📚 Year 9 CIE Engineering: Formula & Theorem Quick Reference Handbook | Year 9 CIE 工程：公式定理速查手册
This quick reference guide is tailored for Year 9 students following the CIE Engineering pathway. It consolidates the core formulas, principles, and definitions you will regularly apply in design, mechanics, and electronics projects. Use it as a portable summary to sharpen your problem-solving skills before lessons, assessments, or practical tasks.
本速查手册专为学习 CIE 工程课程的 Year 9 学生设计。它汇总了你在设计、力学和电子项目中将反复运用的核心公式、定理与定义。你可以把它作为便携式总结，在课前、评估或实践任务前用来打磨解题技巧。
1. Forces and Linear Motion | 力与直线运动
The relationship between resultant force, mass, and acceleration is the bedrock of dynamics. When multiple forces act on an object, you must find the net force along the line of action before applying the equation.
合力、质量与加速度之间的关系是动力学的基础。当多个力作用在一个物体上时，你必须先求出沿作用线的净力，再代入公式。
F = m × a
F = resultant force (newtons, N), m = mass (kilograms, kg), a = acceleration (metres per second squared, m/s²).
F = 合力（牛顿，N），m = 质量（千克，kg），a = 加速度（米每二次方秒，m/s²）。
The acceleration is always in the direction of the resultant force. If the mass is constant, doubling the force doubles the acceleration. Conversely, for the same force, twice the mass gives half the acceleration.
加速度的方向总是与合力的方向相同。如果质量恒定，力加倍则加速度加倍。反过来，当力相同时，质量变为两倍，加速度就减半。
The equation for average speed in uniform motion is a simpler tool for steady-speed problems or for finding a typical value in a journey with changing speeds.
匀速运动中平均速度的公式更简单，适合匀速问题或求变速旅程中的典型值。
v = s / t
v = average speed (m/s), s = distance travelled (metres, m), t = time taken (seconds, s).
v = 平均速度（m/s），s = 行进距离（米，m），t = 所用时间（秒，s）。
2. Moments and Turning Forces | 力矩与转动效应
A moment is the turning effect produced by a force acting at a distance from a pivot. In equilibrium, the total clockwise moments equal the total anticlockwise moments about any pivot. This principle underpins lever design, bridge balances, and robotic arm calculations.
力矩是力在离支点一定距离处产生的转动效应。在平衡状态下，对任一转动轴的顺时针力矩之和等于逆时针力矩之和。这一原理支撑着杠杆设计、桥梁平衡和机械臂计算。
M = F × d
M = moment (newton metres, N·m), F = force applied (N), d = perpendicular distance from the pivot to the line of action of the force (m).
M = 力矩（牛顿·米，N·m），F = 施加的力（N），d = 从支点到力作用线的垂直距离（m）。
When a force is applied at an angle, only the component perpendicular to the lever arm generates a moment. The parallel component produces zero turning effect. For a uniform beam, the weight acts at its centre of mass, which is usually the geometric centre.
当力以某个角度施加时，只有垂直于杠杆臂的分量才产生力矩。平行分量产生的转动效应为零。对于均质梁，重力作用在其质心处，通常就是几何中心。
The principle of moments can be stated as: sum of clockwise moments = sum of anticlockwise moments, for a body in rotational equilibrium. A simple supported beam with a load placed off-centre is a classic application.
力矩原理可陈述为：对于处于转动平衡的物体，顺时针力矩之和等于逆时针力矩之和。一个简支梁上放置偏载的情况就是典型应用。
3. Work, Energy and Power | 功、能与功率
Work is done when a force moves its point of application in the direction of the force. In engineering, understanding work and power enables you to specify motors, estimate fuel consumption, and design efficient mechanisms.
当一个力使其作用点沿力的方向移动时，力就做了功。在工程中，理解功和功率能帮助你选配电动机、估算燃料消耗并设计高效机构。
W = F × d
W = work done (joules, J), F = constant force (N), d = distance moved in the direction of the force (m).
W = 做功（焦耳，J），F = 恒力（N），d = 在力的方向上移动的距离（m）。
Gravitational potential energy gained by an object lifted at constant speed is a special case of work against gravity. The force needed equals the weight, and the distance is the height gained.
物体匀速提升时所获得的引力势能是克服重力做功的一个特例。所需的力等于重量，距离就是上升的高度。
Eₚ = m × g × h
Eₚ = gravitational potential energy (J), m = mass (kg), g = gravitational field strength (≈ 9.8 N/kg on Earth), h = height (m).
Eₚ = 重力势能（J），m = 质量（kg），g = 重力场强度（地球取约 9.8 N/kg），h = 高度（m）。
Kinetic energy is the energy stored in a moving object. The formula reveals that speed has a squared effect, so doubling speed quadruples kinetic energy, which is critical in crash safety design.
动能是运动物体存储的能量。公式显示速度有平方效应，因此速度加倍会使动能变成四倍，这在碰撞安全设计中至关重要。
Eₖ = ½ × m × v²
Eₖ = kinetic energy (J), m = mass (kg), v = velocity (speed in a given direction, m/s).
Eₖ = 动能（J），m = 质量（kg），v = 速度（特定方向上的速率，m/s）。
Power is the rate of doing work or transferring energy. A powerful actuator completes the same job in a shorter time. The standard unit is the watt (W), where 1 W = 1 J/s.
功率是做功或传递能量的速率。一个强劲的执行器能在更短时间内完成相同的工作。标准单位是瓦特（W），1 W = 1 J/s。
P = W / t and P = F × v (for constant velocity)
P = power (W), W = work done (J), t = time (s); for constant velocity: F = force (N), v = velocity (m/s).
P = 功率（W），W = 功（J），t = 时间（s）；匀速运动时：F = 力（N），v = 速度（m/s）。
4. Efficiency | 效率
Engineering systems always lose some energy through friction, heat, sound, or vibration. Efficiency compares the useful energy or power output to the total input, guiding material selection and lubrication strategies.
工程系统总会因摩擦、热、声音或振动损失一部分能量。效率将有用的输出能量或功率与总输入进行比较，从而指导材料选择和润滑策略。
Efficiency = (Useful energy output / Total energy input) × 100%
Efficiency has no units and is often expressed as a percentage. It can also be calculated using power: (Useful power output / Total power input) × 100%.
效率 没有单位，通常以百分比表示。也可用功率计算：（有用输出功率 / 总输入功率）× 100%。
For a pulley or gearbox, the mechanical efficiency is always less than 100%. High-efficiency designs reduce wasted energy, which saves running costs and reduces overheating. Even a well-lubricated chain drive seldom exceeds 98% efficiency.
对于滑轮组或变速箱，机械效率始终低于 100%。高效率设计能减少浪费的能量，从而节省运行成本并降低过热风险。即便润滑良好的链传动，效率也极少超过 98%。
5. Simple Machines and Mechanical Advantage | 简单机械与机械效益
Simple machines such as levers, inclined planes, and pulleys make tasks easier by altering the magnitude or direction of a force. Their effectiveness is described by mechanical advantage (MA) and velocity ratio (VR).
杠杆、斜面和滑轮等简单机械通过改变力的大小或方向使任务变得更轻松。它们的效能用机械效益（MA）和速比（VR）来描述。
Mechanical Advantage (MA) = Load / Effort
Load is the output force the machine overcomes (N), Effort is the input force applied (N). If MA > 1, the machine amplifies force.
Load（荷载）是机器克服的输出力（N），Effort（力效）是施加的输入力（N）。若 MA > 1，机器放大出力。
Velocity Ratio (VR) = Distance moved by effort / Distance moved by load
The velocity ratio is a purely geometric property depending on the lengths of lever arms, number of rope segments, or slope of an inclined plane. Efficiency links these two: Efficiency = MA / VR × 100%.
速比是一个纯几何属性，取决于杠杆臂长、绳索根数或斜面坡度。效率将这两者联系起来：效率 = MA / VR × 100%。
A Class 1 lever (fulcrum between effort and load) can have MA > 1, equal to 1, or < 1. A wheelbarrow is a Class 2 lever with MA always > 1. Tweezers are a Class 3 lever with MA < 1, sacrificing force for increased speed and control at the load end.
第一类杠杆（支点在力效和荷载之间）的 MA 可以大于、等于或小于 1。手推车是第二类杠杆，MA 始终大于 1。镊子是第三类杠杆，MA < 1，牺牲力来换取荷载端更高的速度和控制精度。
6. Electricity Basics | 电学基础
Ohm’s law is the central rule for resistive circuits. It applies to metallic conductors at constant temperature and to many electronic components within their linear range. The formula links voltage, current, and resistance in a directly proportional relationship for a fixed resistance.
欧姆定律是阻性电路的核心法则。它适用于恒温下的金属导体以及许多在其线性范围内的电子元件。对于固定电阻，该公式将电压、电流和电阻以正比关系联系起来。
V = I × R
V = potential difference or voltage (volts, V), I = current (amperes, A), R = resistance (ohms, Ω).
V = 电势差或电压（伏特，V），I = 电流（安培，A），R = 电阻（欧姆，Ω）。
Electrical power dissipated in a resistor can be expressed in three useful forms by substituting Ohm’s law. The choice depends on which two quantities you know.
电阻器上耗散的电功率可通过代入欧姆定律表达为三种有用的形式。选择哪种取决于你已知哪两个量。
P = V × I or P = I² × R or P = V² / R
P = electrical power (W), V = voltage (V), I = current (A), R = resistance (Ω).
P = 电功率（W），V = 电压（V），I = 电流（A），R = 电阻（Ω）。
Resistors in series and parallel behave differently. These rules are essential when you design sensor circuits or decide how to split voltage for a control system.
串联和并联电阻的特性不同。当你设计传感器电路或决定如何为控制系统分压时，这些规则至关重要。
| Configuration (配置) | Total Resistance (总电阻) | Current / Voltage rule (电流/电压规则) |
|---|---|---|
| Series (串联) | Rtotal = R₁ + R₂ + R₃ + … | Same current through all; voltage divides. |
| Parallel (并联) | 1/Rtotal = 1/R₁ + 1/R₂ + 1/R₃ + … | Same voltage across each branch; current divides. |
For two parallel resistors, a quick formula is: Rtotal = (R₁ × R₂) / (R₁ + R₂). The total resistance is always smaller than the smallest individual resistance in the parallel set.
对于两个并联电阻，一个快捷公式是：Rtotal = (R₁ × R₂) / (R₁ + R₂)。总电阻总是小于并联组中最小的单个电阻。
7. Material Properties: Stress and Strain | 材料性质：应力与应变
When a structural member is loaded, it experiences internal forces that resist deformation. The intensity of these internal forces is called stress. The resulting deformation relative to the original dimensions is called strain. Together they define the stiffness of a material.
当构件受载时，其内部会产生抵抗变形的力。这些内力的密集程度称为应力。由此产生的相对于原始尺寸的变形称为应变。二者共同定义了材料的刚度。
Stress (σ) = F / A
σ = tensile or compressive stress (pascals, Pa, or N/m²), F = applied force (N), A = cross-sectional area (m²).
σ = 拉伸或压缩应力（帕斯卡，Pa 或 N/m²），F = 施加的力（N），A = 横截面积（m²）。
Strain (ε) = ΔL / L
ε = strain (dimensionless; often given as a percentage), ΔL = change in length (m), L = original length (m).
ε = 应变（无量纲，常以百分比给出），ΔL = 长度变化量（m），L = 原始长度（m）。
Young’s modulus (E) measures a material’s elastic stiffness. It applies only in the linear, elastic region of the stress–strain curve where the material returns to its original shape when unloaded.
杨氏模量（E）衡量材料的弹性刚度。它仅适用于应力-应变曲线的线弹性区域，在该区域内卸载后材料能恢复原状。
E = σ / ε
E = Young’s modulus (Pa or N/m²). A high E means the material is stiff and stretches very little under load (e.g., steel). A low E indicates flexibility (e.g., rubber).
E = 杨氏模量（Pa 或 N/m²）。E 值高意味着材料刚硬，受载时拉伸极小（如钢）。E 值低则表示柔韧（如橡胶）。
8. Pressure in Solids and Fluids | 固体与流体中的压力
Pressure is the amount of force acting perpendicularly on a unit area. Designing foundations, pneumatic tyres, and hydraulic jacks all rely on controlling pressure to avoid sinking, bursting, or slipping.
压力是垂直作用在单位面积上的力的大小。设计地基、充气轮胎和液压千斤顶都依赖于控制压力，以避免下陷、爆裂或滑移。
p = F / A
p = pressure (pascals, Pa, where 1 Pa = 1 N/m²), F = force acting perpendicular to the surface (N), A = area over which the force is spread (m²).
p = 压力（帕斯卡，Pa，1 Pa = 1 N/m²），F = 垂直作用在表面上的力（N），A = 力分布的面积（m²）。
In a liquid, pressure increases with depth due to the weight of the fluid above. This hydrostatic pressure acts equally in all directions at a given depth and is the reason dam walls are thicker at the bottom.
在液体中，由于上方流体自重，压力随深度增加。这种静水压力在给定深度处向各个方向均等作用，这也是水坝底部更厚的原因。
p = ρ × g × h
ρ = density of the fluid (kg/m³), g = gravitational field strength (N/kg), h = height (or depth) of the fluid column (m).
ρ = 流体密度（kg/m³），g = 重力场强度（N/kg），h = 液柱高度（或深度）（m）。
Pascal’s principle states that a pressure change applied to an enclosed fluid is transmitted undiminished to every part of the fluid and to the walls of the container. This is the working principle behind hydraulic brakes and car lifts.
帕斯卡原理指出，施加在密闭流体上的压强变化会大小不变地传递到流体的每一部分及容器壁上。这是液压制动和汽车升降机的工作原理。
9. Fluid Flow Continuity | 流体流动连续性
For an incompressible fluid (most liquids) flowing through a closed pipe, the mass flow rate remains constant. When the pipe narrows, the fluid speed must increase, and vice versa. This principle is used in carburettors and flow meters.
对于在闭合管道中流动的不可压缩流体（大多数液体），质量流量保持恒定。当管道变窄时，流体速度必然增加，反之亦然。化油器和流量计就运用了这一原理。
A₁ × v₁ = A₂ × v₂
A₁, A₂ = cross-sectional areas at points 1 and 2 (m²), v₁, v₂ = average flow velocities at those points (m/s).
A₁、A₂ = 点 1 和点 2 处的横截面积（m²），v₁、v₂ = 相应点的平均流速（m/s）。
The volumetric flow rate Q = A × v is measured in m³/s. In a heating or cooling system, matching the flow rate to the pipe diameter helps maintain efficient heat transfer without wasting pumping power.
体积流量 Q = A × v，单位为 m³/s。在供暖或冷却系统中，使流量与管径匹配有助于维持高效的热传递，同时避免浪费泵送功率。
10. Gears and Mechanical Drives | 齿轮与机械传动
Gears transmit rotational motion and torque between shafts. The gear ratio determines the speed and torque conversion, making it a fundamental design choice in vehicles, clocks, and industrial machinery. For two meshing spur gears, the number of teeth is directly proportional to the pitch circle diameter.
齿轮在轴之间传递旋转运动和扭矩。齿轮比决定了转速与扭矩的转换，使其成为车辆、时钟和工业机械中的基本设计选择。对于两个啮合的直齿轮，齿数与节圆直径成正比。
Gear Ratio = (Number of teeth on driven gear) / (Number of teeth on driver gear)
Also: Gear Ratio = Speed of driver / Speed of driven = Torque of driven / Torque of driver (ignoring friction).
同样：齿轮比 = 主动轮转速 / 从动轮转速 = 从动轮扭矩 / 主动轮扭矩（忽略摩擦）。
A gear ratio greater than 1 means the driven gear turns more slowly but with higher torque (speed reducer). A ratio less than 1 gives higher output speed but lower torque (overdrive). Compound gear trains multiply ratios, allowing large speed reductions in a compact space.
齿轮比大于 1 意味着从动轮转速更慢但扭矩更大（减速器）。齿轮比小于 1 则提供更高的输出转速但扭矩更低（超速传动）。复式齿轮系将各级传动比相乘，从而在紧凑空间内实现大幅减速。
For a simple belt-and-pulley system, the velocity ratio is given by the diameter ratio: VR = Diameter of driven pulley / Diameter of driver pulley. The mechanical advantage then follows from the efficiency as before.
对于简单的带轮传动系统，速比由直径比给出：VR = 从动轮直径 / 主动轮直径。然后如前所述，机械效益再通过效率导出。
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