描述物体运动的快慢，定义为单位时间内通过的距离。
v = s / t
Here v is speed, s is distance and t is time. The SI unit of speed is metre per second (m/s or m·s⁻¹).
式中 v 为速度，s 为距离，t 为时间。速度的国际单位是米每秒（m/s 或 m·s⁻¹）。
Average speed can be calculated by total distance divided by total time, while instantaneous speed is the speed at a specific moment.
平均速度可用总距离除以总时间计算，瞬时速度则是某一特定时刻的速度。
In engineering, speed calculations are fundamental for vehicle design and transport systems.
在工程中，速度计算是车辆设计和运输系统的基础。
2. Force and Acceleration | 力与加速度
Newton’s second law states that the force acting on an object equals its mass multiplied by its acceleration.
牛顿第二定律指出，作用在物体上的力等于物体的质量乘以加速度。
F = m × a
Force F is measured in newtons (N), mass m in kilograms (kg) and acceleration a in metres per second squared (m/s²).
力 F 的单位是牛顿（N），质量 m 的单位是千克（kg），加速度 a 的单位是米每二次方秒（m/s²）。
Weight is a special force caused by gravity, calculated as the mass of an object times gravitational field strength.
重量是重力产生的特殊力，等于物体的质量乘以重力场强。
W = m × g
On Earth, g ≈ 9.8 N/kg (often rounded to 10 N/kg for estimation).
在地球上，g ≈ 9.8 N/kg（估算时通常取10 N/kg）。
3. Pressure | 压强
Pressure measures how concentrated a force is over an area. It is a key concept in hydraulics and structural engineering.
压强衡量力在面积上的集中程度，是液压和结构工程中的关键概念。
P = F / A
Pressure P is in pascals (Pa), force F in newtons, and area A in square metres (m²).
压强 P 的单位是帕斯卡（Pa），力 F 的单位是牛顿，面积 A 的单位是平方米（m²）。
A larger area reduces pressure for the same force, which is why broad tyres are used on heavy vehicles.
在相同力的情况下，较大的面积会减小压强，因此重型车辆使用宽轮胎。
Pressure acts evenly in a fluid, enabling hydraulic systems to multiply force.
压强在流体中均匀传递，从而使液压系统能够放大作用力。
4. Levers and Moments | 杠杆与力矩
A moment is the turning effect of a force about a pivot. It depends on the force magnitude and the perpendicular distance from the pivot.
力矩是力对支点的转动效应，取决于力的大小和支点到力作用线的垂直距离。
M = F × d
Moment M is in newton-metres (N·m), F is force in newtons, and d is the perpendicular distance in metres.
力矩 M 的单位是牛顿·米（N·m），F 是力（牛顿），d 是垂直距离（米）。
For a lever to be in balance, the sum of clockwise moments must equal the sum of anticlockwise moments.
杠杆平衡时，顺时针力矩之和必须等于逆时针力矩之和。
F₁ × d₁ = F₂ × d₂
A lever can multiply force if the effort is applied further from the pivot than the load, giving a mechanical advantage greater than one.
如果动力臂大于阻力臂，杠杆可以放大作用力，从而获得大于1的机械效益。
5. Work and Energy | 功与能量
Work is done when a force moves an object through a distance in the direction of the force. Energy is the capacity to do work.
当力使物体沿力的方向移动一段距离时，就做了功。能量是做功的能力。
W = F × d
Work W is in joules (J), force F in newtons, and distance d in metres.
功 W 的单位是焦耳（J），力 F 的单位是牛顿，距离 d 的单位是米。
Gravitational potential energy (GPE) is stored when an object is raised against gravity.
重力势能（GPE）是物体反抗重力被抬高时储存的能量。
GPE = m × g × h
m is mass (kg), g is gravitational field strength (N/kg), and h is height (m).
m 为质量（kg），g 为重力场强（N/kg），h 为高度（m）。
Kinetic energy (KE) is the energy of a moving object.
动能（KE）是运动物体具有的能量。
KE = ½ × m × v²
Here v is the speed of the object in m/s.
v 为物体速度，单位为 m/s。
6. Power | 功率
Power is the rate at which work is done or energy is transferred. It tells us how quickly a machine can perform a task.
功率是做功或能量转移的速率，反映机器完成任务的快慢。
P = W / t
Power P is in watts (W), work W in joules, and time t in seconds.
功率 P 的单位是瓦特（W），功 W 的单位是焦耳，时间 t 的单位是秒。
High-power engines do the same amount of work in less time, which is essential for accelerating vehicles quickly.
高功率发动机可以在更短时间内完成相同的功，这对于快速加速车辆至关重要。
7. Efficiency | 效率
Efficiency measures how much of the input energy is converted into useful output energy. No real machine is 100% efficient due to friction and other losses.
效率衡量输入能量中有多少转化为有用的输出能量。由于摩擦等损耗，实际机器的效率不可能达到100%。
η = (Useful output energy ÷ Total input energy) × 100%
Efficiency can also be calculated using useful power output and total power input.
效率也可用有用输出功率与总输入功率计算。
η = (Useful power output ÷ Total power input) × 100%
Engineers aim to maximise efficiency to save energy and reduce operating costs.
工程师力求最大化效率，以节约能源并降低运行成本。
8. Ohm’s Law and Resistance | 欧姆定律与电阻
Ohm’s law relates voltage, current, and resistance in an electrical circuit. It is fundamental to analysing almost all electronic and power systems.
欧姆定律描述了电路中电压、电流和电阻之间的关系，是分析几乎所有电子与电力系统的基础。
V = I × R
Voltage V is in volts (V), current I in amperes (A), and resistance R in ohms (Ω).
电压 V 的单位是伏特（V），电流 I 的单位是安培（A），电阻 R 的单位是欧姆（Ω）。
Resistance describes how much a component opposes the flow of electric current; higher resistance results in a smaller current for a given voltage.
电阻描述元件对电流阻碍作用的大小；在电压一定时，电阻越大，电流越小。
9. Series and Parallel Circuits | 串联与并联电路
Components in a circuit can be arranged in series or parallel, and the total resistance is calculated differently for each configuration.
电路中的元件可以串联或并联，总电阻的计算方法因连接方式而异。
In a series circuit, the total resistance is simply the sum of all individual resistances.
在串联电路中，总电阻等于各电阻之和。
Rₜₒₜₐₗ = R₁ + R₂ + R₃ + …
In a parallel circuit, the reciprocal of the total resistance equals the sum of the reciprocals of individual resistances.
在并联电路中，总电阻的倒数等于各电阻倒数之和。
1 / Rₜₒₜₐₗ = 1 / R₁ + 1 / R₂ + 1 / R₃ + …
For two resistors in parallel, a convenient form is Rₜₒₜₐₗ = (R₁ × R₂) / (R₁ + R₂).
对于两个并联电阻，常用公式为 Rₜₒₜₐₗ = (R₁ × R₂) / (R₁ + R₂)。
10. Density and Material Properties | 密度与材料性质
Density describes how much mass is packed into a given volume. It helps engineers select materials for lightweight or heavy-duty designs.
密度表示单位体积内包含的质量，帮助工程师为轻量化或重型设计选择材料。
ρ = m / V
Density ρ is in kg/m³, mass m in kg, and volume V in m³.
密度 ρ 的单位是 kg/m³，质量 m 的单位是 kg，体积 V 的单位是 m³。
An object will float if its average density is less than the density of the fluid it displaces – a principle used in ship and submarine design.
物体的平均密度小于所排开流体的密度时，物体会漂浮——这一原理用于船舶和潜艇设计。
Simple material properties such as stiffness and strength are often introduced through the concepts of stress (σ = F / A) and strain (ε = ΔL / L₀).
简单的材料特性（如刚度和强度）常通过应力（σ = F / A）和应变（ε = ΔL / L₀）的概念引入。
- Stress is the internal force per unit area.
- 应力是单位面积承受的内力。
- Strain is the extension divided by the original length.
- 应变是伸长量除以原始长度。
11. Mechanical Advantage and Gear Ratios | 机械效益与传动比
Mechanical advantage (MA) tells us how much a machine multiplies the effort force put into it.
机械效益（MA）表示机器将输入力放大的倍数。
MA = Load force / Effort force
In gear systems, the velocity ratio (gear ratio) is the ratio of the number of teeth on the driven gear to the number of teeth on the driver gear.
在齿轮系统中，转速比（传动比）是从动齿轮齿数与主动齿轮齿数之比。
Gear ratio = Teeth on driven gear / Teeth on driver gear
A high gear ratio reduces output speed but increases torque, useful for climbing steep slopes or lifting heavy loads.
大传动比降低输出转速但增大扭矩，适用于爬坡或提升重物。
Simple machines such as pulleys, inclined planes and wheel-and-axle also provide mechanical advantage, allowing engineers to design efficient lifting and moving systems.
滑轮、斜面、轮轴等简单机械同样提供机械效益，帮助工程师设计高效的提升和运输系统。
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