Year 8 CAIE Engineering: Formula & Theorem Quick Reference Handbook

📚 Year 8 CAIE Engineering: Formula & Theorem Quick Reference Handbook | 公式定理速查手册

This quick reference handbook compiles the essential formulas and theorems required for Year 8 CAIE Engineering. It covers foundational topics in mechanics, electricity, materials and geometry. Use it to reinforce your understanding and as a revision aid before assessments.

本速查手册汇编了 Year 8 CAIE 工程中关键的基础公式与定理，涵盖力学、电学、材料与几何等核心主题。你可以用它来巩固所学内容，并在考试前进行高效复习。

1. Force, Pressure and Area | 力、压强与面积

Pressure is the force applied per unit area. The formula links how concentrated a force is on a surface.

压强表示单位面积上承受的作用力，描述力在一个表面上的集中程度。

P = F / A

where P is pressure in pascals (Pa), F is force in newtons (N), and A is area in square metres (m²). A larger area reduces pressure for the same force.

其中 P 为压强（帕斯卡 Pa），F 为力（牛顿 N），A 为面积（平方米 m²）。在作用力相同时，面积越大，压强越小。

1 Pa = 1 N/m²

压强单位：1 帕斯卡等于 1 牛顿/平方米。

2. Density and Mass | 密度与质量

Density measures how much mass is contained in a given volume. It helps identify materials and predict whether objects float or sink.

密度衡量单位体积内所含物质的质量，常用于材料鉴别以及判断物体在流体中的浮沉。

ρ = m / V

where ρ (Greek letter rho) is density in kg/m³, m is mass in kg, V is volume in m³. Water has a density of about 1000 kg/m³.

其中 ρ 为密度（千克/立方米 kg/m³），m 为质量（kg），V 为体积（m³）。水的密度约为 1000 kg/m³。

To find mass from density and volume: m = ρ × V.

若已知密度和体积，可通过 m = ρ × V 计算质量。

3. Speed, Distance and Time | 速度、距离与时间

Speed describes how fast an object moves, calculated as the distance travelled per unit time.

速度描述物体运动的快慢，即单位时间内通过的距离。

v = s / t

Here v is average speed in m/s, s is distance in metres (m), t is time in seconds (s). Rearranged forms: s = v × t and t = s / v.

其中 v 为平均速度（米/秒 m/s），s 为距离（m），t 为时间（s）。变形公式：s = v × t，t = s / v。

For objects moving at constant speed, this formula gives the exact value. For changing speeds, it gives the average.

当物体匀速运动时，该公式给出瞬时速度；速度变化时，所得为平均速度。

4. Acceleration | 加速度

Acceleration is the rate of change of velocity. A positive value means speeding up, negative (deceleration) means slowing down.

加速度表示速度改变的快慢。正值表示加速，负值（又称减速度）表示减速。

a = (v – u) / t

where a is acceleration in m/s², v is final velocity (m/s), u is initial velocity (m/s), t is time (s).

其中 a 为加速度（m/s²），v 为末速度（m/s），u 为初速度（m/s），t 为时间（s）。

Gravity on Earth accelerates objects downward at approximately 9.8 m/s².

地球表面重力加速度约为 9.8 m/s²，方向竖直向下。

5. Work, Energy and Power | 功、能与功率

Work is done when a force moves an object. Energy is the capacity to do work, and power is the rate of doing work.

力使物体发生位移时就做了功。能量是做功的本领，功率则表示做功的快慢。

W = F × d

Work done W in joules (J) equals force F (N) multiplied by distance d (m) moved in the direction of the force.

功 W 的单位是焦耳（J），等于力 F（N）乘以沿力方向移动的距离 d（m）。

P = W / t

Power P in watts (W) is work done divided by time t. Also P = F × v (force × velocity) for constant speed.

功率 P 的单位是瓦特（W），等于功 / 时间；当物体匀速运动时也可写作 P = F × v（力 × 速度）。

Gravitational potential energy: Eₚ = mgh (mass × gravity × height). Kinetic energy: Eₖ = ½mv².

重力势能：Eₚ = mgh（质量 × 重力加速度 × 高度）；动能：Eₖ = ½mv²。

6. Levers and Moments | 杠杆与力矩

A moment is the turning effect of a force about a pivot. Levers use moments to magnify forces or distances.

力矩是力使物体绕支点转动的效应。杠杆通过力矩原理达到省力或增大距离的目的。

M = F × d

where M is the moment in newton-metres (N m), F is the applied force (N), d is the perpendicular distance from the pivot to the line of action (m).

其中 M 为力矩（牛顿·米 N·m），F 为作用力（N），d 为支点到力的作用线的垂直距离（m）。

Principle of moments: For a balanced lever, total anticlockwise moments = total clockwise moments.

杠杆平衡条件（力矩原理）：顺时针力矩之和等于逆时针力矩之和。

7. Electrical Quantities: Ohm’s Law and Power | 电量：欧姆定律与电功率

Ohm’s law connects voltage, current and resistance. Electrical power formulas reveal how quickly energy is converted in a circuit.

欧姆定律联系了电压、电流与电阻。电功率公式则表明电路中能量转换的速率。

V = I × R

Voltage V in volts (V), current I in amperes (A), resistance R in ohms (Ω).

电压 V（伏特），电流 I（安培），电阻 R（欧姆）。

P = I × V

Electrical power P in watts (W). Combining with Ohm’s law gives P = I²R and P = V²/R.

电功率 P（瓦特 W）。结合欧姆定律可得 P = I²R 与 P = V²/R。

8. Series and Parallel Circuits | 串联与并联电路

Components in series share the same current; in parallel they share the same voltage. The rules for total resistance differ.

串联电路中各元件电流相同；并联电路中各元件电压相同。两者总电阻的计算法则不同。

Series: R_total = R₁ + R₂ + R₃ + …

串联总电阻等于各电阻之和。

Parallel: 1/R_total = 1/R₁ + 1/R₂ + …

并联总电阻的倒数等于各电阻倒数之和。对两个并联电阻：R_total = (R₁ × R₂) / (R₁ + R₂)。

Current rules: I_total = I₁ = I₂ (series); I_total = I₁ + I₂ (parallel). Voltage: V_total = V₁ + V₂ (series); V_total = V₁ = V₂ (parallel).

电流规律：串联 I₁ = I₂ = I_total；并联 I_total = I₁ + I₂。电压规律：串联 V_total = V₁ + V₂；并联 V_total = V₁ = V₂。

9. Thermal Expansion | 热膨胀

Most materials expand when heated and contract when cooled. Linear expansion can be quantified for engineering design.

大多数材料热胀冷缩。在工程设计中须考虑线膨胀量。

ΔL = α × L₀ × ΔT

where ΔL is change in length (m), α is coefficient of linear expansion (/°C), L₀ is original length (m), ΔT is temperature change (°C).

其中 ΔL 为长度变化（m），α 为线膨胀系数（/°C），L₀ 为原长（m），ΔT 为温度变化（°C）。

Bridges and railway tracks include expansion gaps to prevent buckling.

桥梁和铁轨都设有伸缩缝，以防温度变化引起弯曲破坏。

10. Stress, Strain and Young’s Modulus | 应力、应变与杨氏模量

Stress measures internal force per area; strain measures deformation. Young’s modulus indicates material stiffness.

应力衡量单位面积所受的内力；应变表征形变程度。杨氏模量反映材料的刚性。

Stress σ = F / A (N/m² or Pa)

Strain ε = ΔL / L₀ (dimensionless)

Young’s Modulus E = σ / ε (Pa)

Stress σ: force F over cross‑sectional area A. Strain ε: extension ΔL divided by original length L₀. E is the ratio stress/strain in the elastic region.

应力 σ = 力 / 横截面积；应变 ε = 伸长量 / 原长；杨氏模量 E = 应力 / 应变（弹性范围内）。

11. Geometry and Area of Common Shapes | 常见形状的几何与面积

Accurate area calculations are vital in engineering for material estimation, force distribution and design.

精确的面积计算在工程中至关重要，用于材料估算、力分布分析和设计。

Shape / 形状	Area Formula / 面积公式	Perimeter / 周长
Rectangle / 矩形	A = l × w	P = 2(l + w)
Triangle / 三角形	A = ½ × b × h	Sum of sides
Circle / 圆	A = πr²	C = 2πr or πd
Trapezium / 梯形	A = ½(a+b)h	Sum of sides

For a triangle, b is base length and h is perpendicular height. For a circle, r is radius, d is diameter, and π ≈ 3.1416.

三角形的面积为 ½ × 底 × 高；圆的面积 πr²；π 约等于 3.1416。

12. Volume and Surface Area | 体积与表面积

Volume and surface area formulas are essential when designing containers, structures or analysing buoyancy.

体积与表面积公式在容器设计、结构设计以及浮力分析中不可或缺。

Solid / 立体	Volume / 体积	Surface Area / 表面积
Cuboid / 长方体	V = l × w × h	SA = 2(lw + lh + wh)
Cylinder / 圆柱体	V = πr²h	SA = 2πr(r + h)
Sphere / 球体	V = ⁴⁄₃πr³	SA = 4πr²
Cone / 圆锥体	V = ⅓πr²h	SA = πr(r + l) (l = slant height)

These relationships help determine capacity, material usage and fluid displacement.

这些公式有助于确定容量、材料用量与流体的置换量。

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Year 8 CAIE Engineering: Formula & Theorem Quick Reference Handbook | 公式定理速查手册