📚 Year 11 CAIE Engineering: Formula & Theorem Quick Reference Handbook | Year 11 CAIE 工程：公式定理速查手册
This quick reference handbook compiles the essential formulas and theorems required for the CAIE Engineering syllabus at Year 11. It covers core topics including mechanics, materials, electricity, thermal physics, and mechanical systems. Keep this guide handy for rapid revision and problem-solving.
本速查手册汇编了Year 11 CAIE工程课程所需的核心公式与定理，涵盖力学、材料、电学、热物理和机械系统等关键模块。请随时查阅以进行快速复习和解题练习。
1. Stress and Strain | 应力和应变
Stress is a measure of the internal force per unit area within a material. It is denoted by the Greek letter sigma (σ) and calculated as the applied force divided by the cross-sectional area.
应力衡量材料内部单位面积上的内力，用希腊字母σ表示，计算公式为施加的力除以截面积。
σ = F / A
Strain is the measure of deformation representing the displacement between particles in the material. It is dimensionless and is the ratio of change in length to the original length.
应变是衡量材料形变的指标，无量纲，定义为长度变化量与原长之比。
ε = ΔL / L₀
Where F is force in newtons (N), A is cross-sectional area in square metres (m²), ΔL is the extension, and L₀ is the original length. Stress has units of pascals (Pa) or N/m².
其中F为力(N)，A为截面积(m²)，ΔL为伸长量，L₀为原长。应力的单位是帕斯卡(Pa)或N/m²。
2. Young’s Modulus | 杨氏模量
Young’s modulus (E) is a measure of the stiffness of an elastic material. It is defined as the ratio of stress to strain within the elastic limit.
杨氏模量(E)是衡量弹性材料刚度的参数，定义为弹性范围内应力与应变的比值。
E = σ / ε = (F × L₀) / (A × ΔL)
This relationship holds only as long as the material obeys Hooke’s law and does not exceed its proportional limit. A high Young’s modulus indicates a stiff material that resists deformation.
该关系仅在材料遵循胡克定律且未超过比例极限时成立。杨氏模量高表示材料坚硬，抵抗变形能力强。
3. Moments and the Principle of Moments | 力矩与力矩原理
A moment is the turning effect of a force about a pivot. It is the product 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，等于力的大小乘以支点到力作用线的垂直距离。
M = F × d
The principle of moments states that for a body in rotational equilibrium, the sum of the clockwise moments about any pivot equals the sum of the anticlockwise moments.
力矩原理表明，当物体处于转动平衡时，对于任一支点，顺时针力矩之和等于逆时针力矩之和。
ΣM_clockwise = ΣM_anticlockwise
This principle is used to calculate unknown forces or distances in levers, beams, and other structures.
该原理用于计算杠杆、横梁和其他结构中的未知力或距离。
4. Mechanical Advantage and Velocity Ratio | 机械利益与速度比
Mechanical advantage (MA) is the ratio of the load force to the effort force in a machine. It indicates how much the machine multiplies the input force.
机械利益(MA)是机器中负载力与施加力之比，表示机器将输入力放大了多少倍。
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. It depends only on the geometry of the machine.
速度比(VR)是在相同时间内，施加力移动的距离与负载移动的距离之比，仅取决于机器的几何结构。
VR = distance moved by effort / distance moved by load
For an ideal frictionless machine, MA would equal VR, but in reality friction reduces MA.
在理想无摩擦的机器中，MA等于VR，但实际中摩擦会降低机械利益。
5. Work, Energy and Power | 功、能与功率
Work done (W) by a constant force is the product of the force and the displacement in the direction of the force.
恒力做的功(W)等于力与沿力方向位移的乘积。
W = F × d
Kinetic energy (KE) is the energy an object possesses due to its motion.
动能(KE)是物体由于运动而具有的能量。
KE = ½ m v²
Gravitational potential energy (GPE) is the energy an object has due to its position in a gravitational field.
重力势能(GPE)是物体因在引力场中的位置而具有的能量。
GPE = m g h
Power is the rate of doing work or transferring energy.
功率是做功或能量传递的速率。
P = W / t
According to the conservation of energy, energy cannot be created or destroyed, only converted from one form to another.
根据能量守恒定律，能量不能被创造或消灭，只能从一种形式转化为另一种形式。
6. Efficiency | 效率
Efficiency (η) measures how well a system converts input energy or power into useful output. It is expressed as a percentage.
效率(η)衡量系统将输入能量或功率转化为有用输出的程度，以百分比表示。
η = (useful output energy / total input energy) × 100%
For mechanical systems, efficiency can also be calculated using mechanical advantage and velocity ratio.
对于机械系统，效率也可通过机械利益和速度比计算。
η = (MA / VR) × 100%
Due to friction, heat loss, and other irreversibilities, efficiency is always less than 100%.
由于摩擦、热损失及其他不可逆因素，效率总是低于100%。
7. Ohm’s Law and Electrical Power | 欧姆定律与电功率
Ohm’s law relates the voltage across a conductor, the current through it, and its resistance. It is valid for ohmic materials at constant temperature.
欧姆定律将导体两端的电压、通过的电流及其电阻联系起来，适用于恒温条件下的欧姆材料。
V = I × R
Electrical power is the rate at which electrical energy is transferred by a circuit. Several equivalent forms can be used depending on the known quantities.
电功率是电路传输电能的速率，可根据已知量使用不同的等价形式。
P = I × V = I² × R = V² / R
Here V is measured in volts (V), I in amperes (A), R in ohms (Ω), and P in watts (W).
其中V以伏特(V)为单位，I以安培(A)为单位，R以欧姆(Ω)为单位，P以瓦特(W)为单位。
8. Resistors in Series and Parallel | 串联与并联电阻
When resistors are connected end-to-end in series, the total resistance is the sum of the individual resistances.
当电阻器首尾相连串联时，总电阻为各个电阻值之和。
R_total = R₁ + R₂ + R₃ + …
For resistors connected in parallel, the reciprocal of the total resistance equals the sum of the reciprocals of the individual resistances.
对于并联连接的电阻器，总电阻的倒数等于各电阻倒数之和。
1 / R_total = 1 / R₁ + 1 / R₂ + 1 / R₃ + …
For two resistors in parallel, a simplified formula is R_total = (R₁ × R₂) / (R₁ + R₂). Parallel circuits always have a total resistance smaller than the smallest individual resistor.
两个电阻并联时，可用简化公式R_total = (R₁ × R₂) / (R₁ + R₂)。并联电路的总电阻总是小于最小的单个电阻。
9. Pressure and Pascal’s Principle | 压强与帕斯卡原理
Pressure is the force applied perpendicular to a surface per unit area. It is a scalar quantity.
压强是垂直施加在单位面积上的力，是一个标量。
p = F / A
Pascal’s principle states that when pressure is applied to an enclosed incompressible fluid, the pressure change is transmitted undiminished to all parts of the fluid and to the walls of its container.
帕斯卡原理指出，对封闭的不可压缩流体施加压强时，该压强变化会无损地传递到流体的各个部分及容器壁上。
This principle is the foundation of hydraulic systems. For two connected pistons, the following relationships hold under ideal conditions.
该原理是液压系统的基础。在理想条件下，对于两个相连活塞，下列关系成立。
F₁ / A₁ = F₂ / A₂
d₁ × A₁ = d₂ × A₂
Where F is the force on each piston, A is the piston area, and d is the distance moved. The hydraulic press multiplies force, and its velocity ratio is A₂ / A₁.
其中F为活塞上的力，A为活塞面积，d为移动距离。液压机能放大作用力，其速度比为A₂ / A₁。
10. Thermal Expansion | 热膨胀
Most materials expand when heated. Linear expansion describes the change in one dimension of a solid, while volume expansion applies to liquids and three-dimensional solids.
大多数材料受热时会膨胀。线性膨胀描述固体一维尺寸的变化，而体积膨胀适用于液体和三维固体。
ΔL = α × L₀ × ΔT
ΔV = β × V₀ × ΔT
Here α is the coefficient of linear expansion, β is the coefficient of volume expansion, L₀ and V₀ are the original length and volume, and ΔT is the change in temperature. For isotropic solids, β ≈ 3α.
其中α为线性膨胀系数，β为体积膨胀系数，L₀和V₀为原长和原体积，ΔT为温度变化。对于各向同性固体，β ≈ 3α。
Thermal expansion must be considered in engineering designs, such as bridges with expansion joints and railway tracks with small gaps.
工程设计中必须考虑热膨胀，例如留有膨胀缝的桥梁以及带小间隙的铁轨。
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