📚 Year 10 CIE Engineering: Formulas and Theorems Quick Reference | Year 10 CIE 工程：公式定理速查手册
This handbook provides a concise reference of essential formulas and theorems for Year 10 CIE Engineering. Keep it handy for quick revision and problem-solving across mechanics, electricity, materials and more.
本手册提供了 Year 10 CIE 工程中必备公式与定理的简明参考。随身携带以便快速复习和解题，涵盖力学、电学、材料等多个领域。
1. Newton’s Laws of Motion | 牛顿运动定律
Newton’s First Law: An object remains at rest or in uniform motion in a straight line unless acted upon by a resultant external force.
牛顿第一定律：物体将保持静止或匀速直线运动状态，除非受到合外力的作用。
Newton’s Second Law: The acceleration of an object is directly proportional to the resultant force and inversely proportional to its mass. The direction of acceleration is the same as the net force.
牛顿第二定律：物体的加速度与所受合外力成正比，与质量成反比，加速度方向与合外力方向相同。
F = m a
Where F = resultant force in newtons (N), m = mass in kilograms (kg), a = acceleration in metres per second squared (m/s²).
其中 F = 合力（牛，N），m = 质量（千克，kg），a = 加速度（米每二次方秒，m/s²）。
Newton’s Third Law: Whenever two objects interact, they exert equal and opposite forces on each other.
牛顿第三定律：两个物体相互作用时，彼此施加的力大小相等、方向相反，且作用在同一直线上。
2. Mechanical Advantage & Velocity Ratio | 机械效益与速比
Mechanical Advantage (MA) is the ratio of load to effort. It indicates how much a machine multiplies the input force.
机械效益 (MA) 是负载与动力的比值，表示机械放大输入力的倍数。
MA = Load / Effort
Velocity Ratio (VR) is the ratio of the distance moved by the effort to the distance moved by the load in the same time.
速比 (VR) 是动力移动的距离与相同时间内负载移动的距离之比。
VR = d_e / d_l
Where d_e = distance moved by effort, d_l = distance moved by load. For an ideal machine with no friction, MA = VR. In reality, MA is always less than VR due to energy losses.
其中 d_e = 动力移动的距离，d_l = 负载移动的距离。理想无摩擦机械中 MA = VR；实际由于能量损耗，MA 恒小于 VR。
3. Force, Mass & Acceleration | 力、质量与加速度
Weight is the gravitational force on an object. It is a vector quantity directed towards the centre of the Earth.
重量是作用在物体上的重力，是一个方向指向地心的矢量。
W = m g
On Earth, g ≈ 9.8 m/s²; for quick estimates, use g = 10 m/s². W is in newtons (N).
在地球表面，g ≈ 9.8 m/s²；快速估算时取 g = 10 m/s²。重量 W 的单位是牛 (N)。
When several forces act on an object, find the resultant force by vector addition. Equilibrium occurs when the resultant force is zero.
多个力作用时，通过矢量加法求合力。合力为零则物体处于平衡状态。
Friction opposes motion and converts kinetic energy into heat. It depends on the nature of surfaces and the normal reaction force.
摩擦力阻碍运动，将动能转化为热能。它取决于接触面的性质与法向反作用力。
4. Moments & Levers | 力矩与杠杆
The moment of a force about a pivot equals the force multiplied by the perpendicular distance from the pivot to the line of action of the force.
力对支点的力矩等于力的大小乘以支点到力作用线的垂直距离。
M = F × d
Unit: newton metre (N m). Moment is a vector and can cause clockwise or anticlockwise rotation.
单位：牛·米 (N m)。力矩是矢量，可产生顺时针或逆时针转动效应。
Principle of Moments: For a system in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any pivot.
力矩原理：对于转动平衡的系统，绕任一支点，顺时针力矩之和等于逆时针力矩之和。
Levers are classified into three types: Class 1 (pivot between load and effort, e.g. seesaw), Class 2 (load between pivot and effort, e.g. wheelbarrow), Class 3 (effort between pivot and load, e.g. fishing rod).
杠杆分为三类：一类（支点在负载与动力之间，如跷跷板），二类（负载在支点与动力之间，如手推车），三类（动力在支点与负载之间，如钓鱼竿）。
5. Work, Energy & Power | 功、能量与功率
Work is done when a force moves its point of application in the direction of the force.
当力使其作用点沿力的方向移动时，力做功。
W = F × d
Work done (W) is measured in joules (J). 1 J = 1 N m.
功 (W) 的单位是焦耳 (J)。1 J = 1 N m。
Kinetic energy is the energy of motion; gravitational potential energy is stored due to height.
动能是物体因运动而具有的能量；重力势能是因高度而储存的能量。
KE = ½ m v²
GPE = m g h
Power is the rate of doing work or transferring energy.
功率是做功或能量转换的速率。
P = W / t = E / t
Unit: watt (W); 1 W = 1 J/s. Larger machines use kilowatts (kW).
单位：瓦特 (W)；1 W = 1 J/s。大型机械常用千瓦 (kW)。
6. Electrical Formulas (Ohm’s Law & Power) | 电学公式（欧姆定律与功率）
Ohm’s Law relates voltage, current and resistance in a metallic conductor at constant temperature.
欧姆定律描述了恒温下金属导体中电压、电流与电阻的关系。
V = I × R
V = voltage (volts, V), I = current (amperes, A), R = resistance (ohms, Ω).
V = 电压（伏特，V），I = 电流（安培，A），R = 电阻（欧姆，Ω）。
Electrical power can be expressed in several forms, depending on the known quantities.
电功率可根据已知量表示为不同形式。
P = V I = I² R = V² / R
Energy consumed by an electrical device is the product of power and time.
用电器消耗的电能等于功率与时间的乘积。
E = P t = V I t
For resistors in series: Rtotal = R₁ + R₂ + R₃ + … For resistors in parallel: 1/Rtotal = 1/R₁ + 1/R₂ + 1/R₃ + …
串联电阻：R总 = R₁ + R₂ + R₃ + …；并联电阻：1/R总 = 1/R₁ + 1/R₂ + 1/R₃ + …
7. Stress, Strain & Young’s Modulus | 应力、应变与杨氏模量
When a material is subjected to external forces, internal resistive forces develop. Stress measures the intensity of these internal forces.
材料受外力时产生内部抵抗力，应力度量该内力的强度。
σ = F / A
σ = stress in pascals (Pa) or N/m², F = applied force (N), A = original cross‑sectional area (m²).
σ = 应力（帕，Pa 或 N/m²），F = 作用力 (N)，A = 原始横截面积 (m²)。
Strain is the fractional change in length caused by the applied stress.
应变是应力引起的长度变化率。
ε = ΔL / L₀
ε = strain (no units), ΔL = change in length, L₀ = original length.
ε = 应变（无量纲），ΔL = 长度变化，L₀ = 原始长度。
Young’s modulus (E) is the ratio of stress to strain within the elastic limit; it characterises material stiffness.
杨氏模量 (E) 为弹性限度内应力与应变之比，表征材料的刚度。
E = σ / ε = (F/A) / (ΔL/L₀)
A steep stress‑strain curve indicates a stiff material; a gradual curve indicates a more flexible material.
应力‑应变曲线陡峭表示材料刚硬，平缓则表示柔韧。
8. Gear Ratios & Pulleys | 齿轮比与滑轮
A gear train transmits rotary motion and torque. The gear ratio determines speed and torque changes.
齿轮系传递旋转运动和扭矩，齿轮比决定转速与扭矩的变化。
Gear Ratio = Tdriven / Tdriver
Where Tdriven = number of teeth on the driven gear, Tdriver =
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