📚 Year 9 Cambridge Engineering: Formula & Theorem Quick Reference Handbook | 九年级剑桥工程：公式定理速查手册
This handbook compiles the essential formulas, theorems, and key relationships you will encounter in Year 9 Cambridge Engineering. Use it as a quick reference for solving problems in mechanics, electronics, materials, and systems.
本手册汇集了九年级剑桥工程课程中需要掌握的基本公式、定理和关键关系。你可以把它用作解答力学、电子学、材料与系统问题的速查工具。
1. Forces and Motion | 力与运动
Newton’s Second Law relates force, mass, and acceleration: F = m x a, where F is force in newtons (N), m mass in kilograms (kg), and a acceleration in metres per second squared (m/s²).
牛顿第二定律描述了力、质量与加速度的关系：F = m x a，其中 F 为力（牛顿 N），m 为质量（千克 kg），a 为加速度（米每二次方秒 m/s²）。
Weight is a special force caused by gravity: W = m x g. On Earth, g ≈ 10 N/kg.
重量是由重力引起的特殊力：W = m x g。在地球上，g 约为 10 N/kg。
Average speed is calculated by s = d / t, where d is distance and t is time. Acceleration is the rate of change of velocity: a = (v – u) / t, with initial velocity u and final velocity v.
平均速度由 s = d / t 计算，其中 d 为距离，t 为时间。加速度是速度变化的速率：a = (v – u) / t，u 为初速度，v 为末速度。
2. Moments and Levers | 力矩与杠杆
The turning effect of a force is called a moment: M = F x d, where d is the perpendicular distance from the pivot to the line of action of the force, measured in metres. The moment is measured in newton-metres (N·m).
力的转动效果称为力矩：M = F x d，其中 d 为从支点到力作用线的垂直距离（米）。力矩的单位是牛·米（N·m）。
For a lever to be in equilibrium, the sum of clockwise moments must equal the sum of anticlockwise moments: F₁ x d₁ = F₂ x d₂. This is the principle of moments.
杠杆平衡时，顺时针力矩之和等于逆时针力矩之和：F₁ x d₁ = F₂ x d₂。这就是力矩原理。
3. Energy, Work and Power | 能量、功与功率
Work is done when a force moves an object: W = F x d, where work W is in joules (J), force F in newtons, and distance d in metres (along the line of force).
力使物体移动时即做功：W = F x d，功 W 的单位为焦耳（J），力 F 为牛顿，距离 d 为米（沿力的方向）。
Gravitational potential energy gained by an object raised through a height h is Eₚ = m x g x h. Kinetic energy of a moving object is Eₖ = ½ x m x v².
物体被举高 h 所增加的引力势能为 Eₚ = m x g x h。运动物体的动能为 Eₖ = ½ x m x v²。
Power is the rate of doing work: P = W / t or P = E / t, where P is in watts (W). In electrical systems, power can also be expressed as P = I x V.
功率是做功的速率：P = W / t 或 P = E / t，P 的单位为瓦特（W）。在电气系统中，功率也可表示为 P = I x V。
4. Electrical Circuits and Quantities | 电路与电学量
Electric current is the flow of charge: I = Q / t, where I is current in amperes (A), Q is charge in coulombs (C), and t is time in seconds.
电流是电荷的流动：I = Q / t，I 为电流（安培 A），Q 为电荷（库仑 C），t 为时间（秒）。
Voltage (potential difference) is the energy transferred per unit charge: V = E / Q. Resistance is a measure of how difficult it is for current to flow, defined by Ohm’s law in the next section.
电压（电位差）是每单位电荷传递的能量：V = E / Q。电阻是电流流动难易程度的度量，将在下一节通过欧姆定律定义。
5. Ohm’s Law and Resistance | 欧姆定律与电阻
Ohm’s law states that the current through a resistor at constant temperature is directly proportional to the voltage across it: V = I x R, where R is resistance in ohms (Ω).
欧姆定律表明，在恒温条件下，通过电阻的电流与电阻两端的电压成正比：V = I x R，R 为电阻（欧姆 Ω）。
The resistance of a component can be found from R = V / I. Using Ohm’s law, power dissipated in a resistor can also be written as P = I² x R or P = V² / R.
元件的电阻可由 R = V / I 求出。结合欧姆定律，电阻消耗的功率也可表示为 P = I² x R 或 P = V² / R。
6. Mechanical Advantage and Efficiency | 机械利益与效率
Mechanical advantage (MA) is the ratio of the load force to the effort force: MA = Load / Effort. Velocity ratio (VR) is the ratio of the distance moved by the effort to the distance moved by the load: VR = d_effort / d_load.
机械利益（MA）是负载力与施力之比：MA = 负载 / 施力。速度比（VR）是施力移动距离与负载移动距离之比：VR = 施力移动距离 / 负载移动距离。
Efficiency compares useful work output to total energy input: Efficiency = (Work output / Energy input) x 100%, or for machines Efficiency = (MA / VR) x 100%.
效率比较有用功输出与总能量输入：效率 = (输出功 / 输入能量) x 100%，或对于机器 效率 = (MA / VR) x 100%。
7. Gear and Pulley Systems | 齿轮与滑轮系统
The gear ratio is given by Gear Ratio = Number of teeth on driven gear / Number of teeth on driver gear. A gear ratio greater than 1 increases torque but reduces speed; a ratio less than 1 increases speed.
齿轮比由 齿轮比 = 从动齿轮齿数 / 主动齿轮齿数 给出。齿轮比大于 1 时增加扭矩但降低转速；小于 1 时提高转速。
For a pulley system, the velocity ratio equals the number of rope sections supporting the load (for a simple block and tackle). Mechanical advantage can be gained using multiple pulleys.
对于滑轮系统，速度比等于支撑负载的绳索段数（以简单滑轮组为例）。通过使用多个滑轮可获得机械利益。
8. Hooke’s Law and Springs | 胡克定律与弹簧
Hooke’s law describes the extension of a spring or elastic material under a load: F = k x x, where F is the applied force, x is the extension (in metres), and k is the spring constant (N/m). This law holds up to the limit of proportionality.
胡克定律描述了弹簧或弹性材料在负载下的伸长：F = k x x，其中 F 为施加的力，x 为伸长量（米），k 为弹簧常数（N/m）。该定律在比例极限内成立。
Elastic potential energy stored in a stretched spring can be found from Eₑ = ½ x k x x².
拉伸弹簧中储存的弹性势能可由 Eₑ = ½ x k x x² 求得。
9. Pressure in Solids and Fluids | 固体与流体内的压力
Pressure is defined as force per unit area: P = F / A, where P is in pascals (Pa), F in newtons, and A in square metres. A smaller area results in a larger pressure for the same force.
压力定义为每单位面积的力：P = F / A，P 的单位为帕斯卡（Pa），F 为牛顿，A 为平方米。相同力下，面积越小压力越大。
In a liquid, pressure increases with depth: P = h x ρ x g, where h is depth, ρ is the density of the liquid, and g is gravitational field strength. This equation is fundamental for hydraulic systems.
在液体中，压力随深度增加：P = h x ρ x g，其中 h 为深度，ρ 为液体密度，g 为重力场强度。该方程是液压系统的基础。
10. Stress, Strain and Material Properties | 应力、应变与材料性质
Engineering materials experience stress when subjected to forces. Tensile stress is σ = F / A, where F is applied force and A is the cross-sectional area. Strain is the fractional change in length: ε = ΔL / L₀.
工程材料在受力时会产生应力。拉
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