📚 Year 9 CIE Engineering: Quick Reference Formula & Theorem Handbook | Year 9 CIE 工程：公式定理速查手册
This handbook provides a concise reference of the key formulas and theorems you will encounter in the Year 9 CIE Engineering course. Keep it handy for quick revision and problem-solving.
本手册为你提供 Year 9 CIE 工程课程中核心公式与定理的简明参考。随时查阅，助你快速复习和解题。
1. Kinematic Equations of Motion | 运动学方程
When an object moves with constant acceleration in a straight line, its motion can be described by the following equations.
当物体沿直线做匀加速运动时，可用以下方程描述其运动。
v = u + a t
This equation links final velocity (v) to initial velocity (u), acceleration (a) and time (t).
该方程将末速度 (v) 与初速度 (u)、加速度 (a) 和时间 (t) 联系起来。
s = u t + ½ a t²
This equation gives the displacement (s) in terms of initial velocity, time and acceleration. The symbol ½ represents one-half.
此方程根据初速度、时间和加速度计算出位移 (s)。符号 ½ 表示二分之一。
v² = u² + 2 a s
This equation relates final velocity squared to initial velocity squared, acceleration and displacement.
该方程将末速度的平方与初速度的平方、加速度和位移关联起来。
2. Force and Newton's Laws | 力与牛顿定律
Newton's laws form the foundation of classical mechanics and are essential for analysing forces in engineering problems.
牛顿定律是经典力学的基础，对于分析工程问题中的力至关重要。
F = m × a
Newton's second law states that the net force (F) acting on an object equals its mass (m) multiplied by its acceleration (a).
牛顿第二定律指出，作用在物体上的净力 (F) 等于物体的质量 (m) 乘以加速度 (a)。
W = m × g
The weight (W) of an object is the force due to gravity, where g is the acceleration of free fall (approximately 9.8 m/s² on Earth).
物体的重量 (W) 是重力施加的力，g 是自由落体加速度（地球表面约为 9.8 m/s²）。
3. Moments, Levers and Equilibrium | 力矩、杠杆与平衡
Moments describe the turning effect of a force about a pivot. The principle of moments is used to analyse levers and structures in equilibrium.
力矩描述力绕支点的转动效果。力矩原理用于分析杠杆和处于平衡状态的结构。
M = F × d
The moment (M) of a force about a pivot is the product of the force (F) and the perpendicular distance (d) from the pivot to the line of action of the force.
力对支点的力矩 (M) 等于力 (F) 与支点到力作用线的垂直距离 (d) 的乘积。
F₁ × d₁ = F₂ × d₂
For a lever in equilibrium, the clockwise moment equals the anticlockwise moment. This is the principle of moments.
对于平衡的杠杆，顺时针力矩等于逆时针力矩。这就是力矩原理。
4. Pressure and Stress | 压力与应力
Pressure and stress both describe force distributed over an area, but they are applied in different contexts: pressure in fluids and stress in solid materials.
压力和应力都描述分布在一定面积上的力，但应用情境不同：压力用于流体，应力用于固体材料。
P = F / A
Pressure (P) is defined as the normal force (F) acting per unit area (A). Its SI unit is the pascal (Pa), equivalent to N/m².
压力 (P) 定义为作用在单位面积 (A) 上的法向力 (F)。其国际单位是帕斯卡 (Pa)，即 N/m²。
σ = F / A
Engineering stress (σ, sigma) uses the same formula but refers to the internal force per unit area within a material when an external load is applied.
工程应力 (σ, 西格玛) 公式相同，但指的是在外载荷作用下材料内部单位面积上的内力。
5. Work, Energy and Power | 功、能与功率
Work and energy are scalar quantities closely related to force and motion. Power is the rate at which work is done or energy is transferred.
功和能量是与力和运动密切相关的标量。功率是做功或能量传递的速率。
W = F × s
Work done (W) is calculated by multiplying the constant force (F) in the direction of motion by the displacement (s).
做功 (W) 等于运动方向上的恒力 (F) 乘以位移 (s)。
KE = ½ m v²
Kinetic energy (KE) depends on mass (m) and the square of velocity (v).
动能 (KE) 取决于质量 (m) 和速度的平方 (v²)。
PE = m g h
Gravitational potential energy (PE) near Earth's surface equals mass (m) × gravitational field strength (g) × change in height (h).
近地表的重力势能 (PE) 等于质量 (m) × 引力场强度 (g) × 高度变化 (h)。
P = W / t
Power (P) is the work done or energy transferred divided by the time (t) taken. SI unit: watt (W), equivalent to J/s.
功率 (P) 是做功或传递的能量除以所用时间 (t)。国际单位：瓦特 (W)，即 J/s。
6. Mechanical Advantage and Gear Ratios | 机械利益与传动比
Simple machines and gear trains multiply force or distance. Mechanical advantage (MA) quantifies the force multiplication, while gear ratios describe speed and torque relationships.
简单机械和齿轮系可以放大力量或距离。机械利益 (MA) 量化力的放大倍数，传动比描述转速和转矩的关系。
MA = Fₒᵤₜ / Fᵢₙ
ical advantage is the ratio of output force (load) to input force (effort). (Here Fₒᵤₜ = Fout, Fᵢₙ = Fin.)
机械利益是输出力（负载）与输入力（努力）的比值。
Velocity Ratio (VR) = dᵢₙ / dₒᵤₜ
Velocity ratio is the ratio of the distance moved by the effort (dᵢₙ) to the distance moved by the load (dₒᵤₜ).
速度比是努力移动的距离 (dᵢₙ) 与负载移动的距离 (dₒᵤₜ) 之比。
Gear Ratio = Number of teeth on driven gear / Number of teeth on driver gear
In a gear pair, the gear ratio determines the speed and direction change. A ratio greater than 1 means a reduction in speed but an increase in torque.
在齿轮副中，传动比决定速度和方向的改变。传动比大于1意味着减速增扭。
7. Ohm's Law and Electrical Power | 欧姆定律与电功率
Ohm's law is the fundamental relationship in electric circuits, linking voltage, current and resistance.
欧姆定律是电路中的基本关系，连接了电压、电流和电阻。
V = I × R
The potential difference (V) across a resistor equals the current (I) flowing through it multiplied by its resistance (R).
电阻两端的电势差 (V) 等于流过它的电流 (I) 乘以电阻 (R)。
P = I × V
Electrical power (P) consumed by a component is the product of the current through it and the voltage across it.
元件消耗的电功率 (P) 是流过它的电流与加在其两端电压的乘积。
Using Ohm's law, two alternative forms are often used:
利用欧姆定律，常会用到另外两种形式：
P = I² R and P = V² / R
8. Series and Parallel Circuits | 串联与并联电路
Resistors can be connected in series or parallel, and the total resistance follows distinct rules.
电阻可以串联或并联，总电阻遵循不同的规则。
Rₜₒₜₐₗ = R₁ + R₂ + R₃ + ... (series)
In a series circuit, the total resistance is the sum of the individual resistances. The same current flows through all components.
在串联电路中，总电阻等于各个电阻之和。流过所有元件的电流相同。
1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + 1/R₃ + ... (parallel)
In a parallel circuit, the reciprocal of the total resistance is the sum of the reciprocals of each resistance. The voltage across each branch is the same.
在并联电路中，总电阻的倒数等于各支路电阻的倒数之和。各支路两端的电压相同。
9. Specific Heat Capacity and Thermal Expansion | 比热容与热膨胀
Thermal properties of materials determine how they store and transfer heat, and how their dimensions change with temperature.
材料的热学性质决定了它们如何储存和传递热量，以及尺寸如何随温度变化。
Q = m c Δθ
The heat energy (Q) required to change the temperature of a substance depends on its mass (m), its specific heat capacity (c) and the temperature change (Δθ).
改变物质温度所需的热量 (Q) 取决于其质量 (m)、比热容 (c) 和温度变化 (Δθ)。
ΔL = α L₀ Δθ
The linear thermal expansion (ΔL) of a solid is proportional to its original length (L₀), the coefficient of linear expansion (α, alpha) and the temperature increase (Δθ).
固体的线性热膨胀量 (ΔL) 与其原长 (L₀)、线膨胀系数 (α, 阿尔法) 和温升 (Δθ) 成正比。
10. Stress-Strain and Young's Modulus | 应力-应变与杨氏模量
Engineers need to understand how materials deform under load. Stress, strain and Young's modulus describe elastic behaviour.
工程师需要了解材料在载荷下如何变形。应力、应变和杨氏模量描述了弹性行为。
σ = F / A
Tensile or compressive stress (σ) is the force applied per unit cross-sectional area.
拉伸或压缩应力 (σ) 是施加在单位横截面积上的力。
ε = ΔL / L₀
Strain (ε, epsilon) is the extension (ΔL) per unit original length (L₀); it has no units.
应变 (ε, 艾普西隆) 是单位原长的伸长量 (ΔL)；它没有单位。
E = σ / ε
Young's modulus (E) is the ratio of stress to strain within the elastic limit; it measures the stiffness of a material. Its unit is the pascal (Pa).
杨氏模量 (E) 是在弹性限度内应力与应变的比值；它衡量材料的刚度，单位是帕斯卡 (Pa)。
11. Mechanical Efficiency and Energy Conservation | 机械效率与能量守恒
No machine is perfect; efficiency compares useful output energy to input energy. The conservation of energy principle always applies.
没有完美的机器；效率比较有用输出能量与输入能量。能量守恒原理始终适用。
Efficiency (η) = (Useful work output / Total energy input) × 100%
ficiency is often expressed as a percentage. It can also be written as η = (MA / VR) × 100% for simple machines, where MA is mechanical advantage and VR is velocity ratio.
效率常用百分比表示。对于简单机械，也可写作 η = (MA / VR) × 100%，其中 MA 为机械利益，VR 为速度比。
Total energy input = Useful output + Wasted energy
In any real system, some energy is always dissipated as heat or sound due to friction and other losses.
在任何实际系统中，由于摩擦和其他损耗，总有一部能量以热或声的形式耗散。
12. Common Units and Conversions | 常用单位与换算
Using consistent SI units is vital in engineering calculations. The following table lists the base units and common derived units you should know.
在工程计算中使用一致的国际单位制 (SI) 至关重要。下表列出你该掌握的基本单位和常用导出单位。
| Quantity | SI Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Force | newton | N (kg m/s²) |
| Energy/Work | joule | J (N m) |
| Power | watt | W (J/s) |
| Pressure/Stress | pascal | Pa (N/m²) |
| Electric current | ampere | A |
| Voltage | volt | V (J/C) |
| Resistance | ohm | Ω (V/A) |
Prefixes are used to indicate multiples and submultiples: kilo- (k, 10³), mega- (M, 10⁶), centi- (c, 10⁻²), milli- (m, 10⁻³), micro- (μ, 10⁻⁶). Always check your units when substituting values into formulas.
前缀用于表示倍数和分数：千 (k, 10³)、兆 (M, 10⁶)、厘 (c, 10⁻²)、毫 (m, 10⁻³)、微 (μ, 10⁻⁶)。代入公式时务必核对单位。
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