📚 Year 9 Cambridge Engineering Formula & Theorem Quick Reference Handbook | 剑桥九年级工程公式定理速查手册
This handbook gathers the essential formulas, principles, and quick-reference notes you need for Year 9 Cambridge Engineering. Use it to revise key concepts from mechanics, electricity, materials, and design processes. Each section presents a concept in plain English followed by its Chinese explanation, so you can check your understanding in both languages and strengthen your bilingual scientific vocabulary.
本手册汇集了剑桥九年级工程课程所需的重要公式、定理和速查要点。内容涵盖力学、电学、材料与设计过程等关键领域。每个小节先用英文讲解概念，再用中文解释，帮助你用双语巩固理解，提升学术英语和中文专业词汇。
1. Motion – Average Speed | 运动 – 平均速度
Average speed is the total distance travelled divided by the total time taken. It is a scalar quantity, which means it only has magnitude and no direction.
平均速度是总路程除以总时间。它是一个标量，即只有大小没有方向。
v = s / t
Where v is average speed (m/s), s is distance (m), and t is time (s).
其中 v 为平均速度（米/秒），s 为路程（米），t 为时间（秒）。
1 km = 1000 m, 1 hour = 3600 s. The triangle method helps rearrange: s = v × t, t = s / v.
牢记单位换算：1 千米 = 1000 米，1 小时 = 3600 秒。可用三角形法变形公式：s = v × t，t = s / v。
2. Acceleration | 加速度
Acceleration is the rate at which an object changes its velocity. It is a vector quantity, meaning it has both magnitude and direction.
加速度是物体速度变化的快慢。它是矢量，既有大小也有方向。
a = (v – u) / t
Where a is acceleration (m/s²), v is final velocity (m/s), u is initial velocity (m/s), and t is time (s).
其中 a 为加速度（米/秒²），v 为末速度（米/秒），u 为初速度（米/秒），t 为时间（秒）。
Acceleration due to gravity near Earth’s surface is approximately 9.8 m/s², often rounded to 10 m/s² in quick estimates. Negative acceleration is called deceleration or retardation.
地球表面附近的重力加速度约为 9.8 米/秒²，快速估算时常取 10 米/秒²。负加速度称为减速。
3. Newton’s Second Law | 牛顿第二定律
The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This law links force, mass, and acceleration.
物体的加速度与作用在其上的合力成正比，与物体的质量成反比。该定律将力、质量和加速度联系在一起。
F = m × a
Where F is net force (N), m is mass (kg), and a is acceleration (m/s²).
其中 F 为合力（牛顿），m 为质量（千克），a 为加速度（米/秒²）。
/s². When several forces act on a body, find the resultant force first, then apply F = ma.
1 牛顿是使 1 千克物体产生 1 米/秒² 加速度所需的力。当多个力作用时，先求合力再代入 F = ma。
4. Weight and Gravitational Field Strength | 重力与引力场强度
Weight is the force acting on an object due to gravity. It depends on the mass of the object and the gravitational field strength of the location.
重力是地球引力对物体的作用力。它与物体的质量和当地的引力场强度有关。
W = m × g
Where W is weight (N), m is mass (kg), and g is gravitational field strength (N/kg or m/s²).
其中 W 为重力（牛顿），m 为质量（千克），g 为引力场强度（牛/千克 或 米/秒²）。
e Moon, g ≈ 1.6 N/kg. Note that mass stays the same everywhere, but weight changes with g.
地球上 g ≈ 9.8 牛/千克；月球上 g ≈ 1.6 牛/千克。质量处处相同，但重力随 g 变化。
5. Work Done and Energy Transfer | 做功与能量转移
Work is done when a force moves an object in the direction of the force. Work is a measure of energy transfer.
当一个力使物体沿力的方向移动时，该力做了功。功是能量转移的量度。
W = F × d
Where W is work done (J), F is force (N), and d is distance moved in the direction of the force (m).
其中 W 为做功（焦耳），F 为力（牛顿），d 为沿力方向移动的距离（米）。
o the displacement does work. Lifting an object vertically against gravity: W = m × g × h.
如果力与运动方向不一致，只有沿位移方向的分力做功。竖直提升物体：W = m × g × h。
6. Power | 功率
Power is the rate at which work is done or energy is transferred. A higher power means the same amount of work is completed in less time.
功率是做功或能量转移的快慢。功率越大，完成相同功所需时间越短。
P = W / t
Where P is power (W, watts), W is work done or energy transferred (J), and t is time (s).
其中 P 为功率（瓦特），W 为做功或能量（焦耳），t 为时间（秒）。
: when a constant force moves an object at constant speed, P = F × v, where v is velocity.
1 瓦 = 1 焦/秒。另一种常用形式：恒力使物体匀速运动时 P = F × v，v 为速度。
7. Moment of a Force | 力矩
The moment of a force is its turning effect about a pivot. It depends on the size of the force and the perpendicular distance from the pivot to the line of action of the force.
力矩是力对支点的转动效应。它取决于力的大小以及支点到力作用线的垂直距离。
M = F × d
Where M is moment (Nm), F is force (N), and d is perpendicular distance from pivot (m).
其中 M 为力矩（牛·米），F 为力（牛顿），d 为支点到力的垂直距离（米）。
For a system in rotational equilibrium (no turning), the sum of clockwise moments equals the sum of anticlockwise moments about any pivot. This principle is used in levers, gears, and structural design.
转动平衡系统中，顺时针力矩之和等于逆时针力矩之和。该原理应用于杠杆、齿轮和结构设计。
8. Stress, Strain, and Young’s Modulus | 应力、应变与杨氏模量
Stress measures the internal force per unit area within a material. Strain measures the deformation relative to original dimensions. Young’s modulus describes a material’s stiffness.
应力衡量材料内部单位面积上的内力。应变是相对于原始尺寸的变形量。杨氏模量反映材料的刚度。
Stress: σ = F / A (unit: Pa or N/m²)
Strain: ε = ΔL / L₀ (no unit, ratio)
Young’s modulus: E = σ / ε (Pa)
应力：σ = F / A（单位帕斯卡或牛/米²）
应变：ε = ΔL / L₀（无单位，比值）
杨氏模量：E = σ / ε（帕斯卡）
Where F is force (N), A is cross-sectional area (m²), ΔL is change in length (m), L₀ is original length (m). E is only valid in the elastic region (Hooke’s law).
F 为力（牛），A 为横截面积（米²），ΔL 为长度变化（米），L₀ 为原始长度（米）。E 仅在弹性阶段（胡克定律范围内）有效。
9. Ohm’s Law and Electrical Power | 欧姆定律与电功率
Ohm’s law states that the current through a conductor is directly proportional to the voltage across it, provided temperature remains constant.
欧姆定律表明，在温度不变的情况下，通过导体的电流与其两端的电压成正比。
V = I × R
Where V is voltage (V), I is current (A), and R is resistance (Ω).
其中 V 为电压（伏特），I 为电流（安培），R 为电阻（欧姆）。
m’s law, P = I² × R = V² / R. Energy transferred: E = P × t (J or kWh for larger amounts).
电功率：P = V × I。结合欧姆定律可得 P = I² × R = V² / R。电能：E = P × t（焦耳或千瓦时用于较大电量）。
10. Mechanical Advantage and Velocity Ratio | 机械效益与速度比
Simple machines like levers, pulleys, and gears make work easier by multiplying force or changing direction. Mechanical advantage (MA) compares output force to input force.
杠杆、滑轮和齿轮等简单机械通过放大作用力或改变方向使工作更省力。机械效益（MA）是输出力与输入力的比值。
MA = Fₒᵤₜ / Fᵢₙ
distance moved by the load: VR = dₑ / dₗ (or number of rope segments in a pulley system).
速度比（VR）是动力移动距离与负载移动距离之比：VR = dₑ / dₗ（或在滑轮组中为承重绳段数）。
Efficiency = (MA / VR) × 100%. Friction always causes VR > MA, so efficiency is less than 100%.
效率 = (MA / VR) × 100%。摩擦使 VR 总大于 MA，因此效率低于 100%。
11. Pressure in Fluids | 流体压强
Pressure is the force exerted per unit area. In fluids, pressure acts equally in all directions and increases with depth.
压强是单位面积上承受的压力。流体中压强向各个方向等大传递，并随深度增加而增大。
p = F / A
p = h × ρ × g
Where p is pressure (Pa), F is force (N), A is area (m²), h is depth (m), ρ is fluid density (kg/m³), g is 9.8 N/kg. The second formula is for pressure due to a liquid column.
其中 p 为压强（帕），F 为力（牛），A 为面积（米²），h 为深度（米），ρ 为液体密度（千克/米³），g 取 9.8 牛/千克。第二式为液体柱压强公式。
Hydraulic systems use Pascal’s principle: pressure applied to an enclosed fluid is transmitted undiminished. This allows force multiplication with different piston areas.
液压系统利用帕斯卡原理：加在密闭液体上的压强能大小不变地向各个方向传递，可通过不同活塞面积实现力的放大。
12. Key Trigonometry for Engineering | 工程常用三角学
Right‑angled triangle ratios are essential for resolving forces, analysing structures, and calculating dimensions in design.
直角三角形边角关系是分解力、分析结构和设计尺寸计算的基础。
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
where c is the hypotenuse. These relationships help find unknown forces in trusses or slopes.
运用勾股定理：c² = a² + b²，c 为斜边。这些关系可用于求解桁架或斜面中的未知力。
When a force F acts at an angle θ to the horizontal, its horizontal component is F cos θ and vertical component is F sin θ.
当力 F 与水平方向成 θ 角时，水平分量为 F cos θ，竖直分量为 F sin θ。
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